Journal of Differential Equations. Smoothing effects for classical solutions of the relativistic Landau Maxwell system

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1 J. Differential Equations Contents lists available at ScienceDirect Journal of Differential Equations Smoothing effects for classical solutions of the relativistic andau Maxwell system Hongjun Yu a,b, a School of Mathematical Sciences, South China Normal University, Guangzhou 51631, China b SISSA, Via Beirut 4, Trieste 3414, Italy article info abstract Article history: Received 5 November 7 Revised February 9 Availableonline5March9 MSC: rimary 35Q99 secondary 35A5 Keywords: Relativistic andau Maxwell system Classical solutions Smoothing effects The relativistic andau Maxwell system is one of the most fundamental and comlete models for describing the dynamics of a dilute hot lasma in which articles interact through Coulomb collisions and their self-consistent electromagnetic field. In this work, we rove that the classical solutions obtained by Strain and Guo become immediately smooth with resect to all variable under the extra assumtion of the electromagnetic field. As a by-roduct, we also rove that the classical solutions to the relativistic andau Poisson system and the relativistic andau equation have the same roerty without any extra assumtion. 9 Elsevier Inc. All rights reserved. 1. Introduction The relativistic andau Maxwell system is one of the most fundamental and comlete models for describing the dynamics of a dilute hot lasma in which articles interact through Coulomb collisions and their self-consistent electromagnetic field. For notational simlicity, we consider the following normalized andau Maxwell system: t F x F E B F = CF, F CF, F, 1.1 t F x F E B F = CF, F CF, F, 1. * Address for corresondence: School of Mathematical Sciences, South China Normal University, Guangzhou 51631, China. address: yuhj@sina.com. -396/$ see front matter 9 Elsevier Inc. All rights reserved. doi:1.116/j.jde.9..1

2 H. Yu / J. Differential Equations with initial condition F ±, x, = F,± x,. HereF ± t, x, are the satially eriodic number density functions for ions and electrons at time t, osition x T 3 =[ π, π] 3 and momentum = 1,, 3. The energy of a article is given by = 1. To comletely describe a dilute lasma, the electromagnetic field Et, x and Bt, x is couled with F ± t, x, through the celebrated Maxwell system: t E x B = t B x E =, x E = {F F } d, x B =, 1.3 {F F } d, 1.4 with initial condition E, x = E x and B, x = B x. The collision between articles is modeled by the relativistic andau collision oerator and its normalized form is given by { { Cg, h = ΦP, Q ghq g q hq } } dq, 1.5 where the four-vectors are P =, 1,, 3 and Q = q, q 1, q, q 3. Moreover, the collision kernel is given the 3 3 non-negative matrix with ΦP, Q = ΛP, Q q SP, Q, 1.6 ΛP, Q = P Q { P Q 1 } 3/, SP, Q = { P Q 1 } I 3 q q { P Q 1 } q q. Here the orentz inner roduct is P Q = q q. From the relativistic andau Maxwell system, we can obtain: d d F t = F t =, dt dt T 3 T 3 { d F t F t dt T 3 T 3 { d 1 F t F t dt T 3 T 3 } Et Bt =, Et Bt } =. From the Maxwell system and the eriodic boundary condition of Et, x, there exists a constant B 1 such that T 3 T Bt, x 3 dx = B. We introduce the normalized relativistic Maxwellian J = e, and the standard erturbation f ± t, x, to J as F ± = J Jf ±.

3 3778 H. Yu / J. Differential Equations We introduce the notation F =[F, F ]. By assuming that [F, E, B ] has the same mass, total momentum and total energy as the steady state [ J,, B], thenwehave Jf t = Jf t =, 1.7 { d 1 dt { d dt T 3 T 3 T 3 T 3 Jf t Jf t Jf t Jf t T 3 T 3 } Et Bt =, 1.8 Et Bt B } =. 1.9 For notational simlicity, we omit the integrating domains T 3 and, which corresond to variable x and variable resectively. For examle, we sometimes write x, instead of x T3 ; R3. Forany integer N, we define the Sobolev sace { Hx N = f x: N { x f < }, H Nx, = f x, : x, β N x β } f <, x, and for N, s, we define the weighted Sobolev sace { Hx, N,s = f x, : β N x β f 1 s/ } <, x, where the multi-index =,, 3, = 3 and x = x 1 x 3 x 3 with x = x 1, x, x 3. The notations for β are the same. It is obvious that Hx, N, = Hx, N.WealsodefineH x, and H,s x, by H x, = N H N x,, H,s x, = Hx, N,s. N The global classical solution of the relativistic andau Maxwell system around the above steady state in the eriodic box has been obtained in [17]. The authors in [13] obtained the global solution of the relativistic andau equation near relativistic Maxwellian in the whole sace. Recently, author in [] also roved that the relativistic andau Maxwell system near steady state in the whole sace has a global classical solution in the Sobolev sace H N x, R3 N 4 by the techniques in [1,13, 16,17]. By the results in [17,], we easily obtain the following existence results of the global classical solution. Theorem 1.1. See Strain and Guo [17]. et N 4 and F,± x, = J Jf,± x,. Assume that [ f, E, B ] satisfies the conservation laws 1.7, 1.8 and 1.9. There exists κ > such that if f H N x, [E, B ] H N x κ, then there exists a unique global solution [F t, x,, Et, x, Bt, x] to the relativistic andau Maxwell system Moreover, there is a constant C > such that f t [,;H N x, [E, B] t [,;H N x κ.

4 H. Yu / J. Differential Equations It is obvious that for any s, there exist constant C 1 > deending on C and s and C > deending on s such that F t [,;H N,s x, [E, B] t [,;H N x 1κ C. Our main theorem states that the classical solutions obtained thanks to Theorem 1.1 satisfy the following smoothness. Theorem 1.. Assume that [ f, E, B ] satisfies the conservation laws 1.7, 1.8 and 1.9 and that F,± x, = J Jf,± x,.thereexistc 3 > and ɛ [, κ ] such that f H 4 x, [E, B ] H 4 x ɛ, the unique classical solution to the system given by Theorem 1.1 satisfies for any < τ 1 < τ < T < and s : F C t [τ, T ]; H,s x, T 3, 1.1 under the assumtions that Eτ1 H x Bτ1 H x Remark 1.3. If Et, x =, Bt, x = and F = F, the system 1.1 and 1. turn into the simle relativistic andau equation. Theorem 1. says that classical solutions to the relativistic andau equation given by Theorem 1.1 become immediately smooth with resect to all variables. Remark 1.4. If Et, x = x φt, x and Bt, x =, the system turn into the relativistic andau Poisson system. Theorem 1. tells us that classical solutions to the relativistic andau Poisson system given by Theorem 1.1 also become immediately smooth with resect to all variables by the ellitic estimates see Remark 3.6 without any extra assumtion. This fact shows that nature that the Maxwell system is hyerbolic may lead us to imose the assumtion Whether this assumtion is removed or not will be considered for the future study. The smoothness of the solutions to the satially homogeneous andau equation has been investigated in [1]. Desvillettes and Villani [8] roved global existence, uniqueness and smoothness of classical solutions to the satially homogeneous andau equation for hard otentials and a large class of initial data. Guo [11] constructed global classical solutions near Maxwellians for a general andau equation in a eriodic box by the nice energy method. Recently, Chen, Desvillettes and He [6] roved that the classical solutions to the andau equation obtained by Guo [11] become immediately smooth with resect to all variables by some hints of [,7] and [8]: roughly seaking, smoothness in the velocity variable is roduced by the ellitic roerty of the diffusive matrix to the andau oerator in [8] and smoothness in the osition variable is roduced by the classical averaging lemma [,3,9]. Although the classical non-relativistic andau equation has been heavily studied [1,6,8,11,14,19], the relativistic version has received scant attention [13,15,17]. emou studied the linearized relativistic equation in [15]. In [17], Strain and Guo resented the global existence of classical solutions to the relativistic andau Maxwell system near steady state. Thanks to this breakthrough, we ossibly imrove smoothness of the solutions to the relativistic andau Maxwell system in this work. Although our main results are roved by using the novel idea of [6], there are several major new difficulties in this aer. The first new difficulty is due to the comlexity of the relativistic andau kernel 1.5. Unlike the classical andau kernel, the relativistic version is not convolution-like and we use the technique of the novel integration by arts introduced in [17] to overcome this difficulty. The second one is the lower regularity of the solutions N 4 in Theorem 1.1 and the N-derivative of

5 378 H. Yu / J. Differential Equations the vector b ± is bigger than N. We have to analyze detailedly the diffusive matrix a ± and the i ij vector b ± i and obtain the corresonding estimates. astly, the matrix a ± ij is different to the classical version due to the relativistic kernel and the roof of the elliticity roerty of the classical version in [6] cannot be alied here. We use the imortant findings of emou [15] to obtain the key elliticity roerty.. Basic estimates In this section, we deduce some basic estimates which will be used reeatedly in the roof of Theorem 1.. If we write a ± ij t, x, = Φ ij P, Q F ± t, x, q dq, b ± i t, x, = then the system 1.1 and 1. can be rewritten as 3 j=1 t F x F E B F Φ ij P, Q q j F ± t, x, q dq, = a F b F a F b F,.1 t F x F E B F = a F b F a F b F.. Here a ± resectively b ± is the matrix resectively the vector with coefficients a ± ij ij resectively b ± i i. In the following we will rove some estimates of a ± and b ±.Wefirstdefinetherelativistic ij i differential oerator: Θ, q = 1 q 1 q1 q q 3 q 3 q3,.3 which is originally introduced in [17] and lays a key role in roving estimates of a ± ij and b ± i.we begin with the following integration by arts lemma. emma.1. Given β >,wehave β Φ ij P, Q f q dq = β 1 β β Θ β1 Φ ij P, Q β q f qφ β β 1,β, q dq,.4 where φ β β 1,β, q is a smooth function which satisfies for all multi-indices ν 1 and ν. ν 1 q ν φ β β 1,β, q q β ν 1 β 1 β ν,.5 Proof. This lemma is similar as emma 3 in [17]. We rove.4 by an induction over the number of derivatives β. Assume β = e i i = 1,, 3. Wewrite i = q qi i q qi = q qi Θ ei.

6 H. Yu / J. Differential Equations Instead of hitting Φ ij P, Q with i, we aly the r.h.s. term above and integrate by arts over q qi to obtain β Φ ij P, Q f q dq = Θ ei Φ ij P, Q f q dq We can write the above in the form.4 with the coefficients given by Φ ij P, Q q qi f q dq. φ e i e i,, q = 1, φe i,e i, q = q. Note that these coefficients satisfy the decay.5. This establishes the fist ste in the induction. Assume the result holds for all β n. Fix an arbitrary β such that β =n 1 and write β = m β for some multi-index β and m = max { j: β j > }. By the induction assumtion, we have β Φ ij P, Q f q dq = β 1 β β m Θ β1 Φ ij P, Q β q f qφ β, q dq. β 1,β We aly the last derivative the same as the case β =1above.Wehave = β 1 β β β 1 β β Θ Θ em β 1 Φ ij P, Q β q f qφ β, q dq.6 β 1,β β 1 β β We collect all the terms above and write it to the following form = β 1 β β Θ β1 Φ ij P, Q q qm β q f qφ β, q dq.7 β 1,β Θ β1 Φ P, Q β q f q m q qm φ β, q dq. β 1,β.8 Θ β1 Φ ij P, Q β q f qφ β β 1,β, q dq. We will check.5 by the terms.6,.7 and.8. For.6, the order of differentiation is and by the induction assumtion, we have β 1 = β 1 e m, β = β, ν 1 q ν φ β β 1,β, q q β ν 1 β 1 β ν q β ν 1 β 1 β ν. For.7, the order of differentiation is β 1 = β 1, β = β e m,

7 378 H. Yu / J. Differential Equations and by the induction assumtion, we have ν 1 q ν q φ β, q q β 1 ν 1 β 1,β β 1 β 1 ν = Cq β ν 1 β 1 β ν. For.8, the order of differentiation is β 1 = β 1, β = β, and by the induction assumtion and eibnitz rule, we have { ν 1 q ν m q qm φ β, q} β 1,β q β 1 ν 1 β 1 β 1 ν q β ν 1 β 1 β ν. emma.. et F t, x, be the non-negative solution of the system in Theorem 1.1. Then there exists a constant C > such that for any multi-indices and β,wehave x β a ± ij t, x, x β 1 F ± 1 β 1/ t, x,.9 x β b ± i t, x, β 1 β β 1 β 1 Proof. We know from emma.1 that when β >, we have x β a ± ij t, x, = β F ± 1 β 1/ t, x..1 Φ ij P, Q x F ±q dq = β 1 β β Θ β1 Φ ij P, Q β q x F ±qφ β β 1,β, q dq, where φ β β 1,β, q satisfies φ β β 1,β, q q β β 1 β. We will use the following slitting: A = { q q [ 1 ] / }, B = { q q [ 1 ] / }..11 From emma in [17], on the set A we have the estimate Θ β1 Φ ij P, Q β 1 q 6,.1 and on the set B we have the estimate Θ β1 Φ ij P, Q β 1 q 7 q Thus on the set A, the Hölder inequality imlies

8 A Θ β1 Φ ij P, Q β q β β β q β 6 H. Yu / J. Differential Equations x F ±qφ β β 1,β, q dq β q x F ±q dq 1/ 1 q 4 dq β β q x F ± 1/ 1 q β 1 dq. q x F ± 1/ 1 q β 1 dq On the set B, We use.13 to obtain Θ β1 Φ ij P, Q β q x F ±qφ β β 1,β, q dq B β q β 7 q 1 β q x F ±q dq β β q 1 q 1/ 3 dq β q x F ± 1/ 1 q β 1 dq β q x F ± 1/ 1 q β 1 dq, where we have used the inequality Therefore, we can obtain that We know that q 1 q 3 dq. x β a ± ij t, x, β 1 x ± F 1 β 1/ t, x. β 1 β x β b ± i t, x, = 3 β j=1 Φ ij P, Q x q j F ± t, x, q dq. This form is similar as the form of a ±. By the arguments similar as the above, we can rove the ij second relation. emma.3. et F t, x, be the non-negative solution of the system in Theorem 1.1. Then there exists a constant C > such that for any multi-indices,and β >,wehave

9 3784 H. Yu / J. Differential Equations x β b ± i t, x, β 1 β 1 F ± t, x, 1/ β 1 x β 1 F ± 1 β 1 1/ t, x. Proof. Byemma4in[17],weknow i Φ ij P, Q q j f q dq = 4 ΦP, Q f q dq κ f,.14 where κ = 7/ π π 1 sin θ 3/ sin θ dθ and By the similar arguments as emma.1, we have β ΦP, Q f q dq = ΦP, Q = P Q q { P Q 1 } 1/. β 1 β β β 4 Θ β1 ΦP, Q q f qφ β β 1,β, q dq,.15 where φ β β 1,β, q is a smooth function which satisfies ν 1 q ν φ β β 1,β, q q β ν 1 β 1 β ν,.16 for all multi-indices ν 1 and ν. On the other hand, we also know Θ β1 ΦP, Q = P Q { P Q 1 } 1/ Θβ1 1 We will use the following inequality in [1] and [17], On the set A, wehave q q q P Q 1 1 q q..17 q q q q By.17 and the last dislay we have. P Q 1 P Q From the Cauchy Schwartz inequality we also have q..18 P Q 1 P Q q q q..19 We use these last two inequalities to get

10 H. Yu / J. Differential Equations Θβ1 ΦP, Q = P Q { P Q 1 } 1/ Θβ1 1 q q {P Q 1} 1 q 1 1 β 1 1 β 1 q.. By.17 we can obtain q q P Q 1 1 q..1 On the set B, wehave Θβ1 ΦP, Q = P Q { P Q 1 } 1/ Θβ1 1 q q {P Q 1} 1/ {P Q 1} 1/ q 1 1 β 1 1/ β 1 q 1/ q 1.. Recalling., the integral.15 on the set A is bounded by A Θ β1 ΦP, Q β q 1 β 1 β 1 β f qφ β β 1,β, q dq 1 β q β q f q dq 1/ 1 q 9 dq β q f q 1 q β 1 dq 1/. β q f q 1 q β 1 dq 1/ Recalling., the integral.15 on the set B is bounded by B Θ β1 ΦP, Q β q 1/ β 1/ β 1/ β f qφ β β 1,β, q dq 1 q β q 1 β q f q dq q 1 q 1/ 9 dq β q f q 1/ 1 q β 1 dq. β q f q 1 q β 1 dq 1/

11 3786 H. Yu / J. Differential Equations By the above inequalities, we can obtain Θ β1 ΦP, Q β q f qφ β β 1,β, q dq 1/ β By the similar arguments as the above, we also obtain β q f q 1 q β 1 dq 1/. ΦP, Q f q 1/ f q 1 q 1 dq 1/. Finally, we can obtain x β b ± i t, x, β 1 β 1 F ± t, x, 1/ β 1 x β 1 F ± 1 β 1 1/ t, x. Here we have used the fact that β κ. emma.4. et F t, x, be the non-negative solution of the system in Theorem 1.1. Then there exists a constant C > such that for any multi-indices,wehave x b ± t, x, 9/ i x F ± t, x, 1 5. Proof. By the integral by arts, we have x b± t, x, = i Φ ij P, Q x q j F ± t, x, q dq = q Φ ij P, Q j x F ±t, x, q dq. On the other hand, we know from emma 3 in [17] that j q j Φ ij P, Q = ΛP, Q q P Q i q i. On the set A, by.18 and.19, we get We see that ΛP, Q P Q i q i q P Q { P Q } 3/ 1 1 q {P Q 1} 3 q q 3 = Cq 5.

12 H. Yu / J. Differential Equations Φ ij P, Q x q j F ± t, x, q dq A 1/ C 1 q 5 dq x F ± t, x, q 1 q 1 dq 1/. x F ± t, x, q 1 q 1 dq 1/ On the set B, by.19 and.1, we get We see that Φ ij P, Q x q j F ± t, x, q dq B 9/ 9/ ΛP, Q P Q i q i q P Q { P Q } 3/ 1 1 q {P Q 1} 3/ q q 3/ q 3 = C 9/ q 7/ q 3. 1 q 7/ 1 q 6 dq 1/ x F ±t, x, q 1 q 1 dq 1/. x F ± t, x, q 1 q 1 dq 1/ Now we will rove the key elliticity roerty for the matrix a ± = a ± ij. emma.5. et F be a non-negative solution of the system given by Theorem 1.1.If f H N x, ɛ for ɛ small enough, then there exists a constant K > such that for any ξ, 3 a ± ij t, x, ξ iξ j K ξ..3 i, j=1 Proof. We will use the idea in [15] to rove this lemma. We first define σ ij = Φ ij P, Q Jq dq..4 For any, σ = σ ij is a symmetric ositive-definite matrix satisfying σ R = Rσ R T for any transformation R belonging to the orthogonal grous O 3 R. et, r, r be an orthogonal

13 3788 H. Yu / J. Differential Equations basis of. Using the invariance of σ under orthogonal transformation of,weeasilydeduce that 3 σ ij i r j = i, j=1 3 σ ij i r j =. i, j=1 This means that, r, r is an orthogonal basis of eigen-vectors of σ. Thus, we get that σ has a simle eigenvalue λ 1 = 1/ σ ij i j associated with the eigen-vector and a double eigenvalue λ = 1/ r σ ij r i r j associated with the eigen-sace, that is, Therefore, we can obtain that λ 1 = 1 λ = 1 r 3 Φ ij P, Q i j Jq dq, i, j=1 3 Φ ij P, Q r i r j Jq dq. i, j=1 σ ij = λ 1 i j { λ δ ij } i j..5 On the other hand, we know from [15] that there are constants c 1, c > such that, as, λ 1 c 1 and λ c and that there exists a constant ν > such that λ = min λ 1, λ ν, for all. By the definition of a ± t, x,, wehave ij a ± ij t, x, = Φ ij P, Q Jq dq Φ ij P, Q Jq f ± t, x, q dq. By Theorem 1.1, we know that there is a constant S > such that f ± t, x, q Sɛ. By the non-negativity of the matrix ΦP, Q, wehave 3 i, j=1 Φ ij P, Q ξ i ξ j Jq f ± t, x, q dq Sɛ 3 i, j=1 Φ ij P, Q ξ i ξ j Jq dq. et σ ij be the integral of the r.h.s. of the above inequality. Then σ ij has the same roerties as σ ij and there exist λ 1, λ such that with σ ij = λ 1 i j { λ δ ij } i j,.6

14 H. Yu / J. Differential Equations λ 1 = 1 λ = 1 r 3 i, j=1 3 i, j=1 Φ ij P, Q i j Jq dq, Φ ij P, Q r i r j Jq dq. By.1,.13 and.4, we know that there is a constant ν 1 > such that λ = min λ 1, λ ν 1, for all. Therefore, we have a ± ij t, x, ξ iξ j σ ij ξ i ξ j Sɛ σ ij ξ i ξ j ν Sɛ ν 1 ξ. i, j=1 i, j=1 i, j=1 If choosing ɛ small enough and K = ν Sɛ ν 1, this comletes the roof of emma.5. Remark.6. By the similar arguments as the above and using the sectrum theory about the classical andau oerator in [7] we can also obtain the ellitic estimate corresonding to the classical andau oerator, that is, emma.1 in [6]. 3. Smoothness effects for classical solutions In this section we shall rove Theorem 1.. The roof will consist in an induction on the number of derivatives in x and which can be controlled. We first rove regularity of variable. emma 3.1. et N 4 be a given integer and F =[F, F ] be a smooth non-negative solution of the system given by Theorem 1.1. We suose that for any T >, s, F t [,T ];H N,s x, K, and F t [,T ];H N,s x, K T 1/, where K = K s, T is a constant. Then there exists a constant C1 >, which deends on N, s, T, and K,suchthat x β F 1 s/ dt dx d C1, 3.1 [,T ] T 3 with β N. Proof. We only consider the regularity of F t, x, about because the regularity of F t, x, can be roved by the similar arguments. By using Eq..1, we know that h = x β F 1 s/ satisfies the following equation using Einstein s convention for the reeated indices and denoting g = x β F : t h x h = I II III, 3.

15 379 H. Yu / J. Differential Equations where I = I = C β 1 < β {,β 1 β E B 1 F 1 s/, for β =, C β 1 β β β 1 x x β 1 F 1 s/ C x E x β F C C β 1 β β 1 x B x β β 1 1 } F s/ s 1 s/ 1 C x E x β F C C β 1 β β 1 x B x β β 1 F, for β >, 1,β 1 β II = i a ij j h sa ij j g 1 s/ 1 i i [ sa ij g 1 s/ 1 j ] III = =, 1 i b i h sb i g 1 s/ 1 i i b i h sb i g 1 s/ 1 i i a ij j h sa ij j g 1 s/ 1 i i [ sa ij g 1 s/ 1 j ], C { [ i x a ij j x F 1 s/] s x a x F 1 s/ 1 i [ i x b i x F 1 s/] s x [ i x a [ i x b i x F 1 s/] s x b i x F 1 s/] s x III = C C β { [ 1 β i a =,β 1 β =β, β 1 1 b i x F 1 s/ 1 i a x F 1 s/ 1 i x F 1 s/ 1 } i, for β =, x β s a x β F 1 s/ 1 [ i i b i s b i x β F 1 s/ 1 [ i i a ij j s a x β F 1 s/ 1 [ i i b i s b i x β F 1 s/ 1 } i, for β >. F 1 s/] x β x β x β F 1 s/] F 1 s/] F 1 s/] We only consider the case β > because the case β = is similar. Multilying 3. by h and integrating on t, x,, we estimate the resulting equation term by term.

16 H. Yu / J. Differential Equations We see that [,T ] T 3 [ t h ] x h hdtdxd= 1 ht h 1 x, x, K. For the first term I 1 of I, the Hölder s inequality entails that [,T ] T 3 I 1 hdtdxd h t [,T ]; x, F t [,T ];Hx, N,s TK. 3.3 For the second term I of I, when N/, integral by arts and the fact that T 3 H T 3 imly that [,T ] T 3 I hdtdxd x [E, B] t,x 1 x β F [,T ] T 3 su x [E, B] F t x t [,T ];H N,s ɛ F ɛ t N,s g 1 s/ [,T ];Hx, t [,T ]; x,, x, x β β 1 F g 1 s dt dx d g 1 s/ t [,T ]; x, where we have used the fact that [E, B] t Hx N. If N/, then β N/ and we have [,T ] T 3 [,T ] I hdtdxd x β F x [E, B] x g 1 s dt d x x su x [E, B] x β F H t 1 x T 3 g 1 s dt d x [,T ] su x [E, B] F t x t [,T ];Hx, N,s g 1 s/ t [,T ]; x, ɛ F ɛ t N,s g 1 s/ [,T ];Hx, t [,T ]; x,. In any case, we can get [,T ] T 3 I hdtdxd ɛ K T ɛ g 1 s/ t [,T ]; x,. 3.4

17 379 H. Yu / J. Differential Equations SimilarlywehavethatthetermI 3 has the following estimate: [,T ] T 3 I 3 hdtdxd ɛ K T ɛ g 1 s/ t [,T ]; x,. For the term containing II, wewriteii = 1 i=1 II i, and then estimate term by term. II 1 hdtdxd= aij j h i h dt dx d [,T ] T 3 [,T ] T 3 = aij j g i g 1 s dt dx d [,T ] T 3 s aij j gg 1 s 1 j dt dx d [,T ] T 3 s aij g 1 s i j dt dx d. [,T ] T 3 Denote the r.h.s. of the above identity as A 1 A A 3. Thanks to emma.5, we get A 1 K [,T ] T 3 g 1 s/ dt dx d. 3.5 By emma. and the Sobolev embedding theorem, we see that su a ij t, x, t,x, x F 1 5 [,T ]; x, K. 3.6 By 3.6 and the Hölder s inequality, we get A K g g 1 s 1/ dt dx d [,T ] T 3 K ɛ g 1 s/ t [,T ]; x, CTK, 3.7 and A 3 K F t N,s [,T ];Hx, CTK3. 3.8

18 H. Yu / J. Differential Equations By 3.6, we obtain that [,T ] T 3 II hdtdxd [,T ] T 3 a ij j g 1 s 1/ g dt dx d K ɛ g 1 s/ t [,T ]; x, CTK. 3.9 For the term containing II 3,wewrite II 3 hdtdxd= s aij j hg 1 s/ 1 j dt dx d [,T ] T 3 [,T ] T 3 = s By 3.6 and the above arguments, we easily get [,T ] T 3 aij j gg 1 s 1 j dt dx d [,T ] T 3 s aij g 1 s i j dt dx d. [,T ] T 3 II 3 hdtdxd CK ɛ g 1 s/ t [,T ]; x, CTK. 3.1 Then we consider the term II 4. Integral by arts imly that [,T ] T 3 1 II 4 hdtdxd= [,T ] T 3 i b i By emma. and the Sobolev embedding theorem, it is easy to know su i b t, x, i t,x, Therefore, we have the following estimate:, β 1 [,T ] T 3 h dt dx d. F t, x, 1 6 t x,. II 4 hdtdxd K3 T. Because a shares the same roerties as a, the remaining terms of II has the similar as the above. ij ij NowweturntothetermIII. WewriteIII = 8 i=1 III i.byintegrationbyarts, [,T ] T 3 = [ i a [,T ] T 3 a x β F 1 s/] hdtdxd x β F i g 1 s dt dx d

19 3794 H. Yu / J. Differential Equations s [,T ] T 3 a ij j x β F g 1 s 1 i dt dx d. For the first term, when 1 β 1 N/ withn 4, emma. imlies that su t,x, a ij,β β 1 x β F 1 β 1 1/ t x, K, so that [,T ] T 3 a x β F i g 1 s dt dx d By emma., we have K Cɛ F ɛ t N,s g 1 s/ [,T ];Hx,. t [,T ]; x, a ij t, x, β β 1 x β F t, x, 1 β 1 1/ When β 1 > N/ and then β < N/, we have the following estimate [,T ] T 3 [,T ] T 3 a β β 1 [,T ] T 3 j a ij x β j x β F i g 1 s dt dx d x β F 1 s/ i g 1 s/ dxdt F t, x, 1 β 1 1/ x β F 1 s/ i g 1 s/ dxdt β β 1 [,T ] β β 1 x β F t, x, 1 β 1 1/ x, x β F 1 s/ x g 1 s 1 s/ dxdt x, x β F 1 s/ t [,T ]; x, ɛ g 1 s 1 s/ t [,T ]; x, C ɛ F t β β [,T ];H N, β 1 x, 1 ɛ K F ɛ t N,s g 1 s/ [,T ];Hx, t [,T ]; x,, 3.1 where we have used the facts that β N and T 3 H T 3. By the similar arguments as the above, we can rove that the second term of III 1 shares the same bound.

20 H. Yu / J. Differential Equations As for the term III 3, integration by arts imlies that [,T ] T 3 = s [ i b i [,T ] T 3 [,T ] T 3 b i b i x β F 1 s/] hdtdxd x β F i g 1 s dt dx d x β F g 1 s 1 i dt dx d. We only consider the first term of III 3 and then the second term is treated similarly. When 1 β 1 N/ 1, by the roof of emma., we easily get b i [,T ]; x, β 1 x F ± 1 β 1 1/ t [,t]; x, β β 1 1 By using the above inequality, we have [,T ] T 3 b i x β C b i [,T ];x, Cɛ F i g 1 s dt dx d x β ɛ g 1 s/ t [,T ]; x, F 1 s/ [,T ]; x, ɛ F ɛ t N,s g 1 s/ [,T ];Hx, t [,T ]; x,. Thanks to emma., we have b i t, x, β β 1 1 x β F t, x, 1 β 1 1/ By using 3.14, the similar arguments as 3.1 imly that, when N/ β 1 N 1 [,T ] T 3 b i x β F i g 1 s dt dx d ɛ K 3 T ɛ g 1 s/ t [,T ]; x,. The case that β 1 =N and β = require more care because the term b i contains the -derivative of F. In this case the total-derivatives of F will be N 1, which is imossible to be controlled by emma. and our assumtion. Thus we devise emmas.3 and.4 in order to treat it.

21 3796 H. Yu / J. Differential Equations By emma.3 and β >, we know that x β b ± i t, x, By the above inequality, we have [,T ] T 3 x β b i β 1 β 1 [,T ] T 3 β 1 β 1 1/ β 1 F i g 1 s dt dx d [,T ] T 3 F ± t, x, F ± t, x, 1 β 1 1/ F ± t, x, F i g 1 s dt dx d 1/ F ± t, x, 1 β 1 1/ F i g 1 s dt dx d C F 1 s/ t [,T ];x, Cɛ F [,T ];Hx, ɛ N g 1 s/ [,T ];Hx, N C F ± t, x, 1 β 1 1/ β 1 β 1 [,T ] T 3 F 1 1s/ i g 1 s dt dx F 1 s/ t [,T ];H 4 x, Cɛ F [,T ];H N x, ɛ g 1 s/ [,T ]; x, C F 1 1s/ [,T ]; x F ɛ [,T ];Hx, N,N4 g 1 s/ [,T ]; x, ɛ K T ɛ g 1 s/ t [,T ]; x,. Here we have used the Sobolev embedding theorem, F 1 s/ t [,T ]; x, F 1 s/ t [,T ];H 4 x, K By the similar arguments as the above, we can rove that the second term of III 3 shares the same bound. The other terms of III can be treated similarly and we omit it. Finally, we obtain that III has the estimate: IIIh dt dx d ɛ K T Cɛ g 1 s/ t [,T ]; x,. [,T ] T 3 By using the above estimates, we see that g 1 s/ t [,T ]; x, K C ɛ K 3 T. K Cɛ If we take ɛ > small enough, we conclude the roof of emma 3.1.

22 H. Yu / J. Differential Equations We now turn to rove the regularity for x variable. From [6] and the case near vacuum about the Boltzmann equation [], we will resort to the averaging lemma of the relativistic version [3,9]. Now we begin with emma 3.. et N 4 be a given integer and F =[F, F ] be a smooth non-negative solution of the system given by Theorem 1.1. We suose that for any T >, s, F t [,T ];H N,s x, K, and F t [,T ];H N,s x, K T 1/, where K = K s, T is a constant. Then there exists a constant C >, which deends on N, s, T, and K, such that [,T ] x β F 1 s/ Ḣ 1/ x dt d C, 3.17 with β N. Proof. et ρt, x, = x β F 1 s4/. To rove 3.17, we only need to rove [,T ] 1 4 m 1/1 ˆρt,m, dt d C, 3.18 m Z 3 where ˆρt,m, is the Fourier coefficient of ρ with resect to the x variable. et χ := χ C c R3 be a cutoff function which satisfies χ and R χ 3 d = 1. We set χ ɛ = ɛ 3 χ ɛ and write ˆρt,m, = [ ˆρt,m, ˆρt,m, χ ɛ ] ˆρt,m, χ ɛ. For the first term of the r.h.s. of the above equality, we get 1 4 ˆρt,m, ˆρt,m, χ ɛ d [ ] ˆρt,m, ˆρt,m, q χɛ q dq d ˆρt,m, ˆρt,m, q 1/ d χ ɛ q dq χ ɛ q q dq ˆρt,m, d ɛ ˆρt,m, d so that

23 3798 H. Yu / J. Differential Equations [,T ] 1 4 [,T ] m 1/1 ˆρt,m, ˆρt,m, χ ɛ dt d m Z 3 m 1/1 ɛ ˆρt,m, dt d m Z 3 Remembering that ρt, x, = h1, we see that ρ satisfies 3. with s relaced by s 4. Then we can write the equation satisfied by ρ under the form t ρ ρ = ρ 1 ρ. 3. Here ρ is the sum of the terms I, II i i = 1, 3, 4, 6, 8, 1 and III i i = 1, 3, 4, 5, while ρ 1 is the sum of the remaining terms. We claim that ρ 1, ρ t [, T ]; x,. We only resent the estimates for the terms I, II 1, II 5, III 1 and III 3 in the case β >, the others being similar. It is obvious that I 1 t [,T ]; x, F t [,T ];H N,s4 x, K T 1/. As for the term I, when N/, we use the Sobolev s inequality and Theorem 1.1 to obtain [,T ] T 3 x E x β F β 1 C x [E, B] t,x [,T ] T 3 x B x β F C x [E, B] t Hx F t N,s4 [,T ];Hx, CK T. x β β 1 F 1 s4 dt dx d x β β 1 F 1 s4 dt dx d When > N/ and then β N/, we use the Sobolev s inequality and Theorem 1.1 to obtain [,T ] T 3 [,T ] x E x β F β 1 x B x β F 1 s4/ x x β β 1 F 1 s4/ x x [E, B] t x [,T ] x β β 1 F 1 s4/ x, dt C x [E, B] t x F t N,s4 [,T ];Hx, CK T. x x β β 1 F 1 s4 dt dx d [E, B] dt x x β F 1 s4/ x,

24 H. Yu / J. Differential Equations For the term I 3, the similar arguments imly that it shares the same bound. As for the term II 1, we know from 3.6 that a ij j ρ t [,T ]; x, K j ρ t [,T ]; x, K C 1, where the last inequality holds thanks to emma 3.1 alied to ρ. As for the term II 5, by emma. and the Sobolev embedding theorem, it is easy to get su b t, x, i t,x, As a consequence, we have, β 1 1 F t, x, 1 6 t x, K. II 5 t [,T ]; x, b i g 1 s/1 i t [,T ]; x, K T 1/. As for the term III 1, when 1 β 1 N/ withn 4, we know from emma. that This imlies that su t,x, a ij,β β 1 a x β x β 1 β 1 1/ t x, K. F 1 s4/ t [,T ]; x, j x β F 1 s4/ t [,T ]; x, K T 1/. From the roof of emma., we see that a ij β β 1 1 x β F t, x, 1 β 1 1/. When β 1 > N/, then β 1 < N/ and we obtain from 3.11 that [,T ] T 3 [,T ] T 3 a a ij β β 1 1 [,T ] T 3 j x β x β F 1 s4/ dt dx d j x β x β F 1 s4/ dt dx F t, x, 1 β 1 1/ F 1 s4/ dt dx j x β F 1 s4/ t [,T ]; x, x β F t, x, 1 β 1 1/ [,T ]; x, F t [,T ];Hx, N,s4 x β F t, x, 1 β 1 1/ [,T ]; x, K4 T.

25 38 H. Yu / J. Differential Equations In any case we know that III 1 has the estimate III 1 t [,T ]; x, K T 1/. For the term III 3, we roceed like for the corresonding term in the roof of emma 3.1. We use 3.13, 3.14 and 3.15 and consider the cases β 1 < N/, N/ β 1 < N and β 1 =N searately in order to get b i x β F 1 s4/ t [,T ]; x, K T 1/. Finally, we indeed see that ρ 1, ρ t [, T ]; x,. Now according to.3 in Theorem.3 averaging lemma of the relativistic version of [], we can rove that T m 1/ ˆρt,m, χ dt C χ ɛ q 1 q χɛ q 1 q q q ˆρ,m, ˆρ,m, t [,T ]; ˆρ 1,m, t [,T ]; ˆρ,m, t [,T ];. Since χ ɛ q1 q q ɛ 3 1, we see that [,T ] 1 4 m Z 3 m m 1/1 ˆρt,m, χ dt d m Z ɛ 6 ɛ 8 ˆρ,m, ˆρ,m, t [,T ]; ˆρ 1,m, t [,T ]; ˆρ,m, t [,T ];. 3.1 If we choose ɛ = m 1/, we can bound 3.19 by using emma 3.1. Then we can bound 3.1 by the facts that ρ,, x, and ρ, ρ 1, ρ t [, T ]; x,. This comletes the roof of emma 3.. The following lemma will imrove the regularity of F about x variable. emma 3.3. et N 4 be a given integer and F =[F, F ] be a smooth non-negative solution of the system given by Theorem 1.1. We suose that for any T >, s, F t [,T ];H N,s x, K, and F t [,T ];H N,s x, K T 1/, where K = K s, T is a constant. Then for any T >, t 1 [, T ] and s, there exists a constant C3 >, which deends on N, s, T, t 1 and K,suchthat with β N. [t 1,T ] x β F 1 s/ Ḣ 1/1 dt d C3, x

26 H. Yu / J. Differential Equations Proof. We denote δ = 1/ for notational simlicity. et g δ,k t, x, = k ht, x, δ 3 for k T 3, where k ht, x, = ht, x k, ht, x, and ht, x, = x β F 1 s/. emma 3. tells us that g δ,k [, T ] so that for some t [, t 1 ], g δ,k t C /t x,,k 1. Noticing that x,,k ĝ δ,k dk = C m δ ĥm. T 3 In order to rove emma 3.3, we only need to rove [t 1,T ] [t 1,T ] T 3 m 4δ ĥm dt d C3, m Z 3 m 4δ ĥm dt d dk C3, m Z 3 or g δ,k dt d C3. 3. Ḣδ x [t 1,T ] In order to rove 3., we first rove that g δ,k dt dx d dk C [t,t ] T 3 T 3 m Z 3 This ste is similar to the roof of 3.1, the main difference being that we now have an additional integration with resect to k. We now write down the equation satisfied by g δ,k : t g δ,k x g δ,k = IV V VI δ 3, 3.4 where { } IV = C 1 k E 1 F k B 1 1 F s/, β =, IV = β 1 < β,β 1 β C β 1 β β β 1 x x β 1 { k F 1 s/ C C β 1 β β 1 k x B x β β 1 F 1 s/ s 1 s/ 1 C k x E x β F C C β 1 β β 1 k x B x β β 1 F, for β >, 1,β 1 β C k x E x β F }

27 38 H. Yu / J. Differential Equations V = i k a ij j h s k a ij j g 1 s/ 1 i i [ s k a ij g 1 s/ 1 j ] i k b i h s k b i g 1 s/ 1 i i k b i h s k b i g 1 s/ 1 i i k a ij j h s k a ij j g 1 s/ 1 i i [ s k a ij g 1 s/ 1 j ], VI = C { [ 1 i k x a x F 1 s/] =, 1 s k x a x F 1 s/ 1 [ i k i x b i x F 1 s/] s k x b i x F 1 s/ 1 [ i i k x s k x a x s k x b i x F 1 s/ 1 } i, for β =, a ij x F 1 s/ 1 i i [ k j x F 1 s/] b i x F 1 s/] VI = C C β { [ 1 β i k a x β F 1 s/] =,β 1 β =β, β 1 1 s k a ij j x β F 1 s/ 1 i [ i k b i x β F 1 s/] s k b i x β F 1 s/ 1 i [ i k a x β F 1 s/] s k a x β F 1 s/ 1 i [ i k b i x β F 1 s/] s k b i x β F 1 s/ 1 i }, for β >. We still consider the case β 1. We multily Eq. 3.4 by g δ,k and then on t, x,, k in the domain [t, T ] T 3 T 3. We only estimate the terms containing the terms of IV, the first and fourth term of V and the first term of VI, denotedbyiv 1, IV, IV 3, V 1, V 4 and VI 1 resectively, since the estimates for the other terms are similar. For the first term IV 1,weget δ IV 1 g δ,k 3 dt dx d dk [t,t ] T 3 T 3 x x β 1 k F 1 s/ δ 3 [t,t ]; x,,k g δ,k. [t,t ]; x,,k 3.5 β 1 < β From 3.17, we easily know for any s and β N [t,t ] T 3 T 3 k x β F 1 s/ dt dx d dk C. 3.6

28 H. Yu / J. Differential Equations By using 3.6 we can rove that the r.h.s. of 3.5 is bounded by some constant C >. For any function f x and gx, k f xgx = f x k k gx gx k f x. For the term IV, we use integral by arts to obtain δ IV g δ,k 3 dt dx d dk [t,t ] T 3 T 3 { = 1 s/ δ 3 k x E x β F,β 1 β } g δ,k dt dx d dk. β 1 k x B x β β 1 F We only estimate the first art of the r.h.s. of the above equation and then the second art is treated similarly. We know that k x E x β F = x Ex k k x β F x β F k x E. We only consider the first art of the r.h.s. of the above equation also because the second art can be estimated in the same way. Then when N/, we know that x Ex k t [t,t ]; x,k x Ex k t [t,t ];. k H x By using the above inequality we can obtain 1 [t,t ] T 3 T 3 x Ex k k x β F 1 s/ δ 3 g δ,k dt dx d dk C k x β F 1 s/ δ 3 t [t,t ]; x,,k ɛ g δ,k t [t,t ]; x,,k ɛ C ɛ g δ,k [t,t ]; 3.7. x,,k When > N/, we can obtain 1 [t,t ] T 3 T 3 [t,t ] T 3 x Ex k k x β F 1 s/ δ 3 g δ,k dt dx d dk x Ex k k x β F 1 s/ δ 3 x C ɛ δ 3 t [t,t ];,k,x ɛ g δ,k t [t,t ];,k,x x Ex k t [t,t ]; x x k x β F 1 s/ g δ,k x dt dx d dk ɛ C ɛ g δ,k [t,t ]; 3.8. x,,k

29 384 H. Yu / J. Differential Equations We will consider the first term V 1 of V. We can write V 1 δ 3 = i a ij x k j g δ,k i k a h δ 3. We denote the r.h.s. of the above equality by B 1 B. After integration by arts, [t,t ] T 3 T 3 B 1 g δ,k dt dx d dk = = aij x k j g δ,k i g δ,k dt dx d dk [t,t ] T 3 T 3 aij x k[ i k g ][ j k g ] 1 s δ 3 dt dx d dk [t,t ] T 3 T 3 s aij x k[ i k g ] k g 1 s 1 j δ 3 dt dx d dk [t,t ] T 3 T 3 s aij x k k g 1 s i j δ 3 dt dx d dk, [t,t ] T 3 T 3 where g = x β F. We write the r.h.s. of the above equality as B 1,1 B 1, B 1,3. Since F is a satially eriodic function, we obtain from emma.5 that B 1,1 K [t,t ] T 3 T 3 [ k g ] 1 s/ δ 3 dt dx d dk. 3.9 Thanks to emma., we know that a ij x k t [,T ]; x,k, x F 1 5 t [,T ]; x, K. Using the Hölder s inequality and emma 3., we get B 1, K [t,t ] T 3 T 3 k g 1 s/ δ 3 k g 1 s1/ δ 3 dt dx d dk [ K ɛ k g ] 1 s/ δ 3 [t,t ]; x,,k C ɛ C. Similarly we can obtain B 1,3 C K. As for the term containing B, we use 3.6, integration by arts and the following inequality k a ij x, k x F 1 5 x, K,

30 H. Yu / J. Differential Equations so that we can obtain B g δ,k dt dx d dk [t,t ] T 3 T 3 [ K ɛ k g ] 1 s/ δ 3 [t,t ]; x,,k C ɛ C. As for the term containing V 4, we use 3.6, integration by arts and the following inequality so that we can obtain k b i x, k x j F 1 5 x, K, δ V 4 g δ,k 3 dt dx d dk [t,t ] T 3 T 3 [ K ɛ k g ] 1 s/ δ 3 [t,t ]; x,,k C ɛ C. As for the terms containing VI 1,wewrite [ i k a x β F 1 s/] δ 3 [ = i a ij x k j x β k F 1 s/] δ 3 [ i k a ij j x β F 1 s/] δ 3, and we denote it by D 1 D.Byintegrationbyarts, [t,t ] T 3 T 3 D 1 g δ,k dt dx d dk = = [t,t ] T 3 T 3 [t,t ] T 3 T 3 [t,t ] T 3 T 3 a ij x k j x β k F i g δ,k 1 s/ δ 3 dt dx d dk a ij x k j x β k F i k g 1 s δ 3 dt dx d dk a ij x k j x β k F k g 1 s 1 i δ 3 dt dx d dk. We write the r.h.s. of the above equality as D 1,1 D 1,.When β 1 N/, emma. ensures that a ij x k t [,T ]; x,k Thanks to 3.6, we get,β β 1 x β F 1 β 1 1/ t [,T ]; x,.

31 386 H. Yu / J. Differential Equations [ D 1,1 K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ j x β k F 1 s/ δ 3 t [t,t ]; x,,k [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. When β 1 > N/, we obtain from emma. that so that D 1,1 a ij t, x, [t,t ] T 3 T 3 β β 1 x β F 1 β 1 1/, a ij x k j x β k F 1 s/ δ 3 k g 1 s/ δ 3 dt dx dk j x β k F 1 s/ δ 3 x β β 1 [t,t ] T 3 F x k 1 β 1 1/ k g 1 s/ δ 3 x, dt dk x,,β β 1 [t,t ] j x β k F 1 s/ δ 3 x,,k x β F x k 1 β 1 1/ k g 1 s/ δ 3 x, dt x,,k [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. Thus in any case we get [ D 1,1 K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. 3.3 Similarly, we can obtain As for D,byintegrationbyarts,weget [t,t ] T 3 T 3 D g δ,k dt dx d dk = = [t,t ] T 3 T 3 [t,t ] T 3 T 3 D 1, K K T C. k a x β k F i g δ,k 1 s/ δ 3 dt dx d dk k a x β F i k g 1 s δ 3 dt dx d dk

32 s [t,t ] T 3 T 3 H. Yu / J. Differential Equations k a ij j x β F k g 1 s 1 i δ 3 dt dx d dk. We denote the r.h.s. of the above equality by D,1 D,.When β 1 N/, emma. ensures that k a ij x,,β β 1 x β k F 1 β 1 1/ x,. Thanks to the above inequality we can obtain D,1 [t,t ] T 3 j k a ij x x β F i k g x 1 s δ 3 dt d dk x,β β 1 [t,t ] T 3 x β k F 1 β 1 1/ δ 3 x, j x β F 1 s/ i k g 1 s/ δ 3 x, dt dk x, [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. When β 1 > N/, we obtain from emma. that k a ij β β 1 x β k F 1 β 1 1/. Thanks to the above inequality we can obtain D,1 [t,t ] T 3 j k a ij x β F 1 s/ i k g 1 s/ δ 3 dt dx dk,β β 1 [t,t ] T 3 x β k F 1 β 1 1/ δ 3 x, j x β F 1 s/ i k g 1 s/ δ 3 x, dt dk x, [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. Similarly we can get Finally the term VI 1 has the following estimate D, K K T C.

33 388 H. Yu / J. Differential Equations δ VI 1 g δ,k 3 dt dx d dk [t,t ] T 3 T 3 [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k ɛ C K T C. As for the terms containing VI 3,wewrite [ i k b i x β F 1 s/] δ 3 [ = i b x k i x β k F 1 s/] δ 3 [ i k b i x β F 1 s/] δ 3, and we denote it by E 1 E.Byintegrationbyarts, [t,t ] T 3 T 3 E 1 g δ,k dt dx d dk = = [t,t ] T 3 T 3 [t,t ] T 3 T 3 [t,t ] T 3 T 3 b i x k x β k F i g δ,k 1 s/ δ 3 dt dx d dk b i x k x β k F i k g 1 s δ 3 dt dx d dk b i x k x β k F k g 1 s 1 i δ 3 dt dx d dk. We write the r.h.s. of the above equality as E 1,1 E 1,.When β 1 N/, emma. ensures that b i x k t [,T ]; x,k,, β β 1 1 x β F 1 β 1 1/ t [,T ]; x, K. Thanks to 3.6 and the Hölder inequality, we get [ E 1,1 K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. When N/ < β 1 N 1, we obtain from emma. that b i x k β β 1 1 x β F x k 1 β 1 1/, so that

34 E 1,1 [t,t ] T 3 T 3 H. Yu / J. Differential Equations b i x k x β k F 1 s/ δ 3 k g 1 s/ δ 3 dt dx dk x β k F 1 s/ δ 3 β β 1 1 [t,t ] T 3 x x β F x k 1 β 1 1/ k g 1 s/ δ 3 x, dt dk x, x β k F 1 s/ δ 3, β β 1 1[t,T ] x,,k x β F x k 1 β 1 1/ k g 1 s δ 3 x, [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. x,,k dt When β 1 =N with β 1 >, we obtain from emma.3 that b i x k β β 1 1 1/ β x β F x k x β F x k 1 β 1 1/. By using the above inequality, we have [t,t ] T 3 T 3 β β 1 1 b x k i k F i k g 1 s δ 3 dt dx d dk [t,t ] T 3 T 3 x β F x k k F 1 s/ δ 3/ k g 1 s/ δ 3 dt dx d dk 1/ C x β F x k 1 β 1 1/ [t,t ] T 3 T 3 k F 1 s/ δ 3/ k g 1 s/ δ 3 dt dx d dk x β F t [t,t ]; x, x β k F 1 s/ β 4, β β 1 1 δ 3 t [t,t ]; x,,k k g 1 s/ δ 3 C β 1 β 1 [t,t ] T 3 k F 1 1s/ δ 3 F x k 1 β 1 1/ x, x t [t,t ]; x,,k k g 1 s/ δ 3 dt dk x,

35 381 H. Yu / J. Differential Equations [ K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C CK ɛ [ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ k F 1 1s/ δ 3 [,T ]; x,,k C C ɛ K ɛ [ k g ] 1 s/ δ 3 t [t,t ];. x,,k Here we have used 3.6 and the Sobolev embedding inequality: F 1 s/ t [,T ]; x, F 1 s/ t [,T ];H 4 x, K. Thus in any case we get [ E 1,1 K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C Similarly, we can obtain As for E,byintegrationbyarts,weget [t,t ] T 3 T 3 E g δ,k dt dx d dk = = [t,t ] T 3 T 3 [t,t ] T 3 T 3 s [t,t ] T 3 T 3 k b i j k b i j k b i j E 1, K K T C. 3.3 x β F i g δ,k 1 s/ δ 3 dt dx d dk x β F i k g 1 s δ 3 dt dx d dk x β F k g 1 s 1 i δ 3 dt dx d dk. We denote the r.h.s. of the above equality by E,1 E,. These two terms can be estimated in the same way. We do not detail the comutation, but we should notice that the cases β 1 N/, N/ < β 1 N 1 and β 1 =N are considered searately. As a consequence, [ E K ɛ k g ] 1 s/ δ 3 t [t,t ]; x,,k C ɛ C. Finally, we can obtain that [ k g ] 1 s/ δ 3 t [t,t ]; x,,k ɛ C K CK 3 T K Cɛ If we take ɛ > small enough, we can rove 3.3.

36 H. Yu / J. Differential Equations In order to rove 3., we will use the method of the roof of emma 3.. We introduce ρ δ,k = g δ,k 1 and write ˆρ δ,k t,m, = [ ˆρ δ,k t,m, ˆρ δ,k t,m, χ ɛ ] ˆρδ,k t,m, χ ɛ, where ˆρt,m, is the Fourier coefficient of ρ with resect to the x variable and the arameter ɛ will be chosen. Following the roof of 3.19, we can obtain [t,t ] T [t,t ] T 3 m 1/1 ˆρ δ,k t,m, ˆρt,m, χ ɛ dt d dk m Z 3 m 1/1 ɛ ˆρt,m, dt d dk m Z 3 Notice that ρ δ,k is the solution of Eq. 3. where s is relaced by s 4. Thus we have t ρ δ,k ρ δ,k = ρ 1 δ,k ρ δ,k δ Here ρ δ,k is the sum of the terms IV, V i i = 1, 3, 4, 6, 8, 1 and VI i i = 1, 3, 4, 6, while ρ 1 δ,k is the sum of the remaining terms. We claim that ρ 1 δ,k δ 3, ρ δ,k δ 3 t [t, T ]; x,,k. We only resent the estimates of IV and VI 3 in the case β 1 here, the others being similar. We write k x E x β F = x Ex k k x β F x β F k x E. We only consider the first art of the r.h.s. of the above equation also because the second art can be estimated in the same way. Then when N/, we know that x Ex k [t,t ]; x,k x Ex k [t,t ]; ɛ. k H x By using 3.6 and the above inequality we can obtain x Ex k k x β F 1 s4/ δ 3 t [t,t ]; x,,k C. When > N/, we can obtain [t,t ] T 3 T 3 [t,t ] T 3 x Ex k k x β F 1 s4/ δ 3 dt dx d dk x Ex k k x β F 1 s/ δ 3 x x dt d dk x Ex k [t,t ]; k 1 x β F 1 s/ δ 3. x t [t,t ];,k,x Now we consider the term VI 3.Wewrite

37 381 H. Yu / J. Differential Equations k b i x β F 1 s4/ δ 3 = b i x k x β k F 1 s4/ δ 3 k b i x β F 1 s4/ δ 3, and we only estimate the first term of r.h.s. of the above equality, the second being similar. When β 1 N/, emma. ensures that b i x k t [,T ]; x,k, β β 1 1 x β F 1 β 1 1/ t [,T ]; x,. Thanks to 3.6, we get b x k i x β k F 1 s4/ δ 3 t [t,t ]; x,,k K C. When N/ < β 1 N 1, we obtain from emma. that b i x k β β 1 1 x β F x k 1 β 1 1/, so that [t,t ] T 3 T 3 b x k i x β k F 1 s4/ δ 3 dt dx d dk β β 1 1 [t,t ] T 3 x β F x k 1 β 1 1/ x, x β k F 1 s4/ δ 3 x dt dx dk x β F 1 β 1 1/ t [t,t ]; x,, β β 1 1 x β k F 1 s4/ δ 3 t [t,t ]; x,,k K C. When β 1 =N with β 1 >, we obtain from emma.3 that b i x k By the above inequality, we have [t,t ] T 3 T 3 β β 1 1 β β 1 1 1/ β x β F x k x β F x k 1 β 1 1/. b x k i k F 1 s4/ δ 3 dt dx d dk [t,t ] T 3 T 3 x β F x k k F 1 s4 δ 3 dt dx d dk

38 C [t,t ] T 3 T 3 H. Yu / J. Differential Equations x β F x k 1 β 1 1/ k F 1 s5 δ 3 dt dx d dk β 4, β β 1 1 x β F [t,t ]; x, x β k F 1 s5/ δ 3 C β 1 β 1 [,T ] T 3 [,T ]; x,,k x β 1 F x k 1 β 1 1/ x, k F 1 s5/ δ 3 x dt dk K C CK x k F 1 s5/ δ 3 [,T ];. x,,k Here we have used 3.6 and the Sobolev embedding inequality: F 1 s/ t [,T ]; x, F 1 s/ t [,T ];H 4 x, K. Proceeding like in the estimate for 3.1, we can get m 1/ [t,t ] T m δ ˆρ δ,k t,m, χ dt d dk m Z 3 m Z 3 m δ ɛ 6 ɛ 8 ˆρ δ,k,m, ˆρ δ,k,m, t [,T ];,k ˆρ 1 δ,k,m, t [,T ];,k ˆρ 1 δ,k,m, t [,T ];,k Choosing ɛ = m δ and using 3.34 and 3.3, we get [t,t ] T m δ ˆρ δ,k t,m, dt d dk C3, m Z 3 which imlies the estimate 3.. This ends the roof of emma 3.3. Now if iterating emma 3.3 twenty times, we end u with a time τ 1 > small enough such that F [τ 1,T ];Hx, N1,s <. We need to transform into estimate. For this, we consider ρ = x β F 1 s/ for β N 1. From the Maxwell system 1.3 and 1.4, we have T 3 x E { x F } x F d { dxd = dt x E x x B }. x Then for any N 1 and t [τ 1, T ], we can obtain from 1.11 that

39 3814 H. Yu / J. Differential Equations x Et t [τ 1,T ]; x, x Bt t [τ 1,T ]; x, x Eτ 1 x x Bτ 1 x CT { x F x F } 1 4 [τ 1,T ]; x, C Remark 3.4. If Et, x = x φt, x and Bt, x =, we use the satially eriodic condition to obtain x x φt t [τ 1,T ]; x C x F t t [τ 1,T ]; x,, x φt t [τ 1,T ]; x F t [τ 1,T ]; x, By the similar arguments as emma 3.1, we easily know ρ t [τ1, T ]; x,. We claim that ρ satisfies an equation of the tye t ρ x ρ = ρ 1 ρ, with ρ 1, ρ t [τ 1, T ]; x,. Indeed, since we know that [E, B] t [τ 1, T ]; Hx N1, F t [, T ]; Hx, N,s and t [τ 1, T ]; Hx, N1,s and ρ t [τ 1, T ]; x,, we can follow the roof of the fact that ρ 1 and ρ in 3. belong to t [τ 1, T ]; x,. Multilying now the above equation by ρ, and integrating on [t, T ] T 3, where t > τ 1 small enough such that ρt,, x,,weget 1 ρt ρt x, ρ1 x, t [t,t]; x, ρ t [t,t]; x, ρ t [t,t]; x, ρ t [t,t]; x,. We thus obtain ρ t [t, t]; x,. By emmas and the above arguments, we can have the following roosition: Proosition 3.5. et N 4 be a given integer and F =[F, F ] be a smooth non-negative solution of the system given by Theorem 1.1. We suose that for any T >, s, F t [,T ];Hx, N,s K,where K = K s, T is a constant. Then for any T >, t [, T ] and s, there exists a constant C6 >, which deends on N, s, T, t and K,suchthat F t [t,t ];H N1,s x, C Proof of Theorem 1.. By using Proosition 3.5 and the assumtion 1.11 reeatedly, we get by induction of N that for any < τ < T < and s, f t [τ, T ]; H,s x,. We now rove by induction on n that t n F t [τ, T ]; Hx,,s. By the above, this is true for n =. et us assume that the induction hyothesis holds for any integer k n. Then for all multi-indices and β, anys,

40 H. Yu / J. Differential Equations [ x β ] [ ] n1 t F 1 s/ x β x n 1 t F s/ = n =,β 1 β =β l= β 1 =,β 1 β =β l= C C β 1 β Cl n t l x E n l t x β F t l x B n l t x β 1 F s/ n C C β [ 1 β Cl n l t a ij n l t x β i j F 1 s/ t l i a ij n l t x β j F 1 s/ t l b i n l t x β i F 1 s/ l t i b i n l t x β F 1 s/ t l a ij n l t x β i j F 1 s/ t l i a ij n l t x β j F 1 s/ t l b i n l t x β i F 1 s/ l t i b i n l t x β F 1 s/] By 1.11, 3.41, t n F t [τ, T ]; Hx,,s and the Maxwell system 1.3 and 1.4, we get, for any multiindex and τ 1 τ t T, that t x B t [τ,t ]; x x x E t [τ,t ]; x C7, t x E t [τ,t ]; x x x B t [τ,t ]; x C { x F x F } 1 4 t [τ,t ]; x, C7. By the above inequalities, we can obtain, for any integer l and any, that l t x [E, B] t [τ,t ]; x C We first estimate the second term of the l.h.s. of We see that [ ] x β x n 1 t F s/ n t x β 1 F 1 s/ t t x,. x, β 1 β By the induction hyothesis the above term is bounded. The third term of the l.h.s. of 3.41 can be bounded by C l t x [E, B] t x n l t x β F t x, C l t x [E, B] t Hx n l t x β F t x,. By the induction hyothesis and 3.4, we know the above term is bounded.

41 3816 H. Yu / J. Differential Equations By emma. and the Sobolev embedding theorem, we know l t a ± ij t x, C β β 1 l t b ± i t x, C β β 1 1 l t x β F ± 1 β 1 1/ t H x,, l t x β F ± 1 β 1 1/ t H x,. By the induction hyothesis, we know the above term is bounded. By using these two bounded terms, we can rove that the other terms of r.h.s. of 3.41 are in t [τ, T ]; x,. Therefore, we know that for some constant C9 >, [ x β n1 t F ] 1 s/ t x, C9. This ends the roof of Theorem 1.. Remark 3.6. By using 3.38 and 3.39, the similar arguments as the above imly that classical solutions to the relativistic andau Poisson system satisfy 1.1 without any extra assumtion. Acknowledgments The author thanks the referee for his/her valuable comments. The research of the author was suorted artially by NSFC Grant and artially by FANEDD. Recently the author was informed that Prof. Chen Y. [4,5] have roved the smoothing effects for classical solutions of the non-relativistic Valsov Maxwell Poisson andau system obtained in [18,19] by the author. References [1] A. Arsen ev, O. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the andau Fokker Planck equation, Math. USSR Sb [] F. Bouchut,. Desvillettes, Averaging lemmas without time Fourier transform and alication to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A [3]. Boudin,. Desvillettes, On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math [4] Y. Chen, Smoothness of classical solutions to Vlasov Poisson andau system, Kinetic Related Models [5] Y. Chen, Smoothness of classical solutions to Vlasov Maxwell andau system near Maxwellians, Discrete Contin. Dyn. Syst [6] Y. Chen,. Desvillettes,. He, Smoothing effects for classical solutions of the full andau equation, Arch. Ration. Mech. Anal., in ress, htt:// [7] P. Degond, M. emou, Disersion relations for the linearized Fokker Planck equation, Arch. Ration. Mech. Anal [8]. Desvillettes, C. Villani, On the satially homogeneous andau equation for hard otentials I II, Comm. Partial Differential Equations [9] R.-J. DiPerna, P.-. ions, Global weak solution of Vlasov Maxwell systems, Comm. Pure Al. Math [1] R.T. Glassey, W. Strauss, Asymtotic stability of the relativistic Maxwellian via fourteen moments, Transort Theory Statist. Phys [11] Y. Guo, The andau equation in eriodic box, Comm. Math. Phys [1] Y. Guo, Boltzmann diffusive limit beyond the Navier Stokes aroximation, Comm. Pure Al. Math [13]. Hsiao, H.-J. Yu, Global classical solutions to the initial value roblem for the relativistic andau equation, J. Differential Equations [14]. Hsiao, H.-J. Yu, On the Cauchy roblem of the Boltzmann and andau equations with soft otentials, Quart. Al. Math [15] M. emou, inearized quantum and relativistic Fokker Planck andau equations, Math. Methods Al. Sci [16] R.M. Strain, The Vlasov Maxwell Boltzmann system in the whole sace, Comm. Math. Phys [17] R.M. Strain, Y. Guo, Stability of the relativistic Maxwellian in a collisional lasma, Comm. Math. Phys