STABILITY FOR RAYLEIGH-BENARD CONVECTIVE SOLUTIONS OF THE BOLTZMANN EQUATION

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1 STABILITY FOR RAYLEIGH-BENARD CONVECTIVE SOLUTIONS OF THE BOLTZMANN EQUATION L.Arkeryd, Chalmers, Goteborg, Sweden, R.Esposito, University of L Aquila, Italy, R.Marra, University of Rome, Italy, A.Nouri, University of Provence, France.

2 The kinetic setting. F t + 1 ε v F x x + 1 ε v F z z G F = 1 Q(F, F), v z ε2 F(, x, z, v) = F (x, z, v), (x, z) ( µπ, µπ) ( π, π), v R 3, F(t, x, π, v) = M (v) w z F(t, x, π, w)dw, t >, v z, for x [ µπ, µπ], where w z F, M = 1 2π e v 2 2, M+ (v) = 1 2π(1 2πελ) 2 e v 2 2(1 2πελ), ε = l d, Q(f, g)(z, v, t) = 1 2 G = 1 dg, ε 2T R 3 dv λ = 1 ε T T + 2πT, µ = h d, S 2 dωb(ω, v v ) { f g +f g f g g f }.

3 The Rayleigh number Ra = 16G(2πλ) is independent of ε and π chosen in [Ra c, (1 + δ)ra c ], for δ small. We construct a stationary solution F s = M + εf s + O(ε 2 ), with M = 1 (2π) v 2 e 3/2 ( 2, fs = M ρ s + u s v + T s v ), where ρ s, u s, T s are expressed in terms of the fluid solution h s = h l + δ h con + O(δ 2 ) to the Oberbeck-Boussinesq system. Moreover, we prove the kinetic non linear stability of F s under suitable initial perturbations.

4 All solutions (stationary or evolutionary) to the Boltzmann equation will be weak L 1 - solutions to the Boltzmann equation. This will be made possible by controlling the solutions in appropriate norms, in particular in the L 2 M norm in the v-variable of the L norm in the space variables.

5 We study the Boltzmann equation for the perturbation Φ = M 1 (F F s ) with the initial datum 5 Φ (x, z, v) = ε n Φ (n) (, x, z, v) + ε 5 p 5, n=1 where dvdxdzmp 5 = and F s + MΦ. The time dependent solution is written 5 Φ(t, x, z, v) = ε n Φ (n) (t, x, z, v) + εr(t, x, z, v), (x, z) Ω µ. n=1 The first term of the expansion in ε is Φ (1) = ρ 1 + u 1 v + θ 1 v 2 3, 2 where the initial data for ρ 1, u 1, θ 1 (t, x, z) are chosen small enough so that the solution (u s (x, z) + u 1 (t, x, z), θ s (x, z) + θ 1 (t, x, z)) of the initial boundary value problem for the O-B equations exists globally in time and converges to (u s, θ s ) when t.

6 Stability : the remainder We construct the rest term R, solution of R t where + 1 ε µ v R x x + 1 ε v R (MR) z GM 1 = 1 z v z ε 2 LR + 1 J(R, R) ε + 1 H(R) + A, ε R(, x, z, v) = R (x, z, v) = ε 4 p 5 (x, z, v), R(t, x, π, v) = M M w z (R(t, x, π, w) + ψ ε (t, x, π, w)) w z Md ψ ε (t, x, π, v), x [ π, π], t >, v z >, H(R) = 1 5 ε J(R, Φ (j) ε j + Φ s ). 1

7 The main result. Theorem There exists a solution R such that lim t Main lines of the proof. [ π,π] 2 R 3 R 2 (t, x, z, v)m(v)dxdzdv =. + [ π,π] 2 R 3 R 2 (t, x, z, v)m(v)dxdzdvdt < cε 7, R 2 (t, x, z, v)m(v)dxdzdv < c ε 2 ( R 2 (, x, z, v)m(v)dxdzdv + + ) A(s) ds.

8 References. L. Arkeryd, R. Esposito, R. Marra, A. Nouri, Stability of the Laminar Solution of the Boltzmann Equation for the Benard Problem, 28. L. Arkeryd, A. Nouri, Asymptotic techniques for kinetic problems of Boltzmann type, 27. R.E.Caflish, The fluid dynamic limit of the nonlinear Boltzmann equation, 198. R. Esposito, R. Marra, J. L. Lebowitz, Solutions to the Boltzmann Equation in the Boussinesq Regime, R. Esposito, M. Pulvirenti, From Particles to Fluids, 24. N. B. Maslova, Nonlinear evolution equations : kinetic approach, N. Masmoudi, Handbook of differential equations : evolutionary equations, 26. Y. Sone, Kinetic Theory and Fluid Dynamics, 22.

9 Three main problems. Avoid exponential growth of R(t,.,.) when t. Indeed, by and (R, LR) C((1 P)R, ν(1 P)R), (R, J(φ H, PR)) C ν 1/2 PR ν 1/2 (1 P)R. it holds that 1 d 2 dt R 2 2,2 C R 2 2,2 + (B, R). Ω µ Take care of the diffuse reflexion boundary conditions. Control the hydrodynamic moments.

10 Fix (x, z) and define 5 L J R = LR + J( ε n Φ (n) + Φ s, PR). n=1 Spectral gap property of L J Lemma There is ε > such that, for < ε < ε, there is c independent of ε and (x, z), for which the following inequalities hold : (L J R, R) c(ν(i P J )R, (I P J )R), (L JR, R) c(ν(i P)R, (I P)R).

11 The following norms are used, ( t π π R 2t,2 = R 2 (s, x, z, v)m(v)dsdxdzdv π π R 3 ( π π R,2 = sup R 2 (t, x, z, v)m(v)dxdzdv t> π π R 3 ( ) 1 R, = sup sup R 2 2 (t, x, z, v)m(v)dv, t> π<x,z<π ( t R 2t,2, = ( t + ( R,2, = sup ( + R 3 π π π v z> π v z< π t> π π sup t> π v z> v z< ) 1 2, ) 1 2, ) 1 v z M(v) R(s, x, π, v) 2 2 dvdxds v z M(v) R(s, x, π, v) 2 dvdxds ) 1 v z M(v) R(t, x, π, v) 2 2 dxdv v z M(v) R(t, x, π, v) 2 dxdv ) 1 2 ) 1 2.,

12 Lemma Let ϕ( τ, x, z, v) be solution to ϕ τ + v ϕ x x + v ϕ (Mϕ) z εgm 1 = 1 z v z ε L Jϕ + g, (1) periodic in x of period 2π, with zero initial and ingoing boundary values at z = π, π, and g x-periodic of period 2π. Set ϕ = ϕ < ϕ >= ϕ (2π) 2 ϕdxdz. Then, if ε ε, δ δ, for ε, δ small enough, there exists η small such that, ( ϕ,2 c ε 1 2 ν 1 2 (I P)g 2,2 +ε 1 2 Pg 2,2 ν 1 2 (I P)ϕ 2,2 +ηε 1 2 < Pϕ > 2,2 ), ( c ε ν 1 2 (I P)g 2,2 + Pg 2,2 +ηε < Pϕ > 2 ), ( 1 Kinetic stability for Rayleigh-Benard 1 convection.

13 Proof of the Lemma. Denote by ˆϕ( τ, ξ, v), ξ = (ξ x, ξ z ) Z 2 the Fourier transform of ϕ with respect to space. Then for ξ (, ), ˆϕ τ = 1 ε L (M ˆϕ) Jϕ iξ v ˆϕ + εgm 1 + ĝ v z r( 1) ξz. v z Here r = F x ϕ( τ, ξ x, ±π, v) for v z. Then, ( 1 ( d τ (P ˆϕ) 2 ( τ, ξ, v)mdv C ε 2 d τ ζ s (v) L Jϕ)( τ, ξ, ) ) + (I P) ˆϕ( τ, ξ, ) 2 + d τ ν 1 ĝ 2 ( τ, ξ, v)mdv + d τ v z r 2 δ 1 ξ 2 ).

14 By the Parseval inequality, ( Pϕ) 2 ( τ, x, z, v)mdvdxdzd τ ( 1 c ε 2 ν((i P)ϕ) 2 ( τ, x, z, v)mdvdxdzd τ + ν 1 g 2 ( τ, x, z, v)mdvdxdzd τ+ γ ϕ 2 2,2, +η ϕ 2 2,2 By Green s formula, γ ϕ 2 2 T,2, + ϕ 2 2 T,2 +1 ε ν 1 2 (I P)ϕ 2 2 T,2 c(ε ν 1 2 (I P)g 2 2 T,2 +η 1 Pϕ 2 2 T,2 + 1 η 1 Pg 2 2 T,2 ).

15 Write R as the sum R 1 + R 2, where R 1 and R 2 are solutions of two different problems. R 1 solves R 1 t + 1 ε v x R 1 x + 1 ε v z R 1 z G (MR 1 ) = 1 M v z ε 2 L JR 1 R 1 (, x, z, v) = R (x, z, v), + 1 ε H 1(R 1 ) + 1 ε g, R 1 (t, x, π, v) = 1 ε ψ(t, x, π, v), t >, v z. The non-hydrodynamic part of R 1 is estimated by Green s formula : for every η 1 >, γ R R 2 T,2, 1 ( T ) 2 2,2 +1 ε ν 1 2 (I PJ )R T,2 ( c R 2 2,2 +ε ν 1 2 (I PJ )g 2 2 T,2 + η 1 2 P JR T, η 1 P J g 2 2 T,2 + 1 ε 2 ψ 2 2 T,2,

16 An a priori bound for P J R 1 is obtained in the following lemma based on dual techniques. Lemma Set h := P J R 1. Then h 2 2,2 c( R ν 1 2 (I PJ )g 2 2,2 + 1 ε 2 P Jg 2 2,2 + 1 ε 3 ψ 2 2,2, ).

17 The remaining part R 2 of R satisfies the equation ε R 2 t R 2 + v x x + v R 2 z z ε G (MR 2 ) = 1 M v z ε L JR 2 + H 1 (R 2 ), R 2 (, x, z, v) =, R 2 (t, x, π, v) = M (v) M(v) w z ( R 1 (t, x, π, w) + R 2 (t, x, π, w) + 1 ε ψ(t, ) x, π, w) w z Mdw, t >, v z.

18 By Green s formula, and noting that H 1 (R 2 ) only depends on (I P)R 2, we get ε R 2 (t) γ R 2 2 2t,2, +c ε ν 1 2 (I PJ )R 2 2 2t,2 γ+ R 2 2 2t,2,. Treatment of the diffuse reflexion boundary conditions : ε R 2 2 2,2 (t) + c ε ν 1 2 (I PJ )R 2 2 2t,2 1 εη f 2 2t,2, +Cεη P JR 2 2 2t,2, γ R 2 2 2t,2, 1 ε 2 f 2 2t,2, +C P JR 2 2 2t,2.

19 Hydrodynamic estimates for R 2. Lemma P J R 2 2 2,2 c 1 ε 2 f 2 2,2 +c 2 P J R 1 2 2,2.

20 Lemma Any solution R 2 satisfies the a priori estimates ( ν 1 2 (I PJ )R 2 2 2,2 c ε R 2 2 +ε ν 1 2 (I PJ )g 2 2,2 + 1 ε P Jg 2 2,2 + 1 ε 2 ψ ) 2 2,2,, 1 P J R 2 2 2,2 c( ε ( R ν 1 2 (I PJ )g 2 2,2 ) + 1 ε 3 P Jg 2 2,2 + 1 ) ε 4 ψ 2 2,2,, ν 1 2 R2 2, c( 1 ε 2 R 2 2,2 + γ R 1 2,2, + 1 ε 2 ψ 2 c( 1 ε 3 R ε 3 ν 1 2 (I PJ )g 2 2,2 + 1 ε 5 P Jg 2 2,2 + 1 ε 6 ψ 2 2,2, + R 2,2 +ε 2 ν 1 2 g 2, + 1 ) ε 2 ψ 2,2,.

21 Theorem There exists a solution R to the rest term problem such that + [ π,π] 2 R 3 R(t, x, z, v) 2 M(v)dtdxdzdv < cε 7. Proof. R is obtained as the limit of an approximating sequence.