Generation and propagation of exponential weighted estimates to solutions of

non-linear collisional equations

Irene M. GambaDepartment of Mathematics and ICES

The University of Texas at Austin

Collaborators: A. Bobylev, Karlstad University.

Ricardo Alonso, UT Austin-Rice University,

Vlad Panferov, CSU, Northridge, CA,

Cedric Villani, ENS Lyon, France.

S. Harsha Tharkbushanam, ICES and PROS

more recently J. Canizo, S. Mischler, C. Mouhot (Paris IV)

Statistical transport from collisional kinetic modelsStatistical transport from collisional kinetic models

• Rarefied ideal gases-elastic:Rarefied ideal gases-elastic: classical conservativeclassical conservative Boltzmann Transport eq.Boltzmann Transport eq.

• Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature өb or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc.

•(Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor.

•Emerging applications from stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes: information percolation models, particle swarms in population dynamics,

‘v

‘v*

v

v*

C = number of particle in the box a = diameter of the spheresN=space dimension

η

elastic collisioninelastic collision

u.η the impact velocity

i.e. enough intersitial space

May be extended to multi-linear interactions

A general form statistical transport : The space-homogenous BTE with external heating sources Important examples from mathematical physics and social sciences:

The termmodels external heating sources:

•background thermostat (linear collisions), •thermal bath (diffusion)•shear flow (friction), •dynamically scaled long time limits (self-similar solutions).

Inelastic Collision u’= (1-β) u + β |u| σ , with σ the direction of elastic post-collisional relative velocity

‘v

‘v*

v

v*

η inelastic collision

Exact energy identity for a Maxwell type interaction models

Then f(v,t) → δ0 as t → ∞ to a singular concentrated measure (unless there is ‘source’)

Current issues of interest regarding energy dissipation: Can one tell the shape or classify possible stationary states and their asymptotics, such as self-similarityself-similarity? Non-Gaussian (or Maxwellian) statistics!

Reviewing Reviewing inelastic inelastic propertiesproperties

INELASTIC Boltzmann collision term: No classical H-Theorem if e = constant < 1

It dissipates total energy for e < 1 by Jensen's inequality:

Inelasticity brings loss of micro reversibilitybut keeps time irreversibilitytime irreversibility !!: That is, there are stationary states and, in some particular cases we can show stability to stationary and self-similar states (Multi-linear Maxwell molecule equations of collisional type and variable hard potentials for collisions with a background thermostat) However: Existence of NESSNESS: Non Equilibrium Statistical States (stable stationary states are non-Gaussian pdf’s)

Non-Equilibrium Stationary Statistical States

YesYes

?

(ARMA’09)

generation of moments estimates

generation of exponentially weighted lower bound

B. Wennberg~’98

Sharp Povzner estimatesSharp Povzner estimates

Summability of moments seriesSummability of moments series

VIII) Generation of exponential L1-weighted estimates (Mouhot’06) and better tails (Alonso,Canizo,IG and

Mouhot in progress)

(JMPA’08)

(JPS’04)

(I.G V.Panferov, c. Villani; ARMA’09)

then

Key property: Summability of series of moments of BTE solutionsKey property: Summability of series of moments of BTE solutions

Sharp Povzner estimates: optimal control of weights in `average’Sharp Povzner estimates: optimal control of weights in `average’

Angular Averaging Inequality: (Angular Averaging Inequality: (A.Bobylev, I.G., V.Panferov, JSP’04)A.Bobylev, I.G., V.Panferov, JSP’04)

and of γ (rate of the intramolecular potentials)

with

Our result extends the Bobylev-Povzner-type estimate (JSP'97) for d=3 and γ=1, (i.e. b(σ)=C)

to d > 1 and kernels with monotone angular dependence on its symmetric part satisfying **

(I.G., V.Panferov C.Villani; ARMA’08(I.G., V.Panferov C.Villani; ARMA’08

Elastic case: β=1 d-dimensions **

Corollary 2: In order to study the behavior of mp with p = ks/2 for a good choice of s, take

moments of evolution forced equations:

for r and R depending on the initial mass m0, energy bound m1 and some high order moments mp0

for some p > 1, depending on the 'heating' force coefficient.∗

The choice of s is done by setting: (shown in the pure diffusion case and bounded angular section γ=1 and stationary state)

Corollary: it is possible to choose s, such that

So, in order to control zp+1/2 we need to divide by Γ(a(p+1/2) + b) and find a suitable value of a such that we can get control of a corresponding recursion inequality relation that produces a geometric growth control for zP

Then

a careful choice of a = 4/3 and b < 1 cancels the coefficients for the two terms proportional to zp-1 , and the right hand side term (from the gain term) is controlled by a constant!!

Similarly for the other cases: diffusion with friction: s = 2.Self-similar (homogeneous cooling) s = 1Shear flow: at least s = 1 but anisotropy is admissible, so other direction might decay faster.

∂tzp +Add the time derivative to compute the Corresponding evolution estimate

In the elastic case with no sources, for 0 < γ ≤ 1 and b(θ) integrable: Exponential moments propagation (I.G. Panferov and Villani, arXiv’06, ARMA’09)

(i.e. a=1 and s=2)

moments of

equation

collision operator

Loss op.

Gain op.

Bernoulli type eq. can also “create” moments (Desvillates 93 B. Wennberg~’98) or “generate”or “generate”

In the elastic case, for 0 < γ ≤ 1 and b(θ)integrable: Propagation and generation estimates

1. Same argument holds for controlling moments of the derivatives of f(t,v) by iterative methods (R. Alonso &I.G. JMPA’08)

2. SS solutions to Elastic collisions with a cold thermostatElastic collisions with a cold thermostat for the choice of a= γ /2 and s= γ and existence (Alonso, Canizo, IG, Mischler, Mouhot, in preparation )

3. GenerationGeneration of moments for a= γ /4 and s= γ /2 (Mouhot JSP’06) for initial data with only 2+ moments

4. Improvement in moments generationgeneration by taking a=a(t, γ) to s=γ (Alonso, Canizo, IG, and Mouhot, in preparation )

(i.e. a=1 and s=2)

for r and R depending on the initial mass m0, energy bound m1 and some high order moments mp0

for some p > 1, ∗ but uniform in t ! but uniform in t !

• Summability of the series of moments is uniform intime Summability of the series of moments is uniform intime • Propagation of exponential moments Propagation of exponential moments

(IG, Vlad Panferov, Cedric Villani, arXiv.org’06 - ARMA’08, and R. Alonso and I.G- JMPA’08)

Upper point-wise uniform bounds for large energy tails for elastic hard spheres or γ-variable potentials in d-dimensions

Comparison principleComparison principle: Q is multi-linear, symmetric, conservative, and L1-contractive for its linear restriction (Crandall & Tartar ’80, also Vandenjapin & Bobylev 75, Kaniel & Shimbrot ’84, Lions ’94 )Remark: it also works in the space inhomogeneous case.

STRATEGY : Find a comparison theorem & construct a suitable barrier function ⇒ Compare to obtain point-wise bounds:

and

Crucial point:Crucial point:

In order to find the barrier probability distributionbarrier probability distribution:

we need

That can be obtain by the following key estimatekey estimate:

Tool: Carleman integral representationCarleman integral representation of

Remark: these propagation properties in L1 and L∞ Maxwellians weighted norms also hold for all the derivatives if initial data have all derivatives under such control (Ricardo Alonso and I.G.; ): we use iterative arguments.

Estimates for Existence theory: Estimates for Existence theory: Average angular estimates & weighted Young’s inequalitiesAverage angular estimates & weighted Young’s inequalities

R. Alonso and E. Carneiro’08, and R. Alonso and E. Carneiro, IG, 08

with

Angular average inequality

These two constants C depends linearly of the expression given above for the constant of the angular averaging lemma

Young’s inequality for variable hard potentials : 1 Young’s inequality for variable hard potentials : 1 ≥≥ λ ≥≥ 0 0

Hardy-Littlewood-Sobolev type inequality for soft potentials Hardy-Littlewood-Sobolev type inequality for soft potentials : : 0 > 0 > λ ≥≥ -n-n

by spectral-Lagrangian based methods for nonConservative energy (IG,H.Tharkabhushanam, JCP’09

Maxwell Molecules modelRescaling of spectral modes exponentially by the continuous spectrum with λ(1)=-2/3

Testing: BTE with (Gaussian) hot and (singular) cold Thermostatexplicit solution problem of asymptotics of same mass particles mixture

Moments calculations:Moments calculations:Explicit solutions problem IG & Bobylev and I.G., JSP06

Testing: BTE with cold Thermostat

Rescaling time by the Kinetic energyAnd velocity with the corresponding thermal Speed Moments of orderq ≥ 1.5 will become unbounded in time

Recent work related to the problem:

Cercignani'95(inelastic BTE derivation); Bobylev, JSP 97 (elastic,hard spheres in 3 d: propagation of L1-exponential estimates ); Bobylev, Carrillo and I.M.G., JSP'00 (inelastic Maxwell type interactions- self similarity- mean field); Bobylev, Cercignani , and with Toscani, JSP '02 &'03 (inelastic Maxwell type interactions); Bobylev, I.M.G, V.Panferov, C.Villani, JSP'04, CMP’04 (inelastic + heat sources); Mischler and Mouhout, Rodriguez Ricart JSP '06 (inelastic + self-similar hard spheres); Bobylev and I.M.G. JSP'06 (Maxwell type interactions-inelastic/elastic + thermostat), Bobylev, Cercignani and I.M.G arXiv.org,06 (CMP’09); (generalized multi-linear Maxwell type interactions-inelastic/elastic: global energy dissipation) I.M.G, V.Panferov, C.Villani, arXiv.org’07, ARMA’09 (elastic n-dimensional variable hard potentials Grad cut-off:: propagation of L1 and L∞-exponential estimates)C. Mouhot, CMP’06 (elastic, VHP, bounded angular cross section: creation of L1-exponential )R. Alonso and I.M.G., JMPA’08 (Grad cut-off, propagation of regularity bounds-elastic d-dim VHP)I.M.G. and Harsha Tarskabhushanam JCP’08(spectral-lagrangian solvers-computation of singulatities)R.Alonso, E.Carneiro (ArXiv.org08)(Young’s inequality for collisional integrals with integrable (grad cut-off) angular cross section)R.Alonso, E.Carneiro, I.M.G. ArXiv.org09 (weigthed Young’s inequality and Hardy Sobolev’s inequalities for collisional integrals with integrable (grad cut-fff)angular cross section)R. Alonso and I.M.G. ArXiv.org09, (Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section)Alonso, Canizo, I.M.G.,Mischler, Mouhot, in preparation (The homogeneous Boltzmann eqaution with a cold thermostat for variable hard potentials)Alonso, Canizo, I.M.G., Mouhot, in preparation (sharper decay for moments creation estimates for variable hard potentials)

For recent preprints and reprints see:www.ma.utexas.edu/users/gamba/research and references therein

Thank you very much for your attention!

Comments: . Tails are important to understand evolution of moments (well known….!!!) They depend on the rate of collision as a function of velocity. (Decay rates to equilibrium states depend on the angular cross section as one can get exact and best constant depending on b(θ) )

• Tails control methods to space inhomogeneous problems: may lead to local in x-space, global in v-space contrl of the solution BTE, …. but we do not how to do it yet…

•The use of Young and Hardy Littlewood Sobolev type of inequalities allows to revisit and/or extend the existence and regularity results of the space inhomogeneous BTE with soft potentials and angular cross sections that are just integrable (Grad cut-off assumption), with data between near two different Mawellians.

•Need to adjust hydrodynamic limits for non conservative phenomena: Hydrodynamic limits with energy dissipation lack of exact/local closure formulas-macroscopic equations may not have an accurate closed form.