YAN GUO, JUHI JANG, AND NING JIANG

Transcription

1 LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION YAN GUO, JUHI JANG, AND NING JIANG Abstract. We revisit the classical ork of Caflisch [C] for compressible Euler limit of the Boltzmann equation. By using a ne L 2 -L method, e prove the validity of the Hilbert expansion before shock formathions in the Euler system ith moderate temperature variation. We study the Boltzmann equation. Introduction () t F v x F = Q(F, F ) here F (t, x, v) is the density of particles of velocity v R 3, and position x Ω = R 3 or T 3, a periodic box. Throughout this paper, the collision operator takes the form Q(F, F 2 ) = v u γ F (u )F 2 (v )q (θ) dµ du R (2) 3 S 2 v u γ F (u)f 2 (v)q (θ) dµ du, R 3 S 2 here u = u (v u) ω, v = v (v u) ω, cos θ = (u v) ω/ v u, γ (hard potential) and q (θ) C cos(θ) (angular cutoff). We assume hard-sphere interaction for Q in this paper, i.e. γ =. We believe the result could be generalized to broader class of the collision kernels. We define a special distribution function µ, the local Maxellians by ρ(t, x) [v u(t, x)]2 (3) µ(t,,, v) = exp { [2πT (t, x)] 3/2 2T (t, x) hich are in equilibrium ith the collision process, i.e. (4) Q(µ, µ) =. ρ, u, T are the macroscopic density, bulk velocity and temperature, respectively. If ρ, u, T are constant in x and t, µ is called a global Maxellian. The fluid dynamics description of a gas is given by the compressible Euler equations: (5) t ρ x (ρu) =, t (ρu) x (ρu u) x p =, [ t ρ(e 2 u 2 ) ] x [ρu(e 2 u 2 ) ] x (pu) = ith the equation of state p = ρrt = 2 ρe. These are the local conservation las of mass, 3 momentum, and energy. In [C], Caflisch shoed that any smooth solution to the compressible Euler system (5) can be used to construct the corresponding solution to the Boltzmann equation ().

2 2 Y. GUO, J. JANG, AND N. JIANG The solution as founded as a Hilbert expansion ith remainder hich as decomposed into lo and high velocity components. After the linearized version of the decomposed remainder equation as estimated, the nonlinear equations are solved by iteration. Hoever, a crucial but undesirable assumption as made in [C] that the remainder vanishes initially. The truncated Hilbert expansion ith such a remainder term can lead to physically unreasonable negative data. This assumption as essentially needed in the linear estimates on the remainder equation. In this paper, e basically follo Caflisch s approach, give a ne a priori estimate. Applying the so-called nonlinear energy method developed by the first author, especially some ne L 2 -L interplay estimates in [G2], e can remove the assumption on the initial data of the remainder in [C]. Furthermore, e establish the uniform L 2 -L estimate for the remainder. This key improvement allos us to apply the result of the current paper to the hydrodynamic limits of the Boltzmann equation, for example, acoustic limits. This ork is under preperation currently. The paper is organized as follos: the next section contains the statement of the main theorem and some key lemmas. Section 3 is devoted to the proof of L 2 estimate. In Section 4, e establish the L estimate for the high velocity part. 2. The Main Theorem As in [C], e take the truncated Hilbert expansion ith the form F = 6 n F n 3 FR, n= here F,..., F 6 are the first 6 terms of the Hilbert expansion, independent of, hich solve the equations = Q(F, F ), { t v x F = Q(F, F ) Q(F, F ), { t v x F = Q(F, F 2 ) Q(F 2, F ) Q(F, F ),... { t v x F 5 = Q(F, F 6 ) Q(F 6, F ) ij=6 i5,j5 Q(F i, F j ). Let [ρ(t, x), u(t, x), T (t, x)] be a smooth solution of the Euler equations (5) for t [, τ], x Ω and let F = µ(t, x, v) from the local Maxellian µ(t, x, v) from ρ, u and T as in (3). We further construct smooth F (t, x, v), F 2 (t, x, v),...f 6 (t, x, v) for t τ. For more detailed discussion, see [C]. No e put F = 5 n= n F n 3 FR (notice e drop F 6) into the Boltzmann equation () to derive the equation of the remainder. Recall that in [C], the remainder FR satisfies (6) t F R v x F R {Q(µ, F R) Q(F R, µ) = 2 Q(F R, F R) {Q(F F 2 2 F 3, F R) Q(F R, F F 2 2 F 3 ) 2 A

3 here LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION 3 (7) A = { t v x F 5 ij 6,i,j5 ij 6 Q(F i, F j ). Note that in the above equation, e drop out the higher order term 3 {Q(F 4 F 5, F R ) Q(F R, F 4 F 5 ). We define the linearized Boltzmann operator at µ as Lg = {Q( g, µ) Q(µ, g) = ν(µ)g K µ g, Γ(g, g 2 ) = Q( g, g 2 ). We use, to denote the standard L 2 inner product in R 3 v, hile e use (, ) to denote L 2 inner product in Ω R 3 ith corresponding L 2 norm 2. We denote the standard L norm in Ω R 3 by. We also define a eighted L 2 norm g 2 ν = g 2 (x, v)ν(v) dx dv, Ω R 3 here the collision frequency ν(v) ν(µ)(v) is defined as ν(µ) = v v µ(v ) dv. R 3 Theorem. Assume that the solution to the Euler equations (ρ(t, x), u(t, x), T (t, x)) is smooth and ρ(t, x) has a positive loer bound for t τ. Furthermore, assume that the temperature T (t, x) satisfies the condition: (8) T M < max T (t, x) < 2T M, t [,τ],x Ω here T M = min T (t, x). Let t [,τ],x Ω F (, x, v) = µ(, x, v) 5 n F n (, x, v) 3 FR(, x, v). n= Then there is an > such that for, and for any β 7, there exists a 2 constant C τ (µ, F, F,..F 6 ) such that sup 3 2 ( v 2 ) β FR(t) sup F tτ R(t) tτ 2 C τ { 3 2 ( v 2 ) β FR() µ F R(). 2 We give a fe remarks on the Theorem : First, based on the a priori estimates given in Theorem, folloing the arguments in [C], e can immediately derive the compressible Euler limit as ell as the existence of the solutions to the Boltzmann equation (). We skip the details here. Second, e make the assumption (8) on the temperature, hich seems restrictive. Hoever, one of the main applications of this ne uniform L 2 -L interplay estimate is to derive the hydrodynamic limits from the Boltzmann equation to fluid dynamics hich is in the regime close to constant states. For those cases, the condition (8) is easy to be achieved. Third, as e mentioned in the introduction, the main ne ingredient of the theorem is the removal of the crucial but undesirable assumption FR (, x, v) in [C].

4 4 Y. GUO, J. JANG, AND N. JIANG The solutions to the Boltzmann equation are constructed near the local Maxellian of the compressible Euler system. So it is natural to rerite the remainder (9) F R = f. Because µ is a local Maxellian, the equation of the remainder includes the ne term ( t v x ) f, thus at large velocities, the distribution functions may be groing rapidly due to streaming. To remedy this difficulty, folloing Caflisch, e introduce a global Maxellian µ M = exp { v 2. (2πT M ) 3/2 2T M Note that under the assumption (8), there exist to constants c, c 2 such that for all (t, x, v) [, τ] Ω R 3, the folloing holds c µ µ M c 2 µ. We further define () FR = { v 2 β M h µm h (v) for some β 7/2. It then suffices to estimate f (t) 2 and h (t) to conclude the theorem. Let Pg be the L 2 v projection ith respect to [, v, v 2]. We have that there exists a positive number δ > such that () Lg, g δ { I Pg 2 ν. The proof of Theorem relies on an interplay beteen L 2 and L estimates for the Boltzmann equation [G2]: L 2 norm of f is controlled by the L norm of the high velocity part and vice versa. These uniform L 2 -L estimates are stated in the folloing to lemmas: Lemma 2. (L 2 -Estimate): Let f, h be defined in (9) and (), and δ > be as in the coercivity estimate (). Then there exists > and a positive constant C = C(µ, F, F,, F 6 ) >, such that for all < (2) d dt f 2 2 δ 2 {I Pf 2 ν C{ 3/2 h ( f 2 2 f 2 ). Lemma 3. (L -Estimate): Let f, h and δ > be the same as in Lemma 2. Then there exist > and a positive constant C = C(µ, F, F,, F 6 ) >, such that for all < (3) sup { 3/2 h (s) C{ 3/2 h sup f (s) 2 7/2. sτ sτ The proof of Theorem is an easy consequence of Lemmas 2 and 3. Proof. of Theorem : d dt f 2 2 δ 2 {I Pf 2 ν { [ ] ( f ) C 3/2 h sup f (s) 2 7/2 2 2 f 2. sτ

5 LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION 5 A simple Gronall inequality yields f (t) 2 ( f () 2 )e Ct{2 3/2 h sup sτ f (s) 2. For small, using the Taylor expansion of the exponential function in the above inequality, e have { (4) f 2 C ( f () 2 ) 3/2 h sup f (s) 2. sτ For t τ, letting small, e conclude the proof of our main theorem as: sup f (t) 2 C τ { f () 2 3/2 h. tτ 3. L 2 Estimate For f Proof. of Lemma 2: In terms of f, e obtain t f v x f Lf = { t v x f 2 Γ(f, f ) Γ( F F 2 2 F 3, f ) Γ(f, F F 2 2 F 3 ) 2 Ā µ here Ā = { tv xf 5 ij 6,i5,j5 ij 6 Γ( F i µ, F j ). Taking L 2 inner product ith f on both sides, since { tv x in v, e have, for any κ >, { t v x µ f, f µ = v κ v κ is a cubic polynomial { x ρ 2 x u 2 x T 2 { v 2 3/2 f v κ f 2 { x ρ x u x T { v 2 3/4 f v κ 2 2 C κ 2 h f 2 C { v 2 3/4 Pf v κ 2 2 C { v 2 3/4 {I Pf v κ 2 2 C κ 2 h f 2 C f 2 2 Cκ2 {I Pf 2 ν. Here e have used the fact { v 2 3/2 f { v 2 2 h, for β 7/2 in (), and the fact µ M < Cµ, under the assumption (8). By Lemma 2.3 in [G] and (): 2 Γ(f, f ), f C 2 { ν(µ)f f 2 2 C 3/2 h f 2 2.

6 6 Y. GUO, J. JANG, AND N. JIANG Similarly, by Lemma 2.3 in [G] and ():, Γ( F F 2 2 F 3, f ), f Γ(f, F F 2 2 F 3 ), f C f 2 F F 2 2 F 3 ν dv R µ 3 C{ Pf 2 ν {I Pf 2 ν C{ f 2 2 {I Pf 2 ν. Clearly, 2 Ā, f C f 2. We therefore conclude our lemma by choosing κ small. Proof. of Lemma 3: As in [C], e define 4. L Estimate For h L M g = µm {Q(µ, M g) Q( M g, µ) = {ν(µ) K M g. Letting K M, g K M ( g ), from (6) and (), e obtain t h v x h ν(µ) h K M,h = 2 Q( h M µm, hm 5 ) i {Q(F i, h M µm ) Q(hM, F i) 2 Ã, here à = { tv xf 5 M ij 6,i5,j5 ij 6 µm Q(F i, F j ). By Duhamel s principle, e have h (t, x, v) = (5) νt h (, x vt, v) t t t t Since M Q( h M and since t i= t ( ) K M,h (s, x v(t s), v)ds, h ) µm ) (s, x v(t s), v)ds Q(F i, h ) µ M µm ) (s, x v(t s), v)ds Q( h ) µ M µm, F i) (s, x v(t s), v)ds ( 2 Q( h M µm ( 5 i= ( 5 i= i i 2 Ã(s, x v(t s), v)ds., h M ) C{ν(µ) h (v) h h from Lemma 9 of [G2], ν(µ) = c ν(µ)(t s) ν(µ)ds c t v u µdu v ρ(t, x) ν M (v), cν M(t s) ν M ds = O(),

7 LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION 7 the second line in (5) is bounded by (6) t C 2 ν(µ)(t s) C{ν(µ) h (s, x v(t s), v) h (s) h (s) ds C 3 sup h (s) 2. st From Lemma 9 from [G2] again, 5 i {Q(F i, h M µm ) Q(hM, F i) ν M (v) h i= so that the third and fourth lines in (5) are bounded by (7) C t µm 5 i F i, i= ν(µ)(t s) ν M (v) h (s) ds C sup h (s). st The last line in (5) is clearly bounded by C 3. We shall mainly concentrate on the second term in the right hand side of (5). Let l M (v, v ) be the correspoding kernel associated ith K M in [C]. We have (8) l M (v, v ) C{ v v v v c v v 2 c v 2 v 2 2 v v 2. Since ν(µ) ν M, e bound the second term by t l M, (v, v )h (s, x v(t s), v ) dv ds, R 3 here l M, = (v) l (v ) M. We no use (5) again to evaluate h. By (6) and (7), e can bound the above by t 2 t C C (9) C s sup v ν(v )(s s ) t t t R 3 l M, (v, v ) dv νs h (, x v(t s) v s, v ) ds R 3 R 3 l M, (v, v )l M, (v, v ) h (s, x v(t s) v (s s ), v ) dv dv ds ds ds sup v ds sup v ds sup v st R 3 l M, (v, v ) dv { 3 sup h (s) 2 l M, (v, v ) dv { sup h (s) R 3 st l M, (v, v ) dv { 2 sup Ã. R 3 st Since sup v R 3 l M, (v, v ) dv < from Lemma 7 of [G2], there is an upper bound except for the seond term as C{ h () 3 sup h (s) 2 sup h (s) C 3. st st We no concentrate on the second term in (9), hich ill be estimated as in the proof of Theorem 2 in [G2].

8 8 Y. GUO, J. JANG, AND N. JIANG CASE : For v N. By Lemma 7 in [G2], l M, (v, v )l M, (v, v )dv dv and thus e have the folloing bound C 2 N t s ν(v)(t s) C v C N, ν(v )(s s ) h (s ) ds ds C N sup h (s). st CASE 2: For v N, v 2N, or v 2N, v 3N. Notice that e have either v v N or v v N, and either one of the folloing is valid correspondingly for some η > : (2) l M, (v, v ) e η 8 N 2 l M, (v, v )e η 8 v v 2, l M, (v, v ) e η 8 N 2 l M, (v, v )e η 8 v v 2. From Lemma 7 in [G2], both l M, (v, v )e η 8 v v 2 and l M, (v, v )e η 8 v v 2 are still finite. We use (2) to combine the cases of v v N or v v N as: t s { C t s C η 2 e v N, v 2N, { t η 8 N 2 v N, v 2N, s (2) C η e η 8 N 2 sup { h (s). st (22) v 2N, v 3N l M, (v, v ) dv sup v ν(v)(t s) v 2N, v 3N l M, (v, v ) dv ν(v )(s s ) h (s ) ds ds CASE 3: s s κ, for κ > small. We bound the last term in (9) by 2 t s s κ ν(v)(t s) C ν(v )(s s ) h (s ) ds ds C sup st{ h (s) κc sup { h (s). st t ν(v)(t s) ds s s κ ds CASE 4. s s, and v N, v 2N, v 3N. This is the last remaining case because if v > 2N, it is included in Case 2; hile if v > 3N, either v 2N or v 2N are also included in Case 2. We no can bound the second term in (9) by C t B s κ e ν(v)(t s) e ν(v )(s s ) l M, (v, v )l M, (v, v )h (s, x (s s )v, v ) here B = { v 2N, v 3N. By (8), l M, (v, v ) has possible integrable singularity of, e can choose l v v N(v, v ) smooth ith compact support such that (23) sup l N (p, v ) l M, (p, v ) dv p 3N N. v 3N

9 Splitting LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION 9 l M, (v, v )l M, (v, v ) = {l M, (v, v ) l N (v, v )l M, (v, v ) {l M, (v, v ) l N (v, v )l N (v, v ) l N (v, v )l N (v, v ), e can use such an approximation (23) to bound the above s, s integration by { C (24) N sup { h (s) sup l M, (v, v ) dv sup l N (v, v ) dv st v 2N v 2N t s κ C e ν(v)(t s) e ν(v )(s s ) l N (v, v )l N (v, v ) h (s, x (s s )v, v ). B Since l N (v, v )l N (v, v ) is bounded, e first integrate over v to get C N h (s, x (s s )v, v ) dv C N { v 2N C N κ 3/2 3/2 v 2N { Ω (x (s s )v ) h (s, x (s s )v, v ) 2 dv /2 y x (s s )3N C N{(s s ) 3/2 κ 3/2 3/2 /2 h (s, y, v ) 2 dy { /2 h (s, y, v ) dy 2. Ω Here e have made a change of variable y = x (s s )v, and for s s κ, dy. dv κ 3 3 In the case of Ω = R 3, the factor {(s s ) 3/2 is not needed. By () and (9), e then further control the last term in (24) by: C N,κ 7/2 C N,κ 7/2 C N,κ 3/2 t s κ t s κ sup f (s) 2. st e ν(v)(t s) e ν(v )(s s ) {(s s ) 3/2 e ν(v)(t s) e ν(v )(s s ) {(s s ) 3/2 v 3N { v 3N In summary, e have established, for any κ > and large N >, { /2 h (s, y, v ) dy 2 dv ds ds Ω /2 f (s, y, v ) 2 dydv ds ds sup { 3/2 h (s) {κ C κ st N sup { 3/2 h (s) 7/2 C C,N 3/2 h st C sup { 3/2 h (s) 2 C N,κ sup f (s) 2. st st For sufficiently small >, first choosing κ small, then N sufficiently large so that {κ Cκ Ω <, N 2 sup { 3/2 h (s) C{ 3/2 h sup f (s) 2 7/2. sτ sτ and e conclude our proof.

10 Y. GUO, J. JANG, AND N. JIANG References [C] Caflisch, R. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math., Vol XXXIII, (98). [G] Guo, Y. The Vlasov-Poisson-Boltzmann system near Maxellians. Comm. Pure Appl. Math., Vol LV, 4-35 (22). [G2] Guo, Y. Decay and continuity of Boltzmann equation in bounded domains. Preprint 28. Division of Applied Mathematics, Bron University address: Courant Institute of Mathemtical Sciences address: Courant Institute of Mathemtical Sciences address: