Acta Math., 207 (2011), 29â201DOI: 10.1007/s11511-011-0068-9c© 2011 by Institut Mittag-Leffler. All rights reserved
On Landau damping
by
Clement Mouhot
University of Cambridge
Cambridge, U.K.
Cedric Villani
Institut Henri Poincare & Universite de Lyon
Lyon, France
Dedicated to Vladimir Arnold and Carlo Cercignani.
Contents
1. Introduction to Landau damping . . . . . . . . . . . . . . . . . . . . 321.1. Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.2. Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.3. Range of validity . . . . . . . . . . . . . . . . . . . . . . . . . . 361.4. Conceptual problems . . . . . . . . . . . . . . . . . . . . . . . . 401.5. Previous mathematical results . . . . . . . . . . . . . . . . . . 41
2. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1. Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2. Linear damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3. Non-linear damping . . . . . . . . . . . . . . . . . . . . . . . . . 462.4. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5. Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.6. Main ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . 522.7. About phase mixing . . . . . . . . . . . . . . . . . . . . . . . . 53
3. Linear damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4. Analytic norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1. Single-variable analytic norms . . . . . . . . . . . . . . . . . . 654.2. Analytic norms in two variables . . . . . . . . . . . . . . . . . 714.3. Relations between functional spaces . . . . . . . . . . . . . . . 734.4. Injections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.5. Algebra property in two variables . . . . . . . . . . . . . . . . 794.6. Composition inequality . . . . . . . . . . . . . . . . . . . . . . 804.7. Gradient inequality . . . . . . . . . . . . . . . . . . . . . . . . . 824.8. Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.9. Sobolev corrections . . . . . . . . . . . . . . . . . . . . . . . . . 854.10. Individual mode estimates . . . . . . . . . . . . . . . . . . . . . 864.11. Measuring solutions of kinetic equations in large time . . . . 874.12. Linear damping revisited . . . . . . . . . . . . . . . . . . . . . 88
30 c. mouhot and c. villani
5. Deflection estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1. Formal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6. Bilinear regularity and decay estimates . . . . . . . . . . . . . . . . . 96
6.1. Basic bilinear estimate . . . . . . . . . . . . . . . . . . . . . . . 98
6.2. Short-term regularity extortion by time cheating . . . . . . . 99
6.3. Long-term regularity extortion . . . . . . . . . . . . . . . . . . 101
7. Control of the time-response . . . . . . . . . . . . . . . . . . . . . . . 106
7.1. Qualitative discussion . . . . . . . . . . . . . . . . . . . . . . . 107
7.2. Exponential moments of the kernel . . . . . . . . . . . . . . . 113
7.3. Dual exponential moments . . . . . . . . . . . . . . . . . . . . 117
7.4. Growth control . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8. Approximation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.1. The natural Newton scheme . . . . . . . . . . . . . . . . . . . 135
8.2. Battle plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9. Local-in-time iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10. Global in time iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.1. The statement of the induction . . . . . . . . . . . . . . . . . 144
10.2. Preparatory remarks . . . . . . . . . . . . . . . . . . . . . . . . 146
10.3. Estimates on the characteristics . . . . . . . . . . . . . . . . . 149
10.4. Estimates on the density and distribution alongcharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.5. Convergence of the scheme . . . . . . . . . . . . . . . . . . . . 170
11. CoulombâNewton interaction . . . . . . . . . . . . . . . . . . . . . . . 172
11.1. Estimates on exponentially large times . . . . . . . . . . . . . 172
11.2. Mode-by-mode estimates . . . . . . . . . . . . . . . . . . . . . 173
12. Convergence in large time . . . . . . . . . . . . . . . . . . . . . . . . . 177
13. Non-analytic perturbations . . . . . . . . . . . . . . . . . . . . . . . . 180
14. Expansions and counterexamples . . . . . . . . . . . . . . . . . . . . 185
14.1. Simple excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 186
14.2. General perturbation . . . . . . . . . . . . . . . . . . . . . . . . 188
15. Beyond Landau damping . . . . . . . . . . . . . . . . . . . . . . . . . 191
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.1. Calculus in dimension d . . . . . . . . . . . . . . . . . . . . . . 193
A.2. Multi-dimensional differential calculus . . . . . . . . . . . . . 193
A.3. Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.4. Fixed-point theorem . . . . . . . . . . . . . . . . . . . . . . . . 195
A.5. Plemelj formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Landau damping may be the single most famous mystery of classical plasma physics.For the past sixty years it has been treated in the linear setting at various degrees ofrigor; but its non-linear version has remained elusive, since the only available results [13],[41] prove the existence of some damped solutions, without telling anything about theirgenericity.
on landau damping 31
In the present work we close this gap by treating the non-linear version of Landaudamping in arbitrarily large times, under assumptions which cover both attractive andrepulsive interactions, of any regularity down to CoulombâNewton.
This will lead us to discover a distinctive mathematical theory of Landau damping,complete with its own functional spaces and functional inequalities. Let us make it clearthat this study is not just for the sake of mathematical rigor: indeed, we shall get newinsights into the physics of the problem, and identify new mathematical phenomena.
The plan of the paper is as follows.
In §1 we provide an introduction to Landau damping, including historical commentsand a review of the existing literature. Then in §2 we state and comment on our mainresult about ânon-linear Landau dampingâ (Theorem 2.6).
In §3 we provide a rather complete treatment of linear Landau damping, slightlyimproving on the existing results both in generality and simplicity. This section can beread independently of the rest.
In §4 we define the spaces of analytic functions which are used in the remainder ofthe paper. The careful choice of norms is one of the keys of our analysis; the complexityof the problem will naturally lead us to work with norms having up to five parameters.As a first application, we shall revisit linear Landau damping within this framework.
In §§5â7 we establish four types of new estimates (deflection estimates, short-termand long-term regularity extortion, echo control); these are the key sections containingin particular the physically relevant new material.
In §8 we adapt the Newton algorithm to the setting of the non-linear Vlasov equation.Then in §§9â11 we establish some iterative estimates along this scheme. (§11 is devotedspecifically to a technical refinement allowing us to handle CoulombâNewton interaction.)
From these estimates our main theorem is easily deduced in §12.
An extension to non-analytic perturbations is presented in §13.
Some counterexamples and asymptotic expansions are studied in §14.
Final comments about the scope and range of applicability of these results are pro-vided in §15.
Even though it basically proves one main result, this paper is very long. This isdue partly to the intrinsic complexity and richness of the problem, partly to the needto develop an adequate functional theory from scratch, and partly to the inclusion ofremarks, explanations and comments intended to help the reader to understand the proofand the scope of the results. The whole process culminates in the extremely technicaliteration performed in §10 and §11. A short summary of our results and methods ofproofs can be found in the expository paper [69].
This project started from an unlikely conjunction of discussions of the authors with
32 c. mouhot and c. villani
various people, most notably Yan Guo, Dong Li, Freddy Bouchet and Etienne Ghys.We also got crucial inspiration from the books [9] and [10] by James Binney and ScottTremaine; and [2] by Serge Alinhac and Patrick Gerard. Warm thanks to Julien Barre,Jean Dolbeault, Thierry Gallay, Stephen Gustafson, Gregory Hammett, Donald Lynden-Bell, Michael Sigal, Eric Sere and especially Michael Kiessling for useful exchanges andreferences; and to Francis Filbet and Irene Gamba for providing numerical simulations.We are also grateful to Patrick Bernard, Freddy Bouchet, Emanuele Caglioti, YvesElskens, Yan Guo, Zhiwu Lin, Michael Loss, Peter Markowich, Govind Menon, YannOllivier, Mario Pulvirenti, Jeff Rauch, Igor Rodnianski, Peter Smereka, Yoshio Sone,Tom Spencer, and the team of the Princeton Plasma Physics Laboratory for further con-structive discussions about our results. Finally, we acknowledge the generous hospitalityof several institutions: Brown University, where the first author was introduced to Lan-dau damping by Yan Guo in early 2005; the Institute for Advanced Study in Princeton,who offered the second author a serene atmosphere of work and concentration during thebest part of the preparation of this work; Cambridge University, who provided repeatedhospitality to the first author thanks to the Award No. KUK-I1-007-43, funded by theKing Abdullah University of Science and Technology; and the University of Michigan,where conversations with Jeff Rauch and others triggered a significant improvement ofour results.
Our deep thanks go to the referees for their careful examination of the manuscript.We dedicate this paper to two great scientists who passed away during the elaboration ofour work. The first one is Carlo Cercignani, one of the leaders of kinetic theory, author ofseveral masterful treatises on the Boltzmann equation, and a long-time personal friend ofthe second author. The other one is Vladimir Arnold, a mathematician of extraordinaryinsight and influence; in this paper we shall uncover a tight link between Landau dampingand the theory of perturbation of completely integrable Hamiltonian systems, to whichArnold has made major contributions.
1. Introduction to Landau damping
1.1. Discovery
Under adequate assumptions (collisionless regime, non-relativistic motion, heavy ions,no magnetic field), a dilute plasma is well described by the non-linear VlasovâPoissonequation
âf
ât+v ·âxf+
F
m·âvf = 0, (1.1)
on landau damping 33
where f=f(t, x, v)ïżœ0 is the density of electrons in phase space (x is the position and v thevelocity), m is the mass of an electron, and F =F (t, x) is the mean-field (self-consistent)electrostatic force:
F =âeE, E =âÎâ1(4Ïïżœ). (1.2)
Here e>0 is the absolute electron charge, E=E(t, x) is the electric field, and ïżœ=ïżœ(t, x)is the density of charges
ïżœ = ïżœiâe
â«R3
f dv, (1.3)
ïżœi being the density of charges due to ions. This model and its many variants are oftantamount importance in plasma physics [1], [5], [49], [54].
In contrast to models incorporating collisions [92], the VlasovâPoisson equation istime-reversible. However, in 1946 Landau [52] stunned the physical community by pre-dicting an irreversible behavior on the basis of this equation. This âastonishing resultâ(as it was called in [88]) relied on the solution of the Cauchy problem for the linearizedVlasovâPoisson equation around a spatially homogeneous Maxwellian (Gaussian) equilib-rium. Landau formally solved the equation by means of Fourier and Laplace transforms,and after a study of singularities in the complex plane, concluded that the electric fielddecays exponentially fast; he further studied the rate of decay as a function of the wavevector k. Landauâs computations are reproduced in [54, §34] and [1, §4.2].
An alternative argument appears in [54, §30]: there the thermodynamical formalismis used to compute the amount of heat Q which is dissipated when a (small) oscillatingelectric field E(t, x)=Eei(k·xâÏt) (k is a wave vector and Ï>0 a frequency) is applied toa plasma whose distribution f0 is homogeneous in space and isotropic in velocity space;the result is
Q =â|E|2 Ïme2Ï
|k|2 ÏâČ(
Ï
|k|)
, (1.4)
where
Ï(v1) =â«
R3
â«R3
f0(v1, v2, v3) dv2 dv3.
In particular, (1.4) is always positive (see the last remark in [54, §30]), which means thatthe system reacts against the perturbation, and thus possesses some âactiveâ stabilizationmechanism.
A third argument [54, §32] consists of studying the dispersion relation, or equiva-lently searching for the (generalized) eigenmodes of the linearized VlasovâPoisson equa-tion, now with complex frequency Ï. After appropriate selection, these eigenmodes areall decaying (Im Ï<0) as t!â. This again suggests stability, although in a somewhatweaker sense than the computation of heat release.
34 c. mouhot and c. villani
The first and third arguments also apply to the gravitational VlasovâPoisson equa-tion, which is the main model for non-relativistic galactic dynamics. This equation issimilar to (1.1), but now m is the mass of a typical star (!), and f is the density ofstars in phase space; moreover the first equation of (1.2) and the relation (1.3) shouldbe replaced by
F =âGmE and ïżœ = m
â«R3
f dv, (1.5)
where G is the gravitational constant, E is the gravitational field and ïżœ is the densityof mass. The books [9] and [10] by Binney and Tremaine constitute excellent referencesabout the use of the VlasovâPoisson equation in stellar dynamicsâwhere it is often calledthe âcollisionless Boltzmann equationâ, see footnote on p. 276 in [10]. On âintermediateâtime scales, the VlasovâPoisson equation is thought to be an accurate description of verylarge star systems [28], which are now accessible to numerical simulations.
Since the work of Lynden-Bell [58] it has been recognized that Landau damping, andwilder collisionless relaxation processes generically dubbed âviolent relaxationâ, consti-tute a fundamental stabilizing ingredient of galactic dynamics. Without these still poorlyunderstood mechanisms, the surprisingly short time scales for relaxation of the galaxieswould remain unexplained.
One main difference between the electrostatic and the gravitational interactions isthat in the latter case Landau damping should occur only at wavelengths smaller thanthe Jeans length [10, §5.2]; beyond this scale, even for Maxwellian velocity profiles, theJeans instability takes over and governs planet and galaxy aggregation.(1)
On the contrary, in (classical) plasma physics, Landau damping should hold at allscales under suitable assumptions on the velocity profile; and in fact one is in general notinterested in scales smaller than the Debye length, which is roughly defined in the sameway as the Jeans length.
Nowadays, not only has Landau damping become a cornerstone of plasma physics,(2)but it has also made its way into other areas of physics (astrophysics, but also wind waves,fluids, superfluids, etc.) and even biophysics. One may consult the concise survey papers[76], [81] and [91] for a discussion of its influence and some applications.
(1) Or at least would do, if galactic matter was smoothly distributed; in presence of âmicroscopicâheterogeneities, a phase transition for aggregation can occur far below this scale [48]. In the languageof statistical mechanics, the Jeans length corresponds to a âspinodal pointâ rather than a phase transi-tion [87].
(2) Ryutov [81] estimated in 1998 that âapproximately every third paper on plasma physics andits applications contains a direct reference to Landau dampingâ.
on landau damping 35
1.2. Interpretation
True to his legend, Landau deduced the damping effect from a mathematical-stylestudy,(3) without bothering to give a physical explanation of the underlying mecha-nism. His arguments anyway yield exact formulae, which in principle can be checkedexperimentally, and indeed provide good qualitative agreement with observations [59].
A first set of problems in the interpretation is related to the arrow of time. Inthe thermodynamic argument, the exterior field is awkwardly imposed from time ââonwards; moreover, reconciling a positive energy dissipation with the reversibility ofthe equation is not obvious. In the dispersion argument, one has to arbitrarily imposethe location of the singularities taking into account the arrow of time (via the Plemeljformula); then the spectral study requires some thinking. All in all, the most convincingargument remains Landauâs original one, since it is based only on the study of the Cauchyproblem, which makes more physical sense than the study of the dispersion relation (seethe remark in [9, p. 682]).
A more fundamental issue resides in the use of analytic function theory, with contourintegration, singularities and residue computation, which has played a major role in thetheory of the VlasovâPoisson equation ever since Landau [54, Chapter 32], see also [10,§5.2.4], and helps little, if at all, to understand the underlying physical mechanism.(4)
The most popular interpretation of Landau damping considers the phenomenon froman energetic point of view, as the result of the interaction of a plasma wave with particlesof nearby velocity [36, p. 18], [10, p. 412], [1, §4.2.3], [54, p. 127]. In a nutshell, theargument says that dominant exchanges occur with those particles which are âtrappedâby the wave because their velocity is close to the wave velocity. If the distributionfunction is a decreasing function of |v|, among trapped particles more are acceleratedthan are decelerated, so the wave loses energy to the plasmaâor the plasma surfs on thewaveâand the wave is damped by the interaction.
Appealing as this image may seem, to a mathematically-oriented mind it will prob-ably make little sense at first hearing.(5) A more down-to-earth interpretation emergedin the fifties from the âwave packetâ analysis of van Kampen [45] and Case [14]: Landaudamping would result from phase mixing. This phenomenon, well known in galactic dy-namics, describes the damping of oscillations occurring when a continuum is transported
(3) Not completely rigorous from the mathematical point of view, but formally correct, in contrastto the previous studies by Landauâs fellow physicistsâas Landau himself pointed out without mercy [52].
(4) van Kampen [45] summarizes the conceptual problems posed to his contemporaries by Landauâstreatment, and comments on more or less clumsy attempts to resolve the apparent paradox caused bythe singularities in the complex plane.
(5) Escande [25, Chapter 4, footnote 6] points out some misconceptions associated with the surferimage.
36 c. mouhot and c. villani
in phase space along an anharmonic Hamiltonian flow [10, pp. 379â380]. The mixingresults from the simple fact that particles following different orbits travel at differentangular(6) speeds, so perturbations start âspiralingâ (see Figure 4.27 on p. 379 in [10])and homogenize by fast spatial oscillation. From the mathematical point of view, phasemixing results in weak convergence; from the physical point of view, this is just the con-vergence of observables, defined as averages over the velocity space (this is sometimescalled âconvergence in the meanâ).
At first sight, both points of view seem hardly compatible: Landauâs scenario sug-gests a very smooth process, while phase mixing involves tremendous oscillations. Thecoexistence of these two interpretations did generate some speculation on the nature ofthe damping, and on its relation to phase mixing, see e.g. [46] or [10, p. 413]. There isactually no contradiction between the two points of view: many physicists have rightlypointed out that Landau damping should come with filamentation and oscillations ofthe distribution function [45, p. 962], [54, p. 141], [1, Vol. 1, pp. 223â224], [57, pp. 294â295]. Nowadays these oscillations can be visualized spectacularly due to deterministicnumerical schemes, see e.g. [95], [39, Figure 3] and [27]. In Figure 1.2 we reproduce someexamples provided by Filbet.
In any case, there is still no definite interpretation of Landau damping: as noted byRyutov [81, §9], papers devoted to the interpretation and teaching of Landau dampingwere still appearing regularly fifty years after its discovery; to quote just a couple ofmore recent examples let us mention works by Elskens and Escande [23], [24], [25]. Thepresent paper will also contribute a new point of view.
1.3. Range of validity
The following issues are addressed in the literature [42], [46], [61], [95] and slightly con-troversial:
âą Does Landau damping really hold for gravitational interaction? The case seemsthinner in this situation than for plasma interaction, all the more as there are manyinstability results in the gravitational context; up to now there has been no consensusamong mathematical physicists [79]. (Numerical evidence is not conclusive because ofthe difficulty of accurate simulations in very large timeâeven in one dimension of space.)
âą Does the damping hold for unbounded systems? Counterexamples from [30] and[31] show that some kind of confinement is necessary, even in the electrostatic case. Moreprecisely, Glassey and Schaeffer show that a solution of the linearized VlasovâPoisson
(6) âAngularâ here refers to action-angle variables, and applies even for straight trajectories in atorus.
on landau damping 37
4·10â5
3·10â5
2·10â5
10â5
0
â10â5
â2·10â5
â3·10â5
â4·10â5
â6 â4 â2 0 2 4 6
v
h(v)
t=0.16
4·10â5
3·10â5
2·10â5
10â5
0
â10â5
â2·10â5
â3·10â5
â4·10â5
â6 â4 â2 0 2 4 6
v
h(v)
t=2.00
Figure 1. A slice of the distribution function (relative to a homogeneous equilibrium) forgravitational Landau damping, at two different times.
equation in the whole space (linearized around a homogeneous equilibrium f0 of infinitemass) decays at best like O(tâ1), modulo logarithmic corrections, for f0(v)=c/(1+|v|2);and like O((log t)âα) if f0 is a Gaussian. In fact, Landauâs original calculations alreadyindicated that the damping is extremely weak at large wavenumbers; see the discussionin [54, §32]. Of course, in the gravitational case, this is even more dramatic because ofthe Jeans instability.
âą Does convergence hold in infinite time for the solution of the âfullâ non-linearequation? This is not clear at all since there is no mechanism that would keep thedistribution close to the original equilibrium for all times. Some authors do not believethat there is convergence as t!â; others believe that there is convergence but arguethat it should be very slow [42], say O(1/t). In the first mathematically rigorous studyof the subject, Backus [4] notes that in general the linear and non-linear evolution break
38 c. mouhot and c. villani
â4
â5
â6
â7
â8
â9
â10
â11
â12
â130 10 20 30 40 50 60 70 80 90 100
t
log E(t)
electric energy in log scale
â12
â13
â14
â15
â16
â17
â18
â19
â20
â21
â22
â230 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
t
log E(t)
gravitational energy in log scale
Figure 2. Time-evolution of the norm of the field, for electrostatic (left) and gravitational(right) interactions. Notice the fast Langmuir oscillations in the electrostatic case.
apart after some (not very large) time, and questions the validity of the linearization.(7)OâNeil [75] argues that relaxation holds in the âquasilinear regimeâ on larger time scales,when the âtrapping timeâ (roughly proportional the inverse square root of the size of theperturbation) is much smaller than the damping time. Other speculations and argumentsrelated to trapping appear in many sources, e.g. [61] and [64]. Kaganovich [44] arguesthat non-linear effects may quantitatively affect Landau damping related phenomena byseveral orders of magnitude.
The so-called âquasilinear relaxation theoryâ [54, §49], [1, §9.1.2], [49, Chapter 10]uses second-order approximation of the Vlasov equation to predict the convergence of thespatial average of the distribution function. The procedure is most esoteric, involving av-
(7) From the abstract: âThe linear theory predicts that in stable plasmas the neglected term willgrow linearly with time at a rate proportional to the initial disturbance amplitude, destroying the validityof the linear theory, and vitiating positive conclusions about stability based on it.â
on landau damping 39
eraging over statistical ensembles, and diffusion equations with discontinuous coefficients,acting only near the resonance velocity for particle-wave exchanges. Because of these dis-continuities, the predicted asymptotic state is discontinuous, and collisions are invokedto restore smoothness. Linear FokkerâPlanck equations(8) in velocity space have alsobeen used in astrophysics [58, p. 111], but only on phenomenological grounds (the ad-hocaddition of a friction term leading to a Gaussian stationary state); and this procedurehas been exported to the study of 2-dimensional incompressible fluids [15], [16].
Even if it were more rigorous, quasilinear theory only aims at second-order cor-rections, but the effect of higher-order perturbations might be even worse. Think ofsomething like
eâtâ
nâN0
Δntnân!
(where N0={0, 1, 2, ... }), then truncation at any order in Δ converges exponentially fastas t!â, but the whole sum diverges to infinity.
Careful numerical simulation [95] seems to show that the solution of the non-linearVlasovâPoisson equation does converge to a spatially homogeneous distribution, but onlyas long as the size of the perturbation is small enough. We shall call this phenomenonnon-linear Landau damping. This terminology summarizes the problem well, still it issubject to criticism since (a) Landau himself sticked to the linear case and did not discussthe large-time convergence of the distribution function; (b) damping is expected to holdwhen the regime is close to linear, but not necessarily when the non-linear term domi-nates;(9) and (c) this expression is also used to designate related but different phenomena[1, §10.1.3]. It should be kept in mind that in the present paper, non-linearity does mani-fest itself, not because there is a significant initial departure from equilibrium (our initialdata will be very close to equilibrium), but because we are addressing very large times,and this is all the more tricky to handle, as the problem is highly oscillating.
⹠Is Landau damping related to the more classical notion of stability in orbitalsense? Orbital stability means that the system, slightly perturbed at initial time froman equilibrium distribution, will always remain close to this equilibrium. Even in thefavorable electrostatic case, stability is not granted; the most prominent phenomenonbeing the Penrose instability [77] according to which a distribution with two deep bumpsmay be unstable. In the more subtle gravitational case, various stability and instabilitycriteria are associated with the names of Chandrasekhar, Antonov, Goodman, Doremus,Feix, Baumann, etc. [10, §7.4]. There is a widespread agreement (see e.g. the comments
(8) These equations act on some ensemble average of the distribution; they are different from theVlasovâLandau equation.
(9) Although phase mixing might still play a crucial role in violent relaxation or other unclassifiednon-linear phenomena.
40 c. mouhot and c. villani
in [95]) that Landau damping and stability are related, and that Landau damping cannotbe hoped for if there is no orbital stability.
1.4. Conceptual problems
Summarizing, we can identify three main conceptual obstacles which make Landau damp-ing mysterious, even sixty years after its discovery:
(i) The equation is time-reversible, yet we are looking for an irreversible behavior ast!â (or t!ââ). The value of the entropy does not change in time, which physicallyspeaking means that there is no loss of information in the distribution function. Thespectacular experiment of the âplasma echoâ illustrates this conservation of microscopicinformation [32], [60]: a plasma which is apparently back to equilibrium after an initialdisturbance, will react to a second disturbance in a way that shows that it has notforgotten the first one.(10) And at the linear level, if there are decaying modes, therealso have to be growing modes!
(ii) When one perturbs an equilibrium, there is no mechanism forcing the systemto go back to this equilibrium in large time; so there is no justification in the use oflinearization to predict the large-time behavior.
(iii) At the technical level, Landau damping (in Landauâs own treatment) rests onanalyticity, and its most attractive interpretation is in terms of phase mixing. But bothphenomena are incompatible in the large-time limit : phase mixing implies an irreversibledeterioration of analyticity. For instance, it is easily checked that free transport inducesan exponential growth of analytic norms as t!ââexcept if the initial datum is spatiallyhomogeneous. In particular, the VlasovâPoisson equation is unstable (in large time) inany norm incorporating velocity regularity. (Space-averaging is one of the ingredientsused in the quasilinear theory to formally get rid of this instability.)
How can we respond to these issues?One way to solve the first problem (time-reversibility) is to appeal to van Kampen
modes as in [10, p. 415]; however these are not so physical, as noticed in [9, p. 682]. Asimpler conceptual solution is to invoke the notion of weak convergence: reversibilitymanifests itself in the conservation of the information contained in the density function;but information may be lost irreversibly in the limit when we consider weak convergence.Weak convergence only describes the long-time behavior of arbitrary observables, each ofwhich does not contain as much information as the density function.(11) As a very simple
(10) Interestingly enough, this experiment was suggested as a way to evaluate the strength of ir-reversible phenomena going on inside a plasma, e.g. the collision frequency, by measuring attenuationswith respect to the predicted echo. See [86] for an interesting application and striking pictures.
(11) In Lynden-Bellâs appealing words [57, p. 295], âa system whose density has achieved a steady
on landau damping 41
illustration, consider the time-reversible evolution defined by u(t, x)=eitxui(x), and no-tice that it does converge weakly to 0 as t!±â; this convergence is even exponentiallyfast if the initial datum ui is analytic. (Our example is not chosen at random: althoughit is extremely simple, it may be a good illustration of what happens in phase mixing.) Ina way, microsocopic reversibility is compatible with macroscopic irreversibility, providedthat the âmicroscopic regularityâ is destroyed asymptotically.
Still in respect to this reversibility, it should be noted that the âdualâ mechanismof radiation, according to which an infinite-dimensional system may lose energy towardsvery large scales, is relatively well understood and recognized as a crucial stability mech-anism [3], [85].
The second problem (lack of justification of the linearization) only indicates thatthere is a wide gap between the understanding of linear Landau damping, and that ofthe non-linear phenomenon. Even if unbounded corrections appear in the linearizationprocedure, the effect of the large terms might be averaged over time or other variables.
The third problem, maybe the most troubling from an analystâs perspective, does notdismiss the phase mixing explanation, but suggests that we shall have to keep track of theinitial time, in the sense that a rigorous proof cannot be based on the propagation of somephenomenon. This situation is of course in sharp contrast with the study of dissipativesystems possessing a Lyapunov functional, as do many collisional kinetic equations [92],[93]; it will require completely different mathematical techniques.
1.5. Previous mathematical results
At the linear level, the first rigorous treatments of Landau damping were performed inthe sixties; see Saenz [82] for rather complete results and a review of earlier works. Thetheory was rediscovered and renewed at the beginning of the eighties by Degond [20], andMaslov and Fedoryuk [63]. In all these works, analytic arguments play a crucial role (forinstance for the analytic extension of resolvent operators), and asymptotic expansionsfor the electric field associated with the linearized VlasovâPoisson equation are obtained.
Also at the linearized level, there are counterexamples by GlasseyâSchaeffer [30], [31]showing that there is in general no exponential decay for the linearized VlasovâPoissonequation without analyticity, or without confining.
In a non-linear setting, the only rigorous treatments so far are those by CagliotiâMaffei [13], and later HwangâVelazquez [41]. Both sets of authors work in the 1-dimensional torus and use fixed-point theorems and perturbative arguments to provethe existence of a class of analytic solutions behaving, asymptotically as t!â, and in a
state will have information about its birth still stored in the peculiar velocities of its stars.â
42 c. mouhot and c. villani
strong sense, like perturbed solutions of free transport. Since solutions of free transportweakly converge to spatially homogeneous distributions, the solutions constructed by thisâscatteringâ approach are indeed damped. The weakness of these results is that they saynothing about the initial perturbations leading to such solutions, which could be veryspecial. In other words: damped solutions do exist, but do we ever reach them?
Sparse as it may seem, this list is kind of exhaustive. On the other hand, there is arather large mathematical literature on the orbital stability problem, due to Guo, Rein,Strauss, Wolansky and LemouâMehatsâRaphael. In this respect see for instance [35] forthe plasma case, and [34] and [53] for the gravitational case; these sources contain manyreferences on the subject. This body of works has confirmed the intuition of physicists,although with quite different methods. The gap between a formal, linear treatment anda rigorous, non-linear one is striking: compare the appendix of [34] to the rest of thepaper. In the gravitational case, these works do not consider homogeneous equilibria,but only localized solutions.
Our treatment of Landau damping will be performed from scratch, and will not relyon any of these results.
2. Main result
2.1. Modeling
We shall work in adimensional units throughout the paper, in d dimensions of space andd dimensions of velocity (dâN={1, 2, ... }).
As should be clear from our presentation in §1, to observe Landau damping, weneed to put a restriction on the length scale (anyway plasmas in experiments are usuallyconfined). To achieve this we shall take the position space to be the d-dimensional torusof sidelength L, namely Td
L=Rd/LZd. This is admittedly a bit unrealistic, but it iscommonly done in plasma physics (see e.g. [5]).
In a periodic setting the Poisson equation has to be reinterpreted, since Îâ1ïżœ isnot well defined unless
â«Td
Lïżœ dx=0. The natural solution consists of removing the mean
value of ïżœ, independently of any âneutralityâ assumption. Let us sketch a justificationin the important case of Coulomb interaction: due to the screening phenomenon, wemay replace the Coulomb potential V by a potential V exhibiting a âcutoffâ at largedistances (typically V could be of Debye type [5]; anyway the choice of approximationhas no influence on the result). If âV âL1(Rd), then âV âïżœ makes sense for a periodicïżœ, and moreover
(âV âïżœ)(x) =â«
Rd
âV (xây)ïżœ(y) dy =â«
[0,L]dâV (L)(xây)ïżœ(y) dy,
on landau damping 43
where V(L)(z)=
âlâZd V (z+lL). Passing to the limit as !0 yields
â«[0,L]d
âV (L)(xây)ïżœ(y) dy =â«
[0,L]dâV (L)(xây)(ïżœâăïżœă)(y) dy =ââÎâ1
L (ïżœâăïżœă),
where Îâ1L is the inverse Laplace operator on Td
L.
In the case of galactic dynamics there is no screening; however it is customary toremove the zeroth-order term of the density. This is known as the Jeans swindle, a trickconsidered as efficient but logically absurd. In 2003, Kiessling [47] reopened the caseand acquitted Jeans, on the basis that his âswindleâ can be justified by a simple limitprocedure, similar to the one presented above; however, the physical basis for the limit isless transparent and subject to debate. For our purposes, it does not matter much: sinceanyway periodic boundary conditions are not realistic in a cosmological setting, we mayjust as well say that we adopt the Jeans swindle as a simple phenomenological model.
More generally, we may consider any interaction potential W on TdL, satisfying the
natural symmetry assumption W (âz)=W (z) (that is, W is even), as well as certainregularity assumptions. Then the self-consistent field will be given by
F =ââW âïżœ, ïżœ(x) =â«
Rd
f(x, v) dv,
where now â denotes the convolution on TdL.
In accordance with our conventions from Appendix A.3, we shall write
W (L)(k) =â«
TdL
eâ2iÏk·x/LW (x) dx.
In particular, if W is the periodization of a potential Rd!R (still denoted W by abuseof notation), i.e.,
W (x) =W (L)(x) =âlâZd
W (x+lL),
then
W (L)(k) = W
(k
L
), (2.1)
where
W (Ο) =â«
Rd
eâ2iÏΟ·xW (x) dx
is the original Fourier transform in the whole space.
44 c. mouhot and c. villani
2.2. Linear damping
It is well known that Landau damping requires some stability assumptions on the unper-turbed homogeneous distribution function, say f0(v). In this paper we shall use a verygeneral assumption, expressed in terms of the Fourier transform
f 0(η) =â«
Rd
eâ2iÏη·vf0(v) dv, (2.2)
the length L, and the interaction potential W . To state it, we define, for tïżœ0 and kâZd,
K0(t, k) =â4Ï2W (L)(k)f 0
(kt
L
) |k|2tL2
; (2.3)
and, for any ΟâC, we define a function L via the following FourierâLaplace transform ofK0 in the time variable:
L(Ο, k) =â« â
0
e2ÏΟâ|k|t/LK0(t, k) dt, (2.4)
where Οâ is the complex conjugate to Ο. Our linear damping condition is expressed asfollows:
(L) There are constants C0, λ, >0 such that |f 0(η)|ïżœC0eâ2Ïλ|η| for any ηâRd;
and for any ΟâC with 0ïżœRe Ο<λ,
infkâZd
|L(Ο, k)â1|ïżœ .
We shall prove in §3 that (L) implies Landau damping. For the moment, let us givea few sufficient conditions for (L) to be satisfied. The first one can be thought of as asmallness assumption on either the length, or the potential, or the velocity distribution.The other conditions involve the marginals of f0 along arbitrary wave vectors k:
Ïk(v) =â«
kv/|k|+kâ„f0(w) dw, v âR. (2.5)
All studies known to us are based on one of these assumptions, so (L) appears as aunifying condition for linear Landau damping around a homogeneous equilibrium.
Proposition 2.1. Let f0=f0(v) be a velocity distribution such that f 0 decays ex-ponentially fast at infinity, let L>0 and let W be an even interaction potential on Td
L,WâL1(Td). If any of the following conditions is satisfied :
(a) smallness:
4Ï2(
maxkâZdâ
|W (L)(k)|)
sup|Ï|=1
â« â
0
|f 0(rÏ)|r dr < 1; (2.6)
on landau damping 45
(b) repulsive interaction and decreasing marginals: for all kâZd and vâR,
W (L)(k) ïżœ 0 and ÏâČk(v){
> 0, if v < 0,< 0, if v > 0,
(2.7)
(c) generalized Penrose condition on marginals: for all kâZd,
ÏâČk(w) = 0 =â W (L)(k)
(p.v.â«
R
ÏâČk(v)
vâwdv
)< 1 for all wâR; (2.8)
then (L) holds true for some C0, λ, >0.
Remark 2.2. ([54, problem in §30]) If f0 is radially symmetric and positive, anddïżœ3, then all marginals of f0 are decreasing functions of |v|. Indeed, if
Ï(v) =â«
Rdâ1f(â
v2+|w|2 ) dw,
then after differentiation and integration by parts we find that
ÏâČ(v) =
â§âšâ© â(dâ3)vâ«
Rdâ1f(â
v2+|w|2 ) dw
|w|2 , if d ïżœ 4,
â2Ïvf(|v|), if d = 3.
Example 2.3. Take a gravitational interaction and Mawellian background:
W (k) =â GÏ|k|2 and f0(v) = ïżœ0 eâ|v|2/2T
(2ÏT )d/2.
Recalling (2.1), we see that (2.6) becomes
L<
âÏT
Gïżœ0=:LJ(T, ïżœ0). (2.9)
The length LJ is the celebrated Jeans length [10], [47], so criterion (a) can be applied,all the way up to the onset of the Jeans instability.
Example 2.4. If we replace the gravitational interaction by the electrostatic interac-tion, the same computation yields
L<
âÏT
e2ïżœ0=:LD(T, ïżœ0), (2.10)
and now LD is essentially the Debye length. Then criterion (a) becomes quite restrictive,but because the interaction is repulsive we can use criterion (b) as soon as f0 is a strictlymonotone function of |v|; this covers in particular Maxwellian distributions, independentlyof the size of the box. Criterion (b) also applies if dïżœ3 and f0 has radial symmetry. For agiven L>0, the condition (L) being open, it will also be satisfied if f0 is a small (analytic)perturbation of a profile satisfying (b); this includes the so-called âsmall bump on tailâstability. Then if the distribution presents two large bumps, the Penrose instability willtake over.
46 c. mouhot and c. villani
Example 2.5. For the electrostatic interaction in dimension 1, (2.8) becomes
(f0)âČ(w) = 0 =ââ«
R
(f0)âČ(v)vâw
dv <Ï
e2L2. (2.11)
This is a variant of the Penrose stability condition [77]. This criterion is in general sharpfor linear stability (up to the replacement of the strict inequality by the non-strict one,and assuming that the critical points of f0 are non-degenerate); see [55, Appendix] forprecise statements.
We shall show in §3 that (L) implies linear Landau damping (Theorem 3.1); thenwe shall prove Proposition 2.1 at the end of that section. The general ideas are closeto those appearing in previous works, including Landau himself; the only novelties lie inthe more general assumptions, the elementary nature of the arguments, and the slightlymore precise quantitative results.
2.3. Non-linear damping
As others have done before in the study of the VlasovâPoisson equation [13], we shallquantify the analyticity by means of natural norms involving Fourier transform in bothvariables (also denoted with a tilde in the sequel). So we define
âfâλ,ÎŒ = supkâZd
ηâRd
|f (L)(k, η)|e2Ïλ|η|e2ÏÎŒ|k|/L, (2.12)
where k varies in Zd, ηâRd, λ and ÎŒ are positive parameters, and we recall the dependenceof the Fourier transform on L (see Appendix A.3 for conventions). Now we can state ourmain result as follows.
Theorem 2.6. (Non-linear Landau damping) Let f0: Rd!R+ be an analytic veloc-ity profile. Let L>0 and let W : Td
L!R be an even interaction potential satisfying
|W (L)(k)|ïżœ CW
|k|1+Îłfor all kâZd (2.13)
for some constants CW >0 and Îłïżœ1. Assume that f0 and W satisfy the stability con-dition (L) from §2.2, with some constants λ, >0; further assume that, for the sameparameter λ,
supηâRd
|f 0(η)|e2Ïλ|η| ïżœC0 andâ
nâNd0
λn
n!âân
v f0âL1(Rd) ïżœC0 <â. (2.14)
on landau damping 47
Then for any 0<λâČ<λ, ÎČ>0 and 0<ÎŒâČ<ÎŒ, there is
Δ = Δ(d, L, CW , C0, , λ, λâČ, ÎŒ, ÎŒâČ, ÎČ, Îł)
with the following property : if fi=fi(x, v) is an initial datum satisfying
ÎŽ := âfiâf0âλ,ÎŒ+â«
TdL
â«Rd
|fiâf0|eÎČ|v| dv dx ïżœ Δ, (2.15)
thenâą the unique classical solution f to the non-linear Vlasov equation
âf
ât+v ·âxfâ(âW âïżœ)·âvf = 0, ïżœ=
â«Rd
f dv, (2.16)
with initial datum f(0, ·)=fi, converges in the weak topology as t!±â, with rateO(eâ2ÏλâČ|t|), to a spatially homogeneous equilibrium f±â (that is, it converges to fâas t!â, and to fââ as t!ââ);
âą the distribution function composed with the backward free transport f(t, x+vt, v)converges strongly to f±â as t!±â;
âą the density ïżœ(t, x)=â«
Rd f(t, x, v) dv converges in the strong topology as t!±â,with rate O(eâ2ÏλâČ|t|), to the constant density
ïżœâ =1Ld
â«Td
L
â«Rd
fi(x, v) dv dx;
in particular the force F =ââW âïżœ converges exponentially fast to 0;âą the space average ăfă(t, v)=
â«Td
Lf(t, x, v) dx converges in the strong topology as
t!±â, with rate O(eâ2ÏλâČ|t|), to f±â.More precisely, there are C>0, and spatially homogeneous distributions fâ(v) and
fââ(v), depending continuously on fi and W , such that
suptâR
âf(t, x+vt, v)âf0(v)âλâČ,ÎŒâČ ïżœCÎŽ,
|f±â(η)âf 0(η)|ïżœCÎŽeâ2ÏλâČ|η| for all η âRd
(2.17)
and
|Lâdf (L)(t, k, η)âf±â(η)1k=0|= O(eâ2ÏλâČ|t|/L) as t!±â, for all (k, η)âZdĂRd,
âf(t, x+vt, v)âf±â(v)âλâČ,ÎŒâČ = O(eâ2ÏλâČ|t|/L) as t!±â,
âïżœ(t, ·)âïżœââCr(Td) = O(eâ2ÏλâČ|t|/L) as |t|!â, for all râN, (2.18)
âF (t, ·)âCr(T1d) = O(eâ2ÏλâČ|t|/L) as |t|!â, for all râN, (2.19)
âăf(t, · , v)ăâf±ââCrÏ(Rd
v) = O(eâ2ÏλâČ|t|/L) as t!±â, for all râN and Ï > 0.
(2.20)
48 c. mouhot and c. villani
In this statement Cr stands for the usual norm on r-times continuously differentiablefunctions, and Cr
Ï involves in addition moments of order Ï, namely
âfâCrÏ
= suprâČïżœr
vâRd
|f (râČ)(v)(1+|v|Ï)|.
These results can be reformulated in a number of alternative norms, both for the strongand for the weak topology.
2.4. Comments
Let us start with a list of remarks about Theorem 2.6.âą The decay of the force field, statement (2.19), is the experimentally measurable
phenomenon which may be called Landau damping.âą Since the energy
E =12
â«Td
L
â«Td
L
ïżœ(x)ïżœ(y) W (xây) dx dy+â«
TdL
â«Rd
f(x, v)|v|22
dv dx
(= potential + kinetic energy) is conserved by the non-linear Vlasov evolution, there isa conversion of potential energy into kinetic energy as t!â (kinetic energy goes up forCoulomb interaction and goes down for Newton interaction). Similarly, the entropy
S =ââ«
TdL
â«Rd
f log f dv dx =â(â«
TdL
ïżœ log ïżœ dx+â«
TdL
â«Rd
f logf
ïżœdv dx
)(= spatial + kinetic entropy) is preserved, and there is a transfer of information fromspatial to kinetic variables in large time.
âą Our result covers both attractive and repulsive interactions, as long as the lineardamping condition is satisfied; it covers the NewtonâCoulomb potential as a limit case(Îł=1 in (2.13)). The proof breaks down for Îł<1; this is a non-linear effect, as any Îł>0would work for the linearized equation. The singularity of the interaction at short scaleswill be the source of important technical problems.(12)
âą Condition (2.14) could be replaced by
|f 0(η)|ïżœC0eâ2Ïλ|η| and
â«Rd
f0(v)eÎČ|v| dv ïżœC0. (2.21)
(12) In a related subject, this singularity is also the reason why the VlasovâPoisson equation is stillfar from being established as a mean-field limit of particle dynamics (see [37] for partial results coveringmuch less singular interactions).
on landau damping 49
But condition (2.14) is more general, in view of Theorem 4.20 below. For instance,f0(v)=1/(1+v2) in dimension d=1 satisfies (2.14) but not (2.21); this distribution iscommonly used in theoretical and numerical studies, see e.g. [39]. We shall also establishslightly more precise estimates under slightly more stringent conditions on f0, see (12.1).
âą Our conditions are expressed in terms of the initial datum, which is a considerableimprovement over [13] and [41]. Still it is of interest to pursue the âscatteringâ programstarted in [13], e.g. in hope of better understanding of the non-perturbative regime.
âą The smallness assumption on fiâf0 is expected, for instance in view of the workof OâNeil [75], or the numerical results of [95]. We also make the standard assumptionthat fiâf0 is well localized.
âą No convergence can be hoped for if the initial datum is only close to f0 in the weaktopology: indeed there is instability in the weak topology, even around a Maxwellian [13].
âą The well-posedness of the non-linear VlasovâPoisson equation in dimension dïżœ3was established by Pfaffelmoser [78] and LionsâPerthame [56] in the whole space. Pfaf-felmoserâs proof was adapted to the case of the torus by Batt and Rein [6]; however, likePfaffelmoser, these authors imposed a stringent assumption of uniformly bounded veloc-ities. Building on Schaefferâs simplification [84] of Pfaffelmoserâs argument, Horst [40]proved well-posedness in the whole space assuming only inverse polynomial decay in thevelocity variable. Although this has not been done explicitly, Horstâs proof can easilybe adapted to the case of the torus, and covers in particular the setting which we use inthe present paper. (The adaptation of [56] seems more delicate.) Propagation of ana-lytic regularity is not studied in these works. In any case, our proof will provide a newperturbative existence theorem, together with regularity estimates which are consider-ably stronger than what is needed to prove the uniqueness. We shall not come back tothese issues which are rather irrelevant for our study: uniqueness only needs local-in-timeregularity estimates, while all the difficulty in the study of Landau damping consists inhandling (very) large time.
âą We note in passing that while blow-up is known to occur for certain solutions ofthe Newtonian VlasovâPoisson equation in dimension 4, blow-up does not occur in thisperturbative regime, whatever the dimension. There is no contradiction since blow-upsolutions are constructed with negative energy initial data, and a nearly homogeneous so-lution automatically has positive energy. (Also, blow-up solutions have been constructedonly in the whole space, where the virial identity is available; but it is plausible, althoughnot obvious, that blow-up is still possible in bounded geometry.)
âą f(t, ·) is not close to f0 in analytic norm as t!â, and does not converge toanything in the strong topology, so the conclusion cannot be improved much. Still weshall establish more precise quantitative results, and the limit profiles f±â are obtained
50 c. mouhot and c. villani
by a constructive argument.âą Estimate (2.17) expresses the orbital âtraveling stabilityâ around f0; it is much
stronger than the usual orbital stability in Lebesgue norms [34], [35]. An equivalentformulation is that if (Tt)tâR stands for the non-linear Vlasov evolution operator, and(T 0
t )tâR for the free transport operator, then in a neighborhood of a homogeneous equi-librium satisfying the stability criterion (L), T 0
ât Tt remains uniformly close to Id for allt. Note the important difference: unlike in the usual orbital stability theory, our conclu-sions are expressed in functional spaces involving smoothness, which are not invariantunder the free transport semigroup. This a source of difficulty (our functional spaces aresensitive to the filamentation phenomenon), but it is also the reason for which this âan-alyticâ orbital stability contains much more information, and in particular the dampingof the density.
âą Compared with known non-linear stability results, and even forgetting about thesmoothness, estimate (2.17) is new in several respects. In the context of plasma physics,it is the first one to prove stability for a distribution which is not necessarily a decreasingfunction of |v| (âsmall bump on tailâ); while in the context of astrophysics, it is the firstone to establish stability of a homogeneous equilibrium against periodic perturbationswith wavelength smaller than the Jeans length.
⹠While analyticity is the usual setting for Landau damping, both in mathematicaland physical studies, it is natural to ask whether this restriction can be dispensed with.(This can be done only at the price of losing the exponential decay.) In the linear case,this is easy, as we shall recall later in Remark 3.5; but in the non-linear setting, leavingthe analytic world is much more tricky. In §13, we shall present the first results in thisdirection.
With respect to the questions raised above, our analysis brings the following answers:(a) Convergence of the distribution f does hold for t!â; it is indeed based on phase
mixing, and therefore involves very fast oscillations. In this sense it is right to considerLandau damping as a âwildâ process. But on the other hand, the spatial density (andtherefore the force field) converges strongly and smoothly.
(b) The space average ăfă does converge in large time. However the conclusions arequite different from those of quasilinear relaxation theory, since there is no need for extrarandomness, and the limiting distribution is smooth, even without collisions.
(c) Landau damping is a linear phenomenon, which survives non-linear perturbationdue to the structure of the VlasovâPoisson equation. The non-linearity manifests itselfby the presence of echoes. Echoes were well known to specialists of plasma physics [54,§35], [1, §12.7], but were not identified as a possible source of unstability. Controllingthe echoes will be a main technical difficulty; but the fact that the response appears in
on landau damping 51
this form, with an associated time-delay and localized in time, will in the end explainthe stability of Landau damping. These features can be expected in other equationsexhibiting oscillatory behavior.
(d) The large-time limit is in general different from the limit predicted by the lin-earized equation, and depends on the interaction and initial datum (more precise state-ments will be given in §14); still the linearized equation, or higher-order expansions, doprovide a good approximation. We shall also set up a systematic recipe for approximat-ing the large-time limit with arbitrarily high precision as the strength of the perturbationbecomes small. This justifies a posteriori many known computations.
(e) From the point of view of dynamical systems, the non-linear Vlasov equationexhibits a truly remarkable behavior. It is not uncommon for a Hamiltonian system tohave many, or even countably many heteroclinic orbits (there are various theories forthis, a popular one being the Melnikov method); but in the present case we see thatheteroclinic/homoclinic orbits(13) are so numerous as to fill up a whole neighborhoodof the equilibrium. This is possible only because of the infinite-dimensional nature ofthe system, and the possibility to work with non-equivalent norms; such a behavior hasalready been reported for other systems [50], [51], in relation with infinite-dimensionalKAM (KolmogorovâArnoldâMoser) theory.
(f) As a matter of fact, non-linear Landau damping has strong similarities with theKAM theory. It has been known since the early days of the theory that the linearizedVlasov equation can be reduced to an infinite system of uncoupled Volterra equations,which makes this equation completely integrable in some sense. (Morrison [66] gave amore precise meaning to this property.) To see a parallel with classical KAM theory,one step of our result is to prove the preservation of the phase-mixing property undernon-linear perturbation of the interaction. (Although there is no ergodicity in phasespace, the mixing will imply an ergodic behavior for the spatial density.) The analogyis reinforced by the fact that the proof of Theorem 2.6 shares many features with theproof of the KAM theorem in the analytic (or Gevrey) setting. (Our proof is closeto Kolmogorovâs original argument, exposed in [18].) In particular, we shall invoke aNewton scheme to overcome a loss of âregularityâ in analytic norms, only in a trickiersense than in KAM theory. If one wants to push the analogy further, one can argue thatthe resonances which cause the phenomenon of small divisors in KAM theory find ananalogue in the time-resonances which cause the echo phenomenon in plasma physics.A notable difference is that in the present setting, time-resonances arise from the non-linearity at the level of the partial differential equation, whereas small divisors in KAM
(13) Here we use these words just to designate solutions connecting two distinct/equal equilibria,without any mention of stable or unstable manifolds.
52 c. mouhot and c. villani
theory arise at the level of the ordinary differential equation. Another major differenceis that in the present situation there is a time-averaging which is not present in KAMtheory.
Thus we see that three of the most famous paradoxical phenomena from twentiethcentury classical physics: Landau damping, echoes and the KAM theorem are intimatelyrelated (only in the non-linear variant of Landauâs linear argument!). This relation, whichwe did not expect, is one of the main discoveries of the present paper.
2.5. Interpretation
A successful point of view adopted in this paper is that Landau damping is a relaxationby smoothness and by mixing. In a way, phase mixing converts the smoothness intodecay. Thus Landau damping emerges as a rare example of a physical phenomenon inwhich regularity is not only crucial from the mathematical point of view, but also canbe âmeasuredâ by a physical experiment.
2.6. Main ingredients
Some of our ingredients are similar to those in [13]: in particular, the use of the Fouriertransform to quantify analytic regularity and to implement phase mixing. New ingredi-ents used in our work include the following.
⹠The introduction of a time-shift parameter to keep memory of the initial time(§4 and §5), thus getting uniform estimates in spite of the loss of regularity in largetime. We call this the gliding regularity : it shifts in phase space from low to high modes.Gliding regularity automatically comes with an improvement of the regularity in x, anda deterioration of the regularity in v, as time passes by.
⹠The use of carefully designed flexible analytic norms behaving well with respectto composition (§4). This requires care, because analytic norms are very sensitive tocomposition, contrary to, say, Sobolev norms.
⹠A control of the deflection of trajectories induced by the force field, to reduce theproblem to homogenization of free flow (§5) via composition. The physical meaning isthe following: when a background with gliding regularity acts on (say) a plasma, thetrajectories of plasma particles are asymptotic to free transport trajectories.
⹠New functional inequalities of bilinear type, involving analytic functional spaces,integration in time and velocity variables, and evolution by free transport (§6). Theseinequalities morally mean the following: when a plasma acts (by forcing) on a smoothbackground of particles, the background reacts by lending a bit of its (gliding) regularity
on landau damping 53
to the plasma, uniformly in time. Eventually the plasma will exhaust itself (the forcewill decay). This most subtle effect, which is at the heart of Landauâs damping, will bemathematically expressed in the formalism of analytic norms with gliding regularity.
âą A new analysis of the time response associated with the VlasovâPoisson equation(§7), aimed ultimately at controlling the self-induced echoes of the plasma. For anyinteraction less singular than that of CoulombâNewton, this will be done by analyzingtime-integral equations involving a norm of the spatial density. To treat the CoulombâNewton potential we shall refine the analysis, considering individual modes of the spatialdensity.
⹠A Newton iteration scheme, solving the non-linear evolution problem as a succes-sion of linear ones (§10). Picard iteration schemes still play a role, since they are run ateach step of the iteration process, to estimate the deflection.
It is only in the linear study of §3 that the length scale L will play a crucial role,via the stability condition (L). In all the rest of the paper we shall normalize L to 1 forsimplicity.
2.7. About phase mixing
A physical mechanism transferring energy from large scales to very fine scales, asymp-totically in time, is sometimes called weak turbulence. Phase mixing provides such amechanism, and in a way our study shows that the VlasovâPoisson equation is subject toweak turbulence. But the phase mixing interpretation provides a more precise picture.While one often sees weak turbulence as a âcascadeâ from low to high Fourier modes,the relevant picture would rather be a 2-dimensional figure with an interplay betweenspatial Fourier modes and velocity Fourier modes. More precisely, phase mixing transfersthe energy from each non-zero spatial frequency k, to large velocity frequences η, andthis transfer occurs at a speed proportional to k. This picture is clear from the solutionof free transport in Fourier space, and is illustrated in Figure 3. (Note the resemblancewith a shear flow.) So there is transfer of energy from one variable (here x) to another(here v); homogenization in the first variable going together with filamentation in thesecond one. The same mechanism may also underlie other cases of weak turbulence.
The fact that the high modes are ultimately damped by some ârandomâ micro-scopic process (collisions, diffusion, etc.) not described by the VlasovâPoisson equationis certainly undisputed in plasma physics [54, §41],(14) but is the object of debate ingalactic dynamics; anyway this is a different story. Some mathematical statistical theo-
(14) See [54, Problem 41]: due to Landau damping, collisions are expected to smooth the distributionquite efficiently; this is a hypoellipticity issue.
54 c. mouhot and c. villani
initial configuration (t=0)
âη
(kinetic modes)
k(spatial modes)
t=t1 t=t2 t=t3
Figure 3. Schematic picture of the evolution of energy by free transport, or perturbationthereof; marks indicate localization of energy in phase space. The energy of the spatial modek is concentrated in large time around Î·ïżœâkt.
0
0.1
0.2
0.3
0.4
â6â4
â20
24
6x
v
Figure 4. The distribution function in phase space (position, velocity) at a given time; noticehow the fast oscillations in v contrast with the slower variations in x.
ries of Euler and VlasovâPoisson equations do postulate the existence of some small-scalecoarse graining mechanism, but resulting in mixing rather than dissipation [80], [90].
3. Linear damping
In this section we establish Landau damping for the linearized Vlasov equation. Before-hand, let us recall that the free transport equation
âf
ât+v ·âxf = 0 (3.1)
has a strong mixing property: any solution of (3.1) converges weakly in large time toa spatially homogeneous distribution equal to the space-averaging of the initial datum.
on landau damping 55
Let us sketch the proof.If f solves (3.1) in TdĂRd, with initial datum fi=f(0, ·), then
f(t, x, v) = fi(xâvt, v),
so the space-velocity Fourier transform of f is given by the formula
f(t, k, η) = fi(k, η+kt). (3.2)
On the other hand, if fâ is defined by
fâ(v) = ăfi( · , v)ă=â«
Td
fi(x, v) dx,
then fâ(k, η)=fi(0, η)1k=0. So, by the RiemannâLebesgue lemma, for any fixed (k, η)we have
|f(t, k, η)âfâ(k, η)|! 0 as |t|!â,
which shows that f converges weakly to fâ. The convergence holds as soon as fi ismerely integrable; and by (3.2), the rate of convergence is determined by the decay offi(k, η) as |η|!â, or equivalently the smoothness in the velocity variable. In particular,the convergence is exponentially fast if (and only if) fi(x, v) is analytic in v.
This argument obviously works independently of the size of the box. But when weturn to the Vlasov equation, length scales will matter, so we shall introduce a length L>0,and work in Td
L=Rd/LZd. Then the length scale will appear in the Fourier transform:see Appendix A.3. (This is the only section in this paper where the scale will play anon-trivial role, so in all the rest of the paper we shall take L=1.)
Any velocity distribution f0=f0(v) defines a stationary state for the non-linearVlasov equation with interaction potential W . Then the linearization of that equationaround f0 yields â§âȘâȘâšâȘâȘâ©
âf
ât+v ·âxfâ(âW âïżœ)·âvf0 = 0,
ïżœ =â«
Rd
f dv.(3.3)
Note that there is no force term in (3.3), due to the fact that f0 does not dependon x. This equation describes what happens to a plasma density f which tries to forcea stationary homogeneous background f0; equivalently, it describes the reaction exertedby the background which is acted upon. (Imagine that there is an exchange of matterbetween the forcing gas and the forced gas, and that this exchange exactly compensatesthe effect of the force, so that the density of the forced gas does not change after all.)
56 c. mouhot and c. villani
Theorem 3.1. (Linear Landau damping) Let f0=f0(v), L>0, W : TdL!R be such
that W (âz)=W (z) and ââWâL1 ïżœCW <â and let fi=fi(x, v) be such that(i) condition (L) from §2.2 holds for some constants λ, >0;(ii) for all ηâRd, |f 0(η)|ïżœC0e
â2Ïλ|η| for some constant C0>0;(iii) for all kâZd and all ηâRd, |f (L)
i (k, η)|ïżœCieâ2Ïα|η| for some α, Ci>0.
Then, as t!â, the solution f(t, ·) to the linearized Vlasov equation (3.3) withinitial datum fi converges weakly to fâ=ăfiă defined by
fâ(v) =1Ld
â«Td
L
fi(x, v) dx;
and ïżœ(x)=â«
Rd f(x, v) dv converges strongly to the constant
ïżœâ =1Ld
â«Td
L
â«Rd
fi(x, v) dv dx.
More precisely, for any λâČ<min{λ, α},{âïżœ(t, ·)âïżœââCr = O(eâ2ÏλâČ|t|/L) for all râN,
|f (L)(t, k, η)âf(L)â (k, η)|= O(eâ2ÏλâČ|kt|/L) for all (k, η)âZdĂZd.
Remark 3.2. Even if the initial datum is more regular than analytic, the convergencewill in general not be better than exponential (except in some exceptional cases [38]).See [10, pp. 414â416] for an illustration. Conversely, if the analyticity width α for theinitial datum is smaller than the âLandau rateâ λ, then the rate of decay will not bebetter than O(eâαt). See [7] and [19] for a discussion of this fact, often overlooked in thephysical literature.
Remark 3.3. The fact that the convergence is to the average of the initial datumwill not survive non-linear perturbation, as shown by the counterexamples in §14.
Remark 3.4. Dimension does not play any role in the linear analysis. This can beattributed to the fact that only longitudinal waves occur, so everything happens âinthe direction of the wave vectorâ. Transversal waves arise in plasma physics only whenmagnetic effects are taken into account [1, Chapter 5].
Remark 3.5. The proof can be adapted to the case when f0 and fi are only Câ; thenthe convergence is not exponential, but still O(tââ). The regularity can also be furtherdecreased, down to W s,1, at least for any s>2; more precisely, if f0âW s0,1 and fiâW si,1,there will be damping with a rate O(tâ ) for any <max{s0â2, si}. (Compare with [1,Volume 1, p. 189].) This is independent of the regularity of the interaction.
on landau damping 57
The proof of Theorem 3.1 relies on the following elementary estimate for Volterraequations. We use the notation of §2.2.
Lemma 3.6. Assume that (L) holds true for some constants C0, , λ>0. Let
CW = âWâL1(TdL)
and let K0 be defined by (2.3). Then any solution Ï(t, k) of
Ï(t, k) = a(t, k)+â« t
0
K0(tâÏ, k)Ï(Ï, k) dÏ (3.4)
satisfies, for any kâZd and any λâČ<λ,
suptïżœ0
|Ï(t, k)|e2ÏλâČ|k|t/L ïżœ (1+C0CW C(λ, λâČ, )) suptïżœ0
|a(t, k)|e2Ïλ|k|t/L.
Here C(λ, λâČ, )=C(1+ â1(1+(λâλâČ)â2)) for some universal constant C.
Remark 3.7. It is standard to solve these Volterra equations by Laplace transforms;but, with a view to the non-linear setting, we shall prefer a more flexible and quantitativeapproach.
Proof. If k=0 this is obvious since K0(t, 0)=0; so we assume k ïżœ=0. Consider λâČ<λ,multiply (3.4) by e2ÏλâČ|k|t/L, and write
Ί(t, k) =Ï(t, k)e2ÏλâČ|k|t/L and A(t, k) = a(t, k)e2ÏλâČ|k|t/L;
then (3.4) becomes
Ί(t, k) =A(t, k)+⫠t
0
K0(tâÏ, k)e2ÏλâČ|k|(tâÏ)/LΊ(Ï, k) dÏ. (3.5)
A particular case. The proof is extremely simple if we make the stronger assumptionâ« â
0
|K0(Ï, k)|e2ÏλâČ|k|Ï/L dÏ ïżœ 1â , â (0, 1).
Then from (3.5),
sup0ïżœtïżœT
|Ί(t, k)|ïżœ sup0ïżœtïżœT
|A(t, k)|
+(
sup0ïżœtïżœT
â« t
0
|K0(tâÏ, k)|e2ÏλâČ|k|(tâÏ)/L dÏ
)sup
0ïżœÏïżœT|Ί(Ï, k)|,
58 c. mouhot and c. villani
whence
sup0ïżœÏïżœt
|Ί(Ï, k)|ïżœ sup0ïżœÏïżœt |A(Ï, k)|1ââ«â
0|K0(Ï, k)|e2ÏλâČ|k|Ï/L dÏ
ïżœsup0ïżœÏïżœt |A(Ï, k)|
,
and therefore
suptïżœ0
|Ï(t, k)|e2ÏλâČ|k|t/L ïżœsuptïżœ0 |a(t, k)|e2ÏλâČ|k|t/L
.
The general case. To treat the general case we take the Fourier transform in thetime variable, after extending K, A and Ί by 0 at negative times. (This presentationwas suggested to us by Sigal, and appears to be technically simpler than the use of theLaplace transform.) Denoting the Fourier transform with a hat and recalling (2.4), wehave, for Ο=λâČ+iÏL/|k|,
Ί(Ï, k) = A(Ï, k)+L(Ο, k) Ί(Ï, k).
By assumption L(Ο, k) ïżœ=1, so
Ί(Ï, k) =A(Ï, k)
1âL(Ο, k).
From there, it is traditional to apply the Fourier (or Laplace) inversion transform.Instead, we apply Plancherelâs identity to find (for each k)
âΊâL2(dt) ïżœâAâL2(dt)
.
We then plug this into the equation (3.5) to get
âΊâLâ(dt) ïżœ âAâLâ(dt)+âK0e2ÏλâČ|k|t/LâL2(dt) âΊâL2(dt)
ïżœ âAâLâ(dt)+âK0e2ÏλâČ|k|t/LâL2(dt) âAâL2(dt)
.(3.6)
It remains to bound the second term. On the one hand,
âAâL2(dt) =(â« â
0
|a(t, k)|2e4ÏλâČ|k|t/L dt
)1/2
ïżœ(â« â
0
eâ4Ï(λâλâČ)|k|t/L dt
)1/2
suptïżœ0
|a(t, k)|e2Ïλ|k|t/L
=(
L
4Ï|k|(λâλâČ)
)1/2
suptïżœ0
|a(t, k)|e2Ïλ|k|t/L.
(3.7)
on landau damping 59
On the other hand,
âK0e2ÏλâČ|k|t/LâL2(dt) = 4Ï2|W (L)(k)| |k|2
L2
(â« â
0
e4ÏλâČ|k|t/L
âŁâŁâŁâŁf 0
(kt
L
)âŁâŁâŁâŁ2t2 dt
)1/2
= 4Ï2|W (L)(k)| |k|1/2
L1/2
(â« â
0
e4ÏλâČu|f 0(Ïu)|2u2 du
)1/2
,
(3.8)
where Ï=k/|k| and u=|k|t/L. The estimate follows since
â« â
0
eâ4Ï(λâλâČ)uu2 du = O((λâλâČ)â3/2).
(Note that the factor |k|â1/2 in (3.7) cancels with |k|1/2 in (3.8).)
It seems that we only used properties of the function L in a strip Re ÎŸïżœÎ»; but thisis an illusion. Indeed, we have taken the Fourier transform of Ί without checking thatit belongs to (L1+L2)(dt), so what we have established is only an a-priori estimate. Toconvert it into a rigorous result, one can use a continuity argument after replacing λâČ bya parameter α which varies from âΔ to λâČ. (By the integrability of K0 and Gronwallâslemma, Ï is obviously bounded as a function of t; so Ï(k, t)eâΔ|k|t/L is integrable for anyΔ>0, and continuous as Δ!0.) Then assumption (L) guarantees that our bounds areuniform in the strip 0ïżœRe ÎŸïżœÎ»âČ, and the proof goes through.
Proof of Theorem 3.1. Without loss of generality, we assume tïżœ0. Considering (3.3)as a perturbation of free transport, we apply Duhamelâs formula to get
f(t, x, v) = fi(xâvt, v)+â« t
0
[(âW âïżœ)·âvf0](Ï, xâv(tâÏ), v) dÏ. (3.9)
Integration in v yields
ïżœ(t, x) =â«
Rd
fi(xâvt, v) dv+â« t
0
â«Rd
[(âW âïżœ)·âvf0](Ï, xâv(tâÏ), v) dv dÏ. (3.10)
Of course, â«Td
L
ïżœ(t, x) dx =â«
TdL
â«Rd
fi(x, v) dv dx.
60 c. mouhot and c. villani
For k ïżœ=0, taking the Fourier transform of (3.10), we obtain
ïżœ(L)(t, k) =â«
TdL
â«Rd
fi(xâvt, v)eâ2iÏk·x/L dv dx
+â« t
0
â«Td
L
â«Rd
[(âW âïżœ)·âvf0](Ï, xâv(tâÏ), v)eâ2iÏk·x/L dv dx dÏ
=â«
TdL
â«Rd
fi(x, v)eâ2iÏk·x/Leâ2iÏk·vt/L dv dx
+â« t
0
â«Td
L
â«Rd
[(âW âïżœ)·âvf0](Ï, x, v)eâ2iÏk·x/Leâ2iÏk·v(tâÏ)/L dv dx dÏ
= f(L)i
(k,
kt
L
)+â« t
0
(âW âïżœ)(L)(Ï, k)·âvf0
(k(tâÏ)
L
)dÏ
= f(L)i
(k,
kt
L
)+â« t
0
(2iÏ
k
LW (L)(k)ïżœ(L)(Ï, k)
)·(
2iÏk(tâÏ)
Lf 0
(k(tâÏ)
L
))dÏ.
In conclusion, we have established the closed equation for ïżœ(L):
ïżœ(L)(t, k) = f(L)i
(k,
kt
L
)â4Ï2W (L)(k)
â« t
0
ïżœ(L)(Ï, k)f 0
(k(tâÏ)
L
) |k|2L2
(tâÏ) dÏ. (3.11)
Recalling (2.3), this is the same as
ïżœ(L)(t, k) = f(L)i
(k,
kt
L
)+â« t
0
K0(tâÏ, k)ïżœ(L)(Ï, k) dÏ.
Without loss of generality Î»ïżœÎ±, where α appears in Theorem 3.1. By assumption (L)and Lemma 3.6,
|ïżœ(L)(t, k)|ïżœC0CW C(λ, λâČ, )Cieâ2ÏλâČ|k|t/L.
In particular, for k ïżœ=0 we have
|ïżœ(L)(t, k)|= O(eâ2ÏλâČâČt/Leâ2Ï(λâČâλâČâČ)|k|/L) for all t ïżœ 1;
so any Sobolev norm of ïżœâïżœâ converges to zero like O(eâ2ÏλâČâČt/L), where λâČâČ is arbitrarilyclose to λâČ and therefore also to λ. By Sobolev embedding, the same is true for any Cr
norm.
Next, we go back to (3.9) and take the Fourier transform in both variables x and v,
on landau damping 61
to find
f (L)(t, k, η) =â«
TdL
â«Rd
fi(xâvt, v)eâ2iÏk·x/Leâ2iÏη·v dv dx
+â« t
0
â«Td
L
â«Rd
(âW âïżœ)(Ï, xâv(tâÏ))·âvf0(v)eâ2iÏk·x/Leâ2iÏη·v dv dx dÏ
=â«
TdL
â«Rd
fi(x, v)eâ2iÏk·x/Leâ2iÏk·vt/Leâ2iÏη·v dv dx
+â« t
0
â«Td
L
â«Rd
(âW âïżœ)(Ï, x)·âvf0(v)
Ăeâ2iÏk·x/Leâ2iÏk·v(tâÏ)/Leâ2iÏη·v dv dx dÏ
= f(L)i
(k, η+
kt
L
)+â« t
0
âW(L)
(k)ïżœ(L)(Ï, k)·âvf0
(η+
k(tâÏ)L
)dÏ.
So
f (L)
(t, k, ηâ kt
L
)= f
(L)i (k, η)+
â« t
0
âW(L)
(k)ïżœ(L)(Ï, k)·âvf0
(ηâ kÏ
L
)dÏ. (3.12)
In particular, for any ηâRd,
f (L)(t, 0, η) = f(L)i (0, η); (3.13)
in other words, ăfă=Lâdâ«
TdL
f dx remains equal to ăfiă for all times.On the other hand, if k ïżœ=0, thenâŁâŁâŁâŁf (L)
(t, k, ηâ kt
L
)âŁâŁâŁâŁïżœ |f (L)
i (k, η)|+⫠t
0
|âW(L)
(k)| |ïżœ(L)(Ï, k)|âŁâŁâŁâŁâvf0
(ηâ kÏ
L
)âŁâŁâŁâŁ dÏ
ïżœCieâ2Ïα|η|+
â« t
0
CW C(λ, λâČ, )Cieâ2ÏλâČ|k|Ï/L
(2ÏC0
âŁâŁâŁâŁÎ·â kÏ
L
âŁâŁâŁâŁeâ2Ïλ|ηâkÏ/L|)
dÏ
ïżœC
(eâ2Ïα|η|+
â« t
0
eâ2ÏλâČ|k|Ï/LeâÏ(λâČ+λ)|ηâkÏ/L| dÏ
),
(3.14)
where we have used that λâČ< 12 (λâČ+λ)<λ, and C only depends on CW , Ci, λ, λâČ and .
In the end,â« t
0
eâ2ÏλâČ|k|Ï/LeâÏ(λâČ+λ)|ηâkÏ/L| dÏ ïżœâ« t
0
eâ2ÏλâČ|η|eâÏ(λâλâČ)|ηâkÏ/L| dÏ
ïżœ L
Ï(λâλâČ)eâ2Ï[λâČâ(λâλâČ)/2]|η|.
62 c. mouhot and c. villani
Plugging this back into (3.14), we obtain, with λâČâČ=λâČâ 12 (λâλâČ),âŁâŁâŁâŁf (L)
(t, k, ηâ kt
L
)âŁâŁâŁâŁïżœCeâ2ÏλâČâČ|η|. (3.15)
In particular, for any fixed η and k ïżœ=0,
|f (L)(t, k, η)|ïżœCeâ2ÏλâČâČ|η+kt/L| = O(eâ2ÏλâČâČ|t|/L).
We conclude that f (L) converges pointwise, exponentially fast, to the Fourier transformof ăfiă. Since λâČ and then λâČâČ can be taken as close to λ as wanted, this ends the proof.
We close this section by proving Proposition 2.1.
Proof of Proposition 2.1. First assume (a). Since f 0 decreases exponentially fast,we can find λ, >0 such that
4Ï2 max |W (L)(k)âŁâŁ sup|Ï|=1
â« â
0
|f 0(rÏ)|re2Ïλr dr ïżœ 1â .
Performing the change of variables kt/L=rÏ inside the integral, we deduce thatâ« â
0
4Ï2|W (L)(k)|âŁâŁâŁâŁf 0
(kt
L
)âŁâŁâŁâŁ |k|2tL2e2Ïλ|k|t/L dt ïżœ 1â ,
and this obviously implies (L).The choice w=0 in (2.8) shows that condition (b) is a particular case of (c), so
we only treat the latter assumption. The reasoning is more subtle than for case (a).Throughout the proof we shall abbreviate W (L) by W . As a start, let us assume d=1and k>0 (so kâN). Then we compute: for any ÏâR,â« â
0
e2iÏÏkt/LK0(t, k) dt
= limλ!0+
â« â
0
eâ2Ïλkt/Le2iÏÏkt/L K0(t, k) dt
=â4Ï2W (k) limλ!0+
â« â
0
â«R
f0(v)eâ2iÏkvt/Leâ2Ïλkt/Le2iÏÏkt/L k2
L2t dv dt
=â4Ï2W (k) limλ!0+
â« â
0
â«R
f0(v)eâ2iÏvteâ2Ïλte2iÏÏtt dv dt.
(3.16)
Then by integration by parts, assuming that (f0)âČ is integrable,â«R
f0(v)eâ2iÏvtt dv =1
2iÏ
â«R
(f0)âČ(v)eâ2iÏvt dv.
on landau damping 63
Plugging this back into (3.16), we obtain the expression
2iÏW (k) limλ!0+
â«R
(f0)âČ(v)â« â
0
eâ2Ï[λ+i(vâÏ)]t dt dv.
Next, recall that for any λ>0,â« â
0
eâ2Ï[λ+i(vâÏ)]t dt =1
2Ï[λ+i(vâÏ)];
indeed, both sides are holomorphic functions of z=λ+i(vâÏ) in the half-plane {zâC:Re z>0}, and they coincide on the real half-axis {zâR:z>0}, so they have to coincideeverywhere. We conclude thatâ« â
0
e2iÏÏkt/LK0(t, k) dt = W (k) limλ!0+
â«R
(f0)âČ(v)vâÏâiλ
dv. (3.17)
The celebrated Plemelj formula states that
1zâi0
= p.v.
(1z
)+iÏÎŽ0, (3.18)
where the left-hand side should be understood as the limit, in weak sense, of 1/(zâiλ) asλ!0+. The abbreviation p.v. stands for principal value, that is, simplifying the possiblydivergent part by using compensations by symmetry when the denominator vanishes.Formula (3.18) is proven in Appendix A.5, where the notion of principal value is alsoprecisely defined. Combining (3.17) and (3.18), we end up with the identityâ« â
0
e2iÏÏkt/LK0(t, k) dt = W (k)[(
p.v.
â«R
(f0)âČ(v)vâÏ
dv
)+iÏ(f0)âČ(Ï)
]. (3.19)
Since W is even, W is real-valued, so the above formula yields the decomposition of thelimit into real and imaginary parts. The problem is to check that the real part cannotapproach 1 at the same time as the imaginary part approaches 0.
As soon as (f0)âČ(v)=O(1/|v|), we haveâ«R
(f0)âČ(v)vâÏ
dv = O
(1|Ï|)
as |Ï|!â,
so the real part in the right-hand side of (3.19) becomes small when |Ï| is large, and wecan restrict to a bounded interval |Ï|ïżœÎ©.
Then the imaginary part, W (k)Ï(f0)âČ(Ï), can become small only in the limit k!â(but then also the real part becomes small) or if Ï approaches one of the zeroes of (f0)âČ.Since Ï varies in a compact set, by continuity it will be sufficient to check the condition
64 c. mouhot and c. villani
only at the zeroes of (f0)âČ. In the end, we have obtained the following stability criterion:for any kâN,
(f0)âČ(Ï) = 0 =â W (k)â«
R
(f0)âČ(v)vâÏ
dv ïżœ= 1 for all Ï âR. (3.20)
Now, if k<0, we can restart the computation as follows:â« â
0
e2iÏÏ|k|t/LK0(t, k) dt
=â4Ï2W (k) limλ!0+
â« â
0
â«R
f0(v)eâ2iÏkvt/Leâ2Ïλ|k|t/Le2iÏÏ|k|t/L |k|2L2
t dv dt.
Then the change of variable v !âv brings us back to the previous computation with k
replaced by |k| (except in the argument of W ) and f0(v) replaced by f0(âv). However,it is immediately checked that (3.20) is invariant under reversal of velocities, that is, iff0(v) is replaced by f0(âv).
Finally, let us generalize this to several dimensions. If kâZd\{0} and ΟâC, we canuse the splitting
v =k
|k|r+w, wâ„k, r =k
|k| ·v
and Fubiniâs theorem to rewrite
L(Ο, k)
=â4Ï2W (k)|k|2L2
â« â
0
â«Rd
f0(v)eâ2iÏkt·v/Lte2Ï|k|Οât/L dv dt
=â4Ï2W (k)|k|2L2
â« â
0
â«R
(â«kr/|k|+kâ„
f0
(k
|k|r+w
)dw
)eâ2iÏ|k|rt/Lte2Ï|k|Οât/L dr dt,
where kâ„ is the hyperplane orthogonal to k. So everything is expressed in terms of the1-dimensional marginals of f0. If f is a given function of vâRd, and Ï is a unit vector,let us write Ïâ„ for the hyperplane orthogonal to Ï, and
fÏ(v) =â«
vÏ+Ïâ„f(w) dw for all v âR. (3.21)
Then the computation above shows that the multi-dimensional stability criterion reducesto the 1-dimensional criterion in each direction k/|k|, and the claim is proven.
4. Analytic norms
In this section we introduce some functional spaces of analytic functions on the spacesRd, Td=Rd/Zd and most importantly TdĂRd. (Changing the sidelength of the torus
on landau damping 65
only results in some changes in the constants.) Then we establish a number of functionalinequalities which will be crucial in the subsequent analysis. At the end of this sectionwe shall reformulate the linear study in this new setting.
Throughout the whole section, d is a positive integer. Working with analytic func-tions will force us to be careful with combinatorial issues, and proofs will at times involvesummation over many indices.
4.1. Single-variable analytic norms
Here âsingle-variableâ means that the variable lives in either Rd or Td, but d may begreater than 1.
Among many possible families of norms for analytic functions, two will be of partic-ular interest for us; they will be denoted by Cλ;p and Fλ;p. The Cλ;p norms are definedfor functions on Rd or Td, while the Fλ;p norms are defined only for Td (although wecould easily cook up a variant in Rd). We shall write Nd
0 for the set of d-tuples of integers(the subscript being here to insist that 0 is allowed). If nâNd
0 and Î»ïżœ0 we shall writeλn=λ|n|. Conventions about the Fourier transform and multi-dimensional differentialcalculus are gathered in the appendix.
Definition 4.1. (One-variable analytic norms) For any pâ[1,â] and Î»ïżœ0, we define
âfâCλ;p :=â
nâNd0
λn
n!âf (n)âLp and âfâFλ;p :=
( âkâZd
e2Ïλp|k||f(k)|p)1/p
, (4.1)
the latter expression standing for supkâZd |f(k)|e2Ïλ|k| if p=â. We further write
Cλ := Cλ;â and Fλ :=Fλ;1. (4.2)
Remark 4.2. The parameter λ can be interpreted as a radius of convergence.
Remark 4.3. The norms Cλ and Fλ are of particular interest because they are alge-bra norms.
We shall sometimes abbreviate â · âCλ;p or â · âFλ;p into â · âλ;p when no confusion ispossible, or when the statement works for either norm.
The norms in (4.1) extend to vector-valued functions in a natural way: if f isvalued in Rd or Td or Zd, define f (n)=(f (n)
1 , ..., f(n)d ) and f(k)=(f1(k), ..., fd(k)); then
the formulae in (4.1) make sense provided that we choose a norm on Rd or Td or Zd.Which norm we choose will depend on the context; the choice will always be done in sucha way to get the duality right in the inequality |a·b|ïżœâaâ âbââ. For instance if f is valued
66 c. mouhot and c. villani
in Zd and g in Td, and we have to estimate f ·g, we may norm Zd by |k|=âdj=1 |kj | and
Td by |x|=supj |xj |.(15) This will not pose any problem, and the reader can forget aboutthis issue; we shall just make remarks about it whenever needed. For the rest of thissection, we shall focus on scalar-valued functions for simplicity of exposition.
Next, we define âhomogeneousâ analytic seminorms by removing the zeroth-orderterm. We write Nd
â=Nd0\{0} and Zd
â=Zd\{0}.Definition 4.4. (One-variable homogeneous analytic seminorms) For pâ[1,â] and
Î»ïżœ0 we write
âfâCλ;p =â
nâNdâ
λn
n!âf (n)âLp and âfâFλ;p =
( âkâZdâ
e2Ïλp|k||f(k)|p)1/p
.
It is interesting to note that affine functions x !a·x+b can be included in Cλ=Cλ;â,even though they are unbounded; in particular âa·x+bâCλ =λ|a|. On the other hand,linear forms x !a·x do not naturally belong to Fλ, because their Fourier expansion isnot even summable (it decays like 1/k).
The spaces Cλ;p and Fλ;p enjoy remarkable properties, summarized in Proposi-tions 4.5, 4.8 and 4.10 below. Some of these properties are well known, others not so.
Proposition 4.5. (Algebra property) (i) For any Î»ïżœ0 and p, q, râ[1,â] such that1/p+1/q=1/r, we have
âfgâCλ;r ïżœ âfâCλ;pâgâCλ;q .
(ii) For any Î»ïżœ0 and p, q, râ[1,â] such that 1/p+1/q=1/r+1, we have
âfgâFλ;r ïżœ âfâFλ;pâgâFλ;q .
(iii) As a consequence, for any Î»ïżœ0, Cλ=Cλ;â and Fλ=Fλ;1 are normed algebras:for either space,
âfgâλ ïżœ âfâλâgâλ.
In particular, âfnâÎ»ïżœâfânλ for any nâN0, and âefâÎ»ïżœeâfâλ .
Remark 4.6. Ultimately, property (iii) relies on the fact that Lâ and L1 are normedalgebras for the multiplication and convolution, respectively.
Remark 4.7. It follows from the Fourier inversion formula and Proposition 4.5 thatâfâCλ ïżœâfâFλ (and âfâCλ ïżœâfâFλ); this is a special case of Proposition 4.8 (iv) below.The reverse inequality does not hold, because âfââ does not control âfâL1 .
(15) Of course all norms are equivalent, still the choice is not innocent when the estimates areiterated infinitely many times; an advantage of the supremum norm on Rd is that it has the algebraproperty.
on landau damping 67
Analytic norms are very sensitive to composition; think that if a>0 then
âf (a Id)âCλ;p = aâd/pâfâCaλ;p ;
so we typically lose on the functional space. This is a major difference with more tra-ditional norms used in partial differential equations theory, such as Holder or Sobolevnorms, for which composition may affect constants but not regularity indices. The nextproposition controls the loss of regularity implied by composition.
Proposition 4.8. (Composition inequality) (i) For any λ>0 and any pâ[1,â],
âf HâCλ;p ïżœ â(detâH)â1â1/pâ âfâCÎœ;p , Îœ = âHâCλ ,
where H is possibly unbounded.(ii) For any λ>0, any pâ[1,â] and any a>0,
âf (a Id +G)âCλ;p ïżœ aâd/pâfâCaλ+Îœ;p , Îœ = âGâCλ .
(iii) For any λ>0,
âf (Id +G)âFλ ïżœ âfâFλ+Îœ , Îœ = âGâFλ .
(iv) For any λ>0 and any a>0,
âf (a Id +G)âCλ ïżœ âfâFaλ+Îœ , Îœ = âGâCλ .
Remark 4.9. Inequality (iv), with C on the left and F on the right, will be mostuseful. The reverse inequality is not likely to hold, in view of Remark 4.7.
The last property of interest for us is the control of the loss of regularity involvedby differentiation.
Proposition 4.10. (Control of gradients) For any λ>λ and any pâ[1,â],
ââfâCλ;p ïżœ 1λe log(λ/λ)
âfâCλ;p , (4.3)
ââfâFλ;p ïżœ 12Ïe(λâλ)
âfâF λ;p . (4.4)
The proofs of Propositions 4.5â4.10 will be preparations for the more complicatedsituations considered in the sequel.
68 c. mouhot and c. villani
Proof of Proposition 4.5. (i) Denoting the norm of Cλ;p by â · âλ;p and using themulti-dimensional Leibniz formula from Appendix A.2, we have
âfgâλ;r =âlâNd
0
â(fg)(l)âLr
λl
l!
ïżœâ
l,mâNd0
mïżœl
(l
m
)âf (m)g(lâm)âLr
λl
l!
ïżœâ
l,mâNd0
mïżœl
(l
m
)âf (m)âLpâg(lâm)âLq
λl
l!
=â
l,mâNd0
mïżœl
âf (m)âLpλm
m!âg(lâm)âLqλlâm
(lâm)!
= âfâλ;pâgâλ;q.
(ii) Denoting now the norm of Fλ;p by â · âλ;p and applying Youngâs convolutioninequality, we get
âfgâλ;r =( â
kâZd
|fg(k)|re2Ïλr|k|)1/r
ïżœ( â
kâZd
(âlâZd
|f(l)| |g(kâl)|e2Ïλ|kâl|e2Ïλ|l|)r )1/r
ïżœ( â
kâZd
|f(k)|pe2Ïλp|kâl|)1/p(â
lâZd
|g(l)|qe2Ïλq|l|)1/q
.
Proof of Proposition 4.8. (i) We use the (multi-dimensional) Faa di Bruno formula:
(f H)(n) =â
ânj=1 jmj=n
n!m1! ...mn!
(f (m1+...+mn) H)nâ
j=1
(H(j)
j!
)mj
;
so
â(f H)(n)âLp ïżœâ
ânj=1 jmj=n
n!m1! ...mn!
âf (m1+...+mn) HâLp
nâj=1
â„â„â„â„H(j)
j!
â„â„â„â„mj
â;
on landau damping 69
and thusânïżœ1
λn
n!â(f H)(n)âLp
ïżœ â(detâH)â1â1/pâ
ââk=1
âf (k)âLp
âân
j=1 jmj=nânj=1 mj=k
λn
m1! ...mn!
nâj=1
â„â„â„â„H(j)
j!
â„â„â„â„mj
â
= â(detâH)â1â1/pââkïżœ1
âf (k)âLp
1k!
(â|l|ïżœ1
λl
l!âH(l)ââ
)k,
where the last step follows from the multi-dimensional binomial formula.(ii) We decompose h(x):=f(ax+G(x)) as
h(x) =â
nâNd0
f (n)(ax)n!
G(x)n
and we apply âk:
âkh(x) =â
k1,k2âNd0
k1+k2=k
ânâNd
0
k!ak1
k1!k2!n!âk1+nf(ax)âk2Gn(x).
Then we take the Lp norm, multiply by λk/k! and sum over k:
âhâCλ;p ïżœ |a|âd/pâ
k1,k2,nïżœ0
λk1+k2 |a|k1
k1!k2!n!ââk1+nfâLpââk2Gnââ
= |a|âd/pâ
k1,nïżœ0
λk1 |a|k1
k1!n!ââk1+nfâLpâGnâCλ
ïżœ |a|âd/pâ
k1,nïżœ0
λk1 |a|k1
k1!n!ââk1+nfâLpâGân
Cλ
= |a|âd/pâmïżœ0
(aλ+âGâCλ)m
m!ââmfâLp ,
where Proposition 4.5 (iii) was used in the second-last step.(iii) In this case we write, with G0=G(0),
h(x) = f(x+G(x)) =âkâZd
f(k)e2iÏk·xe2iÏk·G0e2iÏk·(G(x)âG0);
soh(l) =
âkâZd
f(k)e2iÏk·G0 [e2iÏk·(GâG0) ] (lâk).
70 c. mouhot and c. villani
Then, using again Proposition 4.5,âlâZd
|h(l)|e2Ïλ|l| ïżœâ
l,kâZd
|f(k)|e2Ïλ|k|e2Ïλ|lâk||[e2iÏk·(GâG0) ] (lâk)|
=âkâZd
|f(k)|e2Ïλ|k|âe2iÏk·(GâG0)âλ
ïżœâkâZd
|f(k)|e2Ïλ|k|eâ2Ïk·(GâG0)âλ
ïżœâkâZd
|f(k)|e2Ïλ|k|e2Ï|k|âGâG0âλ
= âfâλ+âGâG0âλ
= âfâλ+Îœ ,
where Îœ=âGâFλ .
(iv) We actually have the more precise result
âf HâCλ ïżœâkâZd
|f(k)|e2Ï|k|âHâCλ . (4.5)
Writing f H=â
kâZd f(k)e2iÏk·H , we see that (4.5) follows from
âeihâCλ ïżœ eâhâCλ . (4.6)
To prove (4.6), let Pn be the polynomial in the variables Xm, mïżœn, defined by theidentity
(ef )(n) = Pn((f (m))mïżœn)ef ;
this polynomial (which can be made more explicit from the Faa di Bruno formula) hasnon-negative coefficients, so â(eif )(n)ââïżœPn((âf (m)â)mïżœn). The conclusion will followfrom the identity (between formal series!)
1+â
nâNdâ
λn
n!Pn((Xm)mïżœn) = exp
( âkâNdâ
λk
k!Xk
). (4.7)
To prove (4.7), it is sufficient to note that the left-hand side is the expansion of eg inpowers of λ at 0, where
g(λ) =âkâNdâ
λk
k!Xk.
on landau damping 71
Proof of Proposition 4.10. (a) Writing â · âλ;p=â · âCλ;p , we have
ââjfâλ;p =â
nâNd0
λn
n!âân
x âjfâLp ,
where âj =â/âxj . If 1j is the d-tuple of integers with 1 in position j and 0 elsewhere,then (n+1j)!ïżœ(|n|+1)n!, so
ââjfâλ;p ïżœ supnâNd
0
(|n|+1)λn
λn+1
â|m|ïżœ1
λm
m!ââmfâLp ,
and the proof of (4.3) follows easily.(b) Writing â · âλ;p=â · âFλ;p , we have
ââjfâλ;p =( â
kâZd
|kj |p|f(k)|pe2Ïλp|k|)1/p
ïżœ(
supkâZd
|k|e2Ï(λâλ)|k|)( â
kâZd
|f(k)|pe2Ïλp|k|)1/p
,
and (4.4) follows.
4.2. Analytic norms in two variables
To estimate solutions and trajectories of kinetic equations we will work on the phase spaceTd
xĂRdv, and use three parameters: λ (gliding analytic regularity), ÎŒ (analytic regularity
in x) and Ï (time-shift along the free transport semigroup). The regularity quantified byλ is said to be gliding because for Ï =0 this is an analytic regularity in v, but as Ï growsthe regularity is progressively transferred from velocity to spatial modes, according tothe evolution by free transport. This catch is crucial to our analysis: indeed, the solutionof a transport equation like free transport or Vlasov cannot be uniformly analytic(16) inv as time goes byâexcept of course if it is spatially homogeneous. Instead, the best wecan do is compare the solution at time Ï to the solution of free transport at the sametime.
The parameters λ and ÎŒ will be non-negative; Ï will vary in R, but often be restrictedto R+, just because we shall work in positive time. When Ï is not specified, this meansÏ =0. Sometimes we shall abuse notation by writing âf(x, v)â instead of âfâ, to stressthe dependence of f on the two variables.
Putting aside the time-shift for a moment, we may generalize the norms Cλ and Fλ
in an obvious way.
(16) By this we mean of course that some norm or seminorm quantifying the degree of analyticsmoothness in v will remain uniformly bounded.
72 c. mouhot and c. villani
Definition 4.11. (Two-variables analytic norms) For any λ, ÎŒïżœ0, we define
âfâCλ,ÎŒ =â
mâNd0
ânâNd
0
λn
n!ÎŒm
m!ââm
x ânv fâLâ(Td
xĂRdv), (4.8)
âfâFλ,ÎŒ =âkâZd
â«Rd
|f(k, η)|e2Ïλ|η|e2ÏÎŒ|k| dη. (4.9)
Of course one might also introduce variants based on Lp or ïżœp norms (with twoadditional parameters p and q, since one can make different choices for the space andvelocity variables).
The norm (4.9) is better adapted to the periodic nature of the problem, and isvery well suited to estimate solutions of kinetic equations (with fast decay as |v|!â);but in the sequel we shall also have to estimate characteristics (trajectories) which areunbounded functions of v. We could hope to play with two different families of norms,but this would entail considerable technical difficulties. Instead, we shall mix the tworecipes to get the following hybrid norms.
Definition 4.12. (Hybrid analytic norms) For any λ, ÎŒïżœ0, let
âfâZλ,ÎŒ =âlâZd
ânâNd
0
λn
n!e2ÏÎŒ|l|âân
v f(l, v)âLâ(Rdv). (4.10)
More generally, for any pâ[1,â], we define
âfâZλ,ÎŒ;p =âlâZd
ânâNd
0
λn
n!e2ÏÎŒ|l|âân
v f(l, v)âLp(Rdv). (4.11)
Now let us introduce the time-shift Ï . We denote by (S0Ï )Ïïżœ0 the geodesic semigroup:
(S0Ï )(x, v)=(x+vÏ, v). Recall that the backward free transport semigroup is defined by
(f S0Ï )Ïïżœ0, and the forward semigroup by (f S0
âÏ )Ïïżœ0.
Definition 4.13. (Time-shift pure and hybrid analytic norms)
âfâCλ,ÎŒÏ
= âf S0ÏâCλ,ÎŒ =
âmâNd
0
ânâNd
0
λn
n!ÎŒm
m!ââm
x (âv+Ïâx)nfâLâ(TdxĂRd
v), (4.12)
âfâFλ,ÎŒÏ
= âf S0ÏâFλ,ÎŒ =
âkâZd
â«Rd
|f(k, η)|e2Ïλ|kÏ+η|e2ÏÎŒ|k| dη, (4.13)
âfâZλ,ÎŒÏ
= âf S0ÏâZλ,ÎŒ =
âlâZd
ânâNd
0
λn
n!e2ÏÎŒ|l|â(âv+2iÏÏ l)nf(l, v)âLâ(Rd
v), (4.14)
âfâZλ,ÎŒ;pÏ
=âlâZd
ânâNd
0
λn
n!e2ÏÎŒ|l|â(âv+2iÏÏ l)nf(l, v)âLp(Rd
v). (4.15)
on landau damping 73
This choice of norms is one of the cornerstones of our analysis: first, because of theirhybrid nature, they will connect well to both periodic (in x) estimates on the force field,and uniform (in v) estimates on the âdeflection mapsâ studied in §5. Secondly, theyare well behaved with respect to the properties of free transport, allowing us to keeptrack of the initial time. Thirdly, they will satisfy the algebra property (for p=â), thecomposition inequality and the gradient inequality (for any pâ[1,â]). Before going onwith the proof of these properties, we note the following alternative representations.
Proposition 4.14. The norm Zλ,ÎŒ;pÏ admits the following alternative representa-
tions:
âfâZλ,ÎŒ;pÏ
=âlâZd
ânâNd
0
λn
n!e2ÏÎŒ|l|âân
v (f(l, v)e2iÏÏl·v)âLp(Rdv), (4.16)
âfâZλ,ÎŒ;pÏ
=â
nâNd0
λn
n!|||(âv+Ïâx)nf |||ÎŒ;p, (4.17)
where|||g|||Ό;p =
âlâZd
e2ÏÎŒ|l|âg(l, v)âLp(Rdv). (4.18)
4.3. Relations between functional spaces
The next propositions are easily checked.
Proposition 4.15. With the notation from §4.2, for any ÏâR,(i) if f is a function only of x then
âfâCλ,ÎŒÏ
= âfâCλ|Ï|+ÎŒ and âfâFλ,ÎŒÏ
= âfâZλ,ÎŒÏ
= âfâFλ|Ï|+ÎŒ ;
(ii) if f is a function only of v then
âfâCλ,ÎŒ;pÏ
= âfâZλ,ÎŒ;pÏ
= âfâCλ;p and âfâFλ,ÎŒÏ
= âfâFλ ;
(iii) for any function f=f(x, v), if ă ·ă stands for spatial average then
âăfăâCλ;p ïżœ âfâZλ,ÎŒ;pÏ
;
(iv) for any function f=f(x, v),â„â„â„â„â«Rd
f dv
â„â„â„â„Fλ|Ï|+ÎŒ
ïżœ âfâZλ,ÎŒ;1Ï
.
74 c. mouhot and c. villani
Remark 4.16. Note, in Proposition 4.15 (i) and (iv), how the regularity in x is im-proved by the time-shift.
Proof. Only (iv) requires some explanations. Let
ïżœ(x) =â«
Rd
f(x, v) dv.
Then for any kâZd,
ïżœ(k) =â«
Rd
f(k, v) dv;
so for any nâNd0,
(2iÏtk)nïżœ(k) =â«
Rd
(2iÏtk)nf(k, v) dv =â«
Rd
(âv+2iÏtk)nf(k, v) dv.
Recalling the conventions from Appendix A.1 we deduceâkâZd
nâNd0
e2ÏÎŒ|k| |2Ïλtk|nn!
|ïżœ(k)|ïżœâkâZd
nâNd0
e2ÏÎŒ|k|λn
n!
â«Rd
|(âv+2iÏtk)nf(k, v)| dv = âfâZλ,ÎŒ;1t
.
Proposition 4.17. With the notation from §4.2,
λ ïżœÎ»âČ and ÎŒïżœÎŒâČ =â âfâZλ,ÎŒÏ
ïżœ âfâZλâČ,ÎŒâČÏ
.
Moreover, for Ï, ÏâR and any pâ[1,â],
âfâZλ,ÎŒ;pÏ
ïżœ âfâZλ,ÎŒ+λ|ÏâÏ|;pÏ
. (4.19)
Remark 4.18. Note carefully that the spaces Zλ,ÎŒÏ are not ordered with respect to
the parameter Ï , which cannot be thought of as a regularity index. We could dispendwith this parameter if we were working in time O(1); but (4.19) is of course of absolutelyno use. This means that errors on the exponent Ï should remain somehow small, in orderto be controllable by small losses on the exponent ÎŒ.
Finally we state an easy proposition which follows from the time invariance of thefree transport equation.
Proposition 4.19. For any Xâ{C,F ,Z} and any t, ÏâR,
âf S0t âXλ,ÎŒ
Ï= âfâXλ,ÎŒ
t+Ï.
Now we shall see that the hybrid norms, and certain variants thereof, enjoy propertiesrather similar to those of the single-variable analytic norms studied before. This willsometimes be technical, and the reader who would like to reconnect to physical problemsis advised to go directly to §4.11.
on landau damping 75
4.4. Injections
In this section we relate Zλ,ÎŒ;pÏ norms to more standard norms entirely based on Fourier
space. In the next theorem we write
âfâYλ,ÎŒÏ
:= âfâFλ,ÎŒ;âÏ
= supkâZd
supηâRd
e2ÏÎŒ|k|e2Ïλ|η+kÏ ||f(k, η)|. (4.20)
Theorem 4.20. (Injections between analytic spaces) (i) If λ, ÎŒïżœ0 and ÏâR then
âfâYλ,ÎŒÏ
ïżœ âfâZλ,ÎŒ;1Ï
. (4.21)
(ii) If 0<λ<λ, 0<ÎŒ<ÎŒïżœM and ÏâR, then
âfâZλ,ÎŒÏ
ïżœ C(d, ÎŒ)(λâλ)d(ÎŒâÎŒ)d
âfâYλ,ÎŒÏ
. (4.22)
(iii) If 0<λ<Î»ïżœÎ, 0<ÎŒ<ÎŒïżœM and bïżœÎČïżœB, then there is C=C(Î,M, b,B, d)such that
âfâZλ,ÎŒ;1Ï
ïżœC1/min{λâλ,ÎŒâÎŒ}
Ă(âfâYλ,ÎŒ
Ï+max
{â«Td
â«Rd
|f(x, v)|eÎČ|v| dv dx,
(â«Td
â«Rd
|f(x, v)|eÎČ|v| dv dx
)2}).
Remark 4.21. The combination of (ii) and (iii), plus elementary Lebesgue interpo-lation, enables us to control all norms Zλ,Ό;p
Ï , 1ïżœpïżœâ.
Proof. By the invariance under the action of free transport, it is sufficient to do theproof for Ï =0.
By integration by parts in the Fourier transform formula, we have
f(k, η) =â«
Rd
f(k, v)eâ2iÏη·v dv =â«
Rd
âmv f(k, v)
eâ2iÏη·v
(2iÏη)mdv.
So|f(k, η)|ïżœ 1
(2Ï|η|)m
â«Rd
|âmv f(k, v)| dv;
and therefore
e2ÏÎŒ|k|e2Ïλ|η||f(k, η)|ïżœ e2ÏÎŒ|k| ânâNd
0
(2Ïλ)n
n!|η|n|f(k, η)|
ïżœ e2ÏÎŒ|k| ânâNd
0
λn
n!
â«Rd
|ânv f(k, v)| dv.
76 c. mouhot and c. villani
This establishes (i).Next, by differentiating the identity
f(k, v) =â«
Rd
f(k, η)e2iÏη·v dη,
we get
âmv f(k, v) =
â«Rd
f(k, η)(2iÏη)me2iÏη·v dη. (4.23)
Then we deduce (ii) by writingâkâZd
mâNd0
e2ÏÎŒ|k|λm
m!ââm
v f(k, v)âLâ(dv)
ïżœâkâZd
e2ÏÎŒ|k|â«
Rd
e2Ïλ|η||f(k, η)| dη
ïżœ( â
kâZd
eâ2Ï(ÎŒâÎŒ)|k|)(â«
Rd
eâ2Ï(λâλ)|η| dη
)supkâZd
ηâRd
|f(k, η)|e2Ïλ|η|e2ÏÎŒ|k|.
The proof of (iii) is the most tricky. We start again from (4.23), but now we integrateby parts in the η variable:
âmv f(k, v) = (â1)q
â«Rd
âqη[f(k, η)(2iÏη)m]
e2iÏη·v
(2iÏv)qdv, (4.24)
where q=q(v) is a multi-index to be chosen.We split Rd
v into 2d disjoint regions Î(i1, ..., in), where i1, ..., in are distinct indicesin {1, ..., d}:
Î(I) = {v âRd : |vj |ïżœ 1 for all j â I and |vj |< 1 for all j /â I}.
If vâÎ(i1, ..., in) we apply (4.24) with the multi-index q defined by
qj ={
2, if j â{i1, ..., in},0, otherwise.
This givesâ«Î(i1,...,in)
|âmv f(k, v)| dv
ïżœ(
1(2Ï)2n
â«Î(i1,...,in)
dvi1 ... dvin
|vi1 |2 ... |vin|2)
supkâZd
ηâRd
|âqη[f(k, η)(2iÏη)m]|.
on landau damping 77
Summing up all pieces and using the Leibniz formula, we getâ«Rd
|âmv f(k, v)| dv ïżœC(d)(1+m2d) sup
kâZd
ηâRd
sup|q|ïżœ2d
|âqη f(k, η)| |2Ïη|mâq.
At this point we apply Lemma 4.22 below with
Δ =14
min{
λâλ
λ,ÎŒâÎŒ
Ό
},
and we get, for qïżœ2d,
|âqη f(k, η)|ïżœC(d)max{λ/(λâλ),ÎŒ/(ÎŒâÎŒ)}K(b, B)eâÏ(λ+λ)|η|
(supηâRd
|f(k, η)|e2Ïλ|η|)1âΔ
Ămax
{(suplâNd
0
ηâRd
ÎČlââlη fââl!
)Δ,
(suplâNd
0
ηâRd
ÎČlââlη fââl!
)2Δ}.
Of course,
ÎČl|âlη f(k, η)|l!
ïżœ (2ÏÎČ)l
â«Rd
|f(k, v)| |v|l
l!dv
ïżœâ«
Rd
|f(x, v)|(2ÏÎČ)l |v|ll!
dv ïżœâ«
Rd
|f(x, v)|e2ÏÎČ|v| dv.
So, all in all,âkâZd
mâNd0
e2ÏÎŒ|k|λm
m!
â«Rd
|âmv f(k, v)| dv
ïżœâ
|q|ïżœ2d
C(d, Î,M, b,B)1/min{λâλ,ÎŒâÎŒ}
Ă(
supηâRd
eâÏ(λ+λ)|η| âmâNd
0
λm(1+m)2d|2Ïη|mâq
m!
)
Ă( â
kâZd
eâ2Ï(ÎŒ(1âΔ)âÎŒ)|k|)(
supkâZd
ηâRd
|f(k, η)|e2ÏÎŒ|k|e2Ïλ|η|)1âΔ
Ămax{(â«
Rd
|f(x, v)|eÎČ|v| dv
)Δ,
(â«Rd
|f(x, v)|eÎČ|v| dv
)2Δ}.
Since âmâNd
0
λm(1+m)2d|2Ïη|mâq
m!ïżœC(q, Î)eÏ(λ+λ)|η|
and âkâZd
eâ2Ï(ÎŒ(1âΔ)âÎŒ)|k| ïżœâkâZd
eâÏ(ÎŒâÎŒ)|k| ïżœ C
(ÎŒâÎŒ)d,
we easily end up with the desired result.
78 c. mouhot and c. villani
Lemma 4.22. Let f : Rd!C and let α>0, Aïżœ1 and qâNd0. Let further ÎČ be such
that 0<bïżœÎČïżœB. If |f(x)|ïżœAeâα|x| for all x, then for any Δâ(0, 14
), one has
|âqf(x)|ïżœC(q, d)1/ΔK(b, B)A1âΔeâ(1â2Δ)α|x|
Ă suprâNd
0
max{(
ÎČr âârfââr!
)Δ,
(ÎČr âârfââ
r!
)2Δ}.
(4.25)
Remark 4.23. One may conjecture that the optimal constant in the right-hand sideof (4.25) is in fact polynomial in 1/Δ; if this conjecture holds true, then the constantsin Theorem 4.20 (iii) can be improved accordingly. Mironescu communicated to us aderivation of polynomial bounds for the optimal constant in the related inequality
âf (k)âLâ(R) ïżœC(k)âfâ1/(k+2)L1(R) âf (k+1)â(k+1)/(k+2)
Lâ(R) ,
based on a real interpolation method.
Proof. Let us first see f as a function of x1, and treat xâČ=(x2, ..., xd) as a param-eter. Thus the assumption is |f(x1, x
âČ)|ïżœ(Aeâα|xâČ|)eâα|x1|. By a more or less standardinterpolation inequality [22, Lemma A.1],
|â1f(x1, xâČ)|ïżœ 2
âAeâα|xâČ|
âeâα|x1|ââ2
1f(x1, xâČ)â1/2
â = 2â
Aeâα|x|âââ2
1fââ. (4.26)
Let Cq1,r1 be the optimal constant (not smaller than 1) such that
|âq11 f(x1, x
âČ)|ïżœCq1,r1(Aeâα|x|)1âq1/r1ââr1r f(x1, x
âČ)âq1/r1â . (4.27)
By iterating (4.26), we find Cq1,r1 ïżœ2â
Cq1â1,r1Cq1+1,r1 . It follows by induction that
Cq,r ïżœ 2q(râq).
Next, using (4.27) and interpolating according to the second variable x2 as in (4.26),we get
|âq22 âq1
1 f(x)|ïżœCq2,r2(Cq1,r1(Aeâα|x|)1âq1/r1ââr1
1 fâq1/r1â )1âq2/r2ââr22 âq1
1 fâq2/r2â
ïżœCq1,r1Cq2,r2(Aeâα|x|)(1âq1/r1)(1âq2/r2)ââr11 fâ(q1/r1)(1âq2/r2)â ââr2
2 âq11 fâq2/r2â .
We repeat this until we get
|âqf(x)|ïżœCq1,r1 ... Cqd,rd(Aeâα|x|)(1âq1/r1)...(1âqd/rd)
Ăââr11 fâ(q1/r1)(1âq2/r2)...(1âqd/rd)
â ââq11 âr2
2 fâ(q2/r2)(1âq3/r3)...(1âqd/rd)
... ââq11 âq2
2 ... âqdâ1dâ1 ârd
d fâqd/rd . (4.28)
on landau damping 79
Choose rj , 1ïżœjïżœd, in such a way that
Δ
dïżœ qj
rjïżœ 2Δ
d;
this is always possible for Δ< 14d. Then Cqj ,rj
ïżœ(2dq2j )1/Δ, and (4.28) implies that
|âqf(x)|ïżœ (2d|q|2)1/Δ(Aeâα|x|)1âΔ maxsïżœr+q
{ââsfâΔâ, ââsfâ2Δ
â}.
Then, since 2(r+q)Î”ïżœ3dq, we have, by a crude application of Stirlingâs formula (inquantitative form), for sïżœr+q,
ââsfâΔâ ïżœ(
ÎČsââsfââs!
)Δ(s!ÎČs
)Î”ïżœ(
supnâNd
0
ÎČnâânfâân!
)ΔC(ÎČ, q, d)Δâ3dq,
and the result easily follows.
4.5. Algebra property in two variables
In this section we only consider the norms Zλ,ÎŒ;pÏ ; but similar results would hold true for
the two-variables spaces C and F , and could be proven with the same methods as thoseused for the one-variable spaces Fλ and Cλ, respectively (note that the Leibniz formulastill applies because âx and âv+Ïâx commute).
Proposition 4.24. (i) For any λ, ÎŒïżœ0, any ÏâR and any p, q, râ[1,â] such that1/p+1/q=1/r, we have
âfgâZλ,ÎŒ;rÏ
ïżœ âfâZλ,ÎŒ;pÏ
âgâZλ,ÎŒ;qÏ
.
(ii) As a consequence, Zλ,ÎŒÏ =Zλ,ÎŒ;â
Ï is a normed algebra:
âfgâZλ,ÎŒÏ
ïżœ âfâZλ,ÎŒÏ
âgâZλ,ÎŒÏ
.
In particular, âfnâZλ,ÎŒÏ
ïżœâfânZλ,ÎŒ
Ïfor any nâN0, and âefâZλ,ÎŒ
Ïïżœe
âfâZλ,ÎŒÏ .
Proof. First we note that (with the notation (4.18)) |||·|||ÎŒ;r satisfies the â(p, q, r)propertyâ: whenever p, q, râ[1,â] satisfy 1/p+1/q=1/r, we have
|||fg|||ÎŒ;r =âlâZd
e2ÏÎŒ|l|âfg(l, ·)âLr(Rdv)
=âlâZd
e2ÏÎŒ|l|â„â„â„â„â
kâZd
f(k, ·)g(lâk, ·)â„â„â„â„
Lr(Rdv)
ïżœâlâZd
âkâZd
e2ÏÎŒ|k|e2ÏÎŒ|lâk|âf(k, ·)âLp(Rdv)âg(lâk, ·)âLq(Rd
v)
= |||f |||Ό;p|||g|||Ό;q.
80 c. mouhot and c. villani
Next, we write
|||fg|||Zλ,ÎŒ;rÏ
=â
nâNd0
λn
n!|||(âv+Ïâx)n(fg)|||ÎŒ;r
=â
nâNd0
λn
n!
âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁ âmïżœn
(n
m
)(âv+Ïâx)mf(âv+Ïâx)nâmg
âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁÎŒ;r
ïżœâ
nâNd0
λn
n!
âmïżœn
(n
m
)|||(âv+Ïâx)mf |||ÎŒ;p|||(âv+Ïâx)nâmg|||ÎŒ;q
=( â
mâNd0
λm
m!|||(âv+Ïâx)mf |||ÎŒ;p
)âlâNd
0
λl
l!|||(âv+Ïâx)lf |||ÎŒ;q
= |||f |||Zλ,ÎŒ;pÏ
|||g|||Zλ,ÎŒ;qÏ
.
(We could also reduce to Ï =0 by means of Proposition 4.19.)
4.6. Composition inequality
Proposition 4.25. (Composition inequality in two variables) For any λ, ÎŒïżœ0 andany pâ[1,â], ÏâR, ÏâR, aâR\{0} and bâR,
âf(x+bv+X(x, v), av+V (x, v))âZλ,ÎŒ;pÏ
ïżœ |a|âd/pâfâZα,ÎČ;pÏ
, (4.29)
whereα = λ|a|+âV âZλ,ÎŒ
Ïand ÎČ = ÎŒ+λ|b+ÏâaÏ|+âXâÏV âZλ,ÎŒ
Ï. (4.30)
Remark 4.26. The norms in (4.30) for X and V have to be based on Lâ, not justany Lp. Also note that the fact that the second argument of f has the form av+V (andnot av+cx+V ) is related to Remark 4.7.
Proof. The proof is a combination of the arguments in Proposition 4.8. In a firststep, we do it for the case Ï =Ï=0, and we write â · âλ,ÎŒ;p=â · âZλ,ÎŒ;p
0.
From the expansion f(x, v)=â
kâZd f(k, v)e2iÏk·x we deduce that
h(x, v) := f(x+bv+X(x, v), av+V (x, v))
=âkâZd
f(k, av+V )e2iÏk·(x+bv+X)
=âkâZd
mâNd0
âmv f(k, av)· V
m
m!e2iÏk·xe2iÏk·bve2iÏk·X .
on landau damping 81
Taking the Fourier transform in x, we see that for any lâZd,
h(l, v) =âkâZd
mâNd0
âmv f(k, av)e2iÏk·bv
âjâZd
(V m)(j)m!
(e2iÏk·X ) (lâkâj).
Differentiating n times via the Leibniz formula (here applied to a product of four func-tions), we get
ânv h(l, v) =
âk,jâZd
m,n1,n2,n3,n4âNd0
n1+n2+n3+n4=n
n!an1
n1!n2!n3!n4!âm+n1
v f(k, av)
Ăân2v (V m)(j)
m!ân3
v (e2iÏk·X ) (lâkâj, v)(2iÏbk)n4e2iÏk·bv.
Multiplying by λne2ÏÎŒ|l|/n! and summing over n and l, taking Lp norms and usingâfgâLp ïżœâfâLpâgâLâ , we finally obtain
âhâλ,ÎŒ
ïżœ |a|âd/pâ
k,j,lâZd
m,n,n1,n2,n3,n4âNd0
n1+n2+n3+n4=n
λne2ÏÎŒ|l||a|n1
n1!n2!n3!n4!ââm+n1
v f(k, ·)âLp
â„â„â„â„ân2v (V m)(j)
m!
â„â„â„â„â
Ăâân3v (e2iÏk·X ) (lâkâj)ââ(2Ï|b| |k|)n4
= |a|âd/pâ
k,j,lâZd
m,n1,n2,n3,n4âNd0
λn1+n2+n3+n4e2ÏÎŒ|k|e2ÏÎŒ|j|e2ÏÎŒ|lâkâj||a|n1
n1!n2!n3!n4!ââm+n1
v f(k, ·)âLp
Ăâ„â„â„â„ân2
v (V m)(j)m!
â„â„â„â„ââân3
v (e2iÏk·X ) (lâkâj)ââ(2Ï|b| |k|)n4
ïżœ |a|âd/pâkâZd
n1,mâNd0
λn1 |a|n1
n1!âân1+m
v f(k, ·)âLp e2ÏÎŒ|k|
Ă(
1m!
âjâZd
n2âNd0
λn2
n2!e2ÏÎŒ|j|âân2
v (V m)(j)ââ)
Ă( â
hâZd
n3âNd0
λn3
n3!e2ÏÎŒ|h|âân3
v (e2iÏk·X ) (h)ââ) â
n4âNd0
(2Ïλ|b| |k|)n4
n4!
= |a|âd/pâkâZd
n1,mâNd0
(λ|a|)n1
n1!e2ÏÎŒ|k|âân1+m
v f(k, ·)âLp
âV mâλ,ÎŒ
m!âe2iÏk·Xâλ,ÎŒe2Ïλ|b| |k|
82 c. mouhot and c. villani
ïżœ |a|âd/pâkâZd
n1,mâNd0
(λ|a|)n1
n1!e2Ï(ÎŒ+λ|b|)|k|âân1+m
v f(k, ·)âLp
âV âmλ,ÎŒ
m!e2Ï|k| âXâλ,ÎŒ
= |a|âd/pâkâZd
nâNd0
1n!
(λ|a|+âV âλ,ÎŒ)nâânv f(k, ·)âLp e2Ï|k|(ÎŒ+λ|b|+âXâλ,ÎŒ)
= |a|âd/pâfâλ|a|+âV âλ,ÎŒ,ÎŒ+λ|b|+âXâλ,ÎŒ.
Now we generalize this to arbitrary values of Ï and Ï . By Proposition 4.19,
âf(x+bv+X(x, v), av+V (x, v))âZλ,ÎŒ;pÏ
= âf(x+v(b+Ï)+X(x+vÏ, v), av+V (x+vÏ, v))âZλ,ÎŒ;p
= âf S0Ï S0
âÏ(x+v(b+Ï)+X(x+vÏ, v), av+V (x+vÏ, v))âZλ,ÎŒ;p
= â(f S0Ï)(x+v(b+ÏâaÏ)+(XâÏV )(x+vÏ, v), av+V (x+vÏ, v))âZλ,ÎŒ;p
= â(f S0Ï)(x+v(b+ÏâaÏ)+Y (x, v), av+W (x, v))âZλ,ÎŒ;p ,
whereW (x, v) =V S0
Ï (x, v) and Y (x, v) = (XâÏV ) S0Ï (x, v).
Applying the result for Ï =0, we deduce that the norm of
h(x, v) = f(x+bv+X(x, v), av+V (x, v))
in Zλ,ÎŒÏ is bounded by
âf S0ÏâZα,ÎČ;p = âfâZα,ÎČ;p
Ï,
whereα = λ|a|+âV S0
ÏâZλ,ÎŒ = |a|λ+âV âZλ,ÎŒÏ
and
ÎČ = ÎŒ+λ|b+ÏâaÏ|+â(XâÏV ) S0ÏâZλ,ÎŒ = ÎŒ+λ|b+ÏâaÏ|+âXâÏV âZλ,ÎŒ
Ï.
This establishes the desired bound.
4.7. Gradient inequality
In the next proposition we shall write
âfâZλ,ÎŒÏ
=âlâZdâ
ânâNd
0
λn
n!â(âv+2iÏÏ l)nf(l, v)âLâ(Rd
v) e2ÏÎŒ|l|. (4.31)
This is again a homogeneous (in the x variable) seminorm.
on landau damping 83
Proposition 4.27. For λ>Î»ïżœ0 and ÎŒ>ÎŒïżœ0, we have the functional inequalities
ââxfâZλ,ÎŒ;pÏ
ïżœ C(d)ÎŒâÎŒ
âfâZλ,ÎŒ;pÏ
,
â(âv+Ïâx)fâZλ,ÎŒ;pÏ
ïżœ C(d)λ log(λ/λ)
âfâZλ,ÎŒ;pÏ
.
In particular, for Ï ïżœ0 we have
ââvfâZλ,ÎŒ;pÏ
ïżœC(d)(
1λ log(λ/λ)
âfâZλ,ÎŒ;pÏ
+Ï
ÎŒâÎŒâfâZλ,ÎŒ;p
Ï
).
The proof is similar to that of Proposition 4.10; the constant C(d) arises in thechoice of norm on Rd. As a consequence, if 1<λ/Î»ïżœ2, we have e.g. the bound
ââfâZλ,ÎŒ;pÏ
ïżœC(d)(
1λâλ
+1+Ï
ÎŒâÎŒ
)âfâZλ,ÎŒ;p
Ï.
4.8. Inversion
From the composition inequality follows an inversion estimate.
Proposition 4.28. (Inversion inequality) (i) Let λ, ÎŒïżœ0, ÏâR and consider a func-tion F : TdĂRd!TdĂRd. Then there is Δ=Δ(d) such that if F satisfies
ââ(FâId)âZλâČ,ÎŒâČÏ
ïżœ Δ(d),
whereλâČ = λ+2âFâId âZλ,ÎŒ
Ïand ÎŒâČ = ÎŒ+2(1+|Ï |)âFâId âZλ,ÎŒ
Ï,
then F is invertible and
âFâ1âId âZλ,ÎŒÏ
ïżœ 2âFâId âZλ,ÎŒÏ
. (4.32)
(ii) More generally, if F,G: TdĂRd!TdĂRd are such that
ââ(FâId)âZλâČ,ÎŒâČÏ
ïżœ Δ(d), (4.33)
whereλâČ = λ+2âFâGâZλ,ÎŒ
Ïand ÎŒâČ = ÎŒ+2(1+|Ï |)âFâGâZλ,ÎŒ
Ï,
then F is invertible and
âFâ1 GâId âZλ,ÎŒÏ
ïżœ 2âFâGâZλ,ÎŒÏ
. (4.34)
84 c. mouhot and c. villani
Remark 4.29. The conditions become very stringent as Ï becomes large: basically,FâId (or FâG in case (ii)) should be of order o(1/Ï) for Proposition 4.28 to be appli-cable.
Remark 4.30. By Proposition 4.27, a sufficient condition for (4.33) to hold is thatthere be λâČâČ and ÎŒâČâČ such that Î»ïżœÎ»âČâČïżœ2λ, ÎŒïżœÎŒâČâČ and
âFâId âZλâČâČ,ÎŒâČâČÏ
ïżœ ΔâČ(d)1+Ï
min{λâČâČâλâČ, ÎŒâČâČâÎŒâČ}.However, this condition is in practice hard to fulfill.
Proof. We prove only (ii), being (i) a particular case. Let f=FâId, h=Fâ1 GâIdand g=GâId, so that Id +g=(Id +f) (Id +h), or equivalently
h = gâf (Id +h).
Thus h is a fixed point ofΊ: Z â! gâf (Id +Z).
Note that Ί(0)=gâf . If Ί is 12 -Lipschitz on the ball B(0, 2âfâgâ) in Zλ,ÎŒ
Ï , then (4.34)will follow by fixed-point iteration as in Theorem A.2. (Here B(x, r):={y :âxâyâïżœr}.)
So let Z and Z be given with
âZâZλ,ÎŒÏ
, âZâZλ,ÎŒÏ
ïżœ 2âfâgâZλ,ÎŒÏ
.
We have
Ί(Z)âΊ(Z) = f(Id +Z)âf(Id +Z) = (ZâZ)·⫠1
0
âf(Id +(1âΞ)Z+ΞZ) dΞ.
By Proposition 4.24,
âΊ(Z)âΊ(Z)âZλ,ÎŒÏ
ïżœ âZâZâZλ,ÎŒÏ
â« 1
0
ââf(Id +(1âΞ)Z+ΞZ)âZλ,ÎŒÏ
dΞ.
For any Ξâ[0, 1], by Proposition 4.25,
ââf(Id +(1âΞ)Z+ΞZ)âZλ,ÎŒÏ
ïżœ ââfâZλ,ÎŒÏ
,
whereλ = λ+max{âZâ, âZâ}ïżœÎ»+2âfâgâZλ,ÎŒ
Ï
and, writing Z=(Zx, Zv) and Z=(Zx, Zv),
ÎŒ= ÎŒ+max{âZxâÏZvâ, âZxâÏZvâ}ïżœÎŒ+2(1+|Ï |)âfâgâZλ,ÎŒÏ
.
If F and G satisfy the assumptions of Proposition 4.28, we deduce that
âΊâLip(B(0,2)) ïżœC(d)Δ(d),
and this is bounded above by 12 if Δ(d) is small enough.
on landau damping 85
4.9. Sobolev corrections
We shall need to quantify Sobolev regularity corrections to the analytic regularity, in thex variable.
Definition 4.31. (Hybrid analytic norms with Sobolev corrections) For λ, ÎŒ, Îłïżœ0,ÏâR and pâ[1,â] we define
âfâZλ,(ÎŒ,Îł);pÏ
=âlâZd
ânâNd
0
λn
n!e2ÏÎŒ|l|(1+|l|)Îłâ(âv+2iÏÏ l)nf(l, v)âLp(Rd
v),
âfâFλ,Îł =âkâZd
e2Ïλ|k|(1+|k|)Îł |f(k)|.
Proposition 4.32. Let λ, ÎŒ, Îłïżœ0, ÏâR and pâ[1,â]. We have the following func-tional (in)equalities.
(i) âfâZλ,(ÎŒ,Îł);pt+Ï
=âf S0t âZλ,(ÎŒ,Îł);p
Ï;
(ii) If 1/p+1/q=1/r then âfgâZλ,(ÎŒ,Îł);rÏ
ïżœâfâZλ,(ÎŒ,Îł);pÏ
âgâZλ,(ÎŒ,Îł);qÏ
, and therefore in
particular Zλ,(ÎŒ,Îł)Ï =Zλ,(ÎŒ,Îł);â
Ï is a normed algebra;(iii) If f depends only on x then âfâZλ,(ÎŒ,Îł)
Ï=âfâFλ|Ï|+ÎŒ,Îł ;
(iv) âfâZλ,(ÎŒ,Îł);pÏ
ïżœâfâZλ,(ÎŒ+λ|ÏâÏ|,Îł);pÏ
;(v) For any ÏâR, aâR\{0} and bâR,
âf(x+bv+X(x, v), av+V (x, v))âZλ,(ÎŒ,Îł);pÏ
ïżœ |a|âd/pâfâZα,(ÎČ,Îł);pÏ
,
whereα = λ|a|+âV âZλ,(ÎŒ,Îł)
Ïand ÎČ = ÎŒ+λ|bÏâaÏ|+âXâÏV âZλ,(ÎŒ,Îł)
Ï;
(vi) Gradient inequality:
ââxfâZλ,(ÎŒ,Îł);pÏ
ïżœ C(d)ÎŒâÎŒ
âfâZλ,(ÎŒ,Îł);pÏ
,
ââfâZλ,(ÎŒ,Îł);pÏ
ïżœC(d)(
1λâλ
+1+Ï
ÎŒâÎŒ
)âfâZλ,(ÎŒ,Îł);p
Ï;
(vii) Inversion: if F,G: TdĂRd!TdĂRd are such that
ââ(FâId)âZλâČ,(ÎŒâČ,Îł)Ï
ïżœ Δ(d),
whereλâČ = λ+2âFâGâZλ,(ÎŒ,Îł)
Ïand ÎŒâČ = ÎŒ+2(1+|Ï |)âFâGâZλ,(ÎŒ,Îł)
Ï,
thenâFâ1 GâId âZλ,(ÎŒ,Îł)
Ïïżœ 2âFâGâZλ,(ÎŒ,Îł)
Ï. (4.35)
86 c. mouhot and c. villani
Proof. The proofs are the same as for the âplainâ hybrid norms; the only notablepoint is that for the proof of (ii) we use, in addition to e2Ïλ|k|ïżœe2Ïλ|kâl|e2Ïλ|l|, theinequality
(1+|k|)Îł ïżœ (1+|kâl|)Îł(1+|l|)Îł .
Remark 4.33. Of course, some of the estimates in Proposition 4.32 can be âim-provedâ by taking advantage of Îł; e.g. for Îłïżœ1 we have
ââxfâZλ,ÎŒ;pÏ
ïżœC(d)âfâZλ,(ÎŒ,Îł);pÏ
.
4.10. Individual mode estimates
To handle very singular cases, we shall at times need to estimate Fourier modes individ-ually, rather than full norms. If f=f(x, v), we write
(Pkf)(x, v) = f(k, v)e2iÏk·x. (4.36)
In particular the following estimates will be useful.
Proposition 4.34. For any λ, ÎŒïżœ0, ÏâR, Lebesgue exponents 1/r=1/p+1/q andkâZd, we have the estimate
âPk(fg)âZλ,ÎŒ;rÏ
ïżœâlâZd
âPlfâZλ,ÎŒ;pÏ
âPkâlgâZλ,ÎŒ;qÏ
.
Proposition 4.35. For any λ>0, ÎŒïżœÎŒïżœ0, ÏâR, pâ[1,â] and kâZd, we have theestimate
âPk[f(x+X(x, v), v)]âZλ,ÎŒ;pÏ
ïżœâlâZd
eâ2Ï(ÎŒâÎŒ)|kâl|âPlfâZλ,Îœ;pÏ
, Îœ = ÎŒ+âXâZλ,ÎŒÏ
.
These estimates also have variants with Sobolev corrections. Note that when Ό=Ό,Proposition 4.35 is a direct consequence of Proposition 4.25 with V =0, b=0 and a=1:
âPk[f(x+X(x, v), v)]âZλ,ÎŒ;pÏ
ïżœ âf(x+X(x, v), v)âZλ,ÎŒ;pÏ
ïżœ âfâZλ,Îœ;pÏ
, Îœ = ÎŒ+âXâZλ,ÎŒÏ
.
Proof of Propositions 4.34 and 4.35. The proof of Proposition 4.34 is quite similarto the proof of Proposition 4.24. (It is no restriction to choose Ï =0 because Pk commuteswith the free transport semigroup.) Proposition 4.35 needs a few words of explanation.As in the proof of Proposition 4.25 we let h(x, v)=f(x+X(x, v), v), and readily obtain
âPkhâZλ,ÎŒ;pÏ
=â
nâNd0
λne2ÏÎŒ|k|
n!âân
v h(k, v)âLp(dv)
ïżœâ
nâNd0
lâZd
λne2ÏÎŒ|k|
n!âân
v f(l, v)âLp(dv)
âmâNd
0
λm
m!ââm
v (e2iÏlX ) (kâl, v)âLâ(dv).
on landau damping 87
At this stage we write
e2ÏÎŒ|k| ïżœ e2ÏÎŒ|l|eâ2Ï(ÎŒâÎŒ)|kâl|e2ÏÎŒ|kâl|,
and use the following crude bound, which holds for all lâZd:
e2ÏÎŒ|kâl|ââmv (e2iÏlX ) (kâl, v)âLâ(dv) ïżœ
âjâZd
e2ÏÎŒ|j|ââmv (e2iÏlX ) (j, v)âLâ(dv).
The rest of the proof is as in Proposition 4.25.
4.11. Measuring solutions of kinetic equations in large time
As we already discussed, even for the simplest kinetic equation, namely free transport, wecannot hope to have uniform-in-time regularity estimates in the velocity variable: rather,because of filamentation, we may have ââvf(t, ·)â=O(t), ââ2
vf(t, ·)â=O(t2), etc. Foranalytic norms we may at best hope for exponential growth.
But the invariance of the âglidingâ norms Zλ,ÎŒÏ under free transport (Proposi-
tion 4.19) makes it possible to look for uniform estimates such as
âf(Ï, ·)âZλ,ÎŒÏ
= O(1) as Ï!â. (4.37)
Of course, by Proposition 4.27, (4.37) implies that
ââvf(Ï, ·)âZλâČ,ÎŒâČÏ
= O(Ï) for λâČ <λ and ÎŒâČ <ÎŒ, (4.38)
and nothing better as far as the asymptotic behavior of âvf is concerned; but (4.37) ismuch more precise than (4.38). For instance it implies that
â(âv+Ïâx)f(Ï, ·)âZλâČ,ÎŒâČÏ
= O(1) for λâČ <λ and ÎŒâČ <ÎŒ.
Another way to get rid of filamentation is to average over the spatial variable x, acommon sense procedure which has already been used in physics [54, §49]. Think that, iff evolves according to free transport, or even according to the linearized Vlasov equation(3.3), then its space-average
ăfă(Ï, v) :=â«
Td
f(Ï, x, v) dx (4.39)
is time-invariant. (We used these infinite number of conservation laws to determine thelong-time behavior in Theorem 3.1.)
88 c. mouhot and c. villani
The bound (4.37) easily implies a bound on the space average: indeed,
âăfă(Ï, ·)âCλ = âăfă(Ï, ·)âZλ,ÎŒÏ
ïżœ âf(Ï, ·)âZλ,ÎŒÏ
= O(1) as Ï!â; (4.40)
and in particular, for λâČ<λ,
âăâvfă(Ï, ·)âCλâČ = O(1) as Ï!â. (4.41)
Again, (4.37) contains a lot more information than (4.41).
Remark 4.36. The idea to estimate solutions of a non-linear equation by comparisonwith some unperturbed (reversible) linear dynamics is already present in the definitionof Bourgain spaces Xs,b [12]. The analogy stops here, since time is a dummy variable inXs,b spaces, while in Zλ,Ό
t spaces it is frozen and appears as a parameter, on which weshall play later.
4.12. Linear damping revisited
As a simple illustration of the functional analysis introduced in this section, let us recastthe linear damping (Theorem 3.1) in this language. This will be the first step for thestudy of the non-linear damping. For simplicity we set L=1.
Theorem 4.37. (Linear Landau damping again) Let f0=f0(v), W : Td!R withââWâL1 ïżœCW and fi(x, v) be such that
(i) condition (L) from §2.2 holds for some constants C0, λ, >0;(ii) âf0âCλ;1 ïżœC0;(iii) âfiâZλ,ÎŒ;1 ïżœÎŽ for some ÎŒ, ÎŽ>0;Then for any λâČ<λ and ÎŒâČ<ÎŒ, the solution of the linearized Vlasov equation (3.3)
satisfiessuptâR
âf(t, ·)âZλâČ,ÎŒâČ;1t
ïżœCÎŽ (4.42)
for some constant C=C(d, CW , C0, λ, λâČ, ÎŒ, ÎŒâČ, ). In particular, ïżœ=â«
Rd f dv satisfies
suptâR
âïżœ(t, ·)âFλâČ|t|+ÎŒâČ ïżœCÎŽ. (4.43)
As a consequence, as |t|!â, ïżœ converges strongly to
ïżœâ =â«
Td
â«Rd
fi(x, v) dv dx,
and f converges weakly to ăfiă=â«
Tdfi dx, at rate O(eâλâČâČ|t|) for any λâČâČ<λâČ.
If moreover âf0âCλ;p ïżœC0 and âfiâZλ,ÎŒ;p ïżœÎŽ for all p in some interval [1, p ], then(4.42) can be reinforced into
suptâR
âf(t, ·)âZλâČ,ÎŒâČ;pt
ïżœCÎŽ, 1 ïżœ p ïżœ p. (4.44)
on landau damping 89
Remark 4.38. The notions of weak and strong convergence are the same as thosein Theorem 3.1. With respect to that statement, we have added an extra analyticityassumption in the x variable; in this linear context this is an overkill (as the proof willshow), but later in the non-linear context this will be important.
Proof. Without loss of generality we restrict our attention to tïżœ0. Although (4.43)follows from (4.42) by Proposition 4.15, we shall establish (4.43) first, and deduce (4.42)due to the equation. We shall write C for various constants depending only on theparameters in the statement of the theorem.
As in the proof of Theorem 3.1, we have
ïżœ(t, k) = fi(k, kt)+â« t
0
K0(tâÏ, k)ïżœ(Ï, k) dÏ
for any tïżœ0 and kâZd. By Lemma 3.6, for any λâČ<λ and ÎŒâČ<ÎŒ,
suptïżœ0
( âkâZd
|ïżœ(t, k)|e2Ï(λâČt+ÎŒâČ)|k|)
ïżœC(λ, λâČ, )( â
kâZd
eâ2Ï(ÎŒâÎŒâČ)|k|)
suptïżœ0
supkâZd
|fi(k, kt)|e2Ï(λâČt+ÎŒ)|k|
ïżœ C(λ, λâČ, )(ÎŒâÎŒâČ)d
suptïżœ0
âkâZd
|fi(k, kt)|e2Ï(λt+ÎŒ)|k|.
Equivalently,
suptïżœ0
âïżœ(t, ·)âFλâČt+ÎŒâČ ïżœC suptïżœ0
â„â„â„â„â«Rd
fi S0ât dv
â„â„â„â„Fλt+ÎŒ
. (4.45)
By Propositions 4.15 and 4.19,â„â„â„â„â«Rd
fi S0ât dv
â„â„â„â„Fλt+ÎŒ
ïżœ âfi S0âtâZλ,ÎŒ;1
t= âfiâZλ,ÎŒ;1
0ïżœ ÎŽ.
This and (4.45) imply (4.43).To deduce (4.42), we first write
f(t, ·) = fi S0ât+â« t
0
((âW âïżœÏ ) S0â(tâÏ))·âvf0 dÏ,
where ïżœÏ =ïżœ(Ï, ·). Then for any λâČâČ<λâČ we have, by Propositions 4.24 and 4.15, for alltïżœ0,
âfâZλâČâČ,ÎŒâČ;1t
ïżœ âfi S0âtâZλâČâČ,ÎŒ;1
t+â« t
0
â(âW âïżœÏ ) S0â(tâÏ)âZλâČâČ,ÎŒâČ;â
tââvf0âZλâČâČ,ÎŒ;1
tdÏ
= âfiâZλâČâČ,ÎŒ;1 +ââvf0âCλâČâČ;1
â« t
0
ââW âïżœÏâFλâČâČÏ+ÎŒâČ dÏ. (4.46)
90 c. mouhot and c. villani
Since âW (0)=0, we have, for any Ï ïżœ0,
ââW âïżœÏâFλâČâČÏ+ÎŒ ïżœ eâ2Ï(λâČâČâλâČ)ÏââW âïżœÏâFλâČÏ+ÎŒâČ
ïżœ ââWâL1 eâ2Ï(λâČâČâλâČ)ÏâïżœÏâFλâČÏ+ÎŒâČ
ïżœCW CÎŽeâ2Ï(λâČâČâλâČ)Ï ;
in particular â« t
0
ââW âïżœÏâFλâČâČ+ÎŒâČ dÏ ïżœ CÎŽ
λâČâČâλâČ . (4.47)
Also, by Proposition 4.10, for 1<λâČ/λâČâČïżœ2 we have
ââvf0âCλâČâČ;1 ïżœ C
λâλâČâČ âf0âCλ;1 ïżœ C C0
λâλâČâČ . (4.48)
Plugging (4.47) and (4.48) into (4.46), we deduce (4.42). The end of the proof is an easyexercise if one recalls that ăf(t, ·)ă=ăfiă for all t.
5. Deflection estimates
Let a small time-dependent force field, denoted by ΔF (t, x), be given on TdĂRd, whoseanalytic regularity improves linearly in time. (Think of ΔF as the force created by adamped density.) This force field perturbs the trajectories S0
Ï,t of the free transport (Ï isthe initial time and t the current time) into trajectories SÏ,t. The goal of this section isto get an estimate on the maps Ωt,Ï =St,Ï S0
Ï,t (so that St,Ï =Ωt,Ï S0t,Ï ). These bounds
should be in an analytic class about as good as F , with a loss of analyticity dependingon Δ; they should also be, for 0ïżœÏ ïżœt,
âą uniform in tïżœÏ ,âą small as Ï!â,âą small as Ï!t.We shall call Ωt,Ï the deflection map (from time Ï to time t). The idea is to com-
pare the free (= without interaction) evolution to the true evolution is at the basis of theinteraction representation, wave operators, and most famously the scattering transforms(in which the whole evolution from t!ââ to t!â is replaced by well-chosen asymp-totics). For all these methods which are of constant use in classical and quantum physicsone can consult [21].
Remark 5.1. The order of composition in Ωt,Ï =St,Ï S0Ï,t is the only one which leads
to useful asymptotics: it is easy to check that in general S0Ï,t St,Ï does not converge to
anything as t!â, even if the force is compactly supported in time.
on landau damping 91
5.1. Formal expansion
Before stating the main result, we sketch a heuristic perturbation study. Let us write aformal expansion of V0,t(x, v) as a perturbation series:
V0,t(x, v) = v+Δv(1)(t, x, v)+Δ2v(2)(t, x, v)+... .
Then we deduce that
X0,t(x, v) =x+vt+Δ
â« t
0
v(1)(s, x, v) ds+Δ2
â« t
0
v(2)(s, x, v) ds+...,
with v(i)(t=0)=0.So
â2X0,t
ât2= Δ
âv(1)
ât+Δ2 âv(2)
ât+... .
On the other hand,
ΔF (t, X0,t) = ΔâkâZd
F (t, k)e2iÏk·xe2iÏk·vte2iÏk·(Δ â« t0 v(1) ds+Δ2 â« t
0 v(2) ds+... )
= ΔâkâZd
F (t, k)e2iÏk·xe2iÏk·vt
Ă(
1+2iÏΔk ·⫠t
0
v(1) ds+2iÏΔ2k ·⫠t
0
v(2) dsâ(2Ï)2Δ2
(k ·⫠t
0
v(1) ds
)2+...
).
By successive identification,âv(1)
ât=âkâZd
F (t, k)e2iÏk·xe2iÏk·vt,
âv(2)
ât=âkâZd
F (t, k)e2iÏk·xe2iÏk·vt2iÏk ·⫠t
0
v(1) ds,
âv(3)
ât=âkâZd
F (t, k)e2iÏk·xe2iÏk·vt
(2iÏk ·
â« t
0
v(2) dsâ(2Ï)2Δ2
(k ·⫠t
0
v(1) ds
)2),
etc.In particular notice that âŁâŁâŁâŁâv(1)
ât
âŁâŁâŁâŁïżœ âkâZd
|F (t, k)|,
so â« â
0
âŁâŁâŁâŁâv(1)
ât
âŁâŁâŁâŁ dt ïżœâ« â
0
âkâZd
|F (t, k)| dt
ïżœâ« â
0
âkâZd
|F (t, k)|e2ÏÎŒteâ2ÏÎŒt dt ïżœCF
â« â
0
eâ2ÏÎŒt dt =CF
2ÏÎŒ.
Thus, under our uniform analyticity assumptions we expect V0,t(x, v) to be a uniformlybounded analytic perturbation of v.
92 c. mouhot and c. villani
5.2. Main result
On Tdx we consider the dynamical system
d2X
dt2= ΔF (t, X);
its phase space is TdĂRd. Although this system is reversible, we shall only consider tïżœ0.The parameter Δ is here only to recall the perturbative nature of the estimate.
For any (x, v)âTdĂRd and any two times Ï, tâR+, let SÏ,t be the transform mappingthe state of the system at time Ï to the state of the system at time t. In more preciseterms, SÏ,t is described by the equations
SÏ,t(x, v) = (XÏ,t(x, v), VÏ,t(x, v)), XÏ,Ï (x, v) =x, VÏ,Ï (x, v) = v,
d
dtXÏ,t(x, v) =VÏ,t(x, v) and
d
dtVÏ,t(x, v) = Δ F (t, XÏ,t(x, v)). (5.1)
From the definition we have the composition identity
St2,t3 St1,t2 = St1,t3 ; (5.2)
in particular St,Ï is the inverse of SÏ,t.We also write S0
Ï,t for the same transform in the case of the free dynamics (Δ=0); inthis case there is an explicit expression:
S0Ï,t(x, v) = (x+v(tâÏ), v), (5.3)
where x+v(tâÏ) is evaluated modulo Zd. Finally, we define the deflection map associatedwith ΔF :
Ωt,Ï = St,Ï S0Ï,t. (5.4)
(There is no simple semigroup property for the transforms Ωt,Ï .)In this section we establish the following estimates.
Theorem 5.2. (Analytic estimates on deflection in hybrid norms) Let Δ>0 and letF =F (t, x) on R+ĂTd satisfy
F (t, 0) = 0 and suptïżœ0
(âF (t, ·)âFλt+ÎŒ +ââxF (t, ·)âFλt+ÎŒ) ïżœCF (5.5)
for some parameters λ, ÎŒ>0 and CF >0. Let tïżœÏ ïżœ0 and let
Ωt,Ï = (ΩXt,Ï , ΩVt,Ï )
on landau damping 93
be the deflection associated with ΔF . Let 0ïżœÎ»âČ<λ, 0ïżœÎŒâČ<ÎŒ and Ï âČïżœ0 be such that
λâČ(Ï âČâÏ) ïżœ 12 (ÎŒâÎŒâČ). (5.6)
Let {R1(Ï, t) =CF eâ2Ï(λâλâČ)Ï min{tâÏ, (2Ï(λâλâČ))â1},R2(Ï, t) =CF eâ2Ï(λâλâČ)Ï min
{12 (tâÏ)2, (2Ï(λâλâČ))â2}.
Assume thatΔR2(Ï, t) ïżœ 1
4 (ÎŒâÎŒâČ) for all 0 ïżœ Ï ïżœ t, (5.7)
andΔCF ïżœ 2Ï2(λâλâČ)2. (5.8)
ThenâΩXt,Ï âId âZλâČ,ÎŒâČ
ÏâČïżœ 2ΔR2(Ï, t) for all 0 ïżœ Ï ïżœ t, (5.9)
andâΩVt,Ï âId âZλâČ,ÎŒâČ
ÏâČïżœ ΔR1(Ï, t) for all 0 ïżœ Ï ïżœ t. (5.10)
Remark 5.3. The proof of Theorem 5.2 is easily adapted to include Sobolev correc-tions. It is important to note that the deflection map is smooth uniformly in time, notjust in gliding regularity (Ï âČ=0 is admissible in (5.6)).
Proof. For a start, let us make the ansatz
St,Ï (x, v) = (xâv(tâÏ)+ΔZt,Ï (x, v), v+ΔâÏZt,Ï (x, v)),
withZt,t(x, v) = 0 and âÏZt,Ï |Ï=t(x, v) = 0.
Then it is easily checked that
Ωt,Ï âId = Δ (Z, âÏZ) S0tâÏ ;
in particularâΩt,Ï âId âZλâČ,ÎŒâČ
ÏâČ= Δ â(Z, âÏZ)âZλâČ,ÎŒâČ
t+ÏâČâÏ
.
To estimate this we shall use a fixed-point argument based on the equation for St,Ï ,namely
d2Xt,Ï
dÏ2= ΔF (Ï,Xt,Ï ),
or equivalentlyd2Zt,Ï
dÏ2= F (Ï, xâv(tâÏ)+ΔZt,Ï ).
94 c. mouhot and c. villani
So let us fix t and define
Κ: (Wt,Ï )0ïżœÏïżœt â! (Zt,Ï )0ïżœÏïżœt
such that (Zt,Ï )0ïżœÏïżœt is the solution ofâ§âȘâȘâȘâšâȘâȘâȘâ©â2Zt,Ï
âÏ2= F (Ï, xâv(tâÏ)+ΔWt,Ï ),
Zt,t = 0,
(âÏZt,Ï )|Ï=t = 0.
(5.11)
What we are after is an estimate of the fixed point of Κ. We do this in two steps.
Step 1. Estimate of Κ(0). Let Z0=Κ(0). By integration of (5.11) (for W =0) wehave
Z0t,Ï =
â« t
Ï
(sâÏ)F (s, xâv(tâs)) ds.
Let Ï be such that λâČÏïżœ 12 (ÎŒâÎŒâČ). Applying the ZλâČ,ÎŒâČ
t+Ï norm and using Proposi-tion 4.19, we get
âZ0t,ÏâZλâČ,ÎŒâČ
t+Ïïżœâ« t
Ï
(sâÏ)âF (s, ·)âZλâČ,ÎŒâČs+Ï
ds =â« t
Ï
(sâÏ)âF (s, ·)âFλâČs+λâČÏ+ÎŒâČ ds.
Of course λâČÏ+ÎŒâČïżœÎŒ, so in particular
λâČs+λâČÏ+ÎŒâČ ïżœâ(λâλâČ)s+λs+ÎŒ.
Combining this with the assumption F (s, 0)=0 yields
âF (s, ·)âFλâČs+λâČÏ+ÎŒâČ ïżœ âF (s, ·)âFλs+ÎŒeâ2Ï(λâλâČ)s ïżœCF eâ2Ï(λâλâČ)s.
So
âZ0t,ÏâZλâČ,ÎŒâČ
t+ÏïżœCF
â« t
Ï
(sâÏ)eâ2Ï(λâλâČ)s ds
ïżœCF eâ2Ï(λâλâČ)Ï min{
(tâÏ)2
2,
1(2Ï(λâλâČ))2
}ïżœR2(Ï, t).
With t still fixed, we define the norm
||||(Zt,Ï )0ïżœÏïżœt|||| := sup
{âZt,ÏâZλâČ,ÎŒâČt+Ï
R2(Ï, t): 0 ïżœ Ï ïżœ t, Ï+t ïżœ 0 and λâČÏ ïżœ ÎŒâÎŒâČ
2
}. (5.12)
on landau damping 95
The above estimates show that ||||Κ(0)||||ïżœ1. (Since t+(Ï âČâÏ)ïżœtâÏ ïżœ0, we may assumethat t+Ïïżœ0, and we aim at finally choosing Ï=Ï âČâÏ .)
Step 2. Lipschitz constant of Κ. We shall prove that under our assumptions, Κis 1
2 -Lipschitz on the ball B(0, 2) in the norm |||| · ||||. Let W, WâB(0, 2), Z=Κ(W ) andZ=Κ(W ). By solving the differential inequality for ZâZ, we get
Zt,Ï âZt,Ï = Δ(Wt,sâWt,s)·⫠1
0
â« t
Ï
(sâÏ)âxF (s, xâv(tâs)+Δ(ΞWt,s+(1âΞ)Wt,s)) ds dΞ.
We divide by R2(Ï, t), take the Z norm and note that R2(s, t)ïżœR2(Ï, t) to obtain
||||(Zt,Ï âZt,Ï )0ïżœÏïżœt||||ïżœ Δ||||(Wt,sâWt,s)0ïżœsïżœt||||A(t)
with
A(t) = supÏ,Ï
â« 1
0
â« t
Ï
(sâÏ)ââxF (s, xâv(tâs)+Δ(ΞWt,s+(1âΞ)Wt,s))âZλâČ,ÎŒâČt+Ï
ds dΞ.
By Proposition 4.25 (composition inequality),
A(t) ïżœâ« t
Ï
(sâÏ)ââxF (s, ·)âZλâČ,ÎŒâČ+e(t,s,Ï)s+Ï
ds =â« t
Ï
(sâÏ)ââxF (s, ·)âFλâČs+λâČÏ+ÎŒâČ+e(t,s,Ï) ds,
withe(t, s, Ï) := ΔâΞWt,s+(1âΞ)Wt,sâZλâČ,ÎŒâČ
t+Ïïżœ 2ΔR2(s, t) ïżœ 2ΔR2(Ï, t).
Using (5.7), we get
λâČs+λâČÏ+ÎŒâČ+e(s, t, Ï) ïżœÎ»âČs+λâČÏ+ÎŒâČ+2ΔR2(Ï, t) ïżœÎ»âČs+ÎŒ = (λs+ÎŒ)â(λâλâČ)s.
By again using the bound on âxF and the assumption F (s, 0)=0, we deduce that
A(t) ïżœ supÏ
â« t
Ï
(sâÏ)CF eâ2Ï(λâλâČ)s ds ïżœR2(0, t) ïżœ CF
4Ï2(λâλâČ)2.
Using (5.8), we conclude that
||||(Zt,Ï âZt,Ï )0ïżœÏïżœt||||ïżœ 12 ||||(Wt,sâWt,s)0ïżœsïżœt||||.
So Κ is 12 -Lipschitz on B(0, 2), and we can conclude the proof of (5.9) by applying
Theorem A.2 and choosing Ï=Ï âČâÏ .
It remains to control the velocity component of Ω, i.e., establish (5.10); this willfollow from the control of the position component. Indeed, if we write
Qt,Ï = Δâ1(ΩVt,Ï âId)(x, v),
96 c. mouhot and c. villani
we have
Qt,Ï =â« t
Ï
F (s, xâv(tâs)+ΔWt,s) ds,
so we can estimate as before
âQt,ÏâZλâČ,ÎŒâČt+(ÏâČâÏ)
ïżœâ« t
Ï
âF (s, ·)âFλâČs+λâČ(ÏâČâÏ)+ÎŒâČ+e(t,s,ÏâČâÏ) ds
to get
âQt,ÏâZλâČ,ÎŒâČt+(ÏâČâÏ)
ïżœâ« t
Ï
CF eâ2Ï(λâλâČ)s ds ïżœR1(Ï, t).
Thus the proof is complete.
Remark 5.4. Instead of directly studying St,Ï S0Ï,t, one can study S0
t,Ï SÏ,t and theninvert with the help of Proposition 4.28 (inversion inequality); in the end the results aresimilar.
Remark 5.5. Loss and Bernard independently suggested to compare the estimatesin the present section with the Nekhoroshev theorem in dynamical systems theory [71],[72]. The latter theorem roughly states that for a perturbation of a completely integrablesystem, trajectories remain close to those of the unperturbed system for a time growingexponentially in the inverse of the size of the perturbation (unlike KAM theory, thisresult is not global in time; but it is more general in the sense that it also applies outsideinvariant tori). In the present setting the situation is better since the perturbation decays.
6. Bilinear regularity and decay estimates
To introduce this crucial section, let us reproduce and improve a key computation from §3.Let G be a function of v, and R a time-dependent function of x with R(0)=0; both G
and R are vector-valued in Rd. (Think of G(v) as âvf(v) and of R(Ï, x) as âW âïżœ(Ï, x).)From now on we shall always use the supremum norm over the coordinates in the followingsense
âFâZ := max1ïżœjïżœd
âFjâZ and âFâF := max1ïżœjïżœd
âFjâF .
Let further
Ï(t, x) =â« t
0
â«Rd
G(v)·R(Ï, xâv(tâÏ)) dv dÏ.
on landau damping 97
Then
Ï(t, k) =â« t
0
â«Td
â«Rd
G(v)·R(Ï, xâv(tâÏ))eâ2iÏk·x dv dx dÏ
=â« t
0
â«Td
â«Rd
G(v)·R(Ï, x)eâ2iÏk·xeâ2iÏk·v(tâÏ) dv dx dÏ
=â« t
0
G(k(tâÏ))·R(Ï, k) dÏ.
Let us assume that G has a âhighâ gliding analytic regularity λ, and estimate Ï inregularity λt, with λ<λ. Let α=α(t, Ï) satisfy
0 ïżœÎ±(t, Ï) ïżœ (λâλ) (tâÏ).
Then, with Zdâ=Zd\{0},
âÏ(t)âFλt ïżœ dâ
kâZdâ
â« t
0
e2Ïλt|k||G(k(tâÏ))| |R(Ï, k)| dÏ
ïżœ d
â« t
0
(supkâZdâ
|G(k(tâÏ))|e2Ï(λ(tâÏ)+α)|k|)( â
kâZd
e2Ï(λÏâα)|k||R(Ï, k)|)
dÏ
ïżœ d(
supηâZd
e2Ïλ|η||G(η)|)(
sup0ïżœÏïżœt
âR(Ï, ·)âFλÏâα
)â« t
0
eâ2Ï((λâλ)(tâÏ)âα) dÏ,
where we have used that
kâZdâ =â 2Ï(λ(tâÏ)+α)|k|ïżœ 2Ïλ|k|(tâÏ)â((λâλ)(tâÏ)âα).
Let us chooseα(t, Ï) = 1
2 (λâλ) min{1, tâÏ}.Then â« t
0
eâ2Ï((λâλ)(tâÏ)âα) dÏ ïżœâ« t
0
eâÏ(λâλ)(tâÏ) dÏ ïżœ 1Ï(λâλ)
.
So in the end
âÏ(t)âFλt ïżœ dâGâX λ
Ï(λâλ)sup
0ïżœÏïżœtâR(Ï)âFλÏâα(t,Ï) ,
where âGâX λ =supηâZd |G(η)|e2Ïλ|η|.In the preceding computation there are three important things to notice, which lie
at the heart of Landau damping:âą The natural index of analytic regularity of Ï in x increases linearly in time: this
is an automatic consequence of the gliding regularity, already observed in §4.
98 c. mouhot and c. villani
âą A bit α(t, Ï) of analytic regularity of G was transferred from G to R, however nomore than a fraction of (λâλ)(tâÏ). We call this the regularity extortion: if f forcesf , i.e. if it satisfies an equation of the form âtf+v ·âxf+F [f ]·âv f=S, then f will giveaway some (gliding) smoothness to ïżœ=
â«Rd f dv.
âą The combination of higher regularity of G and the assumption R(0)=0 has beenconverted into a time-decay, so that the time-integral is bounded, uniformly as t!â.Thus there is decay by regularity.
The main goal of this section is to establish quantitative variants of these effects insome general situations when G is not only a function of v and R not only a function oft and x. Note that we shall have to work with regularity indices depending on t and Ï !
Regularity extortion is related to velocity-averaging regularity, well known in kinetictheory [43]; what is unusual though is that we are working in analytic regularity, and inlarge time, while velocity-averaging regularity is mainly a short-time effect. In fact weshall study two distinct mechanisms for the extortion: the first one will be well suitedfor short times (tâÏ small), and will be crucial later to get rid of small deteriorations inthe functional spaces due to composition; the second one will be well adapted to largetimes (tâÏ!â) and will ensure convergence of the time-integrals.
The estimates in this section lead to a serious twist on the popular view on Landaudamping, according to which the wave gives energy to the particles that it forces; instead,the picture here is that the wave gains regularity from the background, and regularity isconverted into decay.
For the sake of pedagogy, we shall first establish the basic, simple bilinear estimate,and then discuss the two mechanisms once at a time.
6.1. Basic bilinear estimate
Proposition 6.1. (Basic bilinear estimate in gliding regularity) Let G=G(Ï, x, v)and R=R(Ï, x, v) be valued in Rd,
ÎČ(Ï, x) =â«
Rd
(G·R)(Ï, xâv(tâÏ), v) dv,
Ï(t, x) =â« t
0
ÎČ(Ï, x) dÏ.
ThenâÎČ(Ï, ·)âFλt+ÎŒ ïżœ dâGâZλ,ÎŒ;1
ÏâRâZλ,ÎŒ
Ï; (6.1)
and
âÏ(t, ·)âFλt+ÎŒ ïżœ d
â« t
0
âGâZλ,ÎŒ;1Ï
âRâZλ,ÎŒÏ
dÏ. (6.2)
on landau damping 99
Proof. Obviously (6.2) follows from (6.1). To prove (6.1) we apply successivelyPropositions 4.15, 4.19 and 4.24, obtaining
âÎČ(Ï, ·)âFλt+ÎŒ ïżœâ„â„â„â„â«
Rd
(G·R) S0Ïât dv
â„â„â„â„Fλt+ÎŒ
ïżœ â(G·R) S0ÏâtâZλ,ÎŒ;1
t
= âG·RâZλ,ÎŒ;1Ï
ïżœ dâGâZλ,ÎŒ;1Ï
âRâZλ,ÎŒÏ
.
6.2. Short-term regularity extortion by time cheating
Proposition 6.2. (Short-term regularity extortion) Let G=G(x, v) and R=R(x, v)be valued in Rd, and
ÎČ(x) =â«
Rd
(G·R)(xâv(tâÏ), v) dv.
Then for any λ, ÎŒ, tïżœ0 and any b>â1, we have
âÎČâFλt+ÎŒ ïżœ dâGâZλ(1+b),ÎŒ;1Ïâbt/(1+b)
âRâZλ(1+b),ÎŒÏâbt/(1+b)
. (6.3)
Moreover, if Pk stands for the projection on the k-th Fourier mode as in (4.36), one has
e2Ï(λt+ÎŒ)|k||ÎČ(k)|ïżœ dâlâZd
âPlGâZλ(1+b),ÎŒ;1Ïâbt/(1+b)
âPkâlRâZλ(1+b),ÎŒÏâbt/(1+b)
. (6.4)
Remark 6.3. If R only depends on t and x, then the norm of R in the right-handside of (6.3) is âRâFÎœ with
Îœ = λ(1+b)âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+ÎŒ = (Î»Ï +ÎŒ)âb(tâÏ),
as soon as Ï ïżœbt/(1+b). Thus, some regularity has been gained with respect to Propo-sition 6.1. Even if R is not a function of t and x alone, but rather a function of t andx composed with a function depending on all the variables, this gain will be preservedthrough the composition inequality.
Proof. By applying successively Propositions 4.15 (iv), 4.19 and 4.24 (i), we getâ„â„â„â„â«Rd
(G·R)(xâv(tâÏ), v) dv
â„â„â„â„Fλt+ÎŒ
ïżœ â(G·R)(xâv(tâÏ, v))âZλ(1+b),ÎŒ;1t/(1+b)
= âG·RâZλ(1+b),ÎŒ;1Ïâbt/(1+b
ïżœ dâGâZλ(1+b),ÎŒ;1Ïâbt/(1+b)
âRâZλ(1+b),ÎŒ;âÏâbt/(1+b)
,
which is the desired result (6.3).Inequality (6.4) is obtained in a similar way with the help of Proposition 4.34.
100 c. mouhot and c. villani
Remark 6.4. Let us sketch an alternative proof of Proposition 6.2, which is longer buthas the interest to rely on commutators involving âv, âx and the transport semigroup, allof them classically related to hypoelliptic regularity and velocity averaging. Let S=S0
Ïât,so that R S(x, v)=R(xâv(tâÏ), v); and let
D = DÏ,t,b := (Ïâb(tâÏ))âx+(1+b)âv.
Then, by direct computation,
tâx(R S) = (DR) Sâ(1+b)âv(R S); (6.5)
and since âx commutes with âv and D, and with the composition by S as well, this canbe generalized by induction into
(tâx)n(R S) =â
mâNd0
mïżœn
(n
m
)[â(1+b)âv]m((DnâmR) S). (6.6)
Applying this formula with R replaced by G·R and integrating in v yields
(tâx)n
â«Rd
(G·R) S0Ïât dv =
â«Rd
Dn(G·R) S0Ïât dv =
â«Rd
Dn(G·R) dv.
It follows by taking Fourier transform that
(2iÏtk)nÎČ(k) =â«
Rd
[Dn(G·R)] dv =â«
Rd
((1+b)âv+2iÏ(Ïâb(tâÏ))k)n(G·R)(k, v) dv,
whenceâkâZd
nâNd0
e2ÏÎŒ|k| |2Ïλtk|nn!
|ÎČ(k)|
ïżœâkâZd
nâNd0
e2ÏÎŒ|k| (λ(1+b))n
n!
â„â„â„â„[âv+2iÏ
(Ïâ bt
1+b
)k
]n(G·R)(k, v)
â„â„â„â„L1(dv)
= âG·RâZλ(1+b),ÎŒ;1Ïâbt/(1+b)
,
and then (6.3) follows by Proposition 4.24.
Let us conclude this subsection with some comments on Proposition 6.2. When wewish to apply it, what constraints on b(t, Ï) (assumed to be non-negative to fix the ideas)does this presuppose? First, b should be small, so that λ(1+b)ïżœÎ». But most importantly,we have estimated GÏ in a norm ZÏ âČ instead of ZÏ (this is the time cheating), where
on landau damping 101
|Ï âČâÏ |=bt/(1+b). To compensate for this discrepancy, we may apply (4.19), but for thisto work bt/(1+b) should be small, otherwise we would lose a large index of analyticityin x, or at best we would inherit an undesirable exponentially growing constant. So allwe are allowed is b(t, Ï)=O(1/(1+t)). This is not enough to get the time-decay whichwould lead to Landau damping. Indeed, if R=R(x) with R(0)=0, then
âRâZλ(1+b),ÎŒÏâbt/(1+b)
= âRâFλÏ+ÎŒâλb(tâÏ) ïżœ eâλb(tâÏ)âRâFλÏ+ÎŒ ;
so we gain a coefficient eâλb(tâÏ), but thenâ« t
0
eâλb(tâÏ) dÏ ïżœâ« t
0
eâλΔ(tâÏ)/t dÏ =(
1âeâλΔ
λΔ
)t,
which of course diverges in large time.To summarize: Proposition 6.2 is helpful when tâÏ =O(1), or when some extra
time-decay is available. This will already be very useful; but for long-time estimates weneed another, complementary mechanism.
6.3. Long-term regularity extortion
To search for the extra decay, let us refine the computation of the beginning of thissection. Assume that GÏ =âvgÏ , where (gÏ )Ïïżœ0 solves a transport-like equation, soG(Ï, k, η)=2iÏηg(Ï, k, η), and
|G(Ï, k, η)|ïżœ 2Ï|η|eâ2ÏÎŒ|k|eâ2Ïλ|η+kÏ |.
Up to slightly increasing λ and Ό, we may assume that
|G(Ï, k, η)|ïżœ (1+Ï)eâ2ÏÎŒ|k|eâ2Ïλ|η+kÏ |. (6.7)
Let then ïżœ(Ï, x)=â«
Rd f(Ï, x, v) dv, where also f solves a transport equation, but has alower analytic regularity; and R=âW âïżœ. Assuming that |âW (k)|=O(|k|âÎł) for someÎłïżœ0, we have
|R(Ï, k)|ïżœ eâ2Ï(λÏ+ÎŒ)|k|1k ïżœ=0
1+|k|Îł . (6.8)
Let again
Ï(t, x) =â« t
0
â«Rd
G(Ï, xâv(tâÏ), v)·R(Ï, xâv(tâÏ)) dv dÏ.
102 c. mouhot and c. villani
As t!â, G in the integrand of Ï oscillates wildly in phase space, so it is not clear thatit will help at all. But let us compute
Ï(t, k) =â« t
0
â«Td
â«Rd
G(Ï, xâv(tâÏ), v)·R(Ï, xâv(tâÏ))eâ2iÏk·x dv dx dÏ
=â« t
0
â«Td
â«Rd
G(Ï, x, v)·R(Ï, x)eâ2iÏk·xeâ2iÏk·v(tâÏ) dv dx dÏ
=â« t
0
â«Rd
G·R(Ï, k, v)eâ2iÏk·v(tâÏ) dv dÏ
=â« t
0
â«Rd
âlâZd
G(Ï, l, v)·R(Ï, kâl)eâ2iÏk·v(tâÏ) dv dÏ
=â« t
0
âlâZd
G(Ï, l, k(tâÏ))·R(Ï, kâl) dÏ.
At this level, the difference with respect to the beginning of this section lies in the factthat there is a summation over lâZd, instead of just choosing l=0. Note that
Ï(t, 0) =â« t
0
â«Td
â«Rd
G(Ï, x, v)·R(Ï, x) dv dx dÏ = 0,
because G is a v-gradient.From (6.7) and (6.8) we deduce thatâ
kâZd
|Ï(t, k)|e2Ï(λt+ÎŒ)|k|
ïżœâ« t
0
(1+Ï)â
k,lâZd
0 ïżœ=k ïżœ=l
e2ÏÎŒ|k|e2Ïλt|k|eâ2ÏÎŒ|l|eâ2Ïλ|k(tâÏ)+lÏ |eâ2ÏÎŒ|kâl| eâ2ÏÎ»Ï |kâl|
1+|kâl|Îł dÏ.
Using the inequalities
eâ2ÏÎŒ|kâl|e2ÏÎŒ|k|eâ2ÏÎŒ|l| ïżœ eâ2Ï(ÎŒâÎŒ)|l|
and
eâ2ÏÎ»Ï |kâl|e2Ïλt|k|eâ2Ïλ|k(tâÏ)+lÏ | ïżœ eâ2Ï(λâλ)|k(tâÏ)+lÏ |,
we end up with
âÏ(t)âFλt+ÎŒ ïżœâ
k,lâZd
0 ïżœ=k ïżœ=l
eâ2Ï(ÎŒâÎŒ)|l|
1+|kâl|Îłâ« t
0
eâ2Ï(λâλ)|k(tâÏ)+lÏ |(1+Ï) dÏ.
on landau damping 103
If it were not for the negative exponential, the time-integral would be O(t2) as t!â.The exponential helps only a bit: its argument vanishes e.g. for d=1, k>0, l<0 andÏ =kt/(k+|l|). Thus we have the essentially optimal boundsâ« t
0
eâ2Ï(λâλ)|k(tâÏ)+lÏ | dÏ ïżœ 1Ï(λâλ)|kâl| (6.9)
and â« t
0
eâ2Ï(λâλ)|k(tâÏ)+lÏ | Ï dÏ ïżœ 12Ï2(λâλ)2|kâl|2 +
1Ï(λâλ)
|k|t|kâl| . (6.10)
From this computation we conclude that:âą The higher regularity of G has allowed us to reduce the time-integral due to a factor
eâα|k(tâÏ)+lÏ |; but this factor is not small when Ï/t is equal to k/(kâl). As discussed inthe next section, this reflects an important physical phenomenon called (plasma) echo,which can be assimilated to a resonance.
âą If we had (in âglidingâ norm) âGÏâ=O(1) this would ensure a uniform bound onthe integral, as soon as Îł>0, due to (6.9) andâ
k,lâZd
eâα|l|
(1+|kâl|)1+Îł<â.
âą But GÏ is a velocity gradient, so (unless of course G depends only on v) âGÏâdiverges like O(Ï) as Ï!â, which implies a divergence of our bounds in large time, ascan be seen from (6.10). If Îłïżœ1 this comes with a divergence in the k variable, since inthis case â
k,lâZd
eâα|l||k|(1+|kâl|)1+Îł
=â.
(The Coulomb case corresponds to Îł=1, so in this respect it has a borderline divergence.)The following estimate adapts this computation to the formalism of hybrid norms,
and at the same time allows for a time cheating similar to the one in Proposition 6.2.Fortunately, we shall only need to treat the case when R=R(Ï, x); the more general casewith R=R(Ï, x, v) would be much more tricky.
Theorem 6.5. (Long-term regularity extortion) Let G=G(Ï, x, v), R=R(Ï, x) and
Ï(t, x) =â« t
0
â«Rd
G(Ï, xâv(tâÏ), v)·R(Ï, xâv(tâÏ)) dv dÏ.
Let λ, λ, ÎŒ, ÎŒ, ÎŒâČ=ÎŒâČ(t, Ï) and Mïżœ1 be such that (1+M)Î»ïżœÎ»>λ>0 and ÎŒïżœÎŒâČ>ÎŒ>0,and let Îłïżœ0 and b=b(t, Ï)ïżœ0. Then
âÏ(t, ·)âFλt+ÎŒ ïżœâ« t
0
KG0 (t, Ï)âRÏâFÎœ dÏ +
â« t
0
KG1 (t, Ï)âRÏâFÎœ,Îł dÏ, (6.11)
104 c. mouhot and c. villani
where
Îœ = max{Î»Ï +ÎŒâČâ 1
2λb(tâÏ), 0}, (6.12)
KG0 (t, Ï) = deâÏ(λâλ)(tâÏ)
â„â„â„â„â«Td
G(Ï, x, ·) dx
â„â„â„â„Cλ(1+b);1
, (6.13)
KG1 (t, Ï) =
(sup
0ïżœÏïżœt
âGÏâZλ(1+b),ÎŒÏâbt/(1+b)
1+Ï
)K1(t, Ï), (6.14)
K1(t, Ï)
= (1+Ï)d supk,lâZdâ
eâÏ(ÎŒâÎŒ)|l|eâÏ(λâλ)|k(tâÏ)+lÏ |/Meâ2Ï[ÎŒâČâÎŒ+λb(tâÏ)/2]|kâl|
1+|kâl|Îł . (6.15)
Remark 6.6. It is essential in (6.11) to separate the contribution of G(Ï, 0, v) fromthe rest. Indeed, if we removed the restriction l ïżœ=0 in (6.15) the kernel K1 would be toolarge to be correctly controlled in large time. What makes this separation reasonable isthat, although in cases of application G(Ï, x, v) is expected to grow like O(Ï) in largetime, the spatial average
â«Td G(Ï, x, v) dx is expected to be bounded. Also, we will not
need to take advantage of the parameter Îł to handle this term.
Proof. Without loss of generality we may assume that G and R are scalar-valued.(This explains the constant d in front of the right-hand sides of (6.13) and (6.15).) Firstwe assume that G(Ï, 0, v)=0, and we write as before
Ï(t, k) =â« t
0
(âlâZdâ
â«Rd
G(Ï, l, v)R(Ï, kâl)eâ2iÏk·v(tâÏ) dv
)dÏ,
|Ï(t, k)|ïżœâ« t
0
(âlâZdâ
âŁâŁâŁâŁâ«Rd
G(Ï, l, v)eâ2iÏk·v(tâÏ) dv
âŁâŁâŁâŁ|R(Ï, kâl)|)
dÏ. (6.16)
Next we let Ï âČ=Ïâb(tâÏ) and write
e2Ï(λt+ÎŒ)|k| ïżœ eâ2Ï(ÎŒâÎŒ)|l|eâ2Ïλ(ÏâÏ âČ)|kâl|eâ2Ï(ÎŒâČâÎŒ)|kâl|eâ2Ï(λâλ)|k(tâÏ âČ)+lÏ âČ|
Ăe2ÏÎŒ|l|e2Ï(λÏ+ÎŒâČ)|kâl|e2Ïλ|k(tâÏ âČ)+lÏ âČ|.(6.17)
Since 0ïżœÎ»âÎ»ïżœMλ, we have
eâ2Ï(λâλ)|k(tâÏ âČ)+lÏ âČ| ïżœ eâÏ(λâλ)|k(tâÏ)+lÏ |/MeÏλ(ÏâÏ âČ)|kâl|;
so (6.17) implies that
e2Ï(λt+ÎŒ)|k| ïżœ eâ2Ï(ÎŒâÎŒ)|l|eâÏ(λâλ)|k(tâÏ)+lÏ |/Meâ2Ï[ÎŒâČâÎŒ+λ(ÏâÏ âČ)/2]|kâl|
Ăe2ÏÎŒ|l|e2Ï(λÏ+ÎŒâČ)|kâl| ânâNd
0
|2iÏλ(k(tâÏ âČ)+lÏ âČ)|nn!
.(6.18)
on landau damping 105
For each nâNd0,
|2iÏλ(k(tâÏ âČ)+lÏ âČ)|nn!
âŁâŁâŁâŁâ«Rd
G(Ï, l, v)eâ2iÏk·v(tâÏ) dv
âŁâŁâŁâŁ=
λn
n!
âŁâŁâŁâŁâ«Rd
G(Ï, l, v)[2iÏ(k(tâÏ âČ)+lÏ âČ)]neâ2iÏk·v(tâÏ) dv
âŁâŁâŁâŁ=
λn
n!
(tâÏ âČ
tâÏ
)nâŁâŁâŁâŁâ«Rd
G(Ï, l, v)[2iÏ
(k(tâÏ)+lÏ âČ
(tâÏ
tâÏ âČ
))]neâ2iÏk·v(tâÏ) dv
âŁâŁâŁâŁ=
λn
n!
(tâÏ âČ
tâÏ
)nâŁâŁâŁâŁâ«Rd
G(Ï, l, v)[ââv+2iÏlÏ âČ
(tâÏ
tâÏ âČ
)]neâ2iÏk·v(tâÏ) dv
âŁâŁâŁâŁ=
λn
n!
(tâÏ âČ
tâÏ
)nâŁâŁâŁâŁâ«Rd
[âv+2iÏlÏ âČ
(tâÏ
tâÏ âČ
)]nG(Ï, l, v)eâ2iÏk·v(tâÏ) dv
âŁâŁâŁâŁïżœ λn
n!
(tâÏ âČ
tâÏ
)nâ„â„â„â„(âv+2iÏlÏ âČ(
tâÏ
tâÏ âČ
))nG(Ï, l, v)
â„â„â„â„L1(dv)
=λn(1+b)n
n!
â„â„â„â„(âv+2iÏl
(Ïâ bt
1+b
))nG(Ï, l, v)
â„â„â„â„L1(dv)
.
Combining this with (6.16) and (6.18), and summing over k, we deduce that
âÏ(t, ·)âFλt+ÎŒ
=âkâZdâ
e2Ï(λt+ÎŒ)|k||Ï(t, k)|
ïżœâ« t
0
âk,lâZd
nâNd0
eâ2Ï(ÎŒâÎŒ)|l|eâÏ(λâλ)|k(tâÏ)+lÏ |/Meâ2Ï[ÎŒâČâÎŒ+λ(ÏâÏ âČ)/2]|kâl|
1+|kâl|Îł
Ăe2ÏÎŒ|l|e2Ï[λÏ+ÎŒâČâλb(tâÏ)/2]|kâl| λn(1+b)n
n!|R(Ï, kâl)|
Ăâ„â„â„â„(âv+2iÏl
(Ïâ bt
1+b
))nG(Ï, l, v)
â„â„â„â„L1(dv)
dÏ,
and the desired estimate follows readily.Finally we consider the contribution of
G(Ï, 0, v) =â«
Td
G(Ï, x, v) dx.
This is done in the same way, noting that
supkâZdâ
eâ2Ï(λâλ)|k|(tâÏ)
1+|k|Îł ïżœ eâ2Ï(λâλ)(tâÏ).
106 c. mouhot and c. villani
To conclude this section we provide a âmode-by-modeâ variant of Theorem 6.5; thiswill be useful for very singular interactions (Îł=1 in Theorem 2.6).
Theorem 6.7. Under the same assumptions as in Theorem 6.5, for all kâZd wehave the estimate
e2Ï(λt+ÎŒ)|k||Ï(t, k)|ïżœâ« t
0
KG0 (t, Ï)(e2ÏÎœ|k||R(Ï, k)|) dÏ
+â« t
0
âlâZd
KGk,l(t, Ï)e2ÏÎœ|kâl|(1+|kâl|Îł)|R(Ï, kâl)| dÏ,
(6.19)
where KG0 is defined by (6.13), Μ by (6.12) and
KGk,l(t, Ï) =
(sup
0ïżœÏïżœt
âGâZλ(1+b),ÎŒ;1Ïâbt/(1+b)
1+Ï
)Kk,l(t, Ï),
Kk,l(t, Ï) =(1+Ï)deâ2Ï(ÎŒâÎŒ)|l|eâÏ(λâλ)|k(tâÏ)+lÏ |/Meâ2Ï[ÎŒâČâÎŒ+λb(tâÏ)/2]|kâl|
1+|kâl|Îł .
Proof. The proof is similar to that of Theorem 6.5, except that k is fixed and weuse, for each l, the crude bound
e2ÏÎŒ|l|â„â„â„â„(âv+2iÏl
(Ïâ bt
1+b
))nG(Ï, l, v)
â„â„â„â„L1(dv)
ïżœâjâZd
e2ÏÎŒ|j|â„â„â„â„(âv+2iÏj
(Ïâ bt
1+b
))nG(Ï, j, v)
â„â„â„â„L1(dv)
.
7. Control of the time-response
To motivate this section, let us start from the linearized equation (3.3), but now assumethat f0 depends on t, x and v, and that there is an extra source term S, decaying intime. Thus the equation is
âf
ât+v ·âxfâ(âW âïżœ)·âvf0 = S,
and the equation for the density ïżœ, as in the proof of Theorem 3.1, is
ïżœ(t, x) =â«
Rd
fi(xâvt, v) dv
+â« t
0
â«Rd
âvf0(Ï, xâv(tâÏ), v)·(âW âïżœ)(Ï, xâv(tâÏ)) dv dÏ
+â« t
0
â«Rd
S(Ï, xâv(tâÏ), v) dv dÏ.
(7.1)
on landau damping 107
Hopefully we may apply Theorem 6.5 to deduce from (7.1) an integral inequality onÏ(t):=âïżœ(t)âFλt+ÎŒ , which will look like
Ï(t) ïżœA+c
â« t
0
K(t, Ï)Ï(Ï) dÏ, (7.2)
where A is the contribution of the initial datum and the source term, and K(t, Ï) is akernel looking like, say, (6.15).
How do we proceed from (7.2)? Assume for a start that a smallness condition of theform (a) in Proposition 2.1 is satisfied. Then the simple and natural way, as in §3, wouldbe to write
Ï(t) ïżœA+c
(â« t
0
K(t, Ï) dÏ
)sup
0ïżœÏïżœtÏ(Ï)
and deduce that
Ï(t) ïżœ A
1âcâ« t
0K(t, Ï) dÏ
(7.3)
(assuming of course the denominator to be positive). However, if K is given by (6.15), itis easily seen that
â« t
0K(t, Ï) dÏ ïżœ t as t!â, where >0; then (7.3) is useless. In fact
(7.2) does not prevent Ï from tending to â as t!â. Nevertheless, its growth may becontrolled under certain assumptions, as we shall see in this section. Before embarkingon cumbersome calculations, we shall start with a qualitative discussion.
7.1. Qualitative discussion
The kernel K in (6.15) depends on the choice of ÎŒâČ=ÎŒ(t, Ï). How large ÎŒâČâÎŒ can bedepends in turn on the amount of regularization offered by the convolution with theinteraction âW . We shall distinguish several cases according to the regularity of theinteraction.
7.1.1. Analytic interaction
If âW is analytic, there is Ï>0 such that
âïżœââWâFÎœ+Ï ïżœCâïżœâFÎœ for all Îœ ïżœ 0;
then in (6.15) we can afford to choose, say, ÎŒâČâÎŒ=Ï and Îł=0. Thus, assuming that
b =B
1+t
108 c. mouhot and c. villani
with B so small that (ÎŒâČâÎŒ)âλb(tâÏ)ïżœ 12Ï, K is bounded by
K(α)(t, Ï) = (1+Ï) supl,kâZdâ
eâα|l|eâα|kâl|eâα|k(tâÏ)+lÏ |, (7.4)
where α= 12 min{λâλ, ÎŒâÎŒ, Ï}. To fix ideas, let us work in dimension d=1. The goal is
to estimate solutions of
Ï(t) ïżœ a+c
â« t
0
K(α)(t, Ï)Ï(Ï) dÏ. (7.5)
Whenever Ï/t is a rational number distinct from 0 or 1, there are k, lâZ such that|k(tâÏ)+lÏ |=0, and the size of K(α)(t, Ï) mainly depends on the minimum admissiblevalues of k and kâl. Looking at values of Ï/t of the form 1/(n+1) or n/(n+1) suggeststhe approximation
K(α) ïżœ (1+Ï) min{eâαÏ/(tâÏ)eâ2α, eâ2α(tâÏ)/Ïeâα}. (7.6)
But this estimate is terrible: the time-integral of the right-hand side is much larger thanthe integral of K(α). In fact, the fast variation and âwigglingâ behavior of K(α) areessential to get decent estimates.
To get a better feeling for K(α), let us only retain the term for k=1 and l=â1; thisseems reasonable since we have an exponential decay as k or l tend to infinity (anyway,throwing away all other terms can only improve the estimates). So we look at
K(α)(t, Ï) = (1+Ï)eâ3αeâα|tâ2Ï |.
Let us time-rescale by setting kt(Ξ)=tK(α)(t, tΞ) for Ξâ[0, 1] (the t factor appears becausedÏ =t dΞ); then it is not hard to see that
kt
t! eâ3α
2αΎ1/2.
This suggests the following baby model for (7.5):
Ï(t) ïżœ a+ctÏ(
12 t). (7.7)
The important point in (7.7) is that, although the kernel has total mass O(t), thismass is located far from the endpoint Ï =t; this is what prevents the fast growth of Ï.Compare with the inequality Ï(t)ïżœa+ctÏ(t), which implies no restriction at all on Ï.
To be slightly more quantitative, let us look for a power series
Ί(t) =ââ
k=0
aktk
on landau damping 109
1.4
1.2
1
0.8
0.6
0.4
0.2
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=10
exact
approx
12
10
8
6
4
2
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=100
exact
approx
120
100
80
60
40
20
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=1000
exact
approx
Figure 5. The kernel K(α)(t, Ï), together with the approximate upper bound in (7.6), forα=0.5 and t=10, t=100 and t=1000, respectively.
110 c. mouhot and c. villani
achieving equality in (7.7). This yields a0=a and ak+1=cak2âk, so
Ί(t) = a
ââk=0
cktk
2k(kâ1)/2. (7.8)
The function Ί exhibits a truly remarkable behavior: it grows faster than any polynomial,but slower than any fractional exponential exp(ctÎœ), Îœâ(0, 1); essentially it behaves likeA(log t)2 (as can also be seen directly from (7.7)). One may conjecture that solutions of(7.5) exhibit a similar kind of growth.
Let us interpret these calculations. Typically, the kernel K controls the time vari-ation of (say) the spatial density ïżœ which is due to binary interaction of waves. Whentwo waves of distinct frequencies interact, the effect over a long time-period is most ofthe time very small; this is a consequence of the oscillatory nature of the evolution, andthe resulting time-averaging. But at certain particular times, the interaction becomesstrong: this is known in plasma physics as the plasma echo, and can be thought of asa kind of resonance. Spectacular experiments by Malmberg and collaborators are basedon this effect [32], [60]. Namely, if one starts a wave at frequency l at time 0, and forcesit at time Ï by a wave of frequency kâl, a strong response is obtained at time t andfrequency k such that
k(tâÏ)+lÏ = 0 (7.9)
(which of course is possible only if k and l are parallel to each other, with oppositedirections).
In the present non-linear setting, whatever variation the density function is subjectto, will result in echoes at later times. Even if each echo in itself will eventually decay,the problem is whether the accumulation of echoes will trigger an uncontrolled growth(unstability). As long as the expected growth is eaten by the time-decay coming from thelinear theory, non-linear Landau damping is expected. In the present case, the growth of(7.8) is very slow in regard of the exponential time-decay due to the analytic regularity.
7.1.2. Sobolev interaction
If âW only has Sobolev regularity, we cannot afford in (6.15) to take ÎŒâČ(t, Ï) larger thanÎŒ+η(tâÏ)/t (because the amount of regularity transferred in the bilinear estimates isonly O((tâÏ)/t), recall the discussion at the end of §6.2). On the other hand, we haveÎł>0 such that
ââW âïżœâFÎœ,Îł ïżœCâïżœâFÎœ for all Îœ ïżœ 0.
on landau damping 111
and then we can choose this Îł in (6.15). So, assuming b=B/(1+t) with B so small that(ÎŒâČâÎŒ)âλb(tâÏ)ïżœÎ·(tâÏ)/2t, K in (6.15) will be controlled by
K(α),Îł(t, Ï) = (1+Ï)d supl,kâZâ
eâα|l|eâα(tâÏ)|kâl|/teâα|k(tâÏ)+lÏ |
1+|kâl|Îł , (7.10)
where α= 12 min{λâλ, ÎŒâÎŒ, η}. The equation we are considering now is
Ï(t) ïżœ a+â« t
0
K(α),Îł(t, Ï)Ï(Ï) dÏ. (7.11)
For, say, Ï ïżœ 12 t, we have K(α)ïżœK(α/2), and the discussion is similar to that in §7.1.1.
But when Ï aproaches t, the term eâα(tâÏ)|kâl|/t hardly helps. Keeping only k>0 andl=â1 (because of the exponential decay in l) leads us to consider the kernel (for simplicitywe set d=1 without loss of generality)
K(α)(t, Ï) = (1+Ï) supkâZâ
eâα|ktâ(k+1)Ï |
1+(k+1)Îł.
Once again we perform a time-rescaling, setting kt(Ξ)=tK(α)(t, tΞ), and let t!â. Inthis limit each exponential eâα|ktâ(k+1)Ï | becomes localized in a neighborhood of sizeO(t/k) around Ξ=k/(k+1), and contributes a Dirac mass at Ξ=k/(k+1), with amplitude2/α(k+1). We get that
kt
t! 2
α
âkâZ
11+(k+1)Îł
k
(k+1)2ÎŽ1â1/(k+1) as t!â.
This leads us to the following baby model for (7.11):
Ï(t) ïżœ a+ctââ
k=1
1k1+Îł
Ï
((1â 1
k
)t
). (7.12)
If we search forââ
n=0 antn achieving equality, this yields
a0 = a, an+1 = c
( ââk=1
1k1+Îł
(1â 1
k
)n)an.
To estimate the behavior of theâ
k above, we compare it withâ« â
1
1t1+Îł
(1â 1
t
)ndt =â« t
0
uÎłâ1(1âu)n du = B(Îł, n+1) =Î(Îł)Î(n+1)Î(n+Îł+1)
= O
(1nÎł
),
where B(Îł, n+1) is the beta function. All in all, we may expect Ï in (7.11) to behavequalitatively like
Ί(t) = a
âân=0
cntn
(n!)Îł.
Notice that Ί is subexponential for γ>1 (it grows essentially like the fractional expo-nential exp(t1/γ)) and exponential for γ=1. In particular, as soon as γ>1, we expectnon-linear Landau damping again.
112 c. mouhot and c. villani
7.1.3. CoulombâNewton interaction (Îł=1)
When Îł=1, as is the case for Coulomb or Newton interaction, the previous analysisbecomes borderline, since we expect (7.12) to be compatible with an exponential growth,and the linear decay is also exponential. To handle this more singular case, we shall workmode-by-mode, rather than on just one norm. Starting again from (7.1), we consider,for each kâZd,
Ïk(t) = e2Ï(λt+ÎŒ)|k||ïżœ(t, k)|,and hope to get, via Theorem 6.7, an inequality which will roughly take the form
Ïk(t) ïżœAk+c
â« t
0
âlâZd
Kk,l(t, Ï)Ïkâl(Ï) dÏ. (7.13)
(Note that summing over k would yield an inequality worse than (7.11).) To fix theideas, let us work in dimension d=1, and set kïżœ1 and l=â1. Reasoning as in §7.1.2, weobtain the baby model
Ïk(t) ïżœAk+ct
(k+1)1+ÎłÏk+1
(kt
k+1
). (7.14)
The gain with respect to (7.12) is clear: for different values of k, the âdominant timesâare distinct. From the physical point of view, we are discovering that, in some sense,echoes occurring at distinct frequencies are asymptotically well separated.
Let us search again for power series solutions: we set
Ïk(t) =ââ
m=0
ak,mtm, ak,0 = Ak.
By identification,
ak,m = ak+1,mâ1c(k+1)â(1+Îł)
(k
k+1
)mâ1
,
and by induction
ak,m = Ak+mcm
(k!
(k+m)!
)1+Îłkmâ1cm
(k+1)(k+2) ... (k+m)ïżœAk+m
(k!
(k+m)!
)Îł+2
kmâ1cm.
We may expect that Ak+mïżœAeâa(k+m); then
ak,m ïżœA(keâak)kmcm eâam
(m!)Îł+2,
and in particular
Ïk(t) ïżœAeâak/2ââ
m=0
(ckt)m
(m!)Îł+2ïżœAe(1âα)(ckt)α
, α =1
Îł+2.
This behaves like a fractional exponential even for Îł=1, and we can now believe in non-linear Landau damping for such interactions! (The argument above works even for moresingular interactions; but in the proof later the condition Îłïżœ1 will be required for otherreasons, see pp. 152 and 161.)
on landau damping 113
7.2. Exponential moments of the kernel
Now we start to estimate the kernel K(α),Îł from (7.10), without any approximation thistime. Eventually, instead of proving that the growth is at most fractionally exponential,we shall compare it with a slow exponential eΔt. For this, the first step consists inestimating exponential moments of the kernel eâΔt
â« t
0K(t, Ï)eÎ”Ï dÏ . (To get more precise
estimates, one can study eâΔtα⫠t
0K(t, Ï)eΔÏα
dÏ , but such a refinement is not needed forthe proof of Theorem 2.6.)
The first step consists of estimating exponential moments.
Proposition 7.1. (Exponential moments of the kernel) Let Îłâ[1,â) be given. Forany αâ(0, 1), let K(α),Îł be defined by (7.10). Then for any Îł<â there is α=α(Îł)>0such that if Î±ïżœÎ± and Δâ(0, 1), then for any t>0,
eâΔt
â« t
0
K(α),Îł(t, Ï)eÎ”Ï dÏ
ïżœC
(1
αΔγtÎłâ1+
1αΔγtγ
log1α
+1
α2Δ1+γt1+γ+(
1α3
+1
α2Δlog
1α
)eâΔt/4+
eâαt/2
α3
),
where C=C(Îł). In particular,âą if Îł>1 and Î”ïżœÎ±, then
eâΔt
â« t
0
K(α),Îł(t, Ï)eÎ”Ï dÏ ïżœ C(Îł)α3Δ1+ÎłtÎłâ1
;
âą if Îł=1 then
eâΔt
â« t
0
K(α),Îł(t, Ï)eÎ”Ï dÏ ïżœ C
α3
(1Δ+
1Δ2t
).
Remark 7.2. Much stronger estimates can be obtained if the interaction is ana-lytic; that is, when K(α),Îł is replaced by K(α) defined in (7.4). A notable point aboutProposition 7.1 is that for Îł=1 we do not have any time-decay as t!â.
Proof. To simplify notation we shall not recall the dependence of K on Îł, and weshall set d=1 without loss of generality. We first assume that Îł<â, and consider Ï ïżœ 1
2 t,which is the favorable case. We write
K(α)(t, Ï) ïżœ (1+Ï) suplâZ
kâZâ
eâα|l|eâα|kâl|/2eâα|k(tâÏ)+lÏ |.
Since we got rid of the condition l ïżœ=0, the right-hand side is now a non-increasing functionof d. (To see this, pick up a non-zero component of k, and recall our norm conventionsfrom Appendix A.1.) So we assume d=1. By symmetry we may also assume that k>0.
114 c. mouhot and c. villani
Explicit computations yield
â« t/2
0
eâα|k(tâÏ)+lÏ |(1+Ï) dÏ ïżœ
â§âȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâšâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâ©
1α(lâk)
+1
α2(lâk)2, if l > k,
eâαkt
(t
2+
t2
8
), if l = k,
eâα(k+l)t/2
α|kâl|(
1+t
2
), if âk ïżœ l < k,
2α|kâl|+
2kt
α|kâl|2 +1
α2|kâl|2 , if l <âk.
In all cases,â« t/2
0
eâα|k(tâÏ)+lÏ |(1+Ï) dÏ
ïżœ(
3α|kâl|+
1α2|kâl|2 +
2t
α|kâl|)
1k ïżœ=l+eâαkt
(t
2+
t2
8
)1l=k.
So
eâΔt
â« t/2
0
eâα|k(tâÏ)+lÏ |(1+Ï)eÎ”Ï dÏ
ïżœ eâΔt/2
(3
α|kâl|+1
α2|kâl|2 +2t
α|kâl|)
1k ïżœ=l+eâαkt
(t
2+
t2
8
)1l=k
ïżœ eâΔt/4
(3
α|kâl|+1
α2|kâl|2 +8z
αΔ|kâl|)
1k ïżœ=l+eâtα/2
(z
α+
8z2
α2
)1l=k,
where z=supx xeâx=eâ1. Then
eΔt
â« t/2
0
K(α)(t, Ï)eÎ”Ï dÏ ïżœ eâΔt/4â
l,kâZ
0 ïżœ=k ïżœ=l
eâα|l|eâα|kâl|/2
(3
α|kâl|+1
α2|kâl|2 +8z
αΔ|kâl|)
+eâtα/2âlâZ
eâα|l|(
z
α+
8z2
α2
).
Using the bounds (for αâŒ0+)âlâZ
eâαl = O
(1α
),âlâZ
eâαl
l= O
(log
1α
)and
âlâZ
eâαl
l2= O(1),
we end up, for Î±ïżœ 14 , with a bound like
CeâΔt/4
(1α2
log1α
+1α3
+1
α2Δlog
1α
)+Ceâαt/2
(1α2
+1α3
)ïżœC
[eâΔt/4
(1α3
+1
α2Δlog
1α
)+
eâαt/2
α3
].
on landau damping 115
(Note that the last term is O(tâ3), so it is anyway negligible in front of the other termsif Îłïżœ4; in this case the restriction Î”ïżœÎ± can be dispended with.)
âą Next we turn to the more delicate contribution of Ï ïżœ 12 t. For this case we write
K(α)(t, Ï) ïżœ (1+Ï) suplâZdâ
eâα|l| supkâZd
eâα|k(tâÏ)+lÏ |
1+|kâl|Îł , (7.15)
and the upper bound is a non-increasing function of d, so we assume that d=1. Withoutloss of generality, we restrict the supremum to l>0.
The function x !(1+|xâl|Îł)â1eâα|x(tâÏ)+lÏ | is decreasing for xïżœl, increasing forxïżœâlÏ/(tâÏ); and on the interval [âlÏ/(tâÏ), l] its logarithmic derivative goes
from(âα+
Îł/lt
1+((tâÏ)/lt)Îł
)(tâÏ) to âα(tâÏ).
So, if tïżœÎł/α, there is a unique maximum at x=âlÏ/(tâÏ), and the supremum in (7.15)is achieved for k equal to either the lower integer part, or the upper integer part ofâlÏ/(tâÏ). Thus a given integer k occurs in the supremum only for some times Ï
satisfying kâ1<âlÏ/(tâÏ)<k+1. Since only negative values of k occur, let us changethe sign so that k is non-negative. The equation
kâ1 <lÏ
tâÏ<k+1
is equivalent tokâ1
k+lâ1t < Ï <
k+1k+l+1
t.
Moreover, Ï > 12 t implies that kïżœl. Thus, for tïżœÎł/α, we have
eâΔt
â« t
t/2
K(α)(t, Ï)eÎ”Ï dÏ
ïżœ eâΔtââ
l=1
eâαlââ
k=l
â« (k+1)t/(k+l+1)
(kâ1)t/(k+lâ1)
(1+Ï)eâα|k(tâÏ)âlÏ |eΔÏ
1+(k+l)ÎłdÏ.
(7.16)
For tïżœÎł/α we have the trivial bound
eâΔt
â« t
t/2
K(α)(t, Ï)eÎ”Ï dÏ ïżœ Îł
2α;
so in the sequel we shall just focus on the estimate of (7.16).
116 c. mouhot and c. villani
To evaluate the integral in the right-hand side of (7.16), we separate according towhether Ï is smaller or larger than kt/(k+l); we use trivial bounds for eÎ”Ï inside theintegral, and in the end we get the explicit bounds
eâΔt
â« kt/(k+l)
(kâ1)t/(k+lâ1)
(1+Ï)eâα|k(tâÏ)âlÏ |eÎ”Ï dÏ ïżœ eâΔlt/(k+l)
(1
α(k+l)+
kt
α(k+l)2
),
eâΔt
â« (k+1)t/(k+l+1)
kt/(k+l)
(1+Ï)eâα|k(tâÏ)âlÏ |eÎ”Ï dÏ ïżœ eâΔlt/(k+l+1)
Ă(
1α(k+l)
+kt
α(k+l)2+
1α2(k+l)2
).
All in all, there is a numeric constant C such that (7.16) is bounded above by
C
ââl=1
eâαlââ
k=l
(1
α2(k+l)2+γ+
1α(k+l)1+γ
+kt
α(k+l)2+γ
)eâΔlt/(k+l), (7.17)
together with an additional similar term where eâΔlt/(k+l) is replaced by eâΔlt/(k+l+1),and which will satisfy similar estimates.
We consider separately the three contributions in the right-hand side of (7.17). Thefirst one is
1α2
ââl=1
eâαlââ
k=l
eâΔlt/(k+l)
(k+l)2+Îł.
To evaluate the behavior of this sum, we compare it to the 2-dimensional integral
I(t) =1α2
â« â
1
eâαx
â« â
x
eâΔxt/(x+y)
(x+y)2+Îłdy dx.
We change variables (x, y) !(x, u), where u(x, y)=Δxt/(x+y). This has Jacobian deter-minant dx dy/dx du=Δxt/u2, and we find that
I(t) =1
α2Δ1+γt1+γ
â« â
1
eâαx
x1+Îłdx
⫠Δt/2
0
eâuuÎł du = O
(1
α2Δ1+γt1+γ
).
The same computation for the second integral in the right-hand side of (7.17) yields
1αΔγtγ
â« â
1
eâαx
xÎłdx
⫠Δt/2
0
eâuuÎłâ1 du = O
(1
αΔγ tγlog
1α
).
(The logarithmic factor arises only for Îł=1.)The third exponential in the right-hand side of (7.17) is the worst. It yields a
contributiont
α
ââl=1
eâαlââ
k=l
eâΔltk/(k+l)
(k+l)2+Îł. (7.18)
on landau damping 117
We compare this with the integral
t
α
â« â
1
eâαx
â« â
x
eâΔxt/(x+y)y
(x+y)2+Îłdy dx,
and the same change of variables as before equates this with
1αΔγtÎłâ1
â« â
1
eâαx
xÎłdx
⫠Δt/2
0
eâuuÎłâ1 duâ 1αΔ1+ÎłtÎł
â« â
1
eâαx
xÎłdx
⫠Δt/2
0
eâuuÎł du
= O
(1
αΔγtÎłâ1log
1α
).
(Again the logarithmic factor arises only for Îł=1.)The proof of Proposition 7.1 follows by collecting all these bounds and keeping only
the worst one.
Remark 7.3. It is not easy to catch (say numerically) the behavior of (7.18), becauseit comes as a superposition of exponentially decaying modes; any truncation in k wouldlead to a radically different time-asymptotics.
From Proposition 7.1 we deduce L2 exponential bounds.
Corollary 7.4. (L2 exponential moments of the kernel) With the same notationas in Proposition 7.1,
eâ2Δt
â« t
0
K(α),Îł(t, Ï)2e2Î”Ï dÏ ïżœ
â§âȘâȘâȘâšâȘâȘâȘâ©C(Îł)
α4Δ1+2Îłt2(Îłâ1), if Îł > 1,
C
(1
α3Δ2+
1α2Δ3t
), if Îł = 1.
(7.19)
Proof. This follows easily from Proposition 7.1 and the obvious bound
K(α),Îł(t, Ï)2 ïżœC(1+t)K(2α),2Îł(t, Ï).
7.3. Dual exponential moments
Proposition 7.5. With the same notation as in Proposition 7.1, for any Îłïżœ1 wehave
supÏïżœ0
eΔÏ
â« â
Ï
eâΔtK(α),Îł(t, Ï) dt ïżœC(Îł)(
1α2Δ
+1
αΔγlog
1α
). (7.20)
Remark 7.6. The corresponding computation for the baby model considered in §7.1.2is
eÎ”Ï 1+Ï
α
ââk=1
eâΔ(k+1)Ï/k
k1+Îłïżœ 1+Ï
α
â« â
1
eâΔÏ/x
x1+Îłdx =
1+Ï
ÏÎł
1αΔγ
⫠ΔÏ
0
eâuuÎłâ1 du.
So we expect the dependence upon Δ in (7.20) to be sharp as γ!1.
118 c. mouhot and c. villani
16
14
12
10
8
6
4
2
00 5000 10000 15000 20000 25000
K=5K=10
K=100K=1000
16
14
12
10
8
6
4
2
00 20 40 60 80 100
K=5K=10
K=100K=1000
Figure 6. The function (7.18) for Îł=1, truncated at l=1 and kïżœK, for K=5, 10, 100, 1000.The decay is slower and slower, but still exponential (picture above); however, the maximumvalue occurs on a much slower time scale and slowly increases with the truncation parameter(picture below, which is a zoom on shorter times).
Proof. We first reduce to d=1, and split the integral as
eΔÏ
â« â
Ï
eâΔtK(α),Îł(t, Ï) dt = eΔÏ
â« â
2Ï
eâΔtK(α),Îł(t, Ï) dtïžž ïž·ïž· ïžž:=I1
+ eΔÏ
â« 2Ï
Ï
eâΔtK(α),Îł(t, Ï) dt.ïžž ïž·ïž· ïžž:=I2
The first term I1 is easy: for 2Ï ïżœtïżœâ we have
K(α),Îł(t, Ï) ïżœ (1+Ï)ââ
k=2
âlâZâ
eâα|l|âα|kâl|/2 ïżœ C(1+Ï)α2
,
and thus
eΔÏ
â« â
2Ï
eâΔtK(α),Îł(t, Ï) dt ïżœ C(1+Ï)α2
eâÎ”Ï ïżœ C
Δα2.
on landau damping 119
We treat the second term I2 as in the proof of Proposition 7.1:
eΔÏ
â« 2Ï
Ï
K(α),Îł(t, Ï)eâΔt dt ïżœ eÎ”Ï (1+Ï)ââ
l=1
eâαlââ
k=l
â« (k+lâ1)Ï/(kâ1)
(k+l+1)Ï/(k+1)
eâα|k(tâÏ)âlÏ |
1+(k+l)ÎłeâΔt dt
ïżœ (1+Ï)ââ
l=1
eâαlââ
k=l
eâΔlÏ/k
kÎł
2kα
.
We compare this with
2(1+Ï)α
â« â
1
eâαx
â« â
x
eâΔxÏ/y
y1+Îłdy dx =
2αΔγ
1+Ï
ÏÎł
â« â
1
eâαx
xÎł
⫠ΔÏ
0
eâuuÎłâ1 du dx
ïżœ C
αΔγlog
1α
,
where we used the change of variable u=ΔxÏ/y. The desired conclusion follows. Notethat as before the term log(1/α) only occurs when Îł=1, and that, for Îł>1, one couldimprove the estimate above into a time-decay of the form O(Ïâ(Îłâ1)).
7.4. Growth control
We will now state the main result of this section. For a sequence of functions Ί(k, t),kâZd
â=Zd\{0} and tâR, we set
âΊ(t)âλ =âkâZdâ
|Ί(k, t)|e2Ïλ|k|.
We shall use K(s)Ί(t) as a shorthand for (K(k, s)Ί(k, t))kâZdâ , etc.
Theorem 7.7. (Growth control via integral inequalities) Assume that f0=f0(v)and W =W (x) satisfy condition (L) from §2.2 with constants C0, λ0 and ; in particular|f 0(η)|ïżœC0 eâ2Ïλ0|η|. Let further
CW = max{ â
kâZdâ
|W (k)|, supkâZdâ
|k| |W (k)|}
.
Let Aïżœ0, ÎŒïżœ0 and λâ(0, λâ] with 0<λâ<λ0. Let (Ί(k, t))kâZdâ,tïżœ0 be a continuousfunction of tïżœ0, valued in CZ
dâ , such thatâ„â„â„â„Ί(t)â
â« t
0
K0(tâÏ)Ί(Ï) dÏ
â„â„â„â„λt+ÎŒ
ïżœ A+â« t
0
(K0(t, Ï)+K1(t, Ï)+
c0
(1+Ï)m
)âΊ(Ï)âλÏ+ÎŒ dÏ for all t ïżœ 0,
(7.21)
120 c. mouhot and c. villani
where c0ïżœ0, m>1, and K0(t, Ï) and K1(t, Ï) are non-negative kernels. Let
Ï(t) = âΊ(t)âλt+ÎŒ.
Then, we have the following :(i) Assume that Îł>1 and K1=cK(α),Îł for some c>0, αâ(0, α(Îł)), where K(α),Îł is
defined by (7.10), and α(Îł) appears in Proposition 7.1. Then there are positive constantsC and Ï, depending only on Îł, λâ, λ0, , c0, CW and m, uniform as Îł!1, such that if
suptïżœ0
â« t
0
K0(t, Ï) dÏ ïżœÏ (7.22)
and
suptïżœ0
(â« t
0
K0(t, Ï)2 dÏ
)1/2
+supÏïżœ0
â« â
Ï
K0(t, Ï) dt ïżœ 1, (7.23)
then for any Δâ(0, α),
Ï(t) ïżœCA1+c2
0âΔ
eCc0
(1+
c
αΔ
)eCT eCc(1+T 2)eΔt for all t ïżœ 0, (7.24)
where
T = C max{(
c2
α5Δ2+γ
)1/(Îłâ1)
,( c
α2Δγ+1/2
)1/(Îłâ1)
,
(c20
Δ
)1/(2mâ1)}. (7.25)
(ii) Assume that K1=âN
j=1 cjK(αj),1 for some αjâ(0, α(1)), where α(1) appears in
Proposition 7.1; then there is a numeric constant Î>0 such that whenever
1 ïżœ Δ ïżœ ÎNâ
j=1
cj
α3j
,
one has, with the same notation as in (i),
Ï(t) ïżœCA1+c2
0âΔ
eCc0eCT eCc(1+T 2)eΔt for all t ïżœ 0, (7.26)
where
c =Nâ
j=1
cj and T = C max{
1Δ2
Nâj=1
cj
α3j
,
(c20
Δ
)1/(2mâ1)}.
Remark 7.8. Apart from the term c0/(1+Ï)m which will appear as a technical cor-rection, there are three different kernels appearing in Theorem 7.7: the kernel K0, whichis associated with the linearized Landau damping; the kernel K1, describing non-linearechoes (due to interaction between different Fourier modes); and the kernel K0, describ-ing the instantaneous response (due to interaction between identical Fourier modes).
on landau damping 121
We shall first prove Theorem 7.7 assuming
c0 = 0 (7.27)
and â« â
0
supkâZd
|K0(k, t)|e2Ïλ0|k|t dt ïżœ 1â , â (0, 1), (7.28)
which is a reinforcement of condition (L). Under these assumptions the proof of Theo-rem 7.7 is much simpler, and its conclusion can be substantially simplified too: Ï dependsonly on ; condition (7.23) on K0 can be dropped; and the factor eCT (1+c/αΔ3/2) in(7.24) can be omitted. If W ïżœ0 (as for gravitational interaction) and f 0ïżœ0 (as forMaxwellian background), these additional assumptions do not constitute a loss of gen-erality, since (7.28) becomes essentially equivalent to (L), and for c0 small enough theterm c0(1+Ï)âm can be incorporated inside K0.
Proof of Theorem 7.7 under (7.27) and (7.28). We have
Ï(t) ïżœA+â« t
0
(|K0|(tâÏ)+K0(t, Ï)+K1(t, Ï))Ï(Ï) dÏ, (7.29)
where |K0(t)|=supkâZd |K0(k, t)|. We shall estimate Ï by a maximum principle argu-ment. Let Ï(t)=BeΔt, where B will be chosen later. If Ï satisfies, for some T ïżœ0,â§âȘâšâȘâ©
Ï(t) <Ï(t), for 0ïżœ t ïżœT ,
Ï(t) ïżœA+â« t
0
(|K0|(t, Ï)+K0(t, Ï)+K1(t, Ï))Ï(Ï) dÏ , for t ïżœT ,(7.30)
then u(t):=Ï(t)âÏ(t) is positive for tïżœT , and satisfies u(t)ïżœâ« t
0K(t, Ï)u(Ï) dÏ for tïżœT ,
with K=|K0|+K0+K1>0; this prevents u from vanishing at later times, so uïżœ0 andÏïżœÏ. Thus it is sufficient to establish (7.30).
(i) By Proposition 7.1, and sinceâ« t
0(|K0|+K0) dÏ ïżœ1â 1
2 (for Ïïżœ 12 ),
A+â« t
0
(|K0|(t, Ï)+K0(t, Ï))Ï(Ï) dÏ +c
â« t
0
K(α),Îł(t, Ï)Ï(Ï) dÏ
ïżœA+(
1â2
+cC(Îł)
α3Δ1+ÎłtÎłâ1
)BeΔt.
(7.31)
For tïżœT :=(4cC(α3Δ1+Îł ))1/(Îłâ1), this is bounded above by A+(1â 1
4
)BeΔt, which in
turn is bounded by BeΔt as soon as Bïżœ4A/ .On the other hand, from the inequality
Ï(t) ïżœA+(1â
2
)sup
0ïżœÏïżœtÏ(Ï)+c(1+t)
â« t
0
Ï(Ï) dÏ
122 c. mouhot and c. villani
we deduce thatÏ(t) ïżœ 2A
(1+t)e2c(t+t2/2)/ .
In particular, if4A
ecâČ(T+T 2) ïżœB
with câČ=câČ(c, ) large enough, then for 0ïżœtïżœT we have Ï(t)ïżœ 12Ï(t), and (7.30) holds.
(ii) K1=âN
j=1 cjK(αj),1. We use the same reasoning, replacing the right-hand side
in (7.31) by
A+(
1â2
+C
( Nâj=1
cj
α3jΔ
+Nâ
j=1
cj
α3jΔ
2t
))BeΔt.
To conclude the proof, we may first impose a lower bound on Δ to ensure that
C
Nâj=1
cj
α3jΔ
ïżœ8
, (7.32)
and then choose t large enough to guarantee that
CNâ
j=1
cj
α3jΔ
2tïżœ
8; (7.33)
this yields (ii).
Proof of Theorem 7.7 in the general case. We only treat (i), since the reasoning for(ii) is rather similar; and we only establish the conclusion as an a-priori estimate, skippingthe continuity/approximation argument needed to turn it into a rigorous estimate. Thenthe proof is done in three steps.
Step 1. Crude pointwise bounds. From (7.21) we have
Ï(t) =âkâZdâ
|Ί(k, t)|e2Ï(λt+ÎŒ)|k|
ïżœA+âkâZdâ
â« t
0
|K0(k, tâÏ)|e2Ï(λt+ÎŒ)|k||Ί(t, Ï)| dÏ
+â« t
0
(K0(t, Ï)+K1(t, Ï)+
c0
(1+Ï)m
)Ï(Ï) dÏ
ïżœA+â« t
0
(K0(t, Ï)+K1(t, Ï)+
c0
(1+Ï)m+ sup
kâZdâ
|K0(k, tâÏ)|e2Ïλ(tâÏ)|k|)
Ï(Ï) dÏ.
(7.34)
on landau damping 123
We note that for any kâZdâ and tïżœ0,
|K0(k, tâÏ)|e2Ïλ|k|(tâÏ) ïżœ 4Ï2|W (k)|C0eâ2Ï(λ0âλ)|k|t|k|2t
ïżœ CC0
λ0âλsupkâZdâ
|k| |W (k)|ïżœ CC0CW
λ0âλ,
where (here and below) C stands for a numeric constant which may change from line toline. Assuming that
â« t
0K0(t, Ï) dÏ ïżœ 1
2 , we deduce from (7.34) that
Ï(t) ïżœA+12
sup0ïżœÏïżœt
Ï(Ï)+C
â« t
0
(C0CW
λ0âλ+c(1+t)+
c0
(1+Ï)m
)Ï(Ï) dÏ,
and, by Gronwallâs lemma,
Ï(t) ïżœ 2AeC(C0CW t/(λ0âλ)+c(t+t2)+c0Cm), (7.35)
where Cm=â«â0
(1+Ï)âm dÏ .
Step 2. L2 bound. This is the step where the smallness assumption (7.22) will bemost important. For all kâZd
â and tïżœ0 we define
Κk(t) = eâΔtΊ(k, t)e2Ï(λt+ÎŒ)|k|, (7.36)
K0k(t) = eâΔtK0(k, t)e2Ï(λt+ÎŒ)|k|, (7.37)
Rk(t) = eâΔt
(Ί(k, t)â
â« t
0
K0(k, tâÏ)Ί(k, Ï) dÏ
)e2Ï(λt+ÎŒ)|k| = (ΚkâΚkâK0
k)(t), (7.38)
and we extend all these functions by 0 for negative values of t. Taking Fourier transformin the time-variable yields Rk=(1âK0
k)Κk; since condition (L) implies that |1âK0k|ïżœ ,
we deduce that âΚkâL2 ïżœ â1âRkâL2 , i.e.,
âΚkâL2(dt) ïżœâRkâL2(dt)
. (7.39)
Plugging (7.39) into (7.38), we deduce that
âΚkâRkâL2(dt) ïżœâK0
kâL1(dt) âRkâL2(dt) for all kâZdâ. (7.40)
Then
âÏ(t)eâΔtâL2(dt) =â„â„â„â„â
kâZdâ
|Κk|â„â„â„â„
L2(dt)
ïżœâ„â„â„â„â
kâZdâ
|Rk|â„â„â„â„
L2(dt)
+âkâZdâ
âRkâΚkâL2(dt)
ïżœâ„â„â„â„â
kâZdâ
|Rk|â„â„â„â„
L2(dt)
(1+
1 âlâZdâ
âK0l âL1(dt)
).
(7.41)
124 c. mouhot and c. villani
(Note that we bounded âRlâ byâ„â„â
kâZdâ|Rk|â„â„, which seems very crude; but the decay
of K0k as a function of k will save us.) Next, we note that
âK0kâL1(dt) ïżœ 4Ï2|W (k)|
â« â
0
C0eâ2Ï(λ0âλ)|k|t|k|2t dt ïżœ 4Ï2 |W (k)| C0
(λ0âλ)2,
so âkâZdâ
âK0kâL1(dt) ïżœ 4Ï2
( âkâZdâ
|W (k)|)
C0
(λ0âλ)2.
Plugging this into (7.41) and using (7.21) again, we obtain
âÏ(t)eâΔtâL2(dt)
ïżœ(
1+CC0CW
(λ0âλ)2
)â„â„â„â„âkâZdâ
|Rk|â„â„â„â„
L2(dt)
ïżœ(
1+CC0CW
(λ0âλ)2
)(â« â
0
eâ2Δt
(A+â« t
0
(K1+K0+
c0
(1+Ï)m
)Ï(Ï) dÏ
)2dt
)1/2
.
(7.42)
We separate this (by Minkowskiâs inequality) into various contributions which weestimate separately. First, of course,
(â« â
0
eâ2ΔtA2 dt
)1/2
=Aâ2Δ
. (7.43)
Next, for any T ïżœ1, by Step 1 and
â« t
0
K1(t, Ï) dÏ ïżœ Cc(1+t)α
,
we have
(â« T
0
eâ2Δt
(â« t
0
K1(t, Ï)Ï(Ï) dÏ
)2dt
)1/2
ïżœ(
sup0ïżœtïżœT
Ï(t))(â« T
0
eâ2Δt
(â« t
0
K1(t, Ï) dÏ
)2dt
)1/2
ïżœCAeC(C0CW T/(λ0âλ)+c(T+T 2)) c
α
(â« â
0
eâ2Δt(1+t)2 dt
)1/2
ïżœCAc
αΔ3/2eC(C0CW T/(λ0âλ)+c(T+T 2)).
(7.44)
on landau damping 125
Invoking Jensenâs inequality and Fubiniâs theorem, we also have(â« â
T
eâ2Δt
(â« t
0
K1(t, Ï)Ï(Ï) dÏ
)2dt
)1/2
=(â« â
T
(â« t
0
K1(t, Ï)eâΔ(tâÏ)eâΔÏÏ(Ï) dÏ
)2dt
)1/2
ïżœ(â« â
T
(â« t
0
K1(t, Ï)eâΔ(tâÏ) dÏ
)(â« t
0
K1(t, Ï)eâΔ(tâÏ)eâ2ΔÏÏ(Ï)2 dÏ
)dt
)1/2
ïżœ(
suptïżœT
â« t
0
eâΔtK1(t, Ï)eÎ”Ï dÏ
)1/2
Ă(â« â
T
â« t
0
K1(t, Ï)eâΔ(tâÏ)eâ2ΔÏÏ(Ï)2 dÏ dt
)1/2
=(
suptïżœT
â« t
0
eâΔtK1(t, Ï)eÎ”Ï dÏ
)1/2
Ă(â« â
0
â« â
max{Ï,T}K1(t, Ï)eâΔ(tâÏ)eâ2ΔÏÏ(Ï)2 dt dÏ
)1/2
ïżœ(
suptïżœT
â« t
0
eâΔtK1(t, Ï)eÎ”Ï dÏ
)1/2
Ă(
supÏïżœ0
â« â
Ï
eΔÏK1(t, Ï)eâΔt dt
)1/2(â« â
0
eâ2ΔÏÏ(Ï)2 dÏ
)1/2
.
(7.45)
(Basically we copied the proof of Youngâs inequality.) Similarly,(â« â
0
eâ2Δt
(â« t
0
K0(t, Ï)Ï(Ï) dÏ
)2dt
)1/2
ïżœ(
suptïżœ0
â« t
0
eâΔtK0(t, Ï)eÎ”Ï dÏ
)1/2
Ă(
supÏïżœ0
â« â
Ï
eΔÏK0(t, Ï)eâΔt dt
)1/2(â« â
0
eâ2ΔÏÏ(Ï)2 dÏ
)1/2
ïżœ(
suptïżœ0
â« t
0
K0(t, Ï) dÏ
)1/2(supÏïżœ0
â« â
Ï
K0(t, Ï) dt
)1/2(â« â
0
eâ2ΔÏÏ(Ï)2 dÏ
)1/2
.
(7.46)
The last term is also split, this time according to Ï ïżœT or Ï >T :(â« â
0
eâ2Δt
(â« T
0
c0Ï(Ï)(1+Ï)m
dÏ
)2dt
)1/2
ïżœ c0
(sup
0ïżœÏïżœTÏ(Ï))(â« â
0
eâ2Δt
(â« T
0
dÏ
(1+Ï)m
)2dt
)1/2
ïżœ c0CAâ
ΔeC(C0CW T/(λ0âλ)+c(T+T 2))Cm,
(7.47)
126 c. mouhot and c. villani
and
(â« â
0
eâ2Δt
(â« t
T
c0Ï(Ï) dÏ
(1+Ï)m
)2dt
)1/2
= c0
(â« â
0
(â« t
T
eâΔ(tâÏ) eâΔÏÏ(Ï)(1+Ï)m
dÏ
)2dt
)1/2
ïżœ c0
(â« â
0
(â« t
T
eâ2Δ(tâÏ)
(1+Ï)2mdÏ
)(â« t
T
eâ2ΔÏÏ(Ï)2 dÏ
)dt
)1/2
ïżœ c0
(â« â
0
eâ2ΔtÏ(t)2 dt
)1/2(â« â
0
â« t
T
eâ2Δ(tâÏ)
(1+Ï)2mdÏ dt
)1/2
= c0
(â« â
0
eâ2ΔtÏ(t)2 dt
)1/2(â« â
T
1(1+Ï)2m
â« â
Ï
eâ2Δ(tâÏ) dt dÏ
)1/2
= c0
(â« â
0
eâ2ΔtÏ(t)2 dt
)1/2(â« â
T
dÏ
(1+Ï)2m
)1/2(â« â
0
eâ2Δs ds
)1/2
=C
1/22m c0
Tmâ1/2â
Δ
(â« â
0
eâ2ΔtÏ(t)2 dt
)1/2
.
(7.48)
Gathering estimates (7.43)â(7.48), we deduce from (7.42) that
âÏ(t)eâΔtâL2(dt) ïżœ(
1+CC0CW
(λ0âλ)2
)CAâ
Δ
(1+
c
αΔ+c0Cm
)ĂeC(C0CW T/(λ0âλ)+c(T+T 2))+aâÏ(t)eâΔtâL2(dt),
(7.49)
where
a=(
1+CC0CW
(λ0âλ)2
)[(suptïżœT
â« t
0
eâΔtK1(t, Ï)eΔÏdÏ
)1/2(supÏïżœ0
â« â
Ï
eΔÏK1(t, Ï)eâΔt dt
)1/2
+(
suptïżœ0
â« t
0
K0(t, Ï) dÏ
)1/2(supÏïżœ0
â« â
Ï
K0(t, Ï) dt
)1/2
+C
1/22m c0
Tmâ1/2â
Δ
].
Using Propositions 7.1 (case Îł>1) and 7.5, as well as assumptions (7.22) and (7.23),we see that aïżœ 1
2 for Ï small enough and T satisfying (7.25). Then from (7.49) followsthat
âÏ(t)eâΔtâL2(dt) ïżœ(
1+CC0CW
(λ0âλ)2
)CAâ
Δ
(1+
c
αΔ+c0Cm
)eC(C0CW T/(λ0âλ)+c(T+T 2)).
on landau damping 127
Step 3. Refined pointwise bounds. Let us use (7.21) a third time, now for tïżœT :
eâΔtÏ(t) ïżœAeâΔt+â« t
0
(supkâZdâ
|K0(k, tâÏ)|e2Ïλ(tâÏ)|k|)Ï(Ï)eâÎ”Ï dÏ
+â« t
0
(K0(t, Ï)+
c0
(1+Ï)m
)Ï(Ï)eâÎ”Ï dÏ
+â« t
0
eâΔtK1(t, Ï)eΔÏÏ(Ï)eâÎ”Ï dÏ
ïżœAeâΔt+[(â« t
0
(supkâZdâ
|K0(k, tâÏ)|e2Ïλ(tâÏ)|k|)2
dÏ
)1/2
+(â« t
0
K0(t, Ï)2 dÏ
)1/2
+(â« â
0
c20
(1+Ï)2mdÏ
)1/2
+(â« t
0
eâ2ΔtK1(t, Ï)2e2Î”Ï dÏ
)1/2](â« â
0
Ï(Ï)2eâ2Î”Ï dÏ
)1/2
.
(7.50)
We note that, for any kâZdâ,
(|K0(k, t)|e2Ïλ|k|t)2 ïżœ 16Ï4|W (k)|2 |f 0(kt)|2 |k|4t2e4Ïλ|k|t
ïżœCC20 |W (k)|2eâ4Ï(λ0âλ)|k|t|k|4t2
ïżœ CC20
(λ0âλ)2|W (k)|2eâ2Ï(λ0âλ)|k|t|k|2
ïżœ CC20
(λ0âλ)2C2
W eâ2Ï(λ0âλ)|k|t
ïżœ CC20
(λ0âλ)2C2
W eâ2Ï(λ0âλ)t;
so â« t
0
(supkâZdâ
|K0(k, tâÏ)|e2Ïλ(tâÏ)|k|)2
dÏ ïżœ CC20C2
W
(λ0âλ)3.
Then the conclusion follows from (7.50), Corollary 7.4, conditions (7.25) and (7.23), andStep 2.
Remark 7.9. Theorem 7.7 leads to enormous constants, and it is legitimate to askabout their sharpness, say with respect to the dependence in Δ. We expect the constantto be roughly of the order of
supt
(e(ct)1/Îł
eâΔt)ïżœ exp(Δâ1/(Îłâ1)).
Our bound is roughly like exp(Δâ(4+2Îł)/(Îłâ1)); this is worse, but displays the expectedbehavior as an exponential of an inverse power of Δ, with a power that diverges likeO((1âÎł)â1) as Îł!1.
128 c. mouhot and c. villani
Remark 7.10. Even in the case of an analytic interaction, a similar argument sug-gests constants that are at best like (log(1/Δ))log(1/Δ), and this grows faster than anyinverse power of 1/Δ.
To obtain sharper results, in §11 we shall later âbreak the normâ and work directlyon the Fourier modes of, say, the spatial density. In this subsection we establish theestimates which will be used later; the reader who does not particularly care about thecase Îł=1 in Theorem 2.6 can skip them.
For any Îłïżœ1, α>0, k, lâZdâ and 0ïżœÏ ïżœt, we define
K(α),Îłk,l (t, Ï) =
(1+Ï)deâα|l|eâα(tâÏ)|kâl|/teâα|k(tâÏ)+lÏ |
1+|kâl|Îł . (7.51)
We start by exponential moment estimates.
Proposition 7.11. Let Îłâ[1,â). For any αâ(0, 1) and k, lâZdâ, let K
(α),γk,l be
defined by (7.51). Then there is α=α(Îł)>0 such that if Î±ïżœÎ± and Δâ(0, 14α)
then forany t>0,
supkâZdâ
âlâZdâ
eâΔt
â« t
0
K(α),Îłk,l (t, Ï)eÎ”Ï dÏ ïżœ C(d, Îł)
α1+dΔγ+1tγ, (7.52)
supkâZdâ
âlâZdâ
eâΔt
(â« t
0
K(α),Îłk,l (t, Ï)2e2Î”Ï dÏ
)1/2
ïżœ C(d, Îł)αdΔγ+1/2tÎłâ1/2
, (7.53)
supkâZdâ
âlâZdâ
supÏïżœ0
eΔÏ
â« â
Ï
K(α),Îłk,l eâΔt dt ïżœ C(d, Îł)
α2+dΔ. (7.54)
Proof. We first reduce to the case d=1. Monotonicity cannot be used now, but wenote that
K(α),Îłk,l (t, Ï) ïżœ
dâj=1
eâα|l1|eâα|l2| ... eâα|ljâ1| K(α),Îłkj ,lj
(t, Ï)eâα|lj+1| ... eâα|ld|,
where Kkj ,lj stands for a 1-dimensional kernel. Thus
supkâZdâ
âlâZdâ
â« t
0
eâΔtK(α),Îłk,l (t, Ï)eÎ”Ï dÏ
ïżœ supkâZdâ
( âmâZd
eâα|m|)dâ1 dâ
j=1
âljâZ
â« t
0
eâΔtK(α),Îłkj ,lj
(t, Ï)eÎ”Ï dÏ
ïżœ C(d)αdâ1
sup1ïżœjïżœd
supkjâZ
âljâZ
â« t
0
eâΔtK(α,Îł)kj ,lj
(t, Ï)eÎ”Ï dÏ.
on landau damping 129
In other words, for (7.52) we may just consider the 1-dimensional case, provided we allowfor an extra multiplicative constant C(d)/αdâ1. A similar reasoning holds for (7.53) and(7.54). From now on we focus on the case d=1.
Without loss of generality we assume that k>0, and only treat the worst case l<0.(The other case k, l>0 is simpler and yields an exponential decay in time of the formeâc min{α,Δ}t). For simplicity we also write Kk,l=K
(α),γk,l . An easy computation yields
eâΔt
â« t
0
Kk,l(t, Ï)eÎ”Ï dÏ ïżœ Ceâα|l|
1+|kâl|Îł(
1α|kâl|+
|k|tα|kâl|2 +
1α2|kâl|2
)eâΔ|l|t/|kâl|.
Then, for any kïżœ1, we have (crudely writing α2=O(α))
â1âl=ââ
â« t
0
eâΔtKk,l(t, Ï)eÎ”Ï dÏ ïżœC
( ââl=1
eâαleâΔlt/(k+l)
α2(k+l)1+γ+
ââl=1
eâαleâΔlt/(k+l)
α(k+l)2+γkt
). (7.55)
For the first sum in the right-hand side of (7.55) we writeââ
l=1
eâαleâΔlt/(k+l)
(k+l)1+Îłïżœ
ââl=1
eâαl
l1+Îł
(Δlt
k+l
)1+Îł
eâΔlt/(k+l) 1(Δt)1+Îł
ïżœ C(Îł)(Δt)1+Îł
. (7.56)
For the second sum in the right-hand side of (7.55) we separate according to 1ïżœlïżœk
or lïżœk+1:kâ
l=1
eâαleâΔlt/(k+l)
(k+l)2+Îłktïżœ
kâl=1
eâαleâΔt/(k+1)
(k+1)2+Îłkt
ïżœ C
αeâΔt/(k+1)
(Δt
k+1
)1+Îłk
k+1t
(Δt)1+γ
ïżœ C
αΔ1+γtγ
(7.57)
andââ
l=k+1
eâαleâΔlt/(k+l)
(k+l)2+ÎłktïżœC
ââl=k+1
eâαleâΔt/2
k2+Îłktïżœ C
α
eâΔt/4
Δk1+Îłïżœ C
αΔ1+γtγ. (7.58)
The combination of (7.55)â(7.58) completes the proof of (7.52).Now we turn to (7.53). The estimates are rather similar, since
Kk,l(t, Ï)2 ïżœC(1+t)Kk,l(t, Ï)
with Îł !2Îł and α !2α. So (7.55) should be replaced byâlâZ
eâΔt
(â« t
0
Kk,l(t, Ï)2e2Î”Ï dÏ
)1/2
ïżœC
( ââl=1
eâαleâΔlt/(k+l)(1+t)1/2
α(k+l)γ+1/2+
ââl=1
eâαleâΔlt/(k+l)(kt)1/2(1+t)1/2
α1/2(k+l)1+γ
).
(7.59)
130 c. mouhot and c. villani
For the first sum we use (7.56) with Îł replaced by Îłâ 12 : for tïżœ1,
(1+t)1/2ââ
l=1
eâαleâΔlt/(k+l)
(k+l)1+Îłâ1/2ïżœ C(Îł)t1/2
(Δt)Îł+1/2ïżœ C(Îł)
Δγ+1/2tγ. (7.60)
For the second sum in the right-hand side of (7.59) we write
kâl=1
eâαleâΔlt/(k+l)k1/2t
(k+l)1+ÎłïżœC
ââl=1
eâαleâΔt/(k+1)
(Δt
k+1
)Îł+1/2k1/2
(1+k)1/2
t
(Δt)γ+1/2
ïżœ C
αΔγ+1/2tÎłâ1/2
and
ââl=k+1
eâαleâΔlt/(k+l)k1/2t
(k+l)1+ÎłïżœC
ââl=1
eâαl
lÎł+1/2eâΔt/2t ïżœCeâΔt/4 ïżœ C
(Δt)Îłâ1/2Δ.
With this, (7.53) is readily obtained.Finally we consider (7.54). As in Proposition 7.5, one easily shows that
supkâZâ
âlâZâ
supÏïżœ0
eΔÏ
â« â
2Ï
Kk,l(t, Ï)eâΔt dt ïżœ C
Δα2
âlâZâ
eâα|l| ïżœ C
Δα3.
Then one has
eΔÏ
â« 2Ï
Ï
eâα|k(tâÏ)+lÏ |eâΔt dt ïżœ C
α2k+
C
αΔk+
C
αkeâΔlÏ/k.
So the problem amounts to estimate
âlâZâ
supÏïżœ0
(1+Ï)eâαleâΔlÏ/k
αk(k+l)ÎłïżœâlâZâ
eâαl
(1α
+1
Δl(k+l)ÎłeâΔlÏ/k ΔlÏ
k
)ïżœC
(1α2
+1Δ
),
and the proof is complete.
We conclude this section with a mode-by-mode analogue of Theorem 7.7.
Theorem 7.12. Let f0=f0(v) and W =W (x) satisfy condition (L) from §2.2 withconstants C0, λ0 and ; in particular |f 0(η)|ïżœC0e
â2Ïλ0|η|. Further let
CW = max{â
kâZdâ
|W (k)|, supkâZdâ
|k| |W (k)|}
.
on landau damping 131
Let (Ak)kâZdâ , ÎŒïżœ0 and λâ(0, λâ] with 0<λâ<λ0. Let (Ί(k, t))kâZd,tïżœ0 be a con-tinuous function of tïżœ0, valued in CZ
d
, such that for all tïżœ0, Ί(0, t)=0 and for anykâZd
â,
e2Ï(λt+ÎŒ)|k|âŁâŁâŁâŁÎŠ(k, t)â
â« t
0
K0(k, tâÏ)Ί(k, Ï) dÏ
âŁâŁâŁâŁïżœAk+
â« t
0
K0(t, Ï)e2Ï(λÏ+ÎŒ)|k||Ί(k, Ï)| dÏ
+â« t
0
âlâZdâ
(cK
(α),Îłk,l (t, Ï)+
cl
(1+Ï)m
)e2Ï(λÏ+ÎŒ)|kâl||Ί(kâl, Ï)| dÏ,
(7.61)
where c>0, clïżœ0 (for lâZdâ), m>1, Îłïżœ1, K0(t, Ï) is a non-negative kernel, K
(α),γk,l is
defined by (7.51) and α<α(Îł) is defined in Proposition 7.11. Then there are positiveconstants C and Ï, depending only on Îł, λâ, λ0, ,
c := max{â
lâZdâ
cl,
(âlâZdâ
c2l
)1/2},
CW and m such that if
suptïżœ0
â« t
0
K0(t, Ï) dÏ ïżœÏ (7.62)
and
suptïżœ0
(â« t
0
K0(t, Ï)2 dÏ
)1/2
+supÏïżœ0
â« â
Ï
K0(t, Ï) dt ïżœ 1, (7.63)
then for any Δâ(0, 14α)
and for any tïżœ0,
supkâZdâ
|Ί(k, t)|e2Ï(λt+ÎŒ)|k| ïżœCA(1+c2)â
ΔeCc(1+
c
α2Δ
)eCT eCc(1+T 2)/αeΔt, (7.64)
where A:=supk Ak and
T = C max{(
c2
α3+2dΔγ+2
)1/Îł
,( c
αdΔγ+1/2
)1/(Îłâ1/2)
,
(c2
Δ
)1/(2mâ1)}. (7.65)
Proof. The proof is quite similar to the proof of Theorem 7.7, so we shall only pointout the differences. As in the proof of Theorem 7.7, we start by crude pointwise boundsobtained by Gronwallâs inequality; but this time on the quantity
Ï(t) = supkâZdâ
|Ί(k, t)|e2Ï(λt+ÎŒ)|k|.
132 c. mouhot and c. villani
Since âlâZdâ
Kk,l(t, Ï) =O
(1+Ï
α
),
we find thatÏ(t) ïżœ 2AeC(C0CW t/(λ0âλ)+c(t+t2)/α+cCm). (7.66)
We next define Κk, K0k and Rk as in Step 2 of the proof of Theorem 7.7, and we
deduce (7.39) and (7.40). Let
Ïk(t) = |Ί(k, t)|e2Ï(λt+ÎŒ)|k|. (7.67)
Then
âÏk(t)eâΔtâL2(dt) ïżœ âRkâL2(dt)
(1+
âK0kâL1(dt)
)ïżœ âRkâL2(dt)
(1+
CCW C0
);
whence
âÏk(t)eâΔtâL2(dt) ïżœ(
1+CC0CW
(λ0âλ)2
)[â« â
0
eâ2Δt
(Ak+
â« t
0
K0(t, Ï)Ïk(Ï) dÏ
+âlâZdâ
â« t
0
(cKk,l(t, Ï)+
cl
(1+Ï)m
)Ïkâl(Ï) dÏ
)2dt
]1/2
.
(7.68)
We separate this into various contributions as in the proof of Theorem 7.7. Inparticular, using (7.66) and â« t
0
âlâZdâ
Kk,l dÏ = O
(1+t
α2
),
we find that[â« T
0
eâ2Δt
(â« t
0
âlâZdâ
Kk,l(t, Ï)Ïkâl(Ï) dÏ
)2dt
]1/2
ïżœ(
supkâZdâ
sup0ïżœtïżœT
Ïk(t))(â« T
0
eâ2Δt
(â« t
0
âlâZdâ
Kk,l(t, Ï) dÏ
)2dt
)1/2
ïżœCAc
α2Δ3/2eC(C0CW T/(λ0âλ)+c(T+T 2)/α).
(7.69)
Also, [â« â
T
eâ2Δt
(â« t
0
âlâZdâ
Kk,l(t, Ï)Ïkâl(Ï) dÏ
)2dt
]1/2
ïżœ(
suptïżœT
â« t
0
eâΔtâlâZdâ
Kk,l(t, Ï)eÎ”Ï dÏ
)1/2
Ă(â« â
T
â« t
0
âlâZdâ
Kk,l(t, Ï)eâΔ(tâÏ)eâ2ΔÏÏkâl(Ï)2 dÏ dt
)1/2
,
on landau damping 133
and the last term inside the parentheses isâlâZdâ
â« â
0
(â« â
max{Ï,T}Kk,l(t, Ï)eâΔ(tâÏ) dt
)eâ2ΔÏÏkâl(Ï)2 dÏ
ïżœ(â
lâZdâ
supÏïżœ0
â« â
Ï
Kk,l(t, Ï)eâΔ(tâÏ) dt
)(suplâZdâ
â« â
0
eâ2ΔÏÏl(Ï)2 dÏ
).
The computation for K0 is the same as in the proof of Theorem 7.7, and the termsin (1+Ï)âm are handled in essentially the same way: simple computations yield[â« â
0
eâ2Δt
(â« T
0
âlâZdâ
clÏkâl(Ï)
(1+Ï)mdÏ
)2dt
]1/2
ïżœ(
sup0ïżœÏïżœT
suplâZdâ
Ïl(Ï))[â« â
0
eâ2Δt
(â« T
0
âlâZdâ
cl
(1+Ï)mdÏ
)2dt
]1/2
ïżœ cCmAâ
ΔeC(C0CW T/(λ0âλ)+c(T+T 2)/α)
(7.70)
and[â« â
0
eâ2Δt
(â« t
T
âlâZdâ
clÏkâl(Ï) dÏ
(1+Ï)m
)2dt
]1/2
ïżœ[
suptïżœ0
suplâZdâ
(â« t
T
eâ2ΔÏÏl(Ï)2 dÏ
)(âlâZdâ
cl
)2 â« â
0
â« t
T
eâ2Δ(tâÏ)
(1+Ï)2mdÏ dt
]1/2
ïżœ c
(C2m
ΔT 2mâ1
)1/2(suplâZdâ
â« â
0
eâ2ΔÏÏl(Ï)2 dÏ
)1/2
.
(7.71)
All in all, we end up with
supkâZdâ
âÏk(t)eâΔtâL2(dt)
ïżœ(
1+CC0CW
(λ0âλ)2
)CAâ
Δ
(1+
c
α2Δ+cCm
)eC(C0CW T/(λ0âλ)+c(T+T 2)/α)
+a supkâZdâ
âÏk(t)eâΔtâL2(dt),
(7.72)
where
a=(
1+CC0CW
(λ0âλ)2
)[c2
(suptïżœT
âlâZdâ
â« t
0
eâΔtKk,l(t, Ï)eÎ”Ï dÏ
)1/2
Ă(â
lâZdâ
supÏïżœ0
â« â
Ï
eΔÏKk,l(t, Ï)eâΔt dt
)1/2
+(
suptïżœ0
â« t
0
K0(t, Ï) dÏ
)1/2(supÏïżœ0
â« â
Ï
K0(t, Ï) dt
)1/2
+C
1/22m c0
Tmâ1/2â
Δ
].
134 c. mouhot and c. villani
Applying Proposition 7.11, we see that aïżœ 12 , as soon as T satisfies (7.65), and then
we deduce from (7.72) a bound on supkâZdââÏk(t)eâΔtâL2(dt).
Finally we conclude as in Step 3 of the proof of Theorem 7.7: from (7.61), we get
eâΔtÏk(t) ïżœAkeâΔt+[(â« t
0
(supkâZdâ
|K0(k, tâÏ)|e2Ïλ(tâÏ)|k|)2
dÏ
)1/2
+(â« t
0
K0(t, Ï)2 dÏ
)1/2
+c
(â« â
0
dÏ
(1+Ï)2m
)1/2
+câlâZdâ
(â« t
0
eâ2ΔtKk,l(t, Ï)2e2Î”Ï dÏ
)1/2]
Ă(
supkâZdâ
â« â
0
Ïk(Ï)2eâ2Î”Ï dÏ
)1/2
,
(7.73)
and the conclusion follows by a new application of Proposition 7.11.
8. Approximation schemes
Having defined a functional setting (§4) and identified several mathematical/physicalmechanisms (§§5â7), we are prepared to fight the Landau damping problem. For thatwe need an approximation scheme solving the non-linear Vlasov equation. The problemis not to prove the existence of solutions (this is much easier), but to devise the schemein such a way that it leads to relevant estimates for our study.
The first idea which may come to mind is a classical Picard scheme for quasilinearequations:
âtfn+1+v ·âxfn+1+F [fn]·âvfn+1 = 0. (8.1)
This has two drawbacks: first, fn+1 evolves by the characteristics created by F [fn], andthis will deteriorate the estimates in analytic regularity. Secondly, there is no hope to geta closed (or approximately closed) equation on the density associated with fn+1. Morepromising, and more in the spirit of the linearized approach, would be a scheme like
âtfn+1+v ·âxfn+1+F [fn+1]·âvfn = 0. (8.2)
(Physically, fn+1 forces fn, and the question is whether the reaction will exhaust fn+1
in large time.) But when we write (8.2) we are implicitly treating a higher-order term(âvf) of the equation in a perturbative way; so this has no reason to converge.
To circumvent these difficulties, we shall use a Newton iteration: not only will thisprovide more flexibility in the regularity indices, but at the same time it will yield anextremely fast rate of convergence (something like O(Δ2n
)) which will be most welcometo absorb the large constants coming from Theorem 7.7 or 7.12.
on landau damping 135
8.1. The natural Newton scheme
Let us adapt the abstract Newton scheme to an abstract evolution equation in the formâf
ât= Q(f),
around a stationary solution f0 (so Q(f0)=0). Write the Cauchy problem with initialdatum fiïżœf0 in the form
Ί(f) := (âtfâQ(f), f(0, ·))â(0, fi).
Starting from f0, the Newton iteration consists in solving inductively
Ί(fnâ1)+ΊâČ(fnâ1)·(fnâfnâ1) = 0 for n ïżœ 1.
More explicitly, writing hn=fnâfnâ1, we should solve{âth
1 = QâČ(f0)·h1,
h1(0, ·) = fiâf0,
and, for all nïżœ1, {âth
n+1 = QâČ(fn)·hn+1â[âtfnâQ(fn)],
hn+1(0, ·) = 0.
By induction, for nïżœ1, this is the same as{âth
n+1 = QâČ(fn)·hn+1+[Q(fnâ1+hn)âQ(fnâ1)âQâČ(fnâ1)·hn],hn+1(0, ·) = 0.
This is easily applied to the non-linear Vlasov equation, for which the non-linearityis quadratic. So we define the natural Newton scheme for the non-linear Vlasov equationas follows:
f0 = f0(v) is given (homogeneous stationary state)
andfn = f0+h1+...+hn,
where {âth
1+v ·âxh1+F [h1]·âvf0 = 0,
h1(0, ·) = fiâf0,(8.3)
and, for all nïżœ1,{âth
n+1+v ·âxhn+1+F [fn]·âvhn+1+F [hn+1]·âvfn =âF [hn]·âvhn,
hn+1(0, ·) = 0.(8.4)
Here F [f ] is the force field created by the particle distribution f , namely
F [f ](t, x) =ââ«
Td
â«Rd
âW (xây)f(t, y, w) dw dy. (8.5)
Note also that all the ïżœn=â«
Rd hn dv for nïżœ1 have zero spatial average.
136 c. mouhot and c. villani
8.2. Battle plan
The treatment of (8.3) was performed in §4.12. Now the problem is to handle all equationsappearing in (8.4). This is much more complicated, because for nïżœ1 the backgrounddensity fn depends on t and x, instead of just v. As a consequence,
(a) equation (8.4) cannot be considered as a perturbation of free transport, becauseof the presence of âvhn+1 in the left-hand side;
(b) the reaction term F [hn+1]·âvfn no longer has the simple product structure(function of x)Ă(function of v), so it becomes harder to get hands on the homogenizationphenomenon;
(c) because of spatial inhomogeneities, echoes will appear; they are all the moredangerous that, âvfn is unbounded as t!â, even in gliding regularity. (It grows likeO(t), which is reminiscent of the observation made by Backus [4].)
The estimates in §§5â7 have been designed precisely to overcome these problems;however we still have a few conceptual difficulties to solve before applying these tools.
Recall the discussion in §4.11: the natural strategy is to propagate the bound
supÏïżœ0
âfÏâZλ,ÎŒ;1Ï
<â (8.6)
along the scheme; this estimate contains in particular two crucial pieces of information:âą a control of ïżœÏ =
â«Rd fÏ dv in FλÏ+ÎŒ norm,
âą a control of ăfÏ ă=â«
Td fÏ dx in Cλ;1 norm.So the plan would be to try to inductively get estimates of each hn in a norm like
the one in (8.6), in such a way that hn is extremely small as n!â, and allowing a slightdeterioration of the indices λ and ÎŒ as n!â. Let us try to see how this would work:assuming that
supÏïżœ0
âhkÏâZλk,ÎŒk;1
Ïïżœ ÎŽk for all 0 ïżœ k ïżœn,
we should try to bound hn+1Ï . To âsolveâ (8.4), we apply the classical method of charac-
teristics: as in §5, we define (XnÏ,t, V
nÏ,t) as the solution ofâ§âȘâšâȘâ©
d
dtXn
Ï,t(x, v) =V nÏ,t(x, v),
d
dtV n
Ï,t(x, v) =F [fn](t, XnÏ,t(x, v)),
XnÏ,Ï (x, v) =x, V n
Ï,Ï (x, v) = v.
Then (8.4) is equivalent to
d
dthn+1(t, Xn
0,t, Vn0,t(x, v)) = ÎŁn+1(t, Xn
0,Ï (x, v), V n0,Ï (x, v)), (8.7)
whereÎŁn+1(t, x, v) =âF [hn+1]·âvfnâF [hn]·âvhn. (8.8)
on landau damping 137
Integrating (8.7) in time and recalling that hn+1(0, ·)=0, we get
hn+1(t, Xn0,t(x, v), V n
0,t(x, v)) =â« t
0
ÎŁn+1(Ï,Xn0,Ï (x, v), V n
0,Ï (x, v)) dÏ.
Composing with (Xnt,0, V
nt,0) and using (5.2) yields
hn+1(t, x, v) =â« t
0
ÎŁn+1(Ï,Xnt,Ï (x, v), V n
t,Ï (x, v)) dÏ.
We rewrite this using the deflection map
Ωnt,Ï (x, v) = (Xn
t,Ï , V nt,Ï )(x+v(tâÏ), v) =Sn
t,Ï S0Ï,t;
then we finally obtain
hn+1(t, x, v) =â« t
0
(ÎŁn+1Ï Î©n
t,Ï )(xâv(tâÏ), v) dÏ
=ââ« t
0
[(F [hn+1Ï ] Ωn
t,Ï )·((âvfnÏ ) Ωn
t,Ï )](xâv(tâÏ), v) dÏ
ââ« t
0
[(F [hnÏ ] Ωn
t,Ï )·((âvhnÏ ) Ωn
t,Ï )](xâv(tâÏ), v) dÏ.
(8.9)
Since the unknown hn+1 appears on both sides of (8.9), we need to get a self-consistentestimate. For this we have little choice but to integrate in v and get an integral equationon ïżœ[hn+1]=
â«Rd hn dv, namely
ïżœ[hn+1](t, x) =â« t
0
â«Rd
[((ïżœ[hn+1Ï ]ââW ) Ωn
t,Ï )·GnÏ,t] S0
Ïât(x, v) dv dÏ
+(terms from stage n),(8.10)
where GnÏ,t=âvfn
Ï Î©nt,Ï . By induction hypothesis, Gn
Ï,t is smooth with regularity indicesroughly equal to λn and ÎŒn; so if we accept to lose just a bit more on the regularity wemay hope to apply the long-term regularity extortion and decay estimates from §6, andthen time-response estimates of §7, and get the desired damping.
However, we are facing a major problem: composition of ïżœ[hn+1Ï ]ââW by Ωn
t,Ï im-plies a loss of regularity in the right-hand side with respect to the left-hand side, which isof course unacceptable if one wants a closed estimate. The short-term regularity extor-tion from §6 remedies this, but the price to pay is that Gn should now be estimated attime Ï âČ=Ïâbt/(1+b) instead of Ï , and with index of gliding analytic regularity roughlyequal to λn(1+b) rather than λn. Now the catch is that the error induced by composition
138 c. mouhot and c. villani
by Ωn depends on the whole distribution fn, not just hn. Thus, if the parameter b shouldcontrol this error it should stay of order 1 as n!â, instead of converging to 0.
So it seems we are sentenced to lose a fixed amount of regularity (or rather of radiusof convergence) in the transition from stage n to stage n+1; this is reminiscent of theâNashâMoser syndromâ [2]. The strategy introduced by Nash [70] to remedy such aproblem (in his case arising in the construction of Câ isometric embeddings) consistedof combining a Newton scheme with regularization; his method was later developed byMoser [67] for the Câ KAM theorem (see [68, pp. 19â21] for some interesting historicalcomments). A clear and concise proof of the NashâMoser implicit function theorem,together with its application to the Câ embedding problem, can be found in [83]. TheNashâMoser method is arguably the most powerful perturbation technique known tothis day. However, despite significant effort, we were unable to set up any relevantregularization procedure (in gliding regularity, of course) which could be used in NashâMoser style, because of three serious problems:
âą The convergence of the NashâMoser scheme is no longer as fast as that of theârawâ Newton iteration; instead, it is determined by the regularity of the data, and theresulting rates would be unlikely to be fast enough to win over the gigantic constantscoming from §7.
⹠Analytic regularization in the v variable is extremely costly, especially if we wishto keep a good localization in velocity space, as the one appearing in Theorem 4.20 (iii),that is exponential integrability in v; then the uncertainty principle basically forces usto pay O(eC/Δ2
), where Δ is the strength of the regularization.⹠Regularization comes with an increase of amplitude (there is as usual a trade-
off between size and regularity); if we regularize before composition by Ωn, this willdevastate the estimates, because the analytic regularity of f g depends not only on theregularity of f and g, but also on the amplitude of gâId.
Fortunately, it turned out that a ârawâ Newton scheme could be used; but thisrequired us to give up the natural estimate (8.6), and replace it by the pair of estimatesâ§âȘâšâȘâ©
supÏïżœ0
âïżœÏâFλÏ+ÎŒ <â,
sup0ïżœÏïżœt
âfÏ Î©t,ÏâZλ(1+b),ÎŒ;1Ïâbt/(1+b)
<â.(8.11)
Here b=b(t) takes the form constant/(1+t), and is kept fixed all along the scheme;moreover λ and ÎŒ will be slightly larger than λ and ÎŒ, so that none of the two estimatesin (8.11) implies the other one. Note carefully that there are now two times (t and Ï)explicitly involved, so this is much more complex than (8.6). Let us explain why thisstrategy is nonetheless workable.
on landau damping 139
λn
λn(1+b)
λn+1
λn+1(1+b)
λâ=λâ
...
t
Figure 7. Indices of gliding regularity appearing throughout our Newton scheme, in the norm
of ïżœ[hÏ ] and in the norm of hÏ Î©t,Ï , respectively, plotted as functions of t.
First, the density ïżœn=â«
Rd fn dv determines the characteristics at stage n, and there-fore the associated deflection Ωn. If ïżœn
Ï is bounded in FλnÏ+ÎŒ, then by Theorem 5.2 wecan estimate Ωn
t,Ï in ZλâČn,ÎŒâČ
n
Ï âČ , as soon as (essentially) λâČnÏ âČ+ÎŒâČ
nïżœÎ»nÏ +ÎŒn and λâČn<λn,
and these bounds are uniform in t.
Of course, we cannot apply this theorem in the present context, because λn(1+b)is not bounded above by λn. However, for large times t we may afford λn(1+b(t))<λn,while λn(1+b)(Ïâbt/(1+b))ïżœÎ»nÏ for all times; this will be sufficient to repeat the argu-ments in §5, getting uniform estimates in a regularity which depends on t. (The constantsare uniform in t; but the index of regularity goes decreases with t.) We can also do thiswhile preserving the other good properties from Theorem 5.2, namely exponential decayin Ï , and vanishing near Ï =t.
Figure 7 summarizes schematically the way we choose and estimate the glidingregularity indices.
Besides being uniform in t, our bounds need to be uniform in n. For this we shallhave to stratify all our estimates, that is decompose ïżœ[fn]=ïżœ[h1]+...+ïżœ[hn], and considerseparately the influence of each term in the equations for characteristics. This can workonly if the scheme converges very fast.
Once we have estimates on Ωnt,Ï in a time-varying regularity, we can work with the
kinetic equation to derive estimates on hnÏ Î©n
t,Ï ; and then on all hkÏ Î©n
t,Ï , also in a norm oftime-varying regularity. We can also estimate their spatial average, in a norm Cλ(1+b);1;due to the exponential convergence of the deflection map as Ï!â these estimates willturn out to be uniform in Ï .
Next, we can use all this information, in conjunction with Theorem 6.5, to get anintegral inequality on the norm of ïżœ[hn+1
Ï ] in FλÏ+ÎŒ, where λ and ÎŒ are only slightlysmaller than λn and ÎŒn. Then we can go through the response estimates of §7, this givesus an arbitrarily small loss in the exponential decay rate, at the price of a huge constant
140 c. mouhot and c. villani
which will eventually be wiped out by the fast convergence of the scheme. So we havean estimate on ïżœ[hn+1], and we are in business to continue the iteration. (To ensure thepropagation of the linear damping condition, or equivalently of the smallness of K0 inTheorem 7.7, throughout the scheme, we shall have to stratify the estimates once more.)
9. Local-in-time iteration
Before working out the core of the proof of Theorem 2.6 in §10, we shall need a short-timeestimate, which will act as an âinitial regularity layerâ for the Newton scheme. (This willgive us room later to allow the regularity index to depend on t.) So we run the wholescheme once in this section, and another time in the next section.
Short-time estimates in the analytic class are not new for the non-linear Vlasovequation: see in particular the work of Benachour [8] on the VlasovâPoisson equation.His arguments can probably be adapted for our purpose; also the CauchyâKovalevskayamethod could certainly be applied. We shall provide here an alternative method, basedon the analytic function spaces from §4, but not needing the apparatus from §§5â7.Unlike the more sophisticated estimates which will be performed in §10, these ones areâalmostâ Eulerian (the only characteristics are those of free transport). The main toolis given by the following lemma.
Lemma 9.1. Let f be an analytic function, λ(t)=λâKt and ÎŒ(t)=ÎŒâKt; let T >0be so small that λ(t), ÎŒ(t)>0 for 0ïżœtïżœT . Then for any Ïâ[0, T ] and any pïżœ1,
d+
dt
âŁâŁâŁâŁt=Ï
âfâZλ(t),ÎŒ(t);pÏ
ïżœâ K
1+ÏââfâZλ(Ï),ÎŒ(Ï);p
Ï, (9.1)
where d+/dt stands for the upper right derivative.
Remark 9.2. Time-differentiating Lebesgue integrability exponents is common prac-tice in certain areas of analysis; see e.g. [33]. Time-differentiation with respect to regu-larity exponents is less common; however, as pointed out to us by Strain, Lemma 9.1 isstrongly reminiscent of a method recently used by Chemin [17] to derive local analyticregularity bounds for the NavierâStokes equation. We expect that similar ideas can beapplied to more general situations of CauchyâKovalevskaya type, especially for first-orderequations, and maybe this has already been done.
on landau damping 141
Proof. For notational simplicity, let us assume d=1. The left-hand side of (9.1) isâkâZd
nâNd0
e2ÏÎŒ(Ï)|k|2ÏÎŒ(Ï)|k|λn(Ï)n!
â(âv+2iÏkÏ)nf(k, v)âLp(dv)
+âkâZd
nâNd0
e2ÏÎŒ(Ï)|k|λ(Ï)λnâ1(Ï)(nâ1)!
â(âv+2iÏkÏ)n
f(k, v)âLp(dv)
ïżœ âKâkâZd
nâNd0
e2ÏÎŒ(Ï)|k|2Ï|k|λn(Ï)n!
â(âv+2iÏkÏ)nf(k, v)âLp(dv)
âKâkâZd
nâNd0
e2ÏÎŒ(Ï)|k|λn(Ï)n!
â(âv+2iÏkÏ)n+1
f(k, v)âLp(dv)
ïżœ âKâkâZd
nâNd0
e2ÏÎŒ(Ï)|k|λn(Ï)n!
â(âv+2iÏkÏ)nâxf(k, v)âLp(dv)
+KÏ
1+Ï
âkâZd
nâNd0
e2ÏÎŒ(Ï)|k|λn(Ï)n!
â(âv+2iÏkÏ)nâxf(k, v)
â„â„â„Lp(dv)
â K
1+Ï
âkâZd
nâNd0
e2ÏÎŒ(Ï)|k|λn(Ï)n!
â(âv+2iÏkÏ)nâvf(k, v)âLp(dv),
where in the last step we used that
â(âv+2iÏkÏ)hâïżœ ââvhââÏâ2iÏkhâ1+Ï
.
The conclusion follows.
Now let us see how to propagate estimates through the Newton scheme described in§10. The first stage of the iteration (h1 in the notation of (8.3)) was considered in §4.12,so we only need to care about higher orders. For any kïżœ1, we solve
âthk+1+v ·âxhk+1 = ÎŁk+1,
whereÎŁk+1 =â(F [hk+1]·âvfk+F [fk]·âvhk+1+F [hk]·âvhk)
(note the difference with (8.7)â(8.8)). Recall that fk=f0+h1+...+hk. We define
λk(t) =λkâ2Kt and ÎŒk(t) =ÎŒkâKt,
142 c. mouhot and c. villani
where (λk)âk=1 and (ÎŒk)âk=1 are decreasing sequences of positive numbers.We assume inductively that at stage n of the iteration, we have constructed (λk)n
k=1,(ÎŒk)n
k=1 and (ÎŽk)nk=1 such that
sup0ïżœtïżœT
âhk(t, ·)âZλk(t),ÎŒk(t);1t
ïżœ ÎŽk for all 1 ïżœ k ïżœn
for some fixed T >0. The issue is to construct λn+1, Όn+1 and Ύn+1 so that the inductionhypothesis is satisfied at stage n+1.
At t=0, hn+1=0. Then we estimate the time-derivative of âhn+1âZλn+1(t),ÎŒn+1(t);1t
.Let us first pretend that the regularity indices λn+1 and Όn+1 do not depend on t; then
hn+1(t) =â« t
0
ÎŁn+1 S0â(tâÏ) dÏ,
so, by Proposition 4.19,
âhn+1âZλn+1,ÎŒn+1;1t
ïżœâ« t
0
âÎŁn+1Ï S0
â(tâÏ)âZλn+1,ÎŒn+1;1t
dÏ
ïżœâ« t
0
âÎŁn+1Ï âZλn+1,ÎŒn+1;1
ÏdÏ,
and thusd+
dtâhn+1âZλn+1,ÎŒn+1;1
t
ïżœ âÎŁn+1t âZλn+1,ÎŒn+1;1
t
.
Finally, according to Lemma 9.1, to this estimate we should add a negative multiple ofthe norm of âhn+1 to take into account the time-dependence of λn+1 and ÎŒn+1.
All in all, after application of Proposition 4.24, we get
d+
dtâhn+1(t, ·)âZλn+1(t),ÎŒn+1(t);1
t
ïżœ âF [hn+1t ]âFλn+1t+ÎŒn+1 ââvfn
t âZλn+1,ÎŒn+1;1t
+âF [fnt ]âFλn+1t+ÎŒn+1 ââvhn+1
t âZλn+1,ÎŒn+1;1t
+âF [hnt ]âFλn+1t+ÎŒn+1 ââvhn
t âZλn+1,ÎŒn+1;1t
âKââxhn+1t âZλn+1,ÎŒn+1;1
t
âKââvhn+1t âZλn+1,ÎŒn+1;1
t
,
where K>0, t is sufficiently small, and all exponents λn+1 and Όn+1 in the right-handside actually depend on t.
From Proposition 4.15 (iv) we easily get âF [h]âFλt+ÎŒ ïżœCââhâZλ,ÎŒ;1t
. Moreover, byProposition 4.10,
ââfnâZλn+1,ÎŒn+1;1t
ïżœnâ
k=1
ââhkâZλn+1,ÎŒn+1;1t
ïżœC
nâk=1
âhkâZλk+1,ÎŒk+1;1t
min{λkâλn+1, ÎŒkâÎŒn+1} .
on landau damping 143
We end up with the bound
d+
dtâhn+1(t, ·)âZλn+1(t),ÎŒn+1(t);1
t
ïżœ(
C
nâk=1
ÎŽk
min{λkâλn+1, ÎŒkâÎŒn+1}âK
)ââhn+1âZλn+1(t),ÎŒn+1(t);1
t
+ÎŽ2n
min{λnâλn+1, ÎŒnâÎŒn+1} .
We conclude that if
nâk=1
ÎŽk
min{λkâλn+1, ÎŒkâÎŒn+1} ïżœ K
C, (9.2)
then we may choose
ÎŽn+1 =ÎŽ2n
min{λnâλn+1, ÎŒnâÎŒn+1} . (9.3)
This is our first encounter with the principle of âstratificationâ of errors, which will becrucial in the next section: to control the error at stage n+1, we use not only the smallnessof the error from stage n, but also an information about all previous errors; namely thefact that the convergence of the size of the error is much faster than the convergence of theregularity loss. Let us see how this works. We choose λkâλk+1=ÎŒkâÎŒk+1=Î/k2, whereÎ>0 is arbitrarily small. Then for kïżœn, λkâλn+1ïżœÎ/k2, and therefore ÎŽn+1ïżœÎŽ2
nn2/Î.The problem is to check that
âân=1
n2ÎŽn <â. (9.4)
Indeed, then we can choose K large enough for (9.2) to be satisfied, and then T smallenough that, say, λââ2KT ïżœÎ»ïżœ and ÎŒââKT ïżœÎŒïżœ, where Î»ïżœ<λâ and ÎŒïżœ<ÎŒâ have beenfixed in advance.
If ÎŽ1=ÎŽ, the general term in the series of (9.4) is
n2 ÎŽ2n
În(22)2
nâ1(32)2
nâ2(42)2
nâ2... ((nâ1)2)2n2.
To prove the convergence for ÎŽ small enough, we assume by induction that ÎŽnïżœzan
, wherea is fixed in the interval (1, 2) (say a=1.5); and we claim that this condition propagatesif z>0 is small enough. Indeed,
ÎŽn+1 ïżœ z2an
În2 ïżœ zan+1 z(2âa)an
n2
Î,
144 c. mouhot and c. villani
and this is bounded above by zan+1if z is so small that
z(2âa)an ïżœ În2
for all nâN.
This concludes the iteration argument. Note that the convergence is still extremelyfastâlike O(zan
) for any a<2. (Of course, when a approaches 2, the constants be-come huge, and the restriction on the size of the perturbation becomes more and morestringent.)
Remark 9.3. The method used in this section can certainly be applied to more gen-eral situations of CauchyâKovalevskaya type. Actually, as pointed out to us by Bonyand Gerard, the use of a regularity index which decays linearly in time, combined with aNewton iteration, was used by Nirenberg [73] to prove an abstract CauchyâKovalevskayatheorem. Nirenberg uses a time-integral formulation, so there is nothing in [73] compa-rable to Lemma 9.1, and the details of the proof of convergence differ from ours; but thegeneral strategy is similar. Nirenbergâs proof was later simplified by Nishida [74] with aclever fixed-point argument; in the present section anyway, our final goal is to provideshort-term estimates for the successive corrections arising from the Newton scheme.
10. Global in time iteration
Now let us implement the scheme described in §8, with some technical modifications. Iff is a given kinetic distribution, we write
ïżœ[f ] =â«
Rd
f dv and F [f ] =ââW âïżœ[f ].
We letfn = f0+h1+...+hn, (10.1)
where the successive corrections hk are defined by the natural Newton scheme introducedin §8. As in §5, we define Ωk
t,Ï as the deflection from time t to time Ï , generated by theforce field F [fk]=ââW âïżœ[fk]. (Note that Ω0=Id.)
10.1. The statement of the induction
We shall fix pâ[1,â] and make the following assumptions:âą Regularity of the background : there are λ>0 and C0>0 such that
âf0âCλ;p ïżœC0 for all pâ [1, p ].
on landau damping 145
⹠Linear damping condition: The stability condition (L) from §2.2 holds with pa-rameters C0, λ (the same as above) and >0.
âą Regularity of the interaction: There are Îł>1 and CF >0 such that for any Îœ>0,
ââW âïżœâFÎœ,Îł ïżœCF âïżœâFÎœ . (10.2)
âą Initial layer of regularity (coming from §9): having chosen Î»ïżœ<λ and ÎŒïżœ<ÎŒ, weassume that for all pâ[1, p ],
sup0ïżœtïżœT
(âhkt âZλ,ÎŒ;p +âïżœ[hk
t ]âFÎŒ ) ïżœ ζk for all k ïżœ 1, (10.3)
where T is some positive time, and ζk converges to zero extremely fast: ζk=O(zakI
I ),zI ïżœCÎŽ<1, 1<aI <2 (aI chosen in advance, arbitrarily close to 2).
âą Smallness of the solution of the linearized equation (coming from §4.12): givenλ1<Î»ïżœ and ÎŒ1<ÎŒïżœ, we assume thatâ§âȘâšâȘâ©
supÏïżœ0
âïżœ[h1Ï ]âFλ1Ï+ÎŒ1 ïżœ ÎŽ1,
sup0ïżœÏïżœt
âh1ÏâZλ1(1+b),ÎŒ1;p
Ïâbt/(1+b)ïżœ ÎŽ1 for all pâ [1, p ],
(10.4)
where ÎŽ1ïżœCÎŽ.Then we prove the following induction for any nïżœ1,â§âȘâšâȘâ©
supÏïżœ0
âïżœ[hkÏ ]âFλkÏ+ÎŒk ïżœ ÎŽk,
sup0ïżœÏïżœt
âhkÏ Î©kâ1
t,Ï âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk,for all kâ{1, ..., n} and pâ [1, p ], (10.5)
whereâą (ÎŽk)âk=1 is a sequence satisfying 0<CF ζkïżœÎŽk, and ÎŽk=O(zak
), z<zI , 1<a<aI (aarbitrarily close to aI);
âą (λk, ÎŒk) are decreasing to (λâ, ÎŒâ), where (λâ, ÎŒâ) are arbitrarily close to(λ1, ÎŒ1); in particular we impose
Î»ïżœâλâ ïżœ min{1, 1
2λâ}
and ÎŒïżœâÎŒâ ïżœ min{1, 1
2ÎŒâ}; (10.6)
âą T is some small positive time in (10.3); we impose
λ#T ïżœ 12 (ÎŒïżœâÎŒ1); (10.7)
âą b=b(t)=B
1+t, where Bâ(0, T ) is a (small) constant.
146 c. mouhot and c. villani
10.2. Preparatory remarks
As announced in (10.5), we shall propagate the following âprimaryâ controls on thedensity and distribution:
supÏïżœ0
âïżœ[hkÏ ]âFλkÏ+ÎŒk ïżœ ÎŽk for all kâ{1, ..., n} (En
ïżœ )
and
sup0ïżœÏïżœt
âhkÏ Î©kâ1
t,Ï âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk for all kâ{1, ..., n} and pâ [1, p ]. (Enh)
Estimate (Enïżœ ) obviously implies, via (10.2), up to a multiplicative constant,
supÏïżœ0
âF [hkÏ ]âFλkÏ+ÎŒk,Îł ïżœ ÎŽk for all kâ{1, ..., n}. (En
ïżœ )
Before we can go from there to stage n+1, we need an additional set of estimateson the deflection maps (Ωk)n
k=1, which will be used to(1) update the control on Ωk
t,Ï âId;(2) establish the needed control along the characteristics for the background
(âvfnÏ ) Ωn
t,Ï
(same index for the distribution and the deflection);(3) update some technical controls allowing us to exchange (asymptotically) gradient
and composition by Ωkt,Ï ; this will be crucial to handle the contribution of the zero mode
of the background after composition by characteristics.This set of deflection estimates falls into three categories. The first group expresses
the closeness of Ωk to Id:â§âȘâšâȘâ©sup
0ïżœÏïżœtâΩkXt,Ï âId âZλâ
k(1+b),(ÎŒâ
k,Îł)
Ïâbt/(1+b)
ïżœ 2Rk2(Ï, t),
sup0ïżœÏïżœt
âΩkVt,Ï âId âZλâk(1+b),(ÎŒâ
k,Îł)
Ïâbt/(1+b)
ïżœRk1(Ï, t),
for all kâ{1, ..., n}, (EnΩ)
with λk>λâk>λk+1, ÎŒk>ÎŒâ
k>ÎŒk+1 and
â§âȘâȘâȘâȘâȘâšâȘâȘâȘâȘâȘâ©Rk
1(Ï, t) =( kâ
j=1
ÎŽjeâ2Ï(λjâλâ
j )Ï
2Ï(λjâλâj )
)min{tâÏ, 1}
Rk2(Ï, t) =
( kâj=1
ÎŽjeâ2Ï(λjâλâ
j )Ï
(2Ï(λjâλâj ))2
)min{
(tâÏ)2
2, 1}
.
(10.8)
on landau damping 147
The second group of estimates expresses the fact that ΩnâΩk is very small when k
is large:â§âȘâȘâȘâȘâšâȘâȘâȘâȘâ©sup
0ïżœÏïżœtâΩnXt,Ï âΩkXt,ÏâZλâ
n(1+b),(ÎŒân,Îł)
Ïâbt/(1+b)ïżœ 2Rk,n
2 (Ï, t),
sup0ïżœÏïżœt
âΩnVt,Ï âΩkVt,ÏâZλân(1+b),(ÎŒâ
n,Îł)Ïâbt/(1+b)
ïżœRk,n1 (Ï, t)+Rk,n
2 (Ï, t),
sup0ïżœÏïżœt
â(Ωkt,Ï )â1 Ωn
t,Ï âId âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ 4(Rk,n1 (Ï, t)+Rk,n
2 (Ï, t)),
for all kâ{0, ..., nâ1},
(EnΩ)
with â§âȘâȘâȘâȘâȘâšâȘâȘâȘâȘâȘâ©Rk,n
1 (Ï, t) =( nâ
j=k+1
ÎŽjeâ2Ï(λjâλâ
j )Ï
2Ï(λjâλâj )
)min{tâÏ, 1}
Rk,n2 (Ï, t) =
( nâj=k+1
ÎŽjeâ2Ï(λjâλâ
j )Ï
(2Ï(λjâλâj ))2
)min{
(tâÏ)2
2, 1}
.
(10.9)
(Choosing k=0 brings us back to the previous estimates (EnΩ).)
The last group of estimates expresses the fact that the differential of the deflectionmap is uniformly close to the identity (in a way which is more precise than what wouldfollow from the first group of estimates):â§âȘâšâȘâ©
sup0ïżœÏïżœt
ââΩkXt,Ï â(I, 0)âZλâk(1+b),ÎŒâ
kÏâbt/(1+b)
ïżœ 2Rk2(Ï, t),
sup0ïżœÏïżœt
ââΩkVt,Ï â(0, I)âZλâk(1+b),ÎŒâ
kÏâbt/(1+b)
ïżœRk1(Ï, t)+Rk
2(Ï, t),
for all kâ{1, ..., n},
(EnâΩ)
where â=(âx,âv), and I is the identity matrix.An important property of the functions Rk,n
1 (Ï, t) and Rk,n2 (Ï, t) is their fast decay
as Ï!â and as k!â, uniformly in nïżœk; this is due to the fast convergence of thesequence (ÎŽk)âk=1. Eventually, if râN is given, we shall have
Rk,n1 (Ï, t) ïżœÏr,1
k,n(Ï, t) and Rk,n2 (Ï, t) ïżœÏr,2
k,n(Ï, t) for all r ïżœ 1, (10.10)
with
Ïr,1k,n(Ï, t) := Cr
Ï
( nâj=k+1
ÎŽj
(2Ï(λjâλâj ))1+r
)min{tâÏ, 1}
(1+Ï)r
and
Ïr,2k,n(Ï, t) := Cr
Ï
( nâj=k+1
ÎŽj
(2Ï(λjâλâj ))2+r
)min{
12 (tâÏ)2, 1
}(1+Ï)r
for some absolute constant CrÏ depending only on r (we also denote Ïr,1
0,n=Ïr,1n and
Ïr,20,n=Ïr,2
n ).
148 c. mouhot and c. villani
From the estimates on the characteristics and (Enh) will follow the following âsec-
ondary controlsâ on the distribution function:â§âȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâšâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâ©
sup0ïżœÏïżœt
â(âxhkÏ ) Ωkâ1
t,Ï âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk,
sup0ïżœÏïżœt
ââx(hkÏ Î©kâ1
t,Ï )âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk,
â((âv+Ïâx)hkÏ ) Ωkâ1
t,Ï âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk,
â(âv+Ïâx)(hkÏ Î©kâ1
t,Ï )âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk,
sup0ïżœÏïżœt
1(1+Ï)2
â(ââhkÏ ) Ωkâ1
t,Ï âZλk(1+b),ÎŒk;1Ïâbt/(1+b)
ïżœ ÎŽk,
sup0ïżœÏïżœt
(1+Ï)2â(âhkÏ ) Ωkâ1
t,Ï ââ(hkÏ Î©kâ1
t,Ï )âZλk(1+b),ÎŒk;1Ïâbt/(1+b)
ïżœ ÎŽk.
for all kâ{1, ..., n} and pâ [1, p ].
(Enh)
The transition from stage n to stage n+1 can be summarized as follows:
(Enïżœ )
(An)=â (En
Ω)+(EnΩ)+(En
âΩ),
(Enïżœ )+(En
Ω)+(EnΩ)+(En
âΩ)+(Enh)+(En
h)(Bn)=â (En+1
ïżœ )+(En+1ïżœ )+(En+1
h )+(En+1h ).
The first implication (An) is proven by an amplification of the technique used in §5;ultimately, it relies on repeated application of Picardâs fixed-point theorem in analyticnorms. The second implication (Bn) is the harder part; it uses the machinery from §6and §7, together with the idea of simultaneously propagating a shifted Z norm for thekinetic distribution and an F norm for the density.
In both implications, the stratification of error estimates will prevent the blow upof constants. So we shall decompose the force field Fn generated by fn as
Fn = F [fn] = E1+...+En,
where Ek=F [hk]=ââW âïżœ[hk].The plan of the estimates is as follows. We shall inductively construct a sequence of
constant coefficients
Î»ïżœ >λ1 >λâ1 >λ2 > ...> λn >λâ
n >λn+1 > ...,
ÎŒïżœ >ÎŒ1 >ÎŒâ1 >ÎŒ2 > ...> ÎŒn >ÎŒâ
n >ÎŒn+1 > ...
(where λn and ÎŒn will be fixed in the proof of (An), and λn+1 and ÎŒn+1 in the proofof (Bn)) converging respectively to λâ and ÎŒâ; and a sequence (ÎŽk)âk=1 decreasing veryfast to zero. For simplicity we shall let
Rn(Ï, t) =Rn1 (Ï, t)+Rn
2 (Ï, t) and Rk,n(Ï, t) =Rk,n1 (Ï, t)+Rk,n
2 (Ï, t),
on landau damping 149
and assume that 2Ï(λjâλâj )ïżœ1; so
Rk,n(Ï, t) ïżœ CrÏ
( nâj=k+1
ÎŽj
(2Ï(λjâλâj ))2+r
)min{tâÏ, 1}
(1+Ï)rand R0,n =Rn. (10.11)
It will be sufficient to work with some fixed r, large enough (as we shall see, r=4 will do).To go from stage n to stage n+1, we shall do as follows:
Implication (An) (§10.3):Step 1. Estimate ΩnâId (the bound should be uniform in n).Step 2. Estimate ΩnâΩk (kïżœnâ1; the error should be small when k!â).Step 3. Estimate âΩnâI.Step 4. Estimate (Ωk)â1 Ωn.
Implication (Bn) (§10.4):Step 5. Estimate hk and its derivatives along the composition by Ωn.Step 6. Estimate ïżœ[hn+1], using §6 and §7.Step 7. Estimate F [hn+1] from ïżœ[hn+1].Step 8. Estimate hn+1 Ωn.Step 9. Estimate derivatives of hn+1 composed with Ωn.Step 10. Show that for hn+1, â and composition by Ωn asymptotically commute.
10.3. Estimates on the characteristics
In this subsection, we assume that (Enïżœ ) is proven, and establish (En
Ω)+(EnΩ)+(En
âΩ).Let λâ
n<λn and ÎŒân<ÎŒn to be fixed later on.
10.3.1. Step 1. Estimate of ΩnâId
This is the first and archetypal estimate. We shall bound ΩnXt,Ï âx in the hybrid normZλâ
n(1+b),ÎŒân
Ïâbt/(1+b) . The Sobolev correction Îł will play no role here in the proofs, and forsimplicity we shall forget it in the computations, just recall it in the final results. (UseProposition 4.32 whenever needed.)
Since we expect the characteristics for the force field Fn to be close to the freetransport characteristics, it is natural to write
Xnt,Ï (x, v) =xâv(tâÏ)+Zn
t,Ï (x, v), (10.12)
where Znt,Ï solves â§âšâ©
â2
âÏ2Zn
t,Ï (x, v) =Fn(Ï, xâv(tâÏ)+Znt,Ï (x, v))
Znt,t(x, v) = 0, âÏZn
t,Ï |t=Ï (x, v) = 0.
(10.13)
150 c. mouhot and c. villani
(With respect to §5 we have dropped the parameter Δ, to take advantage of the âstratifiedânature of Fn; anyway this parameter was cosmetic.) So if we fix t>0, (Zn
t,Ï )0ïżœÏïżœt is afixed point of the map
Κ: (Wt,Ï )0ïżœÏïżœt â! (Zt,Ï )0ïżœÏïżœt
defined by â§âšâ©â2
âÏ2Zt,Ï = Fn(Ï, xâv(tâÏ)+Wt,Ï )
Zt,t = 0, âÏZt,Ï |Ï=t = 0.
(10.14)
The goal is to estimate Znt,Ï âx in the hybrid norm Zλâ
n(1+b),ÎŒân
tâbt/(1+b) .We first bound (Zn
0 )t,Ï =Κ(0). Explicitly,
(Zn0 )t,Ï (x, v) =
â« t
Ï
(sâÏ) Fn(s, xâv(tâs)) ds.
By Propositions 4.15 (i) and 4.19,
â(Zn0 )t,ÏâZλâ
n(1+b),ÎŒân
tâbt/(1+b)ïżœâ« t
Ï
(sâÏ)âFn(s, xâv(tâs))âZλân(1+b),ÎŒâ
ntâbt/(1+b)
ds
=â« t
Ï
(sâÏ)âFn(s, ·)âZλân(1+b),ÎŒâ
nsâbt/(1+b)
ds
=â« t
Ï
(sâÏ)âFn(s, ·)âFÎœ(s,t) ds,
(10.15)
where
Îœ(s, t) =λân|sâb(tâs)|+ÎŒâ
n. (10.16)
Case 1. If sïżœbt/(1+b), then
Îœ(s, t) ïżœÎ»âns+ÎŒâ
n ïżœÎ»k s+ÎŒkâ(λkâλân)s, kâ{1, ..., n}. (10.17)
Case 2. If s<bt/(1+b), then necessarily sïżœBïżœT . Taking into account (10.6), wehave
Îœ(s, t) =λân bt+ÎŒâ
nâλân(1+b)s (10.18)
ïżœÎ»ânB+ÎŒâ
nâ(λkâλân)s. (10.19)
(Of course, the assumption Î»ïżœâλâïżœmin{1, 1
2λâ}
implies that λkâλânïżœÎ»â
n.) In partic-ular, by (10.7),
Îœ(s, t) ïżœÎŒïżœâ(λkâλân)s, kâ{1, ..., n}. (10.20)
on landau damping 151
We plug these bounds into (10.15), then use Ek(s, 0)=0 and the bounds (Enïżœ ) and
(10.17) (for large times), and (10.3) and (10.20) (for short times). This yields
â(Zn0 )t,ÏâZλâ
n(1+b),ÎŒân
tâbt/(1+b)
ïżœnâ
k=1
(â« t
Ïâšbt/(1+b)
(sâÏ)âEk(s, ·)âFλks+ÎŒkâ(λkâλân)s ds
+â« Ïâšbt/(1+b)
Ï
(sâÏ)âEk(s, ·)âFÎŒâ(λkâλân)s ds
)ïżœ
nâk=1
(â« t
Ïâšbt/(1+b)
(sâÏ)eâ2Ï(λkâλân)sâEk(s, ·)âFλks+ÎŒk ds
+â« Ïâšbt/(1+b)
Ï
(sâÏ)eâ2Ï(λkâλân)sâEk(s, ·)âFÎŒ ds
)ïżœ
nâk=1
ÎŽk
â« t
Ï
(sâÏ)eâ2Ï(λkâλân)s ds
ïżœnâ
k=1
ÎŽkeâ2Ï(λkâλân)Ï min
{(tâÏ)2
2,
1(2Ï(λkâλâ
n))2
}ïżœRn
2 (Ï, t).
(10.21)
Let us define the norm
||||(Zt,Ï )0ïżœÏïżœt||||n := sup0ïżœÏïżœt
âZt,ÏâZλân(1+b),ÎŒâ
ntâbt/(1+b)
Rn2 (Ï, t)
.
(Note the difference with §5: now the regularity exponents depend on time(s).) Inequality(10.21) means that ||||Κ(0)||||nïżœ1. We shall check that Κ is 1
2 -Lipschitz on the ball B(0, 2)in the norm |||| · ||||n. This will be subtle: the uniform bounds on the size of the force field,coming from the preceding steps, will allow us to get good decaying exponentials, whichin turn will imply uniform error bounds at the present stage.
So let W, WâB(0, 2), and let Z=Κ(W ) and Z=Κ(W ). As in §5, we write
Zt,Ï âZt,Ï =â« 1
0
â« t
Ï
(sâÏ)âxFn(s, xâv(tâs)+ΞWt,s+(1âΞ)Wt,s)·(Wt,sâWt,s) ds dΞ,
and deduce that
||||(Zt,Ï âZt,Ï )0ïżœÏïżœt||||n ïżœA(t)||||(Wt,sâWt,s)0ïżœsïżœt||||n,
where
A(t) = sup0ïżœÏïżœsïżœt
Rn2 (s, t)
Rn2 (Ï, t)
Ăâ« 1
0
â« t
Ï
(sâÏ)ââxFn(s, xâv(tâs)+ΞWt,s+(1âΞ)Wt,s)âZλân(1+b),ÎŒâ
ntâbt/(1+b)
ds dΞ.
152 c. mouhot and c. villani
For Ï ïżœs we have Rn2 (s, t)ïżœRn
2 (Ï, t). Also, by Proposition 4.25 (applied with V =0,b=â(tâs) and Ï=0 in that statement) and Proposition 4.15,
A(t) ïżœ sup0ïżœÏïżœt
â« t
Ï
(sâÏ)ââxFn(s, ·)âFÎœ(s,t)+e(s,t) ds,
where Îœ is defined by (10.16) and the âerrorâ e(s, t) arising from composition is given by
e(s, t) = sup0ïżœÎžïżœ1
âΞWt,s+(1âΞ)Wt,sâZλân(1+b),ÎŒâ
ntâbt/(1+b)
ïżœ 2Rn2 (s, t).
Since
Rn2 (s, t) ïżœÏ1,2
n (s, t) := C1Ï
( nâk=1
ÎŽk
(2Ï(λkâλâk))3
)min{
12 (tâs)2, 1
}1+s
,
we have, for all 0ïżœsïżœt,
2Rn2 (s, t) ïżœ λâ
nb(tâs)2
1sïżœbt/(1+b)+ÎŒïżœâÎŒâ
n
21sïżœbt/(1+b), (10.22)
as soon as
2C1Ï
( nâk=1
ÎŽk
(2Ï(λkâλân))3
)ïżœ min
{λâ
nB
6,ÎŒïżœâÎŒâ
n
2
}for all n ïżœ 1. (C1)
We shall check later in §10.5 the feasibility of condition (C1)âas well as a number ofother forthcoming ones.
The extra error term in the exponent is sufficiently small to be absorbed by whatwe throw away in (10.17) or in (10.18)â(10.20). So we obtain, as in the estimate of Zn
0 ,for any kâ{1, ..., n},
(Îœ+e)(s, t) ïżœ{
λks+ÎŒkâ(λkâλân)s for sïżœ bt/(1+b),
ÎŒïżœâ(λkâλân)s for sïżœ bt/(1+b),
and we deduce (using (Enïżœ ) and Îłïżœ1) that
A(t) ïżœ sup0ïżœÏïżœt
nâk=1
(â« t
Ïâšbt/(1+b)
(sâÏ)ââxEk(s, ·)âFλks+ÎŒkâ(λkâλân) s ds
+â« Ïâšbt/(1+b)
Ï
(sâÏ)ââxEk(s, ·)âFÎŒâ(λkâλân)s ds
)ïżœ sup
0ïżœÏïżœt
nâk=1
ÎŽk
â« t
Ï
(sâÏ)eâ(λkâλân)s ds
ïżœ sup0ïżœÏïżœt
Rn2 (Ï, t) =Rn
2 (0, t)
ïżœnâ
k=1
ÎŽk
(2Ï(λkâλân))2
.
on landau damping 153
If the latter quantity is bounded above by 12 , then Κ is 1
2 -Lipschitz and we may apply thefixed-point result from Theorem A.2. Therefore, under the condition (whose feasibilitywill be checked later)
nâk=1
ÎŽk
(2Ï(λkâλân))2
ïżœ 12
for all n ïżœ 1, (C2)
we deduce thatâZn
t,ÏâZλân(1+b),ÎŒâ
ntâbt/(1+b)
ïżœ 2Rn2 (Ï, t).
After that, the estimates on the deflection map are obtained exactly as in §5: writingΩn
t,Ï =(ΩnXt,Ï , ΩnVt,Ï ), recalling the dependence on Îł again, we end up with
{ âΩnXt,Ï âxâZλân(1+b),(ÎŒâ
n,Îł)Ïâbt/(1+b)
ïżœ 2Rn2 (Ï, t),
âΩnVt,Ï âvâZλân(1+b),(ÎŒâ
n,Îł)Ïâbt/(1+b)
ïżœRn1 (Ï, t).
(10.23)
10.3.2. Step 2. Estimate of ΩnâΩk
In this step our goal is to estimate ΩnâΩk for 1ïżœkïżœnâ1. The point is that the errorshould be small as k!â, uniformly in n, so we cannot just write
âΩnâΩkâïżœ âΩnâId â+âΩkâId â.
Instead, we start again from the differential equation satisfied by Zk and Zn:
â2
âÏ2(Zn
t,Ï âZkt,Ï )(x, v) =Fn(Ï, xâv(tâÏ)+Zn
t,Ï (x, v))âF k(Ï, xâv(tâÏ)+Zkt,Ï (x, v))
= Fn(Ï, xâv(tâÏ)+Znt,Ï )âFn(Ï, xâv(tâÏ)+Zk
t,Ï )
+(FnâF k)(Ï, xâv(tâÏ)+Zkt,Ï ).
This, together with the boundary conditions Znt,tâZk
t,t=0 and âÏ (Znt,Ï âZk
t,Ï )|Ï=t=0, im-plies that
Znt,Ï âZk
t,Ï =â« 1
0
â« t
Ï
(sâÏ)âxFn(s, xâv(tâs)+ΞZkt,s+(1âΞ)Zn
t,s)·(Znt,sâZk
t,s) ds dΞ
+â« t
Ï
(sâÏ)(FnâF k)(s, xâv(tâs)+Zkt,s(x, v)) ds.
We fix t and define the norm
||||(Zt,Ï )0ïżœÏïżœt||||k,n := sup0ïżœÏïżœt
âZt,ÏâZλân(1+b),ÎŒâ
ntâbt/(1+b)
Rk,n2 (Ï, t)
,
154 c. mouhot and c. villani
where Rk,n2 is defined in (10.9). Using the bounds on Zn and Zk in |||| · ||||n (since
|||| · ||||nïżœ|||| · ||||k by using the fact that Rk2ïżœRn
2 ) and proceeding as before, we get that
||||(Znt,Ï âZk
t,Ï )0ïżœÏïżœt||||k,n
ïżœ 12 ||||(Zn
t,Ï âZkt,Ï )0ïżœÏïżœt||||k,n
+âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁ(â« t
Ï
(sâÏ)(FnâF k)(s, xâv(tâs)+Zkt,s) ds
)0ïżœÏïżœt
âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁk,n
.
(10.24)
Next we estimate
â(FnâF k)(s, xâv(tâs)+Zkt,s)âZλâ
n(1+b),ÎŒân
tâbt/(1+b)= â(FnâF k)(s,Xk
t,s)âZλân(1+b),ÎŒâ
ntâbt/(1+b)
= â(FnâF k)(s,Ωkt,s)âZλâ
n(1+b),ÎŒân
sâbt/(1+b)
ïżœ â(FnâF k)(s, ·)âFÎœ(s,t)+e(s,t) ,
where the last inequality follows from Proposition 4.25, Μ is again given by (10.16), and
e(s, t) = âΩkXt,sâId âZλân(1+b),ÎŒâ
nsâbt/(1+b)
ïżœ 2Rk2(s, t) ïżœ 2Rn
2 (s, t).
The same reasoning as in Step 1 yields, under assumptions (C1) and (C2), fork+1ïżœjïżœn,
(Îœ+e)(s, t) ïżœ{
λjs+ÎŒjâ(λjâλân)s for sïżœ bt/(1+b),
ÎŒïżœâ(λjâλân)s for sïżœ bt/(1+b),
and so
âFns âF k
s âFÎœ+e ïżœnâ
j=k+1
ÎŽjeâ2Ï(λjâλâ
n)s.
For any Ï ïżœ0, by integrating in time we find thatâ„â„â„â„â« t
Ï
(sâÏ)(FnâF k)(s, xâv(tâs)+Zkt,s) ds
â„â„â„â„Zλâ
n(1+b),ÎŒân
tâbt/(1+b)
ïżœâ« t
Ï
(sâÏ)nâ
j=k+1
ÎŽjeâ2Ï(λjâλâ
n)s ds ïżœRk,n2 (Ï, t).
Therefore âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁ(â« t
Ï
(sâÏ)(FnâF k)(s, xâv(tâs)+Zkt,s) ds
)0ïżœÏïżœt
âŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁâŁk,n
ïżœ 1
and, by(10.24),||||(Zn
t,Ï âZkt,Ï )0ïżœÏïżœt||||k,n ïżœ 2.
on landau damping 155
Recalling the Sobolev correction, we conclude that
âΩnXt,Ï âΩkXt,ÏâZλân(1+b),(ÎŒâ
n,Îł)Ïâbt/(1+b)
ïżœ 2Rk,n2 (Ï, t). (10.25)
For the velocity component, say U , we write
â
âÏ(Un
t,Ï âUkt,Ï )(x, v) =Fn(Ï, xâv(tâÏ)+Zn
t,Ï (x, v))âF k(Ï, xâv(tâÏ)+Zkt,Ï (x, v))
= Fn(Ï, xâv(tâÏ)+Znt,Ï )âFn(Ï, xâv(tâÏ)+Zk
t,Ï )
+(FnâF k)(Ï, xâv(tâÏ)+Zkt,Ï ),
where Zn and Zk were estimated above, and the boundary conditions are Unt,tâUk
t,t=0.Thus
Unt,Ï âUk
t,Ï =â« 1
0
â« t
Ï
âxFn(s, xâv(tâs)+ΞZkt,s+(1âΞ)Zn
t,s)·(Znt,sâZk
t,s) ds dΞ
+â« t
Ï
(FnâF k)(s, xâv(tâs)+Zkt,s(x, v)) ds,
and from this one easily derives the similar estimates{ âΩnXt,Ï âΩkXt,ÏâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ 2Rk,n2 (t, Ï),
âΩnVt,Ï âΩkVt,ÏâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœRk,n1 (t, Ï)+Rk,n
2 (t, Ï).
10.3.3. Step 3. Estimate of âΩn
We now establish a control on the derivative of the deflection map. Of course, we coulddeduce such a control from the bound on ΩnâId and Proposition 4.32 (vi): for instance,if λââ
n <λân and ÎŒââ
n <ÎŒân, then
ââΩnt,Ï âIâZλââ
n (1+b),(ÎŒâân ,Îł)
Ïâbt/(1+b)ïżœ CRn
2 (Ï, t)min{λâ
nâλâân , ÎŒâ
nâÎŒâân } . (10.26)
But this bound involves very large constants, and is useless in our argument. Betterestimates can be obtained by using again equation (10.13). Writing
(Ωnt,Ï âId)(x, v) = (Zn
t,Ï (x+v(tâÏ), v), Znt,Ï (x+v(tâÏ), v)),
where the dot stands for â/âÏ , we get by differentiation
âxΩnt,Ï â(I, 0) = (âxZn
t,Ï (x+v(tâÏ), v),âxZn
t,Ï (x+v(tâÏ), v)),
âvΩnt,Ï â(0, I) = ((âv+(tâÏ)âx)Zn
t,Ï (x+v(tâÏ), v),
(âv+(tâÏ)âx)Znt,Ï (x+v(tâÏ), v)).
156 c. mouhot and c. villani
Let us estimate for instance âxΩâ(I, 0), or equivalently âxZnt,Ï . By differentiating
(10.13), we obtain
â2
âÏ2âxZn
t,Ï (x, v) =âxFn(Ï, xâv(tâÏ)+Znt,Ï (x, v))·(Id +âxZn
t,Ï ).
So âxZnt,Ï is a fixed point of Κ:W !Q, where W and Q are functions of Ïâ[0, t] satisfyingâ§âšâ©
â2Q
âÏ2=âxFn(Ï, xâv(tâÏ)+Zn
t,Ï )(I+W ),
Q(t) = 0, âÏQ(t) = 0.
We treat this in the same way as in Steps 1 and 2, and find for Qx (the x componentof Q) the same estimates as we had previously on the x component of Ω. For thevelocity component, a direct estimate from the integral equation expressing the velocityin terms of F yields a control by Rn
1 +Rn2 . Finally for âvΩ this is similar, noting that
(âv+(tâÏ)âx)(xâv(tâÏ))=0, the differential equation being for instance:
â2
âÏ2(âv+(tâÏ)âx)Zn
t,Ï (x, v) =âxFn(Ï, xâv(tâÏ)+Znt,Ï (x, v))·((âv+(tâÏ)âx)Zn
t,Ï ).
In the end we obtainâ§âȘâšâȘâ©sup
0ïżœÏïżœtââΩnXt,Ï â(I, 0)âZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)ïżœ 2Rn
2 (Ï, t),
sup0ïżœÏïżœt
ââΩnVt,Ï â(0, I)âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœRn1 (Ï, t)+Rn
2 (Ï, t).(10.27)
10.3.4. Step 4. Estimate of (Ωk)â1 Ωn
We do this by applying Proposition 4.28 with F =Ωk and G=Ωn. (Note that we cannotexchange the roles of Ωk and Ωn in this step, because we have better information on theregularity of Ωk.) Let Δ=Δ(d) be the small constant appearing in Proposition 4.28. If
3Rk2(Ï, t)+Rk
1(Ï, t) ïżœ Δ for all k ïżœ 1, (C3)
then ââΩkt,Ï âIâZλâ
k(1+b),ÎŒâ
kÏâbt/(1+b)
ïżœÎ”; if in addition
2(1+Ï)(1+B)(3Rk,n2 +Rk,n
1 )(Ï, t) ïżœ max{λâkâλâ
n, ÎŒâkâÎŒâ
n}for all kâ{1, ..., nâ1} and all t ïżœ Ï ,
(C4)
then â§âȘâȘâšâȘâȘâ©Î»â
n(1+b)+2âΩnâΩkâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœÎ»âk(1+b),
ÎŒân+2(
1+âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ)âΩnâΩkâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœÎŒâk.
on landau damping 157
(Once again, short times should be treated separately. Further note that the need forthe factor 1+Ï in (C4) ultimately comes from the fact that we are composing also in thev variable, see the coefficient Ï in the last norm of (4.30).) Then Proposition 4.28 (ii)yields
â(Ωkt,Ï )â1 Ωn
t,Ï âId âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ 2âΩkt,Ï âΩn
t,ÏâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ 4(Rk,n1 +Rk,n
2 )(Ï, t).
10.3.5. Partial conclusion
At this point we have established (EnΩ)+(En
Ω)+(EnâΩ).
10.4. Estimates on the density and distribution along characteristics
In this subsection we establish (En+1ïżœ )+(En+1
ïżœ )+(En+1h )+(En+1
h ).
10.4.1. Step 5. Estimate of hk Ωn and (âhk) Ωn, k n.
Let kâ{1, ..., n}. Since
hkÏ Î©n
t,Ï = (hkÏ Î©kâ1
t,Ï ) ((Ωkâ1t,Ï )â1 Ωn
t,Ï ),
the control on hk Ωn will follow from the control on hk Ωkâ1 in (Enh), together with the
control on (Ωkâ1)â1 Ωn in (EnΩ). If
(1+Ï)â(Ωkâ1t,Ï )â1 Ωn
t,Ï âId âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ min{λkâλân, ÎŒkâÎŒâ
n}, (10.28)
then we can apply Proposition 4.25 and get, for any pâ[1, p ], and tïżœÏ ïżœ0,
âhkÏ Î©n
t,ÏâZλân(1+b),ÎŒâ
n;pÏâbt/(1+b)
ïżœ âhkÏ Î©kâ1
t,Ï âZλk(1+b),ÎŒk;pÏâbt/(1+b)
ïżœ ÎŽk. (10.29)
In turn, (10.28) is satisfied if
4(1+Ï)(Rk,n1 (Ï, t)+Rk,n
2 (Ï, t)) ïżœ min{λkâλân, ÎŒkâÎŒâ
n}for all kâ{1, ..., n} and all Ï â [0, t];
(C5)
we shall check later the feasibility of this condition.Then, by the same argument, we also have
sup0ïżœÏïżœt
â(âxhkÏ ) Ωn
t,ÏâZλân(1+b),ÎŒâ
n;pÏâbt/(1+b)
+â((âv+Ïâx)hkÏ ) Ωn
t,ÏâZλân(1+b),ÎŒâ
n;pÏâbt/(1+b)
ïżœ ÎŽk
for all kâ{1, ..., n} and all pâ [1, p ].
158 c. mouhot and c. villani
10.4.2. Step 6. Estimate on ïżœ[hn+1]
This step is the first where we shall use the Vlasov equation. Starting from (8.4), weapply the method of characteristics to get, as in §8,
hn+1(t, Xn0,t(x, v), V n
0,t(x, v)) =â« t
0
ÎŁn+1(Ï,Xn0,Ï (x, v), V n
0,Ï (x, v)) dÏ, (10.30)
whereÎŁn+1 =â(F [hn+1]·âvfn+F [hn]·âvhn).
We compose this with (Xnt,0, V
nt,0) and apply (5.2) to get
hn+1(t, x, v) =â« t
0
ÎŁn+1(Ï,Xnt,Ï (x, v), V n
t,Ï (x, v)) dÏ,
and so, by integration in the v variable,
ïżœ[hn+1](t, x) =â« t
0
â«Rd
ÎŁn+1(Ï,Xnt,Ï (x, v), V n
t,Ï (x, v)) dv dÏ
=ââ« t
0
â«Rd
(Rn+1Ï,t ·Gn
Ï,t)(xâv(tâÏ), v) dv dÏ
ââ« t
0
â«Rd
(RnÏ,t ·Hn
Ï,t)(xâv(tâÏ), v) dv dÏ,
(10.31)
where (with a slight inconsistency in the notation){Rn+1
Ï,t = F [hn+1] Ωnt,Ï , Rn
Ï,t = F [hn] Ωnt,Ï ,
GnÏ,t = (âvfn) Ωn
t,Ï , HnÏ,t = (âvhn) Ωn
t,Ï .(10.32)
Since the free transport semigroup and Ωnt,Ï are measure-preserving, for all 0ïżœÏ ïżœt
we have â«Td
â«Rd
(Rn+1Ï,t ·Gn
Ï,t)(xâv(tâÏ), v) dv dx =â«
Td
â«Rd
Rn+1Ï,t ·Gn
Ï,t dv dx
=â«
Td
â«Rd
F [hn+1]·âvfn dv dx
=â«
Td
â«Rd
âv ·(F [hn+1]fn) dv dx
= 0,
and similarly â«Td
â«Rd
(RnÏ,t ·Hn
Ï,t)(xâv(tâÏ), v) dv dx = 0 for all 0ïżœ Ï ïżœ t.
on landau damping 159
This will allow us to apply the inequalities from §6.
Substep a. Let us first deal with the source term
Ïn,n(t, x) :=â« t
0
â«Rd
(RnÏ,t ·Hn
Ï,t)(xâv(tâÏ), v) dv dÏ. (10.33)
By Proposition 6.2,
âÏn,n(t, ·)âFλânt+ÎŒâ
n ïżœâ« t
0
âRnÏ,tâZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)âHn
Ï,tâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
dÏ. (10.34)
On the one hand, we have from Step 5 that
âHnÏ,tâZλâ
n(1+b),ÎŒân;1
Ïâbt/(1+b)ïżœ 2(1+Ï)ÎŽn.
On the other hand, under condition (C1), we may apply Proposition 4.25 (withÏ=0) to get
âRnÏ,tâZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)ïżœ âF [hn
Ï ]âFÎœn ,
where
Îœn(t, Ï) =ÎŒân+λâ
n(1+b)âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+âΩnXt,Ï âId âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœÎŒân+λâ
n(1+b)âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+2Rn2 (Ï, t).
Proceeding as in Step 1 (treating small times separately), we deduce that
âRnÏ,tâZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)ïżœ âF [hn
Ï ]âFÎœn ïżœ eâ2Ï(λnâλân)ÏâF [hn
Ï ]âF Îœn
ïżœCF eâ2Ï(λnâλân)Ïâïżœ[hn
Ï ]âF Îœn ïżœCF eâ2Ï(λnâλân)ÏÎŽn,
with
Îœn(Ï, t) :=
â§âȘâšâȘâ©ÎŒïżœ, when 0ïżœ Ï ïżœ bt
1+b,
λnÏ +ÎŒn, when Ï ïżœ bt
1+b.
(10.35)
(We have used the gradient structure of the force to convert (gliding) regularity intodecay.) Thusâ« t
0
âRnÏ,tâZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)âHn
Ï,tâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
dÏ ïżœ 2CF ÎŽ2n
â« t
0
eâ2Ï(λnâλân)Ï (1+Ï) dÏ
ïżœ 2CF ÎŽ2n
(Ï(λnâλân))2
.
(10.36)
160 c. mouhot and c. villani
(Note that this is the power 2 which is responsible for the very fast convergence of theNewton scheme.)
Substep b. Now let us handle the term
Ïn,n+1(t, x) :=â« t
0
â«Rd
(Rn+1Ï,t ·Gn
Ï,t)(xâv(tâÏ), v) dv dÏ. (10.37)
This is the focal point of all our analysis, because it is in this term that the self-consistentnature of the Vlasov equation appears. In particular, we will make crucial use of thetime-cheating trick to overcome the loss of regularity implied by composition; and alsothe other bilinear estimates (regularity extortion) from §6, as well as the time-responsestudy from §7. Particular care should be given to the zero spatial mode of Gn, which isassociated with instantaneous response (no echo). In the linearized equation we did notsee this problem because the contribution of the zero mode was vanishing!
We start by introducing
GnÏ,t =âvf0+
nâk=1
âv(hkÏ Î©kâ1
t,Ï ), (10.38)
and we decompose Ïn,n+1 as
Ïn,n+1 = Ïn,n+1+E+E , (10.39)
where
Ïn,n+1(t, x) =â« t
0
â«Rd
F [hn+1Ï ]·Gn
Ï,t(xâv(tâÏ), v) dv dÏ (10.40)
and the error terms E and E are defined by
E(t, x) =â« t
0
â«Rd
((F [hn+1Ï ] Ωn
t,Ï âF [hn+1Ï ])·Gn)(Ï, xâv(tâÏ), v) dv dÏ, (10.41)
E(t, x) =â« t
0
â«Rd
(F [hn+1Ï ]·(GnâGn))(Ï, xâv(tâÏ), v) dv dÏ. (10.42)
We shall first estimate E and E .
Control of E . This is based on the time-cheating trick from §6, and the regularityof the force. By Proposition 6.2,
âE(t, ·)âFλânt+ÎŒâ
n ïżœâ« t
0
âF [hn+1Ï ] Ωn
t,Ï âF [hn+1Ï ]âZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)âGnâZλâ
n(1+b),ÎŒân;1
Ïâbt/(1+b)dÏ. (10.43)
on landau damping 161
From (10.1) and Step 5,
âGnâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
ïżœ ââvf0 Ωnt,ÏâZλâ
n(1+b),ÎŒân;1
Ïâbt/(1+b)+
nâk=1
ââvhkÏ Î©n
t,ÏâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
ïżœC âČ0+( nâ
k=1
ÎŽk
)(1+Ï),
(10.44)
where C âČ0 comes from the contribution of f0.
Next, by Propositions 4.24 and 4.25 (with V =0, Ï =Ï and b=0),
âF [hn+1Ï ] Ωn
t,Ï âF [hn+1Ï ]âZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)
ïżœ âΩnt,Ï âId âZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)
â« 1
0
ââF [hn+1Ï ] (Id +Ξ(Ωn
t,Ï âId))âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
dΞ
ïżœ ââF [hn+1Ï ]âFÎœnâΩn
t,Ï âId âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
,
(10.45)
where
Îœn = ÎŒân+λâ
n(1+b)âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+âΩnXt,Ï âxâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
.
Small times are taken care of, as usual, by the initial regularity layer, so we only focuson the case Ï ïżœbt/(1+b); then
Îœn ïżœÎ»ânÏ +ÎŒâ
nâλânb(tâÏ)+2Rn(Ï, t)
ïżœÎ»ânÏ +ÎŒâ
nâλân
B(tâÏ)1+t
+4C1Ï
( nâk=1
ÎŽk
(2Ï(λkâλâk))3
)min{tâÏ, 1}
1+Ï.
To make sure that ÎœnïżœÎ»ânÏ +ÎŒâ
n, we assume that
4C1Ï
nâk=1
ÎŽk
(2Ï(λkâλâk))3
ïżœ λââB
3, (C6)
and we note thatmin{tâÏ, 1}
1+Ïïżœ 3
tâÏ
1+t.
(This is easily seen by separating four cases: (a) tïżœ2, (b) tïżœ2 and tâÏ ïżœ1, (c) tïżœ2,tâÏ ïżœ1 and Ï ïżœ 1
2 t, (d) tïżœ2, tâÏ ïżœ1 and Ï ïżœ 12 t.)
Then, since Îłïżœ1, we have
ââF [hn+1Ï ]âFÎœn ïżœ ââF [hn+1
Ï ]âFλânÏ+ÎŒâ
n
ïżœ âF [hn+1Ï ]âFλâ
nÏ+ÎŒân,Îł ïżœCF âïżœ[hn+1
Ï ]âFλânÏ+ÎŒâ
n .(10.46)
162 c. mouhot and c. villani
(Note that applying Proposition 4.10 instead of the regularity coming from the interactionwould consume more regularity than we can afford to.)
Plugging this back into (10.45), we get
âF [hn+1Ï ] Ωn
t,Ï âF [hn+1Ï ]âZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)
ïżœ 2Rn(Ï, t)CF âïżœ[hn+1Ï ]âFλâ
nÏ+ÎŒân
ïżœ 2C3ÏCF
( nâk=1
ÎŽk
(2Ï(λkâλâk))5
)1
(1+Ï)3âïżœ[hn+1
Ï ]âFλânÏ+ÎŒâ
n .
Recalling (10.41) and (10.44), applying Proposition 4.24, we conclude that
âE(t, ·)âFλânt+ÎŒâ
n
ïżœ 2C3ÏCF
(C âČ
0+nâ
k=1
ÎŽk
)( nâk=1
ÎŽk
(2Ï(λkâλâk))5
)â« t
0
âïżœ[hn+1Ï ]âFλâ
nÏ+ÎŒân
dÏ
(1+Ï)2.
(10.47)
(We could be a bit more precise; anyway we cannot go further since we do not yet havean estimate on ïżœ[hn+1]. Recall that the latter quantity has zero mean, so the F normabove could be replaced by a F norm.)
Control of E. This will use the control on the derivatives of hk. We start again fromProposition 6.2,
âE(t, ·)âFλânt+ÎŒâ
n ïżœâ« t
0
âGnâGnâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
âF [hn+1Ï ]âFÎČn dÏ, (10.48)
where
ÎČn = λân(1+b)
âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+ÎŒân.
We focus again on the case Ï ïżœbt/(1+b), so that (with crude estimates)
âF [hn+1Ï ]âFÎČn ïżœ âF [hn+1
Ï ]âFλânÏ+ÎŒâ
n ïżœCF âïżœ[hn+1Ï ]âFλâ
nÏ+ÎŒân ,
and the problem is to control GnâGn,
âGnâGnâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
ïżœ â(âvf0) Ωnt,Ï ââvf0âZλâ
n(1+b),ÎŒân;1
Ïâbt/(1+b)
+nâ
k=1
â(âvhkÏ ) Ωn
t,Ï â(âvhkÏ ) Ωkâ1
t,Ï âZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
+nâ
k=1
â(âvhkÏ ) Ωkâ1
t,Ï ââv(hkÏ Î©kâ1
t,Ï )âZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
.
(10.49)
on landau damping 163
By induction hypothesis (Enh), and since the Zλ,Ό
Ï norms are increasing as a functionof λ and ÎŒ,
nâk=1
â(âvhkÏ ) Ωkâ1
t,Ï ââv(hkÏ Î©kâ1
t,Ï )âZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
ïżœ( nâ
k=1
ÎŽk
)1
(1+Ï)2.
It remains to treat the first and second terms in the right-hand side of (10.49). This isdone by inversion/composition as in Step 5; let us consider for instance the contributionof hk, kïżœ1,
ââvhkÏ Î©n
t,Ï ââvhkÏ Î©kâ1
t,Ï âZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
ïżœâ« 1
0
âââvhkÏ ((1âΞ)Ωn
t,Ï +ΞΩkâ1t,Ï )âZλâ
n(1+b),ÎŒân;1
Ïâbt/(1+b)âΩn
t,Ï âΩkâ1t,Ï âZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)dΞ
ïżœ 2âââvhkÏ Î©kâ1
t,Ï âZλâk(1+b),ÎŒâ
k;1
Ïâbt/(1+b)
âΩnt,Ï âΩkâ1
t,Ï âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ 4ÎŽk(1+Ï)2Rkâ1,n(Ï, t)
ïżœ 4C4ÏÎŽk
( nâj=k
ÎŽj
(2Ï(λjâλâj ))6
)1
(1+Ï)2,
where in the second-last step we used (EnΩ), (En
ïżœ ), Propositions 4.24 and 4.28, condi-tion (C5) and the same reasoning as in Step 5.
Summing up all contributions and inserting into (10.48) yields
âE(t, ·)âFλânt+ÎŒâ
n ïżœ 4CF
[C4
Ï
(C âČ
0+nâ
k=1
ÎŽk
)( nâj=1
ÎŽj
(2Ï(λjâλân))6
)+
nâk=1
ÎŽk
]
Ăâ« t
0
âïżœ[hn+1Ï ]âFλâ
nÏ+ÎŒân
dÏ
(1+Ï)2.
(10.50)
Main contribution. Now we consider Ïn,n+1, which we decompose as
Ïn,n+1t = Ïn,n+1
t,0 +nâ
k=1
Ïn,n+1t,k ,
where
Ïn,n+1t,0 (x) =
â« t
0
â«Rd
F [hn+1](Ï, xâv(tâÏ), v)·âvf0(v) dv dÏ,
Ïn,n+1t,k (x) =
â« t
0
â«Rd
(F [hn+1Ï ]·âv(hk
Ï Î©kâ1t,Ï ))(Ï, xâv(tâÏ), v) dv dÏ.
164 c. mouhot and c. villani
Note that their zero modes vanish. For any kïżœ1, we apply Theorem 6.5 (with M=1) toget
âÏn,n+1t,k âFλâ
nt+ÎŒân ïżœâ« t
0
Kn,hk
1 (t, Ï)âF [hn+1Ï ]âFÎœâČ
n,Îł dÏ +â« t
0
Kn,hk
0 (t, Ï)âF [hn+1Ï ]âFÎœâČ
n,Îł dÏ,
where
ÎœâČn = λâ
n(1+b)âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+ÎŒâČn
and
Kn,hk
1 (t, Ï) =Kn,k1 (t, Ï) sup
0ïżœsïżœt
ââv(hks Ωkâ1
t,s )âăâv(hks Ωkâ1
t,s )ăâZλk(1+b),ÎŒksâbt/(1+b)
1+s,
Kn,k1 (t, Ï) = (1+Ï)d sup
l,mâZdâ
eâÏ(ÎŒkâÎŒân)|m| e
â2Ï(ÎŒâČnâÎŒâ
n)|lâm|
1+|lâm|Îł eâÏ(λkâλân)|l(tâÏ)+mÏ |,
Kn,hk
0 (t, Ï) =Kn,k0 (t, Ï) sup
0ïżœsïżœtââvăhk
s Ωkâ1t,s ăâCλk(1+b);1 ,
Kn,k0 (t, Ï) = deâÏ(λkâλâ
n)(tâÏ).
We assume that
ÎŒâČn = ÎŒâ
n+η
(tâÏ
1+t
), η > 0 small, (10.51)
and check that ÎœâČnïżœÎ»â
nÏ +ÎŒân. Leaving apart the small-time case, we assume Ï ïżœbt/(1+b),
so that
ÎœâČn = (λâ
nÏ +ÎŒân)âBλâ
n(tâÏ)1+t
+η
(tâÏ
1+t
),
which is indeed bounded above by λânÏ +ÎŒâ
n as soon as
η ïżœBλââ. (10.52)
Then, with the notation (7.10),
Kn,k1 (t, Ï) ïżœK
(αn,k),Îł1 (t, Ï), (10.53)
withαn,k = Ï min{ÎŒkâÎŒâ
n, λkâλân, 2η}. (10.54)
From the controls on hk (assumption (Enh)) we have that
ââv(hkÏ Î©kâ1
t,Ï )âăâv(hkÏ Î©kâ1
t,Ï )ăâZλk(1+b),ÎŒk;1Ïâbt/(1+b)
ïżœ ââv(hkÏ Î©kâ1
t,Ï )âZλk(1+b),ÎŒk;1Ïâbt/(1+b)
ïżœ ÎŽk(1+Ï)
on landau damping 165
and
âăâv(hkÏ Î©kâ1
t,Ï )ăâCλk(1+b);1 = âă(âv+Ïâx)(hkÏ Î©kâ1
t,Ï )ăâCλk(1+b);1
ïżœ â(âv+Ïâx)(hkÏ Î©kâ1
t,Ï )âZλk(1+b);1Ïâbt/(1+b)
ïżœ ÎŽk.
After controlling F [hn+1] by ïżœ[hn+1], we end up with
âÏn,n+1t,k âFλâ
nt+ÎŒân ïżœCF
â« t
0
( nâk=1
ÎŽkK(αn,k),Îł1 (t, Ï)
)âïżœ[hn+1
Ï ]âFλânÏ+ÎŒâ
n dÏ
+CF
â« t
0
( nâk=1
ÎŽk eâÏ(λkâλân)(tâÏ)
)âïżœ[hn+1
Ï ]âFλânÏ+ÎŒâ
n dÏ,
(10.55)
with αn,k defined by (10.54).
Substep c. Gathering all previous controls, we obtain the following integral inequal-ity for Ï=ïżœ[hn+1]:â„â„â„â„Ï(t, x)â
â« t
0
â«Rd
(âW âÏ)(Ï, xâv(tâÏ))·âvf0(v) dv dÏ
â„â„â„â„Fλâ
nt+ÎŒân
ïżœAn+â« t
0
(Kn
1 (t, Ï)+Kn0 (t, Ï)+
cn0
(1+Ï)2
)âÏ(Ï, ·)âFλâ
nÏ+ÎŒân dÏ,
(10.56)
where, by (10.36), (10.47) and (10.50),
An = suptïżœ0
âÏn,n(t, ·)âFλânt+ÎŒâ
n ïżœ 2CF ÎŽ2n
(Ï(λnâλân))2
, (10.57)
Kn1 (t, Ï) =
(CF
nâk=1
ÎŽk
)K
(αn),Îł1 (t, Ï),
αn = αn,n = Ï min{(ÎŒnâÎŒân), (λnâλâ
n), 2η},
Kn0 (t, Ï) =CF
nâk=1
ÎŽkeâÏ(λkâλân)(tâÏ),
cn0 = 3CF C4
Ï
(C âČ
0+nâ
k=1
ÎŽk
)( nâk=1
ÎŽk
(2Ï(λkâλâk))6
)+
nâk=1
ÎŽk.
(We are cheating a bit when writing (10.56), because in fact one should take into accountsmall times separately; but this does not cause any difficulty.)
We easily estimate Kn0 :â« t
0
Kn0 (t, Ï) dÏ ïżœCF
nâk=1
ÎŽk
Ï(λkâλân)
,
â« â
Ï
Kn0 (t, Ï) dt ïżœCF
nâk=1
ÎŽk
Ï(λkâλân)
,
(â« t
0
Kn0 (t, Ï)2 dÏ
)1/2
ïżœCF
nâk=1
ÎŽkâ2Ï(λkâλâ
n).
166 c. mouhot and c. villani
Let us assume that αn is smaller than α(γ) appearing in Theorem 7.7, and that
3CF C4Ï
(C âČ
0+nâ
k=1
ÎŽk+1)( nâ
k=1
ÎŽk
(2Ï(λkâλâk))6
)ïżœ 1
4, (C7)
CF
nâk=1
ÎŽkâ2Ï(λkâλâ
k)ïżœ 1
2, (C8)
CF
nâk=1
ÎŽk
Ï(λkâλâk)
ïżœ max{
14, Ï
}, (C9)
(note that in these conditions we have strenghtened the inequalities by replacing λkâλân
by λkâλâk where Ï>0 is also defined by Theorem 7.7). Applying Theorem 7.7 with
λ0=λ and λâ=λ1, we deduce that for any Δâ(0, αn) and tïżœ0,
âïżœn+1t âFλâ
nt+ÎŒân ïżœCAn
(1+cn0 )2âΔ
eCcn0
(1+
cn
αnΔ
)eCTΔ,neCcn(1+T 2
Δ,n)eΔt, (10.58)
where
cn = 2CF
nâk=1
ÎŽk
and
TΔ,n = Cγ max{(
c2n
α5nΔ2+γ
)1/(Îłâ1)
,
(cn
α2nΔγ+1/2
)1/(Îłâ1)
,(cn
0 )2/3
Δ1/3
}.
Pick up λâ n<λâ
n such that 2Ï(λânâλâ
n)ïżœÎ±n, and choose Δ=2Ï(λânâλâ
n); recalling thatïżœn+1(t, 0)=0, and that our conditions imply an upper bound on cn and cn
0 , we deducethe uniform control
âïżœn+1t âFλ
â nt+ÎŒâ
nïżœ eâ2Ï(λâ
nâλâ n)tâïżœn+1
t âFλânt+ÎŒâ
n
ïżœCAn
(1+
1
αn(λânâλâ
n)3/2
)eCT 2
n ,(10.59)
where
Tn = C
(1
α5n(λâ
nâλâ n)2+Îł
)1/(Îłâ1)
. (10.60)
10.4.3. Step 7. Estimate on F [hn+1]
As an immediate consequence of (10.2) and (10.59), we have
suptïżœ0
âF [ïżœn+1t ]âFλ
â nt+ÎŒâ
n,ÎłïżœCAn
(1+
1
αn(λânâλâ
n)3/2
)eCT 2
n . (10.61)
on landau damping 167
10.4.4. Step 8. Estimate of hn+1 Ωn
In this step we shall again use the Vlasov equation. We rewrite (10.30) as
hn+1(Ï,Xn0,Ï (x, v), V n
0,Ï (x, v)) =â« Ï
0
ÎŁn+1(s,Xn0,s(x, v), V n
0,s(x, v)) ds;
but now we compose with (Xnt,0, V
nt,0), where tïżœÏ is arbitrary. This gives
hn+1(Ï,Xnt,Ï (x, v), V n
t,Ï (x, v)) =â« Ï
0
ÎŁn+1(s,Xnt,s(x, v), V n
t,s(x, v)) ds.
Then for any pâ[1, p ] and Î»ïżœn<λâ
n, using Propositions 4.19 and 4.24, and the notation(10.32), we get
âhn+1Ï Î©n
t,ÏâZ(1+b)Î»ïżœn,ÎŒâ
n;pÏâbt/(1+b)
= âhn+1Ï (Xn
t,Ï , V nt,Ï )â
Z(1+b)Î»ïżœn,ÎŒâ
n;ptâbt/(1+b)
ïżœâ« Ï
0
âÎŁn+1(s,Xnt,s, V
nt,s)âZ(1+b)Î»ïżœ
n,ÎŒân;p
tâbt/(1+b)
ds
=â« Ï
0
âÎŁn+1(s,Ωnt,s)âZ(1+b)Î»ïżœ
n,ÎŒân;p
sâbt/(1+b)
ds
ïżœâ« Ï
0
âRn+1s,t â
Z(1+b)Î»ïżœn,ÎŒâ
nsâbt/(1+b)
âGns,tâZ(1+b)Î»ïżœ
n,ÎŒân;p
sâbt/(1+b)
ds
+â« Ï
0
âRns,tâZ(1+b)Î»ïżœ
n,ÎŒân
sâbt/(1+b)
âHns,tâZ(1+b)Î»ïżœ
n,ÎŒân;p
sâbt/(1+b)
ds.
Then (proceeding as in Step 6 to check that the exponents lie in the appropriaterange)
âRn+1s,t â
Z(1+b)Î»ïżœn,ÎŒâ
nsâbt/(1+b)
ïżœCF eâ2Ï(λâ nâÎ»ïżœ
n)sâïżœn+1s âF Îœn(s)
andâRn
s,tâZ(1+b)Î»ïżœ,ÎŒân
sâbt/(1+b)
ïżœCF eâ2Ï(λâ nâÎ»ïżœ
n)sâïżœns âF Îœn(s) ïżœCF eâ2Ï(λâ
nâÎ»ïżœn)sÎŽn,
with
Îœn(s, t) :=
â§âȘâȘâšâȘâȘâ©ÎŒïżœ, when sïżœ bt
1+b,
λâ ns+ÎŒâ
n, when sïżœ bt
1+b.
On the other hand, from the induction assumption (Enh)â(En
h) (and again control ofcomposition via Proposition 4.25),
âHns,tâZ(1+b)Î»ïżœ
n,ÎŒân;p
sâbt/(1+b)
ïżœ 2(1+s)ÎŽn
168 c. mouhot and c. villani
and
âGns,tâZ(1+b)Î»ïżœ
n,ÎŒân;p
sâbt/(1+b)
ïżœ 2(1+s)nâ
k=1
ÎŽk.
We deduce thaty(t, Ï) := âhn+1
Ï Î©nt,ÏâZ(1+b)Î»ïżœ
n,ÎŒân;p
Ïâbt/(1+b)
satisfies
y(t, Ï) ïżœ 2CF
( nâk=1
ÎŽk
)â« Ï
0
eâ2Ï(λâ nâÎ»ïżœ
n)sâïżœn+1s âF Îœn(s)(1+s) ds
+2CF ÎŽ2n
â« Ï
0
eâ2Ï(λâ nâÎ»ïżœ
n)s(1+s) ds;
so, for all 0ïżœÏ ïżœt,
âhn+1Ï Î©n
t,ÏâZ(1+b)Î»ïżœn,ÎŒâ
n;pÏâbt/(1+b)
ïżœ4CF max
{ânk=1 ÎŽk, 1
}(2Ï(λâ
nâÎ»ïżœn))2
(ÎŽ2n+sup
sïżœ0âïżœn+1
s âF Îœn(s)
). (10.62)
10.4.5. Step 9. Crude estimates on the derivatives of hn+1
Again we choose pâ[1, p ]. From the previous step and Proposition 4.27 we deduce, forany λâĄ
n such that λâĄn<Î»ïżœ
n<λâĄn, and any ÎŒâĄ
n<ÎŒân, that
ââx(hn+1Ï Î©n
t,Ï )âZλ
âĄn(1+b),ÎŒ
âĄn;p
Ïâbt/(1+b)
+â(âv+Ïâx)(hn+1Ï Î©n
t,Ï )âZλ
âĄn(1+b),ÎŒ
âĄn;p
Ïâbt/(1+b)
ïżœ C(d)
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}âhn+1
Ï Î©nt,ÏâZÎ»ïżœ
n(1+b),ÎŒân;p
Ïâbt/(1+b)
(10.63)
and
ââ(hn+1Ï Î©n
t,Ï )âZλ
âĄn(1+b),ÎŒ
âĄn;p
Ïâbt/(1+b)
ïżœ C(d)(1+Ï)
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}âhn+1
Ï Î©nt,ÏâZÎ»ïżœ
n(1+b),ÎŒân;p
Ïâbt/(1+b)
. (10.64)
Similarly,
âââ(hn+1Ï Î©n
t,Ï )âZλ
âĄn(1+b),ÎŒ
âĄn;p
Ïâbt/(1+b)
ïżœ C(d)(1+Ï)2
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}2âhn+1
Ï Î©nt,ÏâZÎ»ïżœ
n(1+b),ÎŒân;p
Ïâbt/(1+b)
.
(10.65)
10.4.6. Step 10. Chain-rule and refined estimates on derivatives of hn+1
From Step 3 we have
ââΩnt,ÏâZλâ
n(1+b),ÎŒân
Ïâbt/(1+b)+â(âΩn
t,Ï )â1âZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœC(d) (10.66)
on landau damping 169
and (via Proposition 4.27)
âââΩnt,ÏâZλ
âĄn(1+b),ÎŒ
âĄn
Ïâbt/(1+b)
ïżœ C(d)(1+Ï)
min{λânâλâĄ
n, ÎŒânâÎŒâĄ
n}ââΩn
t,ÏâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
ïżœ C(d)(1+Ï)
min{λânâλâĄ
n, ÎŒânâÎŒâĄ
n}.
(10.67)
Combining these bounds with Step 9, Proposition 4.24 and the identities{(âh) Ω = (âΩ)â1â(h Ω),(â2h) Ω = (âΩ)â2â2(h Ω)â(âΩ)â1â2Ω(âΩ)â1(âh Ω),
(10.68)
we get
â(âhn+1Ï ) Ωn
t,ÏâZλâĄn(1+b),ÎŒ
âĄn;1
Ïâbt/(1+b)
ïżœC(d)ââ(hn+1Ï Î©n
t,Ï )âZλ
âĄn(1+b),ÎŒ
âĄn;1
Ïâbt/(1+b)
ïżœ C(d)(1+Ï)
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}âhn+1
Ï Î©nt,ÏâZÎ»ïżœ
n(1+b),ÎŒân;1
Ïâbt/(1+b)
(10.69)
and
â(â2hn+1Ï ) Ωn
t,ÏâZλâĄn(1+b),ÎŒ
âĄn;1
Ïâbt/(1+b)
ïżœC(d)[ââ2(hn+1
Ï Î©nt,Ï )â
ZλâĄn(1+b),ÎŒ
âĄn;1
Ïâbt/(1+b)
+ââ2Ωnt,ÏâZλ
âĄn(1+b),ÎŒ
âĄn;1
Ïâbt/(1+b)
â(âhn+1Ï ) Ωn
t,ÏâZλâĄn(1+b),ÎŒ
âĄn;1
Ïâbt/(1+b)
]ïżœ C(d)(1+Ï)2
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}2âhn+1
Ï Î©nt,ÏâZÎ»ïżœ
n(1+b),ÎŒân;p
Ïâbt/(1+b)
.
(10.70)
This gives us the bounds
â(âhn+1) Ωnâ= O(1+Ï) and â(â2hn+1) Ωnâ= O((1+Ï)2),
which are optimal if one does not distinguish between the x and v variables. We shallnow refine these estimates. We first write
â(hn+1Ï Î©n
t,Ï )â(âhn+1Ï ) Ωn
t,Ï =â(Ωnt,Ï âId)·[(âhn+1
Ï ) Ωnt,Ï ],
and we deduce (via Propositions 4.24 and 4.27)
ââ(hn+1Ï Î©n
t,Ï )â(âhn+1Ï ) Ωn
t,ÏâZλâĄn(1+b),ÎŒ
âĄn;p
Ïâbt/(1+b)
ïżœ ââ(Ωnt,Ï âId)â
ZλâĄn(1+b),ÎŒ
âĄn
Ïâbt/(1+b)
â(âhn+1Ï ) Ωn
t,ÏâZλâĄn(1+b),ÎŒ
âĄn;p
Ïâbt/(1+b)
(10.71)
ïżœC(d)(
1+Ï
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}
)2âΩn
t,Ï âId âZÎ»ïżœ
n(1+b),ÎŒân
Ïâbt/(1+b)
âhn+1Ï Î©n
t,ÏâZÎ»ïżœn(1+b),ÎŒâ
n;pÏâbt/(1+b)
ïżœ C(d)C4Ï
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}2
( nâk=1
ÎŽk
(2Ï(λkâλâk))6
)(1+Ï)â2âhn+1
Ï Î©nt,ÏâZÎ»ïżœ
n(1+b),ÎŒân;p
Ïâbt/(1+b)
.
170 c. mouhot and c. villani
(Note that ΩnâId brings the time-decay, while hn+1 brings the smallness.)This shows that (âhn+1) Ωnïżœâ(hn+1 Ωn) as Ï!â. In view of Step 9, this also
implies the refined gradient estimates
â(âxhn+1Ï ) Ωn
t,ÏâZλâĄn,ÎŒ
âĄn;p
Ïâbt/(1+b)
+â((âv+Ïâx)hn+1Ï ) Ωn
t,ÏâZλâĄn,ÎŒ
âĄn;p
Ïâbt/(1+b)
ïżœCâhn+1Ï Î©n
t,ÏâZÎ»ïżœn(1+b),ÎŒâ
n;pÏâbt/(1+b)
,(10.72)
with
C = C(d)(
C4Ï
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}2
nâk=1
ÎŽk
(2Ï(λkâλâk))6
+1
min{Î»ïżœnâλâĄ
n, ÎŒânâÎŒâĄ
n}
).
10.4.7. Conclusion
Given λn+1<λân and ÎŒn+1<ÎŒâ
n, we define
λn+1 = λâĄn and ÎŒn+1 = ÎŒâĄ
n,
and we impose that
λânâλâ
n = λâ nâÎ»ïżœ
n = Î»ïżœnâλâĄ
n = 13 (λâ
nâλn+1) and ÎŒânâÎŒâĄ
n = ÎŒânâÎŒn+1.
Then from (10.59), (10.61)â(10.63) and (10.70)â(10.72) we see that (En+1ïżœ ), (En+1
ïżœ ),(En+1
h ) and (En+1h ) have all been established in the present subsection, with
ÎŽn+1 =C(d)CF (1+CF )(1+C4
Ï)eC T 2n
min{λânâλn+1, ÎŒâ
nâÎŒn+1}9max{
1,nâ
k=1
ÎŽk,
}(1+
nâk=1
ÎŽk
(2Ï(λkâλâk))6
)ÎŽ2n.
(10.73)
10.5. Convergence of the scheme
For any nïżœ1, we set
λnâλân = λâ
nâλn+1 = ÎŒnâÎŒân = ÎŒâ
nâÎŒn+1 =În2
(10.74)
for some Î>0. By choosing Î small enough, we can make sure that the conditions
2Ï(λkâλâk) < 1 and 2Ï(ÎŒkâÎŒâ
k) < 1
on landau damping 171
are satisfied for all k, as well as the other smallness assumptions made throughout thissection. Moreover, we have λkâλâ
kïżœÎ/k2, so conditions (C1)â(C9) will be satisfied ifnâ
k=1
k12ÎŽk ïżœ Î6Ï andnâ
j=k+1
j6ÎŽj ïżœ Î3Ï
(1k2
â 1n2
)for some small explicit constant Ï>0, depending on the other constants appearing in theproblem. Both conditions are satisfied if
ââk=1
k12ÎŽk ïżœ Î6Ï. (10.75)
Then from (10.60) we have that TnïżœCÎł(n2/Î)(7+Îł)/(Îłâ1), so the induction relationon ÎŽn gives
ÎŽ1 ïżœCÎŽ and ÎŽn+1 = C
(n2
Î
)9eC(n2/Î)(14+2Îł)/(Îłâ1)
ÎŽ2n. (10.76)
To establish this relation we also assumed that Ύn is bounded below by CF ζn, theerror coming from the short-time iteration; but this follows easily by construction, sincethe constraints imposed on Ύn are much worse than those on ζn.
Having fixed Î, we will check that for ÎŽ small enough, (10.76) implies both thefast convergence of (ÎŽk)âk=1, and the condition (10.75), which will justify a posteriorithe derivation of (10.76). (An easy induction is enough to turn this into a rigorousreasoning.)
For this we fix aâ(1, aI), 0<z<zI <1, and we check by induction that
ÎŽn ïżœ Îzan
for all n ïżœ 1. (10.77)
If Î is given, (10.77) holds for n=1 as soon as ÎŽïżœ(Î/C)za. Then, to go from stage n tostage n+1, we should check that
Cn18
Î9eCn(28+4Îł)/(Îłâ1)/Î(14+2Îł)/(Îłâ1)
Î2z2an ïżœ Îzan+1;
this is true if1Î
ïżœ C
Î9supnâN
n19eCn(28+4Îł)/(Îłâ1)/Î(14+2Îł)/(Îłâ1)z(2âa)an
.
Since a<2, the supremum on the right-hand side is finite, and we just have to chooseÎ small enough. Then, reducing Î further if necessary, we can ensure (10.75). Thisconcludes the proof.
Remark 10.1. This argument almost fully exploits the bi-exponential convergenceof the Netwon scheme: a convergence like, say, O(eân1000
) would not be enough to treatvalues of Îł which are close to 1. In §11.2 we shall present a more cumbersome approachwhich is less greedy in the convergence rate, but still needs convergence like O(eânα
) forα large enough.
172 c. mouhot and c. villani
11. CoulombâNewton interaction
In this section we modify the scheme of §10 to treat the case γ=1. We provide twodifferent strategies. The first one is quite simple and will only come close to treat this case,since it will hold on (nearly) exponentially large times in the inverse of the perturbationsize. The second one, somewhat more involved, will hold up to infinite times.
11.1. Estimates on exponentially large times
In this subsection we adapt the estimates of §10 to the case Îł=1, under the additionalrestriction that 0ïżœtïżœA1/ÎŽ(log ÎŽ)2 for some constant A>1.
In the iterative scheme, the only place where we used Îł>1 (and not just Îłïżœ1) is inStep 6, when it comes to the echo response via Theorem 7.7. Now, in the case Îł=1, theformula for Kn
1 should be
Kn1 (t, Ï) =
nâk=1
ÎŽkK(αn,k),11 (t, Ï),
with αn,k=Ï min{ÎŒkâÎŒân, λkâλâ
n, 2η}. By Theorem 7.7 (ii) this induces, in addition toother well-behaved factors, an uncontrolled exponential growth O(eΔnt), with
Δn = Înâ
k=1
ÎŽk
α3n,k
;
in particular Δn will remain bounded and O(Ύ) throughout the scheme.Let us replace (10.74) by
λnâλân = λâ
nâλn+1 = ÎŒnâÎŒân = ÎŒâ
nâÎŒn+1 =Î
n(log(e+n))2,
where Î>0 is very small. (This is allowed since the series
âân=1
1n(log(e+n))2
convergesâthe power 2 could of course be replaced by any r>1.) Then during the firststages of the iteration we can absorb the O(eΔnt) factor by the loss of regularity if, say,
Δn ïżœ Î2n(log(e+n))2
.
Recalling that Δn=O(Ύ), this is satisfied as soon as
n ïżœN :=K
ÎŽ(log(1/ÎŽ))2, (11.1)
on landau damping 173
where K>0 is a positive constant depending on the other parameters of the problem butof course not on Ύ. So during these first stages we get the same long-time estimates asin §10.
For n>N we cannot rely on the loss of regularity any longer; at this stage the erroris about
ÎŽN ïżœCÎŽaN
,
where 1<a<2. To get the bounds for larger values of n, we impose a restriction on thetime-interval, say 0ïżœtïżœTmax. Allowing for a degradation of the rate ÎŽan
into ÎŽan
witha<a, we see that the new factor eΔnTmax can be eaten up by the scheme if
eΔnTmaxÎŽ(aâa)an ïżœ 1 for all n ïżœN .
This is satisfied if
Tmax = O
(aN 1
ÎŽlog
1ÎŽ
).
Recalling (11.1), we see that the latter condition holds true if
Tmax = O
(A1/ÎŽ(log ÎŽ)2 1
ÎŽlog
1ÎŽ
)for some well-chosen constant A>1. Then we can complete the iteration, and end upwith a bound like
âftâfiâZλâČ,ÎŒâČt
ïżœCÎŽ for all tâ [0, Tmax],
where C is another constant independent of ÎŽ. The conclusion follows easily.
11.2. Mode-by-mode estimates
Now we shall change the estimates of §10 a bit more in depth to treat arbitrarily largetimes for γ=1. The main idea is to work mode-by-mode in the estimate of the spatialdensity, instead of looking directly for norm estimates.
Steps 1â5 remain the same, and the changes mainly occur in Step 6.
Substep 6 (a) is unchanged, but we only retain from that substep
e2Ï(λânt+ÎŒâ
n)|l|âŁâŁÏn,nt (l)
âŁâŁïżœ 2CF ÎŽ2n
(Ï(λnâλân))2
for all lâZd. (11.2)
Substep (b) is more deeply changed. Let ÎŒn<ÎŒân.
174 c. mouhot and c. villani
âą First, for each lâZd, we have, by Propositions 4.34 and 4.35, and the last part ofProposition 6.2,
e2Ï(λânt+ÎŒn)|l||E(t, l)|
ïżœâ« t
0
âmâZd
âPm(F [hn+1Ï ] Ωn
t,Ï âF [hn+1Ï ])âZλâ
n(1+b),ÎŒnÏâbt/(1+b)
ĂâPlâmGnÏ,tâZλâ
n(1+b),ÎŒn;1Ïâbt/(1+b)
dÏ
ïżœâ« t
0
âm,mâČâZd
â„â„â„â„PmâmâČ
â« 1
0
âF [hn+1Ï ] (Id +Ξ(Ωn
t,Ï âId)) dΞ
â„â„â„â„Zλâ
n(1+b),ÎŒnÏâbt/(1+b)
ĂâPmâČ(Ωnt,Ï âId)âZλâ
n(1+b),ÎŒnÏâbt/(1+b)
âPlâmGnÏ,tâZλâ
n(1+b),ÎŒn;1Ïâbt/(1+b)
dÏ
ïżœâ« 1
0
â« t
0
âGnÏ,tâZλâ
n(1+b),ÎŒân;1
Ïâbt/(1+b)
â„â„â„â„Ωnt,Ï âId
â„â„â„â„Zλâ
n(1+b),ÎŒân
Ïâbt/(1+b)
âm,mâČâZd
eâ2Ï(ÎŒânâÎŒn)|lâm|
Ăeâ2Ï(ÎŒânâÎŒn)|mâČ|âPmâmâČ(âF [hn+1
Ï ] (Id +Ξ(Ωnt,Ï âId)))âZλâ
n(1+b),ÎŒnÏâbt/(1+b)
dÏ dΞ
ïżœâ« t
0
|GnâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
âΩnt,Ï âId
â„â„â„â„Zλâ
n(1+b),ÎŒân
Ïâbt/(1+b)
Ăâ
m,mâČ,qâZd
eâ2Ï(ÎŒânâÎŒn)|lâm|eâ2Ï(ÎŒâ
nâÎŒn)|mâČ|
Ăeâ2Ï(ÎŒânâÎŒn)|mâmâČâq|âPq(âF [hn+1
Ï ])âF Îœn dÏ,
where
Îœn = ÎŒn+λân(1+b)
âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+âΩnXt,Ï âxâZλân(1+b),ÎŒâ
nÏâbt/(1+b)
.
For Î±ïżœ1 we haveâm,mâČâZd
eâ2Ïα|lâm|eâ2Ïα|mâČ|eâ2Ïα|mâmâČâq| ïżœ C(d)αd
eâÏα|lâq|,
and we can argue as in Substep 6 (b) of §10 to get
e2Ï(λânt+ÎŒâ
n)|l||E(t, l)|
ïżœ C
(ÎŒânâÎŒn)d
(C âČ
0+nâ
k=1
ÎŽk
)( nâk=1
ÎŽk
(2Ï(λkâλâk))5
)ĂâqâZd
eâÏ(ÎŒânâÎŒn)|lâq|
â« t
0
e2Ï(λânÏ+ÎŒn)|q|âŁâŁïżœ[hn+1
Ï ](q)âŁâŁ dÏ
(1+Ï)2.
(11.3)
on landau damping 175
âą Next, we use again Proposition 4.34 and simple estimates to bound E ,
e2Ï(λânt+ÎŒn)|l|âŁâŁE(t, l)
âŁâŁïżœâ« t
0
âGnâGnâZλân(1+b),ÎŒâ
n;1Ïâbt/(1+b)
Ăâ
mâZd
eâ2Ï(ÎŒânâÎŒn)|m|âPlâm(F [hn+1
Ï ])âF ÎČn dÏ,
where
ÎČn = λân(1+b)
âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+ÎŒn.
Reasoning as in Substep 6 (b) of §10, we arrive at
e2Ï(λânt+ÎŒn)|l|âŁâŁE(t, l)
âŁâŁïżœC
(C âČ
0+nâ
k=1
ÎŽk
)( nâj=1
ÎŽj
(2Ï(λjâλân))6
+nâ
k=1
ÎŽk
)
Ăâ
mâZd
eâ2Ï(ÎŒânâÎŒn)|m|
â« t
0
e2Ï(λânÏ+ÎŒn)|lâm|âŁâŁïżœ[hn+1
Ï ](lâm)âŁâŁ dÏ
(1+Ï)2.
(11.4)
âą Then we consider the âmain contributionâ Ïn,n+1, which we decompose as in §10:
Ïn,n+1t = Ïn,n+1
t,0 +nâ
k=1
Ïn,n+1t,k ,
and we write, for kïżœ1,
e2Ï(λânt+ÎŒn)
âŁâŁÏn,n+1t,k (l)
âŁâŁïżœ âmâZd
â« t
0
Kn,hk
l,m (t, Ï)âPlâm(F [hn+1Ï ])âFÎœâČ
n,Îł dÏ
+â« t
0
Kn,hk
0 (t, Ï)âPl(F [hn+1Ï ])âFÎœâČ
n,Îł dÏ,
where
ÎœâČn = λâ
n(1+b)âŁâŁâŁâŁÏâ bt
1+b
âŁâŁâŁâŁ+ÎŒâČn,
Kn,hk
l,m (t, Ï) =Kn,kl,m(t, Ï) sup
0ïżœsïżœt
ââv(hks Ωkâ1
t,s )âăâv(hks Ωkâ1
t,s )ăâZλk(1+b),ÎŒksâbt/(1+b)
1+s,
Kn,kl,m(t, Ï) = (1+Ï)eâÏ(ÎŒkâÎŒn)|m| e
â2Ï(ÎŒâČnâÎŒn)|lâm|
1+|lâm|Îł eâÏ(λkâλân)|l(tâÏ)+mÏ |,
and the formula for Kn,hk
0 is unchanged with respect to §10.
176 c. mouhot and c. villani
Assuming that ÎŒâČn=ÎŒn+η(tâÏ)/(1+t) and reasoning as in Substep 6 (b) of §10, we
end up with the following estimate on the âmain termâ:
e2Ï(λânt+ÎŒn)|l|âŁâŁÏn,n+1
t,k (l)âŁâŁ
ïżœCâ
mâZd
â« t
0
( nâk=1
ÎŽkK(αn,k),Îłl,m (t, Ï)
)e2Ï(λâ
nÏ+ÎŒn)|lâm|âŁâŁïżœ[hn+1Ï ](lâm)
âŁâŁ dÏ
+Câ
mâZd
â« t
0
( nâk=1
ÎŽkeâÏ(λkâλân)(tâÏ)
)e2Ï(λâ
nÏ+ÎŒn)|l|âŁâŁïżœ[hn+1Ï ](lâm)
âŁâŁ dÏ.
(11.5)
Then Substep 6 (c) becomes, with Ί(l, Ï)=ïżœ[hn+1Ï ](l),
e2Ï(λânt+ÎŒn)|l|
âŁâŁâŁâŁÎŠ(l, t)ââ« t
0
K0(l, tâÏ)Ί(l, Ï) dÏ
âŁâŁâŁâŁïżœ CÎŽ2
n
(λnâλân)2
+â
mâZd
â« t
0
(Kn
l,m(t, Ï)+cnm
(1+Ï)2
)e2Ï(λâ
nÏ+ÎŒn)|lâm||Ί(lâm, Ï)| dÏ
+â« t
0
Kn0 (t, Ï)e2Ï(λâ
nÏ+ÎŒn)|l||Ί(l, Ï)| dÏ,
with
Knl,m(t, Ï) =C
nâk=1
ÎŽkK(αn),Îłl,m (t, Ï),
αn = Ï min{ÎŒnâÎŒn, λnâλân, 2η},
Kn0 (t, Ï) =C
nâk=1
ÎŽkeâÏ(λkâλân)(tâÏ),
cnm =
C
(ÎŒânâÎŒn)d
(C âČ
0+nâ
k=1
ÎŽk
)(1+
nâk=1
ÎŽk
(2Ï(λkâλâk))6
)eâÏ(ÎŒâ
nâÎŒn)|m|.
Note that
âmâZd
cnm+( â
mâZd
(cnm)2)1/2
ïżœ C
(ÎŒânâÎŒn)2d
(C âČ
0+nâ
k=1
ÎŽk
)(1+
nâk=1
ÎŽk
(2Ï(λkâλâk))6
).
Then, being Ί(0, t)=0, we can apply Theorem 7.12 and deduce (taking already intoaccount, for the sake of readability of the formula, that
ânk=1 ÎŽk and
ânk=1 ÎŽk/(λkâλâ
k)are uniformly bounded)
e2Ï(λânt+ÎŒn)|l|âŁâŁïżœn+1
t (l)âŁâŁïżœC
ÎŽ2n
(λnâλân)2αnΔ3/2(ÎŒâ
nâÎŒn)2dexp(
C(1+T 2Δ,n)
(ÎŒânâÎŒn)2d
)eΔt,
on landau damping 177
where
TΔ,n = C max{(
1α3+2d
n Δγ+2
)1/Îł
,
(1
αdnΔγ+1/2
)1/(Îłâ1/2)
,
(1Δ
( âmâZd
cnm
)2)1/3}.
If λâ n<λâ
n and ÎŒâ n<ÎŒn are chosen as before and Δ=2Ï(λâ
nâλâ ), this implies a uniformbound on
âïżœn+1t âFλ
â nt+ÎŒ
â n
ïżœ C
(ÎŒnâÎŒâ n)d
suplâZd
e2Ï(λânt+ÎŒn)|l|âŁâŁïżœn+1
t (l)âŁâŁ
obtained with the formula above with
TΔ,n = Tn = C max{
1
λânâλâ
n
,1
λnâλân
,1
ÎŒnâÎŒân
,1
ÎŒânâÎŒn
}a,
where
a = max{
5+Îł+2d
Îł,d+Îł+ 1
2
Îłâ 12
,4d+1
3
}.
Then Steps 7â10 of the iteration can be repeated with the only modification thatÎŒâ
n is replaced by ÎŒâ n.
The convergence (§10.5) works just the same, except that now we need more inter-mediate regularity indices Όn:
ÎŒn+1 = ÎŒâĄn <ÎŒâ
n < ÎŒn <ÎŒân;
the obvious choice being to let ÎŒâ nâÎŒâĄ
n=ÎŒnâÎŒâ n=ÎŒâ
nâÎŒn.Choosing λnâλn+1 and ÎŒnâÎŒn+1 of the order of Î/n2, we arrive in the end at the
induction
ÎŽn+1 ïżœC
(n2
Î
)9+6d
eC(n2/Î)Ο(d,Îł)ÎŽ2n,
where
Ο(d, Îł) := 2d+2 max{
5+Îł+2d
Îł,d+Îł+ 1
2
Îłâ 12
,4d+1
3
}. (11.6)
Then the convergence of the scheme (and a-posteriori justification of all the assump-tions) is done exactly as in §10.
12. Convergence in large time
In this section we prove Theorem 2.6 as a simple consequence of the uniform boundsestablished in §10 and §11.
So let f0, L and W satisfy the assumptions of Theorem 2.6. To simplify the notationwe assume that L=1.
178 c. mouhot and c. villani
The second part of Assumption (2.14) precisely means that f0âCλ;1. We shallactually assume a slightly more precise condition, namely that for some pâ[1,â],â
nâNd0
λn
n!âân
v f0âLp(Rd) ïżœC0 <â for all pâ [1, p ]. (12.1)
(It is sufficient to take p=1 to get Theorem 2.6; but if this bound is available for somep>1 then it will be propagated by the iteration scheme, and will result in more precisebounds.) Then we pick up λâ(0, λ), ÎŒâ(0, ÎŒ), ÎČ>0 and ÎČâČâ(0, ÎČ). By symmetry, weonly consider non-negative times.
If fi is an initial datum satisfying the smallness condition (2.15), then by Theo-rem 4.20, we have a smallness estimate on âfiâf0âZλâČ,ÎŒâČ;p for all pâ[1, p ], λâČ<λ andÎŒâČ<ÎŒ. Then, as in §4.12 we can estimate the solution h1 to the linearized equation{
âth1+v ·âxh1+F [h1]·âvf0 = 0,
h1(0, ·) = fiâf0,(12.2)
and we recover uniform bounds in Z λ,ÎŒ;p spaces, for any λâ(λ, λ) and ÎŒâ(ÎŒ, ÎŒ). Moreprecisely,
suptïżœ0
âïżœ[h1t ]âF λt+ÎŒ +sup
tïżœ0âh1(t, ·)âZλ,ÎŒ;p
t
ïżœCÎŽ, (12.3)
with C=C(d, λâČ, λ, ÎŒâČ, ÎŒ,W, f0) (this is of course assuming Δ in Theorem 2.6 to be smallenough).
We now set λ1=λâČ, and we run the iterative scheme of §§9â11 for all nïżœ2. If Δ issmall enough, up to slightly lowering λ1, we may choose all parameters in such a waythat
λk, λâk!λâ >λ and ÎŒk, ÎŒâ
k!ÎŒâ >ÎŒ as k!â;
then we pick up B>0 such that
ÎŒââλâ(1+B)B ïżœÎŒâČâ >ÎŒ,
and we let b(t)=B/(1+t).As a result of the scheme, we have, for all kïżœ2,
sup0ïżœÏïżœt
âhkÏ Î©kâ1
t,Ï âZλâ(1+b),ÎŒâ;1Ïâbt/(1+b)
ïżœ ÎŽk, (12.4)
whereââ
k=2 ÎŽkïżœCÎŽ and Ωk is the deflection associated with the force field generated byh1+...+hk. Choosing t=Ï in (12.4) yields
suptïżœ0
âhkt âZλâ(1+B),ÎŒâ;1
tâBt/(1+B+t)ïżœ ÎŽk.
on landau damping 179
By Proposition 4.17, this implies that
suptïżœ0
âhkt âZλâ(1+B),ÎŒââλâ(1+B)B;1
tïżœ ÎŽk.
In particular, we have a uniform estimate on hkt in Zλâ,ÎŒâČ
â;1t . Summing up over k yields
for f=f0+ââ
k=1 hk the estimate
suptïżœ0
âf(t, ·)âf0âZλâ,ÎŒâČâ;1t
ïżœCÎŽ. (12.5)
Passing to the limit in the Newton scheme, one shows that f solves the non-linearVlasov equation with initial datum fi. (Once again we do not check the details; to berigorous one would need to establish moment estimates, locally in time, before passingto the limit.) This implies in particular that f stays non-negative at all times.
Applying Theorem 4.20 again, we deduce from (12.5) that
suptïżœ0
âf(t, ·)âf0âYλ,ÎŒt
ïżœCÎŽ;
or equivalently, with the notation used in Theorem 2.6,
suptïżœ0
âf(t, xâvt, v)âf0(v)âλ,ÎŒ ïżœCÎŽ. (12.6)
Moreover, ïżœ=â«
Rd f dv satisfies similarly
suptïżœ0
âïżœ(t, ·)âFλât+ÎŒâ ïżœCÎŽ.
It follows that |ïżœ(t, k)|ïżœCÎŽeâ2Ïλâ|k|teâ2ÏÎŒâ|k| for any k ïżœ=0. On the one hand, bySobolev embedding, we deduce that for any râN,
âïżœ(t, ·)âăïżœăâCr(Td) ïżœCrÎŽeâ2ÏλâČt;
on the other hand, multiplying ïżœ by the Fourier transform of âW , we see that the forceF =F [f ] satisfies
|F (t, k)|ïżœCÎŽeâ2ÏλâČ|k|teâ2ÏÎŒâČ|k| for all t ïżœ 0 and all kâZd, (12.7)
for some λâČ>λ and ÎŒâČ>ÎŒ.Now, from (12.6) we have, for any (k, η)âZdĂRd and any tïżœ0,
|f(t, k, η+kt)âf 0(η)|ïżœCÎŽeâ2ÏÎŒâČ|k|eâ2ÏλâČ|η|; (12.8)
180 c. mouhot and c. villani
so|f(t, k, η)|ïżœ |f 0(η+kt)|+CÎŽeâ2ÏÎŒâČ|k|eâ2ÏλâČ|η+kt|. (12.9)
In particular, for any k ïżœ=0, and any ηâRd,
f(t, k, η) =O(eâ2ÏλâČt). (12.10)
Thus f is asymptotically close (in the weak topology) to its spatial average
g = ăfă=â«
Td
f dx.
Taking k=0 in (12.8) shows that, for any ηâRd,
|g(t, η)âf 0(η)|ïżœCÎŽeâ2ÏλâČ|η|. (12.11)
Also, from the non-linear Vlasov equation, for any ηâRd we have
g(t, η) = fi(0, η)ââ« t
0
â«Td
â«Rd
F (Ï, x)·âvf(Ï, x, v)eâ2iÏη·v dv dx dÏ
= fi(0, η)â2iÏâlâZd
â« t
0
F (Ï, l)·ηf(Ï,âl, η) dÏ.
Using the bounds (12.7) and (12.10), it is easily shown that the above time-integralconverges exponentially fast as t!â, with rate O(eâλâČâČt) for any λâČâČ<λâČ, to its limit
gâ(η) = fi(0, η)â2iÏâlâZd
â« â
0
F (Ï, l)·ηf(Ï,âl, η) dÏ. (12.12)
By passing to the limit in (12.11) we see that
|gâ(η)âf 0(η)|ïżœCÎŽeâ2ÏλâČ|η|,
and this concludes the proof of Theorem 2.6.
13. Non-analytic perturbations
Although the vast majority of studies of Landau damping assume that the perturbationis analytic, it is natural to ask whether this condition can be relaxed. As we noticed inRemark 3.5, this is the case for the linear problem. As for non-linear Landau damping,once the analogy with KAM theory has been identified, it is anybodyâs guess that the
on landau damping 181
answer might come from a Moser-type argument. However, this is not so simple, be-cause the âloss of convergenceâ in our argument is much more severe than the âloss ofregularityâ which Moserâs scheme allows one to overcome.
For instance, the second-order correction h2 satisfies
âth2+v ·âxh2+F [f1]·âvh2+F [h2]·âvf1 =âF [h1]·âvh1.
The action of F [f1] is to curve trajectories, which does not help in our estimates. Dis-carding this effect and solving by Duhamelâs formula and Fourier transform, we obtain,with S=âF [h1]·âvh1 and ïżœ2=
â«Rd h2 dv,
ïżœ2(t, k)ïżœâ« t
0
K0(tâÏ, k)ïżœ2(Ï, k) dÏ
+2iÏ
â« t
0
âlâZd
(kâl)W (kâl)ïżœ2(Ï, kâl)âvh1(Ï, l, k(tâÏ)) dÏ
+â« t
0
S(Ï, k, k(tâÏ)) dÏ.
(13.1)
(The term with K0 includes the contribution of âvf0.)Our regularity/decay estimates on h1 will never be better than those on the solution
of the free transport equation, i.e., hi(xâvt, v), where hi=fiâf0. Let us forget aboutthe effect of K0 in (13.1), replace the contribution of S by a decaying term A(kt). Letus choose d=1 and assume that hi(l, ·)=0 if l ïżœ=±1. For k>0, let us use the long-timeapproximation
hi(â1, k(tâÏ)âÏ) 1[0,t](Ï) dÏ ïżœ c
k+1ÎŽkt/(k+1), c =
â«R
hi(â1, s) ds = hi(â1, 0).
Note that c ïżœ=0 in general. Plugging all these simplifications into estimate (13.1) andchoosing W (k)=1/|k|1+Îł suggests the a-priori simpler equation
Ï(t, k) =A(kt)+ckt
(k+1)Îł+2Ï
(kt
k+1, k+1
). (13.2)
Replacing Ï(t, k) by Ï(t, k)/A(kt) reduces to A=1, and then we can solve this equationby power series as in §7.1.3, obtaining
Ï(t, k)ïżœA(kt)e(ckt)1/(Îł+2). (13.3)
With a polynomial deterioration of the rate, we could use a regularization argument, butthe fractional exponential is much worse.
182 c. mouhot and c. villani
However, our bounds are still good enough to establish decay for Gevrey perturba-tions. Let us agree that a function f=f(x, v) lies in the Gevrey class GÎœ , Îœïżœ1, if
|f(k, η)|= O(exp(âc|(k, η)|1/Îœ)) for some c> 0;
in particular G1 means analytic. (An alternative convention would be to require the nthderivative to be O((n!)Μ).) As we shall explain, we can still get non-linear Landau damp-ing if the initial datum fi lies in GΜ for Μ close enough to 1. Although this is still quitedemanding, it already shows that non-linear Landau damping is not tied to analyticityor quasi-analyticity, and holds for a large class of compactly supported perturbations.
Theorem 13.1. Let λ>0. Let f0=f0(v)ïżœ0 be an analytic homogeneous profilesuch that â
nâNd0
λn
n!âân
v f0âL1(Rd) <â,
and let W =W (x) satisfy |W (k)|=O(1/|k|2). Assume that condition (L) from §2.2 holds.Let Îœâ(1, 1+Ξ) with Ξ=1/Ο(d, Îł), where Ο was defined in (11.6). Let ÎČ>0 and α<1/Îœ.Then there is Δ>0 such that if
ÎŽ := supkâZd
ηâRd
|(fiâf 0)(k, η)|eλ|η|1/Îœ
eλ|k|1/Μ
+â«
Td
â«Rd
|(fiâf0)(x, v)|eÎČ|v| dv dx ïżœ Δ,
then as t!+â the solution f=f(t, x, v) of the non-linear Vlasov equation on TdĂRd
with interaction potential W and initial datum fi satisfies
|f(t, k, η)âfâ(η)|= O(ÎŽeâctα
) for all (k, η)âZdĂRd
and
âF (t, ·)âCr(Td) = O(ÎŽeâctα
) for all râN
for some c>0 and some homogeneous Gevrey profile fâ, where F stands for the self-consistent force.
Remark 13.2. In view of (13.3), one may hope that the result remains true for Ξ=2.Proving this would require much more precise estimates, including among other thingsa qualitative improvement of the constants in Theorem 4.20 (recall Remark 4.23).
Remark 13.3. One could also relax the analyticity of f0, but there is little incentiveto do so.
on landau damping 183
Sketch of proof. We first decompose hi=fiâf0, using truncation by a smooth par-tition of unity in Fourier space,
hi =ââ
n=0
Fâ1(hiÏn)âĄââ
n=0
hni ,
where F is the Fourier transform. Each bump function Ïn should be localized aroundthe domain (in Fourier space)
Dn = {(k, η)âZdĂRd : nK ïżœ |(k, η)|ïżœ (n+1)K},
for some exponent K>1; but at the same time Fâ1(Ïn) should be exponentially decreas-ing in v. To achieve this, we let
Ïn = 1Dn âÎł and Îł(η) = eâÏ|η|2 .
Then Fâ1(Ïn)=Fâ1(1Dn) Îł has Gaussian decay, independently of n; so there is a uni-form bound on â«
Td
â«Rd
|hni (x, v)|eÎČ|v| dv dx
for some ÎČ>0.On the other hand, if (k, η)âDn and (kâČ, ηâČ) /âDnâ1âȘDnâȘDn+1, then
|kâkâČ|+|ηâηâČ|ïżœ cnKâ1
for some c>0. From this one obtains, after simple computations,
|Ïn(k, η)|ïżœ 1(nâ1)Kïżœ|(k,η)|ïżœ(n+2)K +Ceâcn2(Kâ1)eâc(|k|2+|η|2).
So (with constants C and c changing from line to line)
|hni (k, η)|ïżœ Ceâλ|k|1/Îœ
eâλ|η|1/Îœ
1(nâ1)Kïżœ|(k,η)|ïżœ(n+2)K +Ceâcn2(Kâ1)eâc(|k|+|η|)
ïżœ C max{eâλ(nâ1)K/Îœ/2, eâcn2(Kâ1)}
eâλn(|k|+|η|),
where
λn ⌠12λ(n+2)â(1â1/Îœ)K .
If Kïżœ2 then 2(Kâ1)>K/Îœ; so
âhni âYλn,λn ïżœCeâλnK/Îœ/2.
184 c. mouhot and c. villani
Then we may apply Theorem 4.20 to get a bound on hni in the space Z λn,λn;1 with
λn= 12 λn, at the price of a constant exp(C(n+2)(1â1/Îœ)K). Assuming KÎœ>(1â1/Îœ)K,
i.e., Μ<2, we end up with
âhni âZλn,λn;1 = O
(eâcnK/Îœ)
, λn = 12 λn. (13.4)
Then we run the iteration scheme of §8 with the following modifications: (1) insteadof hn(0, ·)=0, choose hn(0, ·)=hn
i , and (2) choose regularity indices λnâŒÎ»n which tendto zero as n tends to infinity. This generates an additional error term of size O
(eâcnK/Îœ)
,and imposes that λnâλn+1 be of order nâ[(1â1/Îœ)K+1]. When we apply the bilinearestimates from §6, we can take λâλ to be of the same order; so α=αn and Δ=Δn canbe chosen proportional to nâ[(1â1/Îœ)K+1]. Then the large constants coming from thetime-response will be, as in §11, of order nqecnr
, with qâN and r=[(1â1/Îœ)K+1]Ο, andthe scheme will still converge like O(eâcns
) for any s<K/Μ, provided that K/Μ>r, i.e.,
Îœâ1+Îœ
K<
1Ο.
The rest of the argument is similar to what we did in §§10â12. In the end the decayrate of any non-zero mode of the spatial density ïżœ is controlled by
âân=0
eâcns
eâλnt ïżœ( ââ
n=0
eâcns
)supnïżœ0
eâcns
eâcnâ(1â1/Îœ)t ïżœCeâcts/K
,
and the result follows since s/K is arbitrarily close to 1/Μ.
Remark 13.4. An alternative approach to Gevrey regularity consists in rewriting thewhole proof with the help of Gevrey norms such as
âfâCλΜ
=ââ
n=0
λnâf (n)âân!Îœ
and âfâFλΜ
=âkâZ
e2Ïλ|k|1/Îœ |f(k)|,
which satisfy the algebra property for any Îœïżœ1. Then one can hybridize these norms,rewrite the time-response in this setting, estimate fractional exponential moments of thekernel, etc. The only part which does not seem to adapt to this strategy is §9 where theanalyticity is crucially used for the local result.
Remark 13.5. In a more general Cr context, we do not know whether decay holdsfor the non-linear VlasovâPoisson equation. Speculations about this issue can be foundin [55] where it is shown that (unlike in the linearized case) one needs more than onederivative on the perturbation. As a first step in this direction, we mention that ourmethods imply a bound like O(ÎŽ/(1+t)râr) for times t=O(1/ÎŽ), where r is a constant
on landau damping 185
and r>r, as soon as the initial perturbation has norm ÎŽ in a functional space Wr involvingr derivatives in a certain sense. The reason why this is non-trivial is that the natural timescale for non-linear effects in the VlasovâPoisson equation is not O(1/ÎŽ), but O(1/
âÎŽ ),
as predicted by OâNeil [75] and very well checked in numerical simulations [61].(17) Letus sketch the argument in a few lines. Assume that (for some positive constants c and C)
hi =ââ
n=0
hni , âhn
i âZλn,λn;1 ïżœ Cn
2rnand λn =
cn
2n. (13.5)
Then we may run the Newton scheme again choosing αnâŒcn/2n, cn=O(ÎŽ2â(râr1)n) andΔn=câČÎŽ. Over a time-interval of length O(1/ÎŽ), Theorem 7.7 (ii) only yields a multiplica-tive constant O(ecÎŽt/α9
n)=O(210n). In the end, after Sobolev injection again, we recovera time-decay on the force F like
ÎŽ
âân=0
2nr22ânreâλnt ïżœCÎŽ supnïżœ0
(2ân(râr3)eâλnt) ïżœ CÎŽ
(1+t)râr4,
as desired. Equation (13.5) means that hi is of size O(ÎŽ) in a functional space Wr whosedefinition is close to the LittlewoodâPaley characterization of a Sobolev space with r
derivatives. In fact, if the conjecture formulated in Remark 4.23 holds true, then itcan be shown that Wr contains all functions in the Sobolev space W r+r0,2 satisfying anadequate moment condition, for some constant r0.
14. Expansions and counterexamples
A most important consequence of the proof of Theorem 2.6 is that the asymptotic be-havior of the solution of the non-linear Vlasov equation can in principle be determined atarbitrary precision as the size of the perturbation tends to 0. Indeed, if we define gk
â(v) asthe large-time limit of hk (say in positive time), then âgkâ=O(ÎŽk), so f0+g1
â+...+gnâ
converges very fast to fâ. In other words, to investigate the properties of the time-asymptotics of the system, we may freely exchange the limits t!â and ÎŽ!0, performexpansions, etc. This at once puts on rigorous grounds many asymptotic expansions usedby various authorsâwho so far implicitly postulated the possibility of this exchange.
With this in mind, let us estimate the first corrections to the linearized theory, inthe regime of a very small perturbation and small interaction strength (which can beachieved by a proper scaling of physical quantities). We shall work in dimension d=1and in a periodic box of length L=1.
(17) Passing from O(1/â
ÎŽ ) to O(1/ÎŽ) is arguably an infinite-dimensional counterpart of Laplaceâsaveraging principle, which yields stability for certain Hamiltonian systems over time-intervals O(1/ÎŽ2)rather than O(1/ÎŽ).
186 c. mouhot and c. villani
14.1. Simple excitation
For a start, let us consider the case where the perturbation affects only the first spatialfrequency. We let
âą f0(v)=eâÏ v2be the homogeneous (Maxwellian) distribution;
âą Î”ïżœi(x)=Δ cos(2Ïx) be the initial space density perturbation;âą Î”ïżœi(x)Ξ(v) be the initial perturbation of the distribution function; we denote by
Ï the Fourier transform of Ξ;⹠αW be the interaction potential, with W (âx)=W (x). We do not specify its form,
but it should satisfy the assumptions in Theorem 2.6.We work in the asymptotic regime Δ!0 and α!0. The parameter Δ measures
the size of the perturbation, while α measures the strength of the interaction; afterdimensional change, if W is an inverse power, α can be thought of as an inverse power ofthe ratio (Debye length)/(perturbation wavelength). We will not write norms explicitly,but all our computations can be made in the norms introduced in §4, with small lossesin the regularity indicesâas we have done throughout the paper.
The first-order correction h1=O(Δ) to f0 is provided by the solution of the linearizedequation (3.3), here taking the form
âth1+v ·âxh1+F [h1]·âvf0 = 0,
with initial datum h1(0, ·)=hi :=fiâf0. As in §3 we get a closed equation for the asso-ciated density ïżœ[h1],
ïżœ[h1](t, k) = hi(k, kt)â4Ï2αW (k)â« t
0
ïżœ[h1](Ï, k)eâÏ(k(tâÏ))2(tâÏ)k2 dÏ.
It follows that ïżœ[h1](t, k)=0 for k ïżœ=±1, so the behavior of ïżœ[h1] is entirely determined byu1(t)=ïżœ[h1](t, 1) and uâ1(t)=ïżœ[h1](t,â1), which satisfy
u1(t) =Δ
2Ï(t)â4Ï2αW (1)
â« t
0
u1(Ï)eâÏ(tâÏ)2(tâÏ) dÏ =Δ
2[Ï(t)+O(α)]. (14.1)
(This equation can be solved explicitly [11, equation (6)], but we only need the expan-sion.) Similarly,
uâ1(t) =Δ
2Ï(ât)â4Ï2αW (1)
â« t
0
uâ1(Ï)eâÏ(tâÏ)2(tâÏ) dÏ =Δ
2[Ï(ât)+O(α)]. (14.2)
The corresponding force, in Fourier transform, is given by
F 1(t, 1) =â2iÏαW (1)u1(t) and F 1(t,â1) = 2iÏαW (1)uâ1(t).
on landau damping 187
From this we also deduce the Fourier transform of h1 itself:
h1(t, k, η) = hi(k, η+kt)â4Ï2αW (k)â« t
0
ïżœ[h1](Ï, k)eâÏ(η+k(tâÏ))2(η+k(tâÏ))·k dÏ.
(14.3)This is 0 if k ïżœ=±1, while
h1(t, 1, η) =Δ
2Ï(η+t)â4Ï2αW (1)
â« t
0
u1(Ï)eâÏ(η+(tâÏ))2(η+(tâÏ)) dÏ
=Δ
2[Ï(η+t)+O(α)]
(14.4)
and
h1(t,â1, η) =Δ
2Ï(ηât)+4Ï2αW (1)
â« t
0
uâ1(Ï)eâÏ(ηâ(tâÏ))2(ηâ(tâÏ)) dÏ
=Δ
2[Ï(ηât)+O(α)].
(14.5)
To get the next order correction, we solve, as in §10,
âth2+v ·âxh2+F [h1]·âvh2+F [h2]·(âvf0+âvh1) =âF [h1]·âvh1,
with zero initial datum. Since h2=O(Δ2), we may neglect the terms F [h1]·âvh2 andF [h2]·âvh1 which are both O(αΔ3). So it is sufficient to solve
âthâČ2+v ·âxhâČ
2+F [hâČ2]·âvf0 =âF [h1]·âvh1 (14.6)
with vanishing initial datum. As t!â, we know that the solution hâČ2(t, x, v) is asymp-
totically close to its spatial average ăhâČ2ă=â«
Td hâČ2 dx. Taking the integral over Td in (14.6)
yieldsâtăhâČ
2ă=âăF [h1]·âvh1ă.Since h1 converges to ăhiă, the deviation of f to ăfiă is given, at order Δ2, by
g(v) =ââ« â
0
ăF [h1]·âvh1ă(t, v) dt =ââ« â
0
âkâZ
F [h1](t,âk)·âvh1(t, k, v) dt.
Applying the Fourier transform and using (14.1), (14.2), (14.4) and (14.5), we deduce
g(η) =ââ« â
0
âkâZ
F [h1](t,âk)·âvh1(t, k, η) dt
=ââ« â
0
F [h1](t,â1)(2iÏη)h1(t, 1, η) dtââ« â
0
F [h1](t, 1)(2iÏη)h1(t,â1, η) dt
= Ï2Δ2αW (1)η(â« â
0
Ï(ât)Ï(η+t) dtââ« â
0
Ï(t)Ï(ηât) dt+O(α))
=âÏ2Δ2αW (1)η(â« â
ââÏ(t)Ï(ηât) sign(t) dt+O(α)
).
188 c. mouhot and c. villani
Summarizing,â§âȘâȘâšâȘâȘâ©limt!â f(t, k, η) = 0, if k ïżœ= 0,
limt!â f(t, 0, η) = fi(t, 0, η)âΔ2α(Ï2W (1))η
(â« â
ââÏ(t)Ï(ηât) sign(t) dt+O(α)
).
(14.7)Since Ï is an arbitrary analytic profile, this simple calculation already shows that
the asymptotic profile is not necessarily the spatial mean of the initial datum.Assuming Î”ïżœÎ±, higher-order expansions in α can be obtained by bootstrap on the
equations (14.1), (14.2), (14.4) and (14.5); for instance,
limt!â f(t, 0, η)
= fi(0, η)âΔ2α(Ï2W (1))η⫠â
ââÏ(t)Ï(ηât) sign(t) dt
âΔ2α2(2Ï2W (1))2η⫠â
0
â« t
0
[(Ï(η+t)Ï(âÏ)âÏ(ηât)Ï(Ï))eâÏ(tâÏ)2(tâÏ)
+Ï(Ï)Ï(ât)eâÏ(η+(tâÏ))2(η+tâÏ)
+Ï(âÏ)Ï(t)eâÏ(ηâ(tâÏ))2(ηâtâÏ)]dÏ dt+O(Δ2α3).
What about the limit in negative time? Reversing time is equivalent to changingf(t, x, v) into f(t, x,âv) and letting time go forward. So we define S(v):=âv and
T (Ï)(η) := Δ2αÏ2W (1)η⫠â
ââÏ(t)Ï(ηât) sign(t) dt;
then T (Ï S)=T (Ï) S, which means that the solutions constructed above are alwayshomoclinic at order O(Δ2α). The same is true for the more precise expansions at orderO(Δ2α2), and in fact it can be checked that the whole distribution f2 is homoclinic;in other words, f is homoclinic up to possible corrections of order O(Δ4). To exhibitheteroclinic deviations, we shall consider more general perturbations.
14.2. General perturbation
Let us now consider a âgeneralâ initial datum fi(x, v) close to f0(v), and expand thesolution f . We write ΔÏk(η)=(fiâf0) (k, η) and ïżœm=ïżœ[hm]. The interaction potentialis assumed to be of the form αW with Î±ïżœ1 and W (x)=W (âx). The first equations ofthe Newton scheme are
ïżœ1(t, k) = ΔÏk(kt)â4Ï2αW (k)â« t
0
ïżœ1(Ï, k)f 0(k(tâÏ))|k|2(tâÏ) dÏ, (14.8)
on landau damping 189
h1(t, k, η) = ΔÏk(η+kt)â4Ï2αW (k)â« t
0
ïżœ1(Ï, k)f 0(η+k(tâÏ))k ·(η+k(tâÏ)) dÏ, (14.9)
h2(t, k, η) =â4Ï2αW (k)â« t
0
ïżœ2(Ï, k)f 0(η+k(tâÏ))k ·(η+k(tâÏ)) dÏ
â4Ï2α
â« t
0
âlâZd
W (l)ïżœ1(Ï, l)h1(Ï, kâl, η+k(tâÏ))l·(η+k(tâÏ)) dÏ
â4Ï2α
â« t
0
âlâZd
W (l)ïżœ2(Ï, l)h1(Ï, kâl, η+k(tâÏ))l·(η+k(tâÏ)) dÏ
â4Ï2α
â« t
0
âlâZd
W (l)ïżœ1(Ï, l)h2(Ï, kâl, η+k(tâÏ))l·(η+k(tâÏ)) dÏ,
(14.10)
ïżœ2(t, k) =â4Ï2αW (k)â« t
0
ïżœ2(Ï, k)f 0(k(tâÏ))|k|2(tâÏ) dÏ
â4Ï2α
â« t
0
âlâZd
W (l)ïżœ1(Ï, l)h1(Ï, kâl, k(tâÏ))l·k(tâÏ) dÏ
â4Ï2α
â« t
0
âlâZd
W (l)ïżœ2(Ï, l)h1(Ï, kâl, k(tâÏ))l·k(tâÏ) dÏ
â4Ï2α
â« t
0
âlâZd
W (l)ïżœ1(Ï, l)h2(Ï, kâl, k(tâÏ))l·k(tâÏ) dÏ.
(14.11)
Here k runs over Zd.From (14.8) and (14.9) we see that ïżœ1 and h1 depend linearly on Δ, and that
ïżœ1(t, k) = Δ[Ïk(kt)+O(α)] and h1(t, k, η) = Δ[Ïk(η+kt)+O(α)]. (14.12)
Then from (14.10) and (14.11), ïżœ2 and h2 are O(Δ2 α); so by plugging (14.12) intothese equations we obtain
ïżœ2(t, k) =â4Ï2Δ2α
â« t
0
âlâZd
W (l)Ïl(lÏ)Ïkâl(ktâlÏ)l·k(tâÏ) dÏ +O(Δ2α2)+O(Δ3α)
(14.13)
and
h2(t, k, η) =â4Ï2Δ2α
â« t
0
âlâZd
W (l)Ïl(lÏ)Ïkâl(η+ktâlÏ)l·(η+k(tâÏ)) dÏ
+O(Δ2α2)+O(Δ3α).
(14.14)
190 c. mouhot and c. villani
We plug these bounds again into the right-hand side of (14.10) to find that
h2(t, 0, η) = IIΔ(t, η)+IIIΔ(t, η)+O(Δ3 α3), (14.15)
where
IIΔ(t, η) =â4Ï2α
â« t
0
âlâZd
(l·η)W (l)ïżœ1(Ï, l)h1(Ï,âl, η) dÏ
is quadratic in Δ, and IIIΔ(t, η) is a third-order correction:
IIIΔ(t, η) = 16Ï4Δ3α2â
m,lâZd
W (l)W (m)
Ăâ« t
0
â« Ï
0
Ïm(ms)[Ïlâm(lÏâms)Ïâl(ηâlÏ)(l·m)(Ïâs)
+Ïl(lÏ)Ïâlâm(ηâlÏâms)m·(ηâl(Ïâs))](l·η) ds dÏ.
(14.16)
If f 0 is even, changing Ïk into Ïk(â·) and η into âη amounts to change k intoâk at the level of (14.8) and (14.9); but then IIΔ is invariant under this operation. Weconclude that f is always homoclinic at second order in Δ, and we consider the influenceof the third-order term (14.16). Let
C[Ï](η) := limt!â IIIΔ(t, η).
After some relabeling, we find that
C[Ï](η) = 16Ï4Δ3α2â
k,lâZd
W (k) W (l)
Ăâ« â
0
â« t
0
Ïl(lÏ)[Ïkâl(ktâlÏ)Ïâk(ηâkt)(k ·l)(tâÏ)
+Ïk(kt)Ïâkâl(ηâktâlÏ)l·(ηâk(tâÏ))](k ·η) dÏ dt.
(14.17)
Now assume that Ïâk=ÏÏk with Ï=±1 (Ï=1 means that the perturbation is even in x,Ï=â1 that it is odd.) Using the symmetry (k, l)$(âk,âl) one can check that
C[Ï S] S = ÏC[Ï],
where S(z)=âz. In particular, if the perturbation is odd in x, then the third-ordercorrection imposes a heteroclinic behavior for the solution, as soon as C[Ï] ïżœ=0.
To construct an example where C[Ï] ïżœ=0, we set d=1, f0=Gaussian,
fiâf0 = sin(2Ïx)Ξ1(v)+sin(4Ïx)Ξ2(v),
on landau damping 191
Ï1=âÏâ1= 12 Ξ1 and Ï2=âÏâ2= 1
2 Ξ2. The six pairs (k, l) contributing to (14.17) are(â1, 1), (1,â1), (1, 2), (2, 1), (â1,â2) and (â2,â1), By playing with the respective sizesof W (1) and W (2) (which amounts in fact to changing the size of the box), it is sufficientto consider the terms with coefficient W (1)2, i.e., the pairs (â1, 1) and (1,â1). Thenthe corresponding bit of C[Ï](η) is
â16Ï4Δ3α2W (1)2η⫠â
0
â« t
0
[Ï1(Ï)Ï1(η+t)Ï2(ât+Ï)(tâÏ)
+Ï1(Ï)Ï1(t)Ï2(η+tâÏ)(η+tâÏ)
+Ï1(âÏ)Ï1(ηât)Ï2(t+Ï)(tâÏ)
+Ï1(âÏ)Ï1(t)Ï2(ηât+Ï)(tâÏâη)]dÏ dt.
If we let Ï1 and Ï2 vary in such a way that they become positive and almost concentratedon R+, the only remaining term is the one in Ï1(Ï)Ï2(η+tâÏ)Ï1(t), and its contributionis negative for η>0. So, at least for certain values of W (1) and W (2) there is a choiceof analytic functions Ï1 and Ï2, such that C[Ï] ïżœ=0. This demonstrates the existence ofheteroclinic trajectories.
To summarize: At first order in Δ, the convergence is to the spatial average; atsecond order there is a homoclinic correction; at third order, if at least three modes withzero sum are excited, there is a possibility for heteroclinic behavior.
Remark 14.1. As pointed out to us by Bouchet, the existence of heteroclinic tra-jectories implies that the asymptotic behavior cannot be predicted on the basis of theinvariants of the equation and the interaction; indeed, the latter do not distinguish be-tween the forward and backward solutions.
15. Beyond Landau damping
We conclude this paper with some general comments about the physical implications ofLandau damping.
Remark 14.1 show in particular that there is no âuniversalâ large-time behavior ofthe solution of the non-linear Vlasov equation in terms of just, say, conservation laws andthe initial datum; the dynamics also have to enter explicitly. One can also interpret thisas a lack of ergodicity: the non-linearity is not sufficient to make the system explore thespace of all âpossibleâ distributions and to choose the most favorable one, whatever thismeans. Failure of ergodicity for a system of finitely many particles was already knownto occur, in relation to the KAM theorem; this is mentioned e.g. in [62, p. 257] for thevortex system. There it is hoped that such behavior disappears as the dimension tends
192 c. mouhot and c. villani
to infinity; but now we see that it also exists even in the infinite-dimensional setting ofthe Vlasov equation.
At first, this seems to be bad news for the statistical theory of the Vlasov equation,pioneered by Lynden-Bell [58] and explored by various authors [16], [65], [80], [89], [94],since even the sophisticated variants of this theory try to predict the likely final statesin terms of just the characteristics of the initial data. In this sense, our results providesupport for an objection raised by Isichenko [42, p. 2372] against the statistical theory.
However, looking more closely at our proofs and results, proponents of the statisticaltheory will have a lot to rejoice about.
To start with, our results are the first to rigorously establish that the non-linearVlasov equation does enjoy some asymptotic âstabilizationâ property in large time, with-out the help of any extra diffusion or ensemble averaging.
Next, the whole analysis is perturbative: each stable spatially homogeneous distri-bution will have its small âbasin of dampingâ, and it may be that some distributions areâmuch more stableâ than others, say in the sense of having a larger basin.
Even more importantly, in §7 we have crucially used the smoothness to overcome thepotentially destabilizing non-linear effects. So any theory based on non-smooth functionsmight not be constrained by Landau damping. This certainly applies to a statisticaltheory, for which smooth functions should be a zero-probability set.
Finally, to overcome the non-linearity, we had to cope with huge constants (evenqualitatively larger than those appearing in classical KAM theory). If one believes in theexplanatory virtues of proofs, these large constants might be the indication that Landaudamping is a thin effect, which might be neglected when it comes to predict the âfinalâstate in a âturbulentâ situation.
Further work needs to be done to understand whether these considerations applyequally to the electrostatic and gravitational cases, or whether the electrostatic case isfavored in these respects; and what happens in âlowâ regularity.
Although the underlying mathematical and physical mechanisms differ, non-linearLandau damping (as defined by Theorem 2.6) may arguably be to the theory of theVlasov equation what the KAM theorem is to the theory of Hamiltonian systems. Likethe KAM theorem, it might be conceptually important in theory and practice, and stillbe severely limited.(18)
Beyond the range of application of KAM theory lies the softer, more robust weakKAM theory developed by Fathi [26] in relation to AubryâMather theory. By a nice co-
(18) It is a well-known scientific paradox that the KAM theorem was at the same time tremendouslyinfluential in the science of the twentieth century, and so restrictive that its assumptions are essentiallynever satisfied in practice.
on landau damping 193
incidence, a Vlasov version of the weak KAM theory has just been developed by Gangboand Tudorascu [29], although with no relation to Landau damping. Making the connec-tion is just one of the many developments which may be explored in the future.
Appendix
In this appendix we gather some elementary tools, our conventions, and some remindersabout calculus. We write N0={0, 1, 2, ... }.
A.1. Calculus in dimension d
For nâNd0 we define
n! = n1! ... nd!
and for n, mâNd0 we set (
n
m
)=(
n1
m1
)...
(nd
md
).
For zâCd and nâZd we let
âzâ= |z1|+...+|zd|, zn = zn11 ... znd
n âC and |z|n = |zn|.
In particular, for zâCd we have
eâzâ = e|z1|+...+|zd| =â
nâNd0
âzân
n!.
We may write e|z| instead of eâzâ.
A.2. Multi-dimensional differential calculus
The Leibniz formula for functions f, g: R!R is
(fg)(n) =nâ
m=0
(n
m
)f (m)g(nâm),
where of course f (n)=dnf/dxn. The expression of derivatives of composed functions isgiven by the Faa di Bruno formula:
(f G)(n) =â
âj jmj=n
n!m1! ...mn!
(f (m1+...+mn) G)nâ
j=1
(G(j)
j!
)mj
.
194 c. mouhot and c. villani
These formulae remain valid in several dimensions, provided that one defines, for amulti-index n=(n1, ..., nd),
f (n) =ân1
âxn11
...ând
âxnd
d
f.
They also remain true if (â1, ..., âd) is replaced by a d-tuple of commuting derivationoperators.
As a consequence, we shall establish the following Leibniz-type formula for operatorswhich are combinations of gradients and multiplications.
Lemma A.1. Let f and g be functions of vâRd and a, bâCd. Then for any nâNd,
(âv+(a+b))n(fg) =â
0ïżœmïżœn
(n
m
)(âv+a)mf(âv+b)nâmg.
Proof. The right-hand side is equal to
â0ïżœqïżœmïżœn
0ïżœrïżœnâm
(n
m
)(m
q
)(nâm
r
)âq
vfârvgamâqbnâmâr.
After changing indices p=q+r and s=mâq, this becomes
â0ïżœrïżœpïżœn
0ïżœsïżœnâp
(n
p
)(p
r
)(nâp
s
)âr
vgâpârv fasbnâpâs =
âp
(n
p
)âp
v(fg)(a+b)nâp
= (âv+(a+b))n(fg).
A.3. Fourier transform
For a function f : Rd!R, we define
f(η) =â«
Rd
eâ2iÏη·vf(v) dv. (A.1)
Then we have the usual formulae
f(v) =â«
Rd
f(η)e2iÏη·v dη and âf(η) = 2iÏηf(η).
Let TdL=Rd/LZd. For a function f : Td
L!R, we define
f (L)(k) =â«
TdL
eâ2iÏk·x/Lf(x) dx. (A.2)
on landau damping 195
Then we have
f(x) =1Ld
âkâZd
f (L)(k)e2iÏk·x/L and âf(L)
(k) = 2iÏk
Lf (L)(k).
For a function f : TdLĂRd!R, we define
f (L)(k, η) =â«
TdL
â«Rd
eâ2iÏk·x/Leâ2iÏη·vf(x, v) dv dx, (A.3)
so that the reconstruction formula reads
f(x, v) =1Ld
âkâZd
â«Rd
f (L)(k, η)e2iÏk·x/Le2iÏη·v dv.
When L=1 we do not specify it: so we just write
f = f (1) and f = f (1).
(There is no risk of confusion since, in that case, (A.3) and (A.1) coincide.)
A.4. Fixed-point theorem
The following theorem is one of the many variants of the Picard fixed-point theorem. Wewrite B(0, R) for the closed ball of center 0 and radius R.
Theorem A.2. (Fixed-point theorem) Let E be a Banach space, F : E!E andR=2âF (0)â. If F : B(0, R)!E is 1
2 -Lipschitz, then it has a unique fixed point in B(0, R).
Proof. Uniqueness is obvious. To prove existence, run the classical Picard iterativescheme initialized at 0: x0=0, x1=F (0), x2=F (F (0)), etc. It is clear that (xn)ân=1 is aCauchy sequence and âxnâïżœâF (0)â(1+...+1/2n)ïżœ2âF (0)â, so xn converges in B(0, R)to a fixed point of F .
A.5. Plemelj formula
The Plemelj formula states that, in DâČ(R),
1xâi0
= p.v.
(1x
)+iÏ ÎŽ0, (A.4)
or, equivalently,1
x+i0= p.v.
(1x
)âiÏ ÎŽ0. (A.5)
Let us give a complete proof of this formula which plays a crucial role in plasma physics.
196 c. mouhot and c. villani
Proof of (A.4). First we recall that
p.v.
(1x
)= lim
Δ!0
1|x|ïżœÎ”
x.
A more explicit expression can be given for this limit. Let Ï be an even function on R
with Ï(0)=1, ÏâLipâ©L1(R); in particularâ«R
1|x|ïżœÎ”
xÏ(x) dx = 0
and (1âÏ(x))/x is bounded. Then for any ÏâLip(R)â©L1(R),â«|x|ïżœÎ”
Ï(x)x
dx =â«|x|ïżœÎ”
Ï(x)x
Ï(x) dx+â«|x|ïżœÎ”
Ï(x)1âÏ(x)
xdx
=â«|x|ïżœÎ”
Ï(x)âÏ(0)x
Ï(x) dx+â«|x|ïżœÎ”
Ï(x)1âÏ(x)
xdx
!â«
R
Ï(x)âÏ(0)x
Ï(x) dx+â«
R
Ï(x)1âÏ(x)
xdx as Δ! 0.
(A.6)
Then (A.4) can be rewritten in the following more explicit way: for any ÏâLip(R)â©L1(R)and any Ï satisfying the above assumptions,
limλ!0+
â«R
Ï(x)xâiλ
dx =â«
R
Ï(x)âÏ(0)x
Ï(x) dx+â«
R
Ï(x)1âÏ(x)
xdx+iÏÏ(0). (A.7)
Now, to prove (A.7), let us pick such a function Ï and write, for λ>0,â«R
Ï(x)xâiλ
dx =â«
R
Ï(x)âÏ(0)xâiλ
Ï(x) dx+â«
R
Ï(x)1âÏ(x)xâiλ
dx+Ï(0)â«
R
Ï(x)xâiλ
dx.
As λ!0, the first two integrals on the right-hand side converge to the right-hand side of(A.6), and there is an extra term proportional to Ï(0); so it only remains to check thatâ«
R
Ï(x)xâiλ
dx! iÏ as λ! 0+. (A.8)
If (A.8) holds for some particular Ï satisfying the requested conditions, then (A.4)follows and it implies that (A.8) holds for any such Ï. So let us pick one particular Ï, sayÏ(x)=eâx2
, which can be extended throughout the complex plane into a holomorphicfunction. Then, since the complex integral is invariant under contour deformation,â«
R
eâx2
xâiλdx =
â«CΔ
eâz2
zâiλdz,
where CΔ is the complex contour made of the straight line (ââ,âΔ), followed by the half-circle DΔ={âΔeiΞ :0ïżœÎžïżœÏ}, followed by the straight line (Δ,â). The contributions ofboth straight lines cancel each other by symmetry, and only the integral on the half-circleDΔ remains. As λ!0 this integral approaches
â«DΔ
eâz2dz/z, which as Δ!0 becomes close
toâ«
DΔdz/z=iÏ, as was claimed.
on landau damping 197
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Clement Mouhot
University of CambridgeDPMMS, Centre for Mathematical SciencesWilberforce RoadCambridge CB3 [email protected]
Cedric Villani
Institut Henri Poincare & Universite de LyonInstitut Camille Jordan, Universite Claude Bernard43 Boulevard du 11 novembre 1918FR-69622 Villeurbanne [email protected]
Received December 10, 2009Received in revised form July 10, 2011