Author manuscript, published in "Kinet. Relat. Models 1, 3 8) 355-368" PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Abstract. By using the energy-type inequality, we obtain, in this paper, the result on propagation of Gevrey regularity for the solution of the spatially homogeneous Landau equation in the cases of Maxwellian molecules and hard potential. hal-4343, version 1 - Nov 9 1. introduction There are many papers concerning the propagation of regularity for the solution of the Boltzmann equation cf. [5, 6, 8, 9, 13] and references therein). In these works, it has been shown that the Sobolev or Lebesgue regularity satisfied by the initial datum is propagated along the time variable. The solutions having the Gevrey regularity for a finite time have been constructed in [15] in which the initial data has the same Gevrey regularity. Recently, the uniform propagation in all time of the Gevrey regularity has been proved in [4] in the case of Maxwellian molecules, which was based on the Wild expansion and the characterization of the Gevrey regularity by the Fourier transform. In this paper, we study the propagation of Gevrey regularity for the solution of Landau equation, which is the limit of the Boltzmann equation when the collisions become grazing, see [3] for more details. Also we know that the Landau equation can be regarded as a non-linear and non-local analog of the hypo-elliptic Fokker-Planck equation, and if we choose a suitable orthogonal basis, the Landau equation in the Maxwellian molecules case will become a non-linear Fokker-Planck equation cf. [17]). Recently, a lot of progress on the Sobolev regularity has been made for the spatially homogeneous and inhomogeneous Landau equations, cf. [, 6, 7, 16] and references therein. On the other hand, in the Gevrey class frame, the local Gevrey regularity for all variables t, x, v is obtained in [1] for some semi-linear Fokker-Planck equations. Let us consider the following Cauchy problem for the spatially homogeneous Landau equation, { } t f= v { av v R n )[ f v ) v f v) f v) v f v )]dv, 1) f, v)= f v), where f t, v) stands for the density of particles with velocity v R n at time t, and a i j ) is a nonnegative symmetric matrix given by a i j v)= δ i j v ) iv j v v γ+,γ [, 1]. ) Here and throughout the paper, we consider only the hard potential case i.e.γ, 1]) and the Maxwellian molecules case i.e. γ = ). Mathematics Subject Classification. Primary: 35B65, 76P5. Key words and phrases. Landau equation, Boltzmann equation, Gevrey regularity. This work is partially supported by the NSFC. 1
HUA CHEN, WEI-XI LI AND CHAO-JIANG XU hal-4343, version 1 - Nov 9 Set b i = n j a i j = v γ v i, i=1,,, n; c= j=1 n i v j a i j = γ+3) v γ, ā i j t, v)= a i j f ) t, v)= a i j v v ) f t, v )dv, b i = b i f, c=c f. R n Then the Cauchy problems 1) can be rewritten in the following form: { t f= n ā i j i v j f c f, f, v)= f v), which is a non-linear diffusion equation with the coefficients ā i j and c depending on the solution f. The motivation for studying the Cauchy problem 3) cf. [1]) comes from the study of the inhomogenous Boltzmann equations without angular cutoff and non linear Vlasov- Fokker-Planck equation see [1, 11]). Throughout the paper, for a mult-indexα=α 1,α,,α n ) and an integer k with k α, the notation D α k is always used to denote γ v with the multi-indexγsatisfying γ α and γ = α k. We denote also by M f t)), E f t)) and H f t)) as the mass, energy and entropy respectively for the function f t, ). That is, M f t))= f t, v) dv, E f t))= 1 f t, v) v dv, R n R n H f t))= f t, v) log f t, v) dv. R n Denote here M = M f )), E = E f )) and H = H f )). It s known that the solutions of the Landau equation satisfy the formal conservation laws: M f t))= M, E f t))=e, H f t)) H, for t. Also in this paper we use the following notations f t, ) L 1 s = f t, v) 1+ v ) s/ dv, R n α v f t, ) = α L s v f t, v) 1+ v ) s/ dv, R n f t, ) Hs m = α v f t). L s α m When there is no risk causing confusion, we write gt) L 1 s for gt, ) L 1 s. Next, let us recall the definition of the Gevrey class function space G σ R N ), where σ 1 is the Gevrey index cf. [14]). Let u C R N ). We say u G σ R N ) if there exists a constant C, called the Gevrey constant, such that for all multi-indicesα N N, α u L R N ) C α +1 α!) σ. We denote by G σ RN ) the space of Gevrey function with compact support. Note that G 1 R N ) is space of all real analytic functions. In the hard potential case, the existence, uniqueness and Sobolev regularity of the weak solution had been studied in [6], in which they proved that, under suitable assumptions on the initial datum e.g. f L 1 +δ withδ>,) there exists a unique weak solution of the 3)
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS 3 hal-4343, version 1 - Nov 9 Cauchy problem 3), which moreover is in the space C R + t ;SR3 ) ). HereR + t andsdenotes the space of smooth functions which decay rapidly at infinity. =,+ ) Assuming the existence of the smooth solution, we state now the main result of the paper as follows: Theorem 1.1. Let f be an initial datum with finite mass, energy and entropy. Suppose f G σ R n ) withσ>1, and f is a solution of the Cauchy problem 3) which satisfies f t, v) L loc [,+ [; H m R n ) ) L loc [,+ [; H m+1 γ R n ) ), for all m. 4) Then f t, ) G σ R n ) for all t> uniformly, namely, the Gevrey constant of f t, ) is independent of t. More precisely, for any fixed T >, there exists a constant C > which is independent of t, such that for any multi-indexα, one has sup α v f t, ) L R n ) C α +1 α!) σ. t [,T] Remark 1. If f L 1 +δ additionally, then by the result of [6], the Cauchy problem 3) admits a solution which satisfies 4). Remark. For simplicity, we shall prove Theorem 1.1 in the case of space dimension n = 3. The conclusion for general cases can be deduced similarly. The plan of the paper is as follows: In Section we prove some lemmas. Section 3 is devoted to the proof of the main result.. Some Lemmas In this section we give some lemmas, which will be used in the proof of the main result. Lemma.1. For anyσ>1, there exists a constant C σ, depending only onσ, such that for all multi-indicesµ N 3, µ 1, 1 β 3 C σ µ σ 1, 5) and 1 β µ 1 1 β µ 1 β µ β ) C σ µ σ 1. 6) Here and throughout the paper, the notation 1 β µ denotes the summation over all the multi-indices satisfyingβ µ and 1 β µ. Also the notation 1 β µ 1 denotes the summation over all the multi-indices satisfyingβ µ and 1 β µ 1. Proof. For each positive integer l, we denote by N{ β = l} the number of the multi-indices β with β =l. In the case when the space dimension is 3, one has It is easy to deduce that N{ β =l}= l+)!! l! 1 β µ 1 β 3 µ l=1 β =l = 1 l+1)l+). 1 l 3= µ l=1 N{ β =l} l 3.
4 HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Combining the estimates above, it holds that 1 β µ 1 β 3 1 µ l=1 l+1)l+) l 3 3 Observing that 3 µ l=1 l 1 C σ µ σ 1 for some constant C σ, we obtain the desired estimate 5). Similarly we can deduce the estimate 6). The following lemma is crucial to the proof of Theorem 1.1. Lemma.. Letσ>1. There exist constant C 1, C >, depending only on M, E, H andγ, such that for all multi-indicesµ N 3 with µ and all t>, we have µ l=1 1 l. hal-4343, version 1 - Nov 9 t µ v f t) + C L 1 v µ v f t) C L µ v D µ 1 f t) γ L γ + C Cµ β v D µ β +1 f t) L γ v D µ 1 f t) L γ [G σ f t)) ] β β µ + C β µ β v f t) L γ v D µ 1 f t) L γ [G σ f t)) ] µ β, where Cµ= β µ! µ β)!β! is the binomial coefficient, and[ G σ f t)) ] ν = ν v f t) L + B ν ν!) σ with B being the constant as given in Lemma.4 below. By the assumption in Theorem 1.1, the solution f t, v) of the Cauchy problem 3) is smooth in v, and so are the coefficients ā i j = a i j f, b i = b i f and c=c f. Here and in what follows, we write C for a constant, depending only on the Gevrey indexσ, and M, E and H the initial mass, energy and entropy), which may be different in different contexts. The proof of Lemma. can be deduced by the following lemmas: Lemma.3. uniformly ellipticity) There exists a constant K, depending only on γ and M, E, H, such that 3 ā i j t, v)ξ i ξ j K1+ v ) γ/ ξ, ξ, andγ [, 1]. 7) Proof. See Proposition 4 of [6] Remark 3. Although the ellipticity of ā i j ) was proved in [6] in the hard potential case γ, 1], it still holds for the Maxwellian caseγ=. This can be seen in the proof of Proposition 4 of [6]. Lemma.4. There exists a constant B, depending only on the Gevrey indexσ>1, such that for all multi-indicesβ with β and all g, h L γ R3 ), one has 3 β [ vā i j t, v))gv)hv)dv C g L γ h L Gσ γ f t)) ] β, for all t>, where [ G σ f t)) ] β ={ D β f t) L + B β [ β )! ] σ }. Proof. Forσ>1, there exists a functionψ G σ R3 ) cf. [14]) with compact support in { v v }, satisfying thatψv)=1 on the ball { v v 1 }, and that for some constant B > 4 depending only on σ, sup λ v ψ B λ λ 1!) σ, for allλ Z 3 +. 8)
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS 5 Write a i j =ψa i j + 1 ψ)a i j. Then ā i j = ψa i j ) f+ [1 ψ)a i j ] f. It is easy to see that β vā i j = [ ν vψa i j ) ] β ν v f )+ { β v[ 1 ψ)ai j ]} f, for ν =. We first treat the term [ ν vψa i j ) ] β ν v f ). A direct computation shows that [ ν vψa i j ) ] β ν v f )v) [ = ν v ψa i j ) ] v v ) β ν v f )v )dv R 3 C β ν v f )v ) dv { v v } C β ν v f t) L. hal-4343, version 1 - Nov 9 Next, for the term { β ]} v[ 1 ψ)ai j f, one has, by using the Leibniz s formula, β ]) v[ 1 ψ)ai j f v) = Cβ λ [ β λ 1 ψ) ] ) v v ) λ va i j v v ) f v )dv λ β β λ v ψ ) v v ) λ v a i j) v v ) f v )dv 1 λ β + =J 1 + J. C λ β { v v 1} {1 v v } [ 1 ψ) ] v v ) β va i j ) v v ) f v )dv In view of ), we can find a constant C, depending only onγ, such that ) λ v a i j v v ) C λ λ )! for 1 v v. And for β, ) β v a i j v v ) C β β )!1+ v γ + v γ ) for 1 v v. From the estimate 8) we know that J 1 + J can be estimated by B β β!) σ f t) L 1 1 λ β C ) λ + C B β β!) σ f t) L 1 1+ v γ ). γ We can take B large enough such that B C. Then we get β ]) v[ 1 ψ)ai j f v) J 1 + J C f t) L 1 γ B β β!) σ 1+ v ) γ/ C B β β!) σ 1+ v ) γ/. In the last inequality we used the fact f t) L 1 γ M + E. Now we choose a constant B such that B β β!) σ B β [ β )! ]σ. It follows immediately that β ]) v[ 1 ψ)ai j f v) CB β [ β )! ]σ 1+ v ) γ/.
6 HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Combining with the estimate on [ ν vψa i j ) ] β ν v f ), we get finally β vā i j v) C { β ν f t) L + B β [ β )! ]σ 1+ v ) γ/} v C [ G σ f t)) ] β 1+ v ) γ/. Thus, combining with Cauchy s inequality, the estimate above gives the proof of Lemma.4. Similar to Lemma.4, we can prove that hal-4343, version 1 - Nov 9 Lemma.5. For all multi-indicesβ with β and all g, h L γ R3 ), one has β v ct, v))gv)hv)dv C g L γ h L γ [G σ f t)) ] β, t. Let us now present the proof of the main result of this section. Proof of Lemma.. Since 3 i ā i j = b j, i=1 3 j b j = c, and f satisfies t f= 3 ā i j i v j f c f, then it holds that t µ v f t) [ = t µ L v f t, v) ] [ µ v f t, v) ] dv 3 = µ vāi j i v j f c f )] [ µ v f t, v) ] dv. [ Moreover, by using Leibniz s formula, we have t µ v f t) L = + + 3 3 3 β µ j=1 ā i j vi v j µ v f ) µ v f ) dv β µ ) β v ā i j vi v j µ β µ β = I)+II)+III)+IV). ) β v ā i j vi v j µ β c ) β v f ) µ v f ) dv f ) µ v f ) dv f ) µ v f ) dv Thus the proof of Lemma. depends on the following estimates. Step 1. Estimate on the term I).
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS 7 Integrating by parts, one has 3 I) = ā i j v j µ v f ) i µ v f ) dv 3 j=1 = I) 1 + I). b j v j µ v f ) µ v f ) dv hal-4343, version 1 - Nov 9 For the term I) 1, one has, by applying the ellipticity property 7), I) 1 K v µ v f 1+ v ) γ/ dv= K v µ v f. L γ For the term I), integrating by parts again, we have I) = I) + c µ v f ) µ v f ) dv. Observing that cv) C f t) L 1 γ 1+ v ) γ/ C1+ v ) γ/, one has This implies I) C µ v f C L v D µ 1 f γ Lγ. I) K v µ v f + C L v D µ 1 f 9) γ Lγ. Step. Upper bound for the term II). Recall that II) = 3 Cµ β ) R β 3 v ā i j vi v j µ β f ) µ v f ) dv. Integrating by parts, we have II)= 3 j=1 3 = II) 1 + II). ) β v b j v j µ β ) β v ā i j v j µ β Observing that β v b j t, v) C1+ v ) γ/ for, one has f ) µ v f ) dv f ) i µ v f ) dv II) 1 C µ v µ β v f t) L γ µ v f t) L γ C µ v D µ 1 f t) L γ. For the term II), if we writeµ=β+µ β), then it holds that II) = 3 ) β v ā i j v j µ β f ) β vi µ β f ) dv.
8 HUA CHEN, WEI-XI LI AND CHAO-JIANG XU hal-4343, version 1 - Nov 9 Since, we can integrate by parts to get 3 II) = Cµ β ) β v ā i j v j µ v f ) i µ β f ) dv Hence Observing that which means + 3 = II) + II) = β+β v 3 3 β+β v β+β v ā i j ) v j µ β β+β v f ) i µ β f ) dv ā i j ) v j µ β v ā i j ) v j µ β v ā i j v) C1+ v ) γ/ for, one has II) C f ) i µ β f ) dv. f ) i µ β f ) dv. Cµ β v µ β f C µ L v D µ 1 f γ Lγ, II) C µ v D µ 1 f L γ. 1) Step 3. Upper bound for the term III) and IV). Recall that 3 III)= Cµ β ) β v ā i j vi v j µ β f ) µ v f ) dv, and that β µ IV)= β µ µ β c ) β v f ) µ v f ) dv. By Lemma.4 and Lemma.5, we have 3 III) C Cµ β vi v j µ β f t) L γ µ [ v f t) L Gσ γ f t)) ] β and C β µ IV) C β µ Cµ β v D µ β +1 f t) L γ v D µ 1 [ f t) L Gσ γ f t)) ] 11) β, β µ Combining with the estimates 9)-1), one has β v f t) L γ v D µ 1 f t) L γ [G σ f t)) ] µ β. 1) t µ v f t) + C L 1 v µ v f t) C L µ v D µ 1 f t) γ L γ + C Cµ β v D µ β +1 f t) L γ v D µ 1 f t) L γ [G σ f t)) ] β β µ + C Cµ β β v f t) L γ v D µ 1 f t) L γ [G σ f t)) ] µ β. β µ
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS 9 This completes the proof of Lemma.. 3. Proof of Theorem 1.1 Theorem 1.1 will be deduced by the following result: Proposition 1. Letσ>1 and f G σ ) be the initial datum with finite mass, energy and entropy, and let f be a smooth solution of the Cauchy problem 3) satisfying 4). Then for any fixed T, <T<+, there exists a constant A, which depends only on M, E, H,γ, T,σand the Gevrey constant of f, such that for any k N, k 1, one has Q) k { T } 1/ sup α v f t) L + v v α f t) dt A k[ k 1)! ] σ L t [,T] γ hal-4343, version 1 - Nov 9 for all multi-indicesα and α with α = α =k. Remark 4. From the estimate Q) k in Proposition 1 we can deduce directly the result of Theorem 1.1. Proof of Proposition 1. We use induction on k to prove the estimate Q) k. Observe that Q) 1 holds if we take A large enough such that A sup f t) H 1 γ + f L [,T];H +. 13) γ) t [,T] Assume that the estimate Q) l holds for 1 l k 1 and k. Then we need to prove that the estimate Q) k is true. Firstly we prove that sup α v f t) L 1 t [,T] A α [ α 1)! ]σ, for all α =k. 14) Applying Lemma. withµ=α, we obtain t α v f t) + C L 1 v α v f C L α v D α 1 f γ L γ + C Cα β v D α β +1 f L γ v D α 1 f L γ [G σ f t)) ] β β α 15) + C Cα β β v f L γ v D α 1 f L γ [G σ f t)) ] α β. β α Next for the last term on the right hand side of the above inequality, one has C Cα β β v f t) L γ v α 1 f t) L γ [G σ f t)) ] α β β α =C f t) L γ v α 1 f t) L γ [G σ f t)) ] α + C Cα β β v f t) L γ v α 1 f t) L γ [G σ f t)) ] α 1 + C β α C β α β v f t) L γ v α 1 v f t) L γ [G σ f t)) ] α β.
1 HUA CHEN, WEI-XI LI AND CHAO-JIANG XU We denote [ G σ f ) ] [ β = sup t [,T] Gσ f t)) ] β. Integrating in both sides of the estimate 15) over the interval [, T], and using the Cauchy inequality, we get T α v f t) α L v f ) C L α v D α 1 f s) Lγds + C β α Cα[ β Gσ f ) ] { T β v D α β +1 f s) Lγds } 1 hal-4343, version 1 - Nov 9 { T + C max t T f t) L γ + C + C β α { T } 1 v D α 1 f s) Lγds { T } [ ] 1{ T G f s)) α ds C β α max t T f t) H 1 γ [G σ f ) ] α 1 C β α[ Gσ f ) ] α β { T } 1 v D α 1 f s) Lγds def = S 1 )+S )+S 3 )+S 4 )+S 5 ). T } 1 β v f s) Lγds } v D α 1 f s) 1 Lγds v D α 1 f s) L γ ds From the induction assumption and the fact β v f L γ v β 1 v f L γ for β 1, we have, respectively, the following estimates: { T { T v α 1 v f s) L γds } 1 α β +1 v f s) Lγds } 1 A α 1[ α )! ]σ ; 16) A α β +1[ α β )! ]σ, β α ; 17) { T } 1 β v f s) Lγds A β 1[ β )! ]σ, β α. 18) We next treat the term [ G σ f ) ] m given by [ Gσ f ) ] m = sup t [,T] [ Gσ f t)) ] m = sup t [,T] m v f t) L + B m m!) σ, Observe that for some C σ >, B m m!) σ C σ B) m m 1)!) σ, for 1 β α 1. Also from the induction assumption, one has max t [,T] m v f t) L A m m 1)!) σ, 1 m α 1=k 1. Thus for A large enough, we have [ Gσ f ) ] m Am m 1)!) σ, 1 m α 1=k 1. 19)
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS 11 hal-4343, version 1 - Nov 9 By using the estimate 16), one has T [ G f s)) ] α ds CA α 1[ α )! ]σ. ) The estimate for the term S 1 ) can be given by 16) directly, S 1 ) C α [ A α 1 α 1)!) σ] C [ A α 1 α!) σ]. 1) We write now the term S ) into two parts, S )=C Cα[ β Gσ f ) ] β = + C 3 β α T C β α[ Gσ f ) ] β { T D α 1 f s) Lγds =S ) + S ). Thus 13) and 16) give that D α 1 f s) L γds { T } 1/ } 1/ D α β +1 f s) Lγds S ) C A α { A α 1[ α )! ] σ } C A { A α 1[ α 1)! ] σ }, and 16), 17) and 19) give that S ) α! C β! α β )! A β [ β 3)! ]σ A α β +1[ α β )! ] σ Observing that 3 β α we have by the estimate 5) A α 1[ α )! ]σ. α! [ ] β 3)! σ [ ] α β )! σ 6 α β! α β )! β 3{ α 1)!}σ, S ) C A α 1 ) [ α )! ] σ[ α 1)! ] σ 3 β α 6C A α 1 ) [ α )! ] σ[ α 1)! ] σ α σ 6 α β 3 { 6C A α 1 [ α 1)! ] σ }. Therefore S )=S ) + S ) 6C A { A α 1[ α 1)! ] σ }. ) Similarly, from the estimates 13), 16) and ), one has S 3 ) C A { A α 1[ α )! ] σ }. 3) From 13), 16) and 19), one can deduce that S 4 ) C A { A α 1[ α 1)! ] σ }. 4)
1 HUA CHEN, WEI-XI LI AND CHAO-JIANG XU It remains to estimate the term S 5 ). From 16), 18) and 19) we have C α! S 5 ) β! α β )! A α β [ α β 1)! ]σ A β 1[ β )! ] σ β α 1 A α 1[ α )! ] σ + C A { A α 1[ α )! ] σ }. hal-4343, version 1 - Nov 9 Since for β α 1, α! [ ] α β 1)! σ [ ] β )! σ β! α β )! then by using the estimate 6), we have α β α β ) S 5 ) C A α 1 ) [ α 1)! ] σ [ α )! ] σ + C A { A α 1[ α )! ] σ } β α 1 3 C A A α 1) [ α 1)! ] σ [ α )! ] σ α σ 3 C A { A α 1[ α 1)! ] σ }. This, combined with 1)-4), implies that [ α 1)! ] σ, α β α β ) α v f t) L α v f ) L C 3 A { A α 1[ α )! ] σ }, t [, T]. Since f )= f G σ ), then there exists a constant L such that α v f ) L { L α [ α 1)! ] σ }. Thus taking A large enough, we can deduce that { 1 α v f t) L A α [ α 1)! ] } σ, for all t [, T], and α =k. This gives the proof of the inequality 14). Finally, we need to prove that T v α v f t) L γdt { 1 A α [ α 1)! ] } σ, α =k. 5) The proof of the estimate 5) is similar to that of 14). Let us apply Lemma. again withµ= α. Then we have t v α f t) + C L 1 v v α f L γ + C C β α v v α β +1 β α + C β α def =Nt). C α v v α 1 f L γ f L γ v v α 1 f L γ [G σ f t)) ] β C β α β v f L γ v v α 1 f L γ [G σ f t)) ] α β
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS 13 Integrating the above inequality over the interval [, T], we then have T C 1 v v α f s) Lγds α v f ) + L By a similar argument as in the proof of 14), one has α v f ) L + T T Ns)ds. Ns)ds C 1 { 1 A α [ α 1)! ] σ}, which gives the estimate 5). The validity of Q) k can be derived directly by the estimates in 14) and 5). References hal-4343, version 1 - Nov 9 [1] H. Chen, W.-X. Li, and C.-J. Xu, The Gevrey Hypoellipticity for linear and non-linear Fokker-Planck equations, accepted by J. Differential Equations 8). [] Y.M. Chen, L. Desvillettes, and L.B. He, Smoothing effects for classic solutions of the full Landau equation, to appear in Arch. Ration. Mech. Anal. [3] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys. 1 199), 59 76. [4] L. Desvillettes, G. Furioli, and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, To appear in Transactions of the American Mathematical Society. [5] L. Desvillettes and C. Mouhot, About L p estimates for the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 5), 17 14. [6] L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations 5 ), 179 59. [7] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys. 31 ), 391 434. [8] T. Gustafsson, L p -estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 9 1986), 3 57. [9] T. Gustafsson, Global L p -properties for the spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 13 1988), 1 38. [1] B. Helffer and Nier F, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, vol. 186, Springer-Verlag, Berlin, 5. [11] F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 4), 151 18. [1] Y. Morimoto and C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto Univ. 47 7), 19 15. [13] C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal. 173 4), 169 1. [14] L. Rodino, Linear Partial Differential Operators in Gevrey spaces, World Scientific Publishing Co. Inc., River Edge, NJ, 1993. [15] S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math. 1 1984). [16] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal. 143 1998), 73 37. [17] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci. 8 1998), 957 983. School of Mathematics and Statistics, Wuhan University, Wuhan 437, P. R. China E-mail address:chenhua@whu.edu.cn School of Mathematics and Statistics, Wuhan University,Wuhan 437, P. R. China E-mail address:weixi.li@yahoo.com
14 HUA CHEN, WEI-XI LI AND CHAO-JIANG XU School of Mathematics and Statistics, Wuhan University, Wuhan 437, P. R. China and Université de Rouen, UMR 685-CNRS, Mathématiques, Avenue de l Université, BR.1, F7681 Saint Etienne du Rouvray, France E-mail address:chao-jiang.xu@univ-rouen.fr hal-4343, version 1 - Nov 9