Daniel Spirn and J. Douglas Wright- Gravity Induced Dispersion for Nearly-Flat Vortex Sheets

8/3/2019 Daniel Spirn and J. Douglas Wright- Gravity Induced Dispersion for Nearly-Flat Vortex Sheets

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GRAVITY INDUCED DISPERSION FOR NEARLY FLATVORTEX SHEETS

DANIEL SPIRNSchool of Mathematics, University of Minnesota,Minneapolis, MN 55455, USA

J. DOUGLAS WRIGHTDepartment of Mathematics, Drexel University,

Philadelphia, PA 19104, USAE-mail: [email protected]

Using techniques from the theory of oscillatory integrals, we prove rigorousestimates which show that th e linearization of the vortex sheet equations ofmotion about a quiescent state disperse under certain circumstances. Such dispersion is only possible only through th e joint effects of surface tension (whichdamps high frequency modes) and gravitation (which damps low frequencymodes).Keywords: vortex sheets; oscillatory integrals; water waves; dispersive esti-mates.

1. Dispersive effects in vortex sheetsConsider the flow of a pair of two-dimensional ideal fluids which shear pastone another along an interface on which surface tension acts - that is, wehave a "vortex sheet" system. Suppose that gravity acts on this system andthat the lighter fluid is above the heavier. Also, suppose that the velocityfield is curl-free at all points in the fluids not on the interface. The fluidscover all of R 2 . The equations of motion for this system are well-known(e.g. [5]):

1U t + (u . \7)u + - \7p + gk = 0 in the fluid domainpu = \7,[u n] = 0[p] =TK

6. = 0 in the fluid domainon the surface

on the surface.


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2Here, u is the velocity field, p is pressure, k = (O,l)t, P is density, g isgravitational acceleration, is the velocity potential, n is the upward unitnormal to the interface, K is curvature of the interface, r is the constant ofsurface tension and [Q] represents the jump of a quantity "Q" across theinterface. The scenario which occurs when upper fluid is replaced with avacuum is called the "water wave" problem.

The vortex sheet and water wave problems are typically referred toas "dispersive systems." One characteristic typical of dispersive systems isthat the amplitude of solutions decays at an algebraic rate, even thoughthe full system may conserves certain norms (e.g. [3]). The main purposeof our work [13] is pin down in full technical detail when this sort of decayshould be expected for linearized vortex sheets and at what rate this decaytakes place. Estimates of this sort can be extremely useful for proving theglobal-in-time existencea of solutions of the nonlinear problem (e.g. [2]), aswell as being of interest in and of themselves.

Since the velocity potential solves Laplace's equation, one can reformulate this problem entirely in terms of functions defined on the interface.There are several ways to do this, and we choose to use a method developedin [7]. The interface can be represented as a curve in R 2 parameterized byarclength. Call this parameter 0:, and suppose that

B(o:, t) = tangent angle w.r.t. horizontal of the interface at (0:, t)and

,(ct, t) = jump in tangential velocity at (ct, t).We omit the exact equations of motion in terms of Band , ; they can befound in [1J. This system is in equilibrium when the two fluids shear pastone another along a perfectly flat interface. That is, when B(ct, t) = 0 and,(ct, t) == "(. Linearizing about this state results in the system:

1atB = '2H(aa,)-2 Aan = 2ra;B + H(aQB) - 2'YaQ' - 2AgB,

where H (the Hilbert tranform) is given by the singular integralHf(o:) := ~ P . v . r f(O:'),do:'.1f JR 0: - 0:

The constant A = (Plower - Pupper) / (Plower + Pupper) is positive.

(1)

aLocal existence of solutions is known for arbitrary initial data , see [1, 8, 14], for instance.


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3Looking for plane wave solutions ei(Ea-w()t)v to (1) yields the disper-

sion relation (see [10])

with

(2)Note that if >.2(0 < 0 at wave number then the dispersion relation isimaginary and we expect exponential growth of that mode. Notice thatthe first term of > . 2 ( ~ ) represents the contribution from surface tension,the second that of shearing and the last comes from gravity. Importantlythe signs on surface tension and gravity terms are positive, while the shearterm is negative. Therefore we see that in the absence of surface tenstion,> . 2 ( ~ ) become negative for large wave numbers, and that in the absence ofgravitation the same is true for low wave numbers. That is to say dispersive decay can only occur if gravitation and surface tension aresufficiently strong relative to the ambient shear.More quantitatively, we have > . 2 ( ~ ) > 0 for all when

-4 ( A2)21-"4 < 4Agr. (3)This condition guarantees there will be no exponential growth of solutions ,but it does not immediately demonstrate that decay occurs. To see this, wefirst compute the solution of (1) by means of the Fourier transform. I t is:

( j ( ~ t) = eic lE t [ ( _ i C l ~ s ~ ~ ~ ~ ( ~ ) t ) + COS(>'(Ot)) ( j o ( ~ ) I ~ I sin(>'(Ot) (C)]

+ >'(0 ')'0 ' ~ ~ ~ ~ ) s i n ( > ' ( ~ ) t ) ~ ( ~ ) + ( - i C l ~ S ~ ~ ~ ~ ( O t ) + COS(>'(Ot)) 1 0 ( ~ ) ] (4)

where Cl = -kY/ 4.


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4Thus estimating the amplitude of the solution boils down to estimating

operators of the form:S(t)f := J[ei>'C)tCJojO] = 1e i ( Q + > ' ( ) t ) C J ( ~ ) [ ( O d ~ .

(Here, the function C J ( ~ ) represents the sundry multipliers that appearin (4).) Such estimates fall within the purview of harmonic analysis; themethod of stationary phase is the typical way of controlling L 00 norms ofsuch oscillatory integrals. We have two concerns when estimating S(t)f.The first is, at what rate does it decay (if at all)? The second is, what isthe least restrictive space in which we can place f?

Roughly speaking, the method of stationary phase says the following(see [11]): Suppose on an interval [a, b] (possibly infinite) we have a point~ s t a t (a stationary point) at which

h ( j ) ( ~ s t a d = 0for j = 1, ... ,n - 1 but that

Then lib e i h ( ) t d ~ 1 ::; eel/noThe constant e is proportional to (min[a,b) Ih(n)l)l/n.

For our problem we need to control, for instance,11 00 e i ( Q + > ' W t ) d ~ 1 = 11 00 e i t ( I < Q + > ' ( O t ) d ~ 1

uniformly in /'i, = x/to Supposing that the dispersive decay condition (3) issatisfied, observe the following facts about A(O:

i) A'(O '" C 1/ 2 for '" O.ii) A ' ( ~ ) '" e/2 as ---> 00.Since X ( ~ ) diverges near the origin and at infinity, there is a minimum

of X (0 some point ~ s t a t . So if we have /'i, = /'i,stat := - X ( ~ s t a d then wehave:

/'i,stat + A ' ( ~ s t a t ) = A " ( ~ s t a t ) = O.I t happens that A/II ( ~ s t a d 'I O. And so the stationary phase argument indicates that this integral should decay like ee l / 3 . (Note that [/'i,stat[ corresponds to the rate of the slowest "ripple" one sees when one throws a pebble


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5into a pond, see [12J.) This is in fact the case, but there is the complicationthat X' (.;) "-' 1';1-1/ 2 for'; -4 00. This means that the constant one getsfrom the stationary phase argument is infinite if we really work with theintegral over all of R + .

We can bypass this problem by truncating our integrals in Fourier space,(see [6]), and this leads us into the second issue: what space is f in? Wehave (if (J = 1)

Here !3 > 0 is a carefully chosen constant. We control the first integralusing the stationary phase argument outlined above. The second term canbe controlled if we assume some regularity of f. That is, if we know f(.;)decays for large';. I f this decay is fast enough, since the second integral isbeing taken over smaller and smaller sets as t increases, this integral decaysas well. Pursuing this course of action shows that (for (J = 1).

IIS(t)fllu>o CC 1/ 3 1IfII H l nL ' .The sundry multipliers (J that appear in (4) may grow as .; goes to infinity,and so their inclusion will correspondingly change the regularity required.They do not change the fact that the rate of decay is C 1 / 3 . We are able tosubstantially reduce the regularity requirement on f by working in Besovspaces, though the technical details are somewhat cumbersome for inclusionhere.

We conclude this note with the following remarks:i) I f one considers the water wave problem with no surface tension, thedecay rate increases from t- 1/ 3 to C 1/ 2 , though the regularity requirementsare more restrictive.ii) Dispersive estimates like the ones described here are the foundation forproving both smoothing estimates (see [4]) and (when combined with a priori energy estimates) Strichartz-type estimates (see [9]), both of importancefor passing to the nonlinear problem.iii) Global existence results for the nonlinear problem typically require adecay rate faster than t - 1/ 2 This seems to spell doom for our long termgoal. However, if one considers three-dimensional fluids, the decay rate increases by C 1/ 2 . That is to say, to t- 5/ 6 with surface tension and to C 1without.


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References[1] Ambrose, D. Well-Posedness of Vortex Sheets with Surface Tension, SIAM

J. Math. Anal. , 35 (2003), 221-244.[2] J. Bourgain, Global solutions of nonlinear Schrdinger equations, AmericanMathematical Society Colloquium Publications, 46. American MathematicalSociety, Providence, RI, (1999).[3] P. Constantin, Decay Estimates for Schrodinger Equations, Commun . Math.Phys., 127 (1990) , 101-108.[4] P. Constantin and J.-C . Saut Local smoothing properties of dispersive equations. J. Amer. Math . Soc. 1 (1988), 413- 439.[5] Crapper, G. D. Introduction to water waves. Ellis Horwood Series: Mathematics and its Applications. Halsted Press [John Wiley & Sons, Inc.], NewYork, (1984) .[6] De Godefroy, A. Nonlinear decay and scattering of solutions to a Bretherton equation in several space dimensions. Electron. J . Differential Equations2005, 141, 17.

[7] Hou, T., Lowengrub, J ., and Shelley, M. Removing the stiffness from interfacial flows with surface tension, J. Compo Phys. 114 (1994).

[8] T. Iguchi, Well-posedness of the initial value problem for capillary-gravitywaves, Funkcial. Ekvac., 44 (2001), 219-24l.[9] J. Shatah and M. Struwe, Geometric Wave Equations, American Mathematical Society, Providence (1998).[10] Siegel, M. A study of singularity formation in the K elvin-Helmholtz instability with surface tension. SIAM J . Appl. Math. 55 (1995) , 865-89l.[11] Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, andoscillatory integrals. Princeton Mathematical Series, 43 . Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, (1993) .

[12] Stoker, J. J. Water waves: The mathematical theory with applications. Pureand Applied Mathematics, IV. Interscience Publishers, Inc., New York,(1957) .

[13] Spirn, D . and Wright, J. Linear dispersive decay estimates for vortex sheetswith surface tension. to appear in Comm. Math. Sci., (2009).

[14] Wu, S. Well-posedness in Sobolev spaces of the full water wave problem in2-D. Invent. Math. 130 (1997), 39-72.

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Daniel Spirn and J. Douglas Wright- Gravity Induced Dispersion for Nearly-Flat Vortex Sheets