Ultra-analytic effect of Cauchy problem for a class of kinetic equations

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1 Ultra-analytic effect of Cauchy problem for a class of kinetic equations Yoshinori Morimoto, Chao-Jiang Xu To cite this version: Yoshinori Morimoto, Chao-Jiang Xu. Ultra-analytic effect of Cauchy problem for a class of kinetic equations. J. Differential Equations, 9, 47, pp <hal-36863> HAL Id: hal Submitted on 17 Mar 9 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Ultra-analytic effect of Cauchy problem for a class of kinetic equations Yoshinori MORIMOTO Graduate School of Human and Environmental Studies Kyoto University, Kyoto, , Japan Chao-Jiang XU Université de Rouen, UMR-685, Mathématiques 7681 Saint Etienne du Rouvray, France and School of Mathematics and Statistics, Wuhan University, Wuhan 437, China Abstract The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem. Key words: Landau equation, Fokker-Planck equation, ultra-analytic effect of Cauchy problem. AMS Classification: 35A5, 35B65, 35D1, 35H, 76P5, 84C4 1. Introduction It is well known that the Cauchy problem of heat equation possesses the ultra-analytic effect phenomenon, namely, if ut, x is the solution of the following Cauchy problem : { t u x u =, x ; t > u t= = u L, then under the uniqueness hypothesis, the solution ut, = e t x u is an ultra-analytic function for any t >. We give now the definition of function spaces A s Ω where Ω is an open subset of. addresses: morimoto@math.h.kyoto-u.ac.jp Yoshinori MORIMOTO, Chao-Jiang.Xu@univ-rouen.fr Chao-Jiang XU Preprint submitted to Elsevier March 17, 9

3 Definition 1.1. For < s < +, we say that f A s Ω, if f C Ω, and there exists C >, N > such that α f L Ω C α +1 α! s, α N d, α N. If the boundary of Ω is smooth, by using Sobolev embedding theorem, we have the same type estimate with L norm replaced by any L p norm for < p +. On the whole space Ω =, it is also equivalent to e c 1 s β f L for some c > and β N d, where e c 1 s e c s 1 ux = F 1 e c ξ 1s ûξ. is the Fourier multiplier defined by If s = 1, it is usual analytic function. If s > 1, it is Gevrey class function. For < s < 1, it is called ultra-analytic function. Notice that all polynomial functions are ultra-analytic for any s >. It is obvious that if u L then, for any t > and any k N, we have ut, = e t xk u A 1 k, namely, there exists C > such that for any m N, t m km x ut, L C k m t x k m ut, L u L C k m m! C k m+1 km! 1 k, where x km = α =km,α N d α x. We say that the diffusion operators x k possess the ultra-analytic effect property if k > 1/, the analytic effect property if k = 1/ and the Gevrey effect property if < k < 1/. We study the Cauchy problem for spatially homogeneous Landau equation { āf ft = Qf, f v v f bff, v, t >, 1.1 f t= = f, where āf = ā ij f and bf = b 1 f,, b d f are defined as follows convolution is w. r.t. the variable v with ā ij f = a ij f, bj f = a ij v = vi a ij f, i, j = 1,, d, i=1 δ ij v iv j v v γ+, γ [ 3, 1]. We consider hereafter only the Maxwellian molecule case which corresponds to γ =. We introduce also the notation, for l R, L p l Rd = {f; 1 + v l/ f L p } is the weighted function space. We prove the following ultra-analytic effect results for the nonlinear Cauchy problem 1.1.

4 Theorem 1.1. Let f L L 1 Rd and < T +. If ft, x > and f L ], T[; L L 1 is a weak solution of the Cauchy problem 1.1, then for any < t < T, we have ft, A 1/, and moreover, for any < T < T, there exists c > such that for any < t T e c t v ft, L e d t f L. 1. In [17], they proved the Gevrey regularity effect of the Cauchy problem for linear spatially homogeneous non-cut-off Boltzmann equation. By a careful revision for the proof of Theorem 1. of [17], one can also prove that the solution of the Cauchy problem 1.1 in [17] belongs to A 1 α for any t >, where < α < 1 is the order of singularity of collision kernel of Boltzmann operator. Hence, if α 1/, there is also the ultra-analytic effect phenomenon. Now the above Theorem 1.1 shows that, for Landau equation, the ultra-analytic effect phenomenon holds in nonlinear case, which is an optimal regularity result. The ultra-analytic effect property is also true for the Cauchy problem of the following generalized Kolmogorov operators { t u + v x u + v α u =, x, v ; t > u t= = u L, where < α <, and the classical Kolmogorov operators is corresponding to α = 1. By Fourier transformation, the explicit solution of the above Cauchy problem is given by ût, η, ξ = e R t ξ+sη α dsû η, ξ + tη. Since there exists c α > see Lemma 3.1 below such that we have c α t ξ α + t α+1 η α t e cαt vα +t α+1 x α ut,, L, ξ + sη α ds, 1.3 i. e. ut,, A 1/α for any t >. Notice that this ultra-analytic if α > 1/ effect phenomenon is similar to heat equations of x, v variables. That is, this means v x + v α is equivalent to x α + v α by time evolution in some sense, though the equation is only transport for x variable. We consider now a more complicate equation, the Cauchy problem for linear Fokker- Planck equation : { ft + v x f = v v f + vf, x, v, t > ; 1.4 f t= = f. This equation is a natural generalization of classical Kolmogorov equation, and a simplified model of inhomogeneous Landau equation see [, 1]. The local property of 3

5 this equation is the same as classical Kolmogorov equation since the add terms v vf is a first order term, but for the studies of kinetic equation, v is velocity variable, and hence it is in whole space v. Then there occurs additional difficulty for analysis of this equation. The definition of weak solution in the function space L ], T[; L x, v L1 1 Rd x, v for the Cauchy problem is standard in the distribution sense, where for 1 p < +, l R { L p l Rd x, v = f S ; 1 + v l/ f L p x, v }. The existence of weak solution is similar to full Landau equation see [1, 13]. We get also the following ultra-analytic effect result. Theorem 1.. Let f L x, v L 1 1 x, v, < T +. Assume that f L ], T[- ; L x, v L1 1 Rd x, v is a weak solution of the Cauchy problem 1.4. Then, for any < t < T, we have ft,, A 1/. Furthermore, for any < T < T there exists c > such that for any < t T, we have e c t v+t 3 x ft,, e d t f L R. 1.5 L d Remark 1.1. The ultra-analyticity results of the above two theorems are optimal for the smoothness properties of solutions. From these results, we obtain a good understanding for the hypoellipticity of kinetic equations see [11, 14], and also the relationship, established by Villani [19] and Desvillettes-Villani [1], between the nonlinear Landau equation with Maxwellian molecules and the linear Fokker-Planck equation. We consider now the spatially inhomogeneous Landau equation { ft + v x f = Qf, f, x, v, t > ; f t= = f x, v. 1.6 The problem is now much more complicate since the solution f is the function of t, x, v variables. We consider it here only in the linearized framework around the normalized Maxwellian distribution µv = π d e v, which is the equilibrium state because Qµ, µ =. Setting f = µ + g, we consider the diffusion part of linear Landau collision operators, Qµ, g = v āµ v g bµg where ā ij µ = a ij µ = δ ij v + 1 v i v j, bj µ = vi a ij µ = vj, i, j = 1,, d. i=1 4

6 In particular, it follows that ā ij µξ i ξ j ξ, for all v, ξ. 1.7 i We then consider the following Cauchy problem { g t + v x g = v āµ v g bµg, x, v, t > ; g t= = g. 1.8 We can also look this equation as a linear model of spatially inhomogeneous Landau equation, which is much more complicate than linear Fokker-Planck equation 1.4, since the coefficients of diffusion part are now variables. The existence and C regularity of weak solution for the Cauchy problem have been considered in [1]. We prove now the following; Theorem 1.3. Let g L x, v L1 Rd x, v, < T +. Assume that g L ], T[- ; L x, v L1 Rd x, v is a weak solution of the Cauchy problem 1.8. Then, for any < t < T, we have gt,, A 1. Furthermore, for any < T < T there exist C, c > such that for any < t T, we have e ct v 1/ +t x 1/ gt,, L ec t g L R. 1.9 d In this theorem, we only consider the analytic effect result for the Cauchy problem 1.8, neglecting the symmetric term Qg, µ in the linearized operators of Landau collision operator cf.,1.15 of [1] because of the technical difficulty, see the remark in the end of section 4. There have been many results about the regularity of solutions for Boltzmann equation without angular cut-off and Landau equation, see [1,, 3, 6, 7, 9, 1, 15, 16] for the C smoothness results, and [4, 5, 8, 18, 17] for Gevrey regularity results for Boltzmann equation and Landau equation in both cases : the spatially homogeneous and inhomogeneous cases. As for the analytic and Gevrey regularities, we remark that the propagation of Gevrey regularities of solutions is investigated in [5] for full nonlinear spatially homogeneous Landau equations, including non-maxwellian molecule case, and the local Gevrey regularity for all variables t, x, v is considered in [4] for some semi-linear Fokker-Planck equations. Comparing those results, the ultra-analyticity for x, v variables showed in Theorem 1.1 is strong although the Maxwellian molecule case is only treated. As a related result for spatially homogeneous Boltzmann equation in the Maxwellian molecule case, we refer [8], where the propagation of Gevrey and ultra-analytic regularity is studied uniformly in time variable t. Throughout the present paper, we focus the smoothing effect of the Cauchy problem, and the uniform smoothness estimate near to t =. Concerning further details of the analytic and Gevrey regularities of solutions for Landau equations and Boltzmann equation without angular cut-off, we refer the introduction of [5] and references therein. 5

7 . Spatially homogeneous Landau equations We consider the Cauchy problem 1.1 and prove Theorem 1.1 in this section. We refer to the works of C. Villani [19, ] for the essential properties of homogeneous Landau equations. We suppose the existence of weak solution ft, v > in L ], T[; L 1 Rd - L. The conservation of mass, momentum and energy reads, d ft, v dt 1 v v dv. Without loss of generality, we can suppose that R ft, vdv = 1, unit mass d ft, vv j dv =, j = 1,,d; zero mean velocity R ft, v v dv = T d, unit temperature ft, vv j v k dv = T j δ jk, T j = T Then we have, T j = ft, vv jdv >, j = 1,,d; directional temperatures. j ā jk f = δ jk v + T T j v j v k ;.1 bj f = v j ;. ā jk fξ j ξ k C 1 ξ, v, ξ..3 j,k where C 1 = min 1 j d {T T j } >. Now for N > d and < δ < 1/N, c >, t >, set G δ t, ξ = e ct ξ 1 + δe c t ξ 1 + δc t ξ N. Since G δ t, L, we can use it as Fourier multiplier, denoted by G δ t, D v ft, v = F 1 G δ t, ξ ˆft, ξ. Then, for any t >, G δ t = G δ t, D v : L H N C b. The object of this section is to prove the uniform bound with respect to δ > of G δ t, D v ft, L. Since ft, L L 1 Rd is a weak solution, we can take G δ t ft, = G δ t, D v ft, H N, 6

8 as test function in the equation of 1.1, whence we have 1 d dt G δtft, L + j,k=1 = 1 t G δ t f, G δ tf LRd + + j,k=1 ā j k f vj G δ tft, v vk G δ tft, v dv vj vj ft, v G δ t ft, v dv { } ā jk f G δ t vj ft, v G δ t ā jk f vj ft, v vk G δ tft, v dv. To estimate the terms in the above equality, we prove the following two propositions. Proposition.1. We have C 1 v G δ tft L Re ā jk f vj G δ t, D v ft, v vk G δ t, D v ft, v dv..4 j,k=1 t G δ t f, G δ tf c v G δ tft L L..5 vj vj ft, v G δ t ft, vdv d G δtft L + c t v G δ tft L..6 Proof : The estimate.4 is exactly the elliptic condition.3. By using the Fourier transformation,.5 is deduced from the following calculus t G δ t, ξ = c ξ 1 Nδ G δ t, ξ 1 + δe ct ξ 1 + δc t ξ = c ξ G δ t, ξ J N,δ where To treat.6, we use J N,δ = 1 Nδ 1 + δe ct ξ 1 + δc t ξ 1. ξj G δ t, ξ = c tξ j G δ t, ξ J N,δ..7 7

9 Then, we have Re vj vj ft, v G δ t, D v ft, vdv = Re v j G δ t, D v ft, v vj G δ t, D v ft, v dv Re [G δ t, D v, v j ]ft, v vj G δ t, D v ft, v dv = d G δtft, L Re Using Fourier transformation and.7, we have that for t >, = = = R 3 R 3 [G δ t, D v, v j ]ft, v vj G δ t, D v ft, v dv. [G δ t, D v, v j ]ft, v vj G δ t, D v ft, v dv G δ t, D v v j ft, v v j G δ t, D v ft, v vj G δ t, D v ft, v dv { i ξj Gδ t, ξ ˆft, ξ G δ t, ξ i ξj ˆft, } ξ G δ t, ξ iξ j ˆft, ξdξ ξj G δ t, ξ ˆft, ξξj G δ t, ξ ˆft, ξdξ = c t ξ G δ t, ξ ˆft, ξ J N,δ dξ c t ξ G δ t, ξ ˆft, ξ dξ, which give.6. The proof of Proposition.1 is now complete. For the commutator term, the special structure of the operator implies Proposition.. j,k=1 { } ā jk f G δ t, D v vj ft, v G δ t, D v ā jk f vj ft, v vk G δ t, D v ft, v dv =. Proof : We introduce now polar coordinates on ξ by setting r = ξ and ω = ξ/ ξ S d 1. Note that / ξ j = ω j / r + r 1 Ω j where Ω j is a vector field on S d 1, and see 8

10 [14], Proposition ω j Ω j =, Ω j ω j = d 1..8 By using Fourier transformation, we have { } ā jk f G δ t, D v vj ft, v G δ t, D v ā jk f vj ft, v = j,k=1 { j,k=1 Noting, in polar coordinates on ξ, we have, denoting by Gr = G δ t, r, = = vk G δ t, D v ft, v dv [ } ξ k δ jk ξ ξk ξj, G δ t, ξ ]ξ j ˆft, ξ G δ t, ξ ˆft, ξdξ. ξ = r + d 1 r r + 1 r Ω j, [ { ω k δ jk r + d 1 } r r j,k=1 { ω k / r + r 1 Ω k ω j / r + r 1 Ω j } [ r + d 1 r r, Gr ] [ ωk / r + r 1 ω k Ω k k=1, Gr ] ω j ωj / r + r 1 Ω j ω j [ r + d 1 r r, Gr ] [ r + d 1 ], Gr r r =, where we have used.8. Then we finish the proof of Proposition.. ], Gr Remark.1. In the above proof of Proposition., we have used the polar coordinates in the dual variable of v, which is essentially related to a form of the Landau operator with Maxwellian molecules. We notice that the same relation in v variable was described by Villani [19] and Desvillettes-Villani [1]. End of proof of Theorem 1.1 : From Propositions.1 and., we get 1 d dt G δtft, L + C 1 1 c c t v G δ tft, L d G δtft, L. 9

11 For any < T < T, choose c small enough such that C 1 1 c c T. Then we get d dt G δtft, L d G δtft, L..9 Integrating the inequality.9 on ], t[, we obtain Take limit δ in.1. Then we get G δ tft, L e d t f L..1 e ct v ft, L e d t f L.11 for any < t T. We have now proved ft, A 1/ and Theorem Linear Fokker-Planck equations In the paper [19], there is an exact solution for spatially homogeneous linear Fokker- Planck equation. In the inhomogeneous case we can also obtain an exact solution of the Cauchy problem 1.4. Denote by ˆft, η, ξ = F x,v ft, x, v, the partial Fourier transformation of f with respect to x, v variable. Then, by Fourier transformation for x, v variables, the linear Fokker-Planck equation 1.4 becomes { t ˆft, η, ξ η ξ ˆft, η, ξ + ξ ξ ˆft, η, ξ = ξ ˆft, η, ξ; ˆf t= = Ff η, ξ. Therefore we obtain the exact solution Note that ˆft, ξ, η = ˆf, ξe t + η1 e t, η exp t ξe τ + η1 e τ dτ = 1 e t ξ + 1 e t ξ η + t ξe τ t + η1 e τ t dτ. t 3 + e t + e t η = X X ξ + X ξ η + log1 X X X η, where X = 1 e t t. We have for < K < /3 t ξe τ + η1 e τ dτ X1 1/K X/ ξ + 1/3 K/X 3 η. Hence for t X < 1/K, we get ft,, A 1/, so that the ultra-analytic effect holds for any t > by means of the semi-group property. But we cannot get the uniform estimate

12 We present now the proof of 1.5 which implies the ultra-analytic effect, by commutator estimates similarly as for homogeneous Landau equation. Set wt, η, ξ = ˆft, η, ξ tη. Then the Cauchy problem 1.4 is equivalent to { t wt, η, ξ = ξ tη wt, η, ξ ξ tη ξ wt, η, ξ; w t= = Ff η, ξ. 3.1 Since we need to study the function t ξ sη ds, we prove the following estimate. Lemma 3.1. For any α >, there exists a constant c α > such that t ξ sη α ds c α t ξ α + t α+1 η α. 3. Remark 3.1. If α =, we can get the above estimate by direct calculation. The following simple proof is due to Seiji Ukai. Proof : Setting s = tτ and η = tη, we see that the estimate is equivalent to 1 ξ τ η α dτ c α ξ α + η α. Since this is trivial when η =, we may assume η. If ξ < η then If ξ η then ξ τ η α dτ η α τ ξ η { ξ / η ξ αdτ 1 = η α η τ + Hence we obtain 3.. Set now η α α + 1 min 1 α+1 α + 1 ξ α + η α. α ξ / η θ 1 θα θ α+1 = dτ } τ ξ αdτ η η α α α ξ τ η α dτ ξ α 1 τ η αdτ 1 ξ α αdτ 1 τ ξ = ξ α α α + 1 ξ α + η α. t φt, η, ξ = c ξ sη ds c t3 η, 11

13 where c > is a small constant to choose later, and c is the constant in 3. with α =. Then 3. implies Let N = d + 1/4. For < δ < 1/4N and t >, set G δ = G δ t, η, ξ = φt, η, ξ c c t ξ + t 3 η. 3.3 e φt,η,ξ 1 + δe φt,η,ξ 1 + δ η + ξ N. 3.4 Since G δ t, L, we can use it as Fourier multiplier, denoted by Gδ t, D x, D v u t, x, v = F 1 η, ξ G δ t, η, ξût, η, ξ. Lemma 3.. Assume that ft, L x, v L1 1 Rd x, v for any t ], T[. Then ξ wt, η, ξ L η,ξ, and ξ tη G δ t, η, ξ wt, η, ξ, η G δ t, η, ξ wt, η, ξ, ξ G δ t, η, ξ wt, η, ξ 3.5 belong to L η,ξ for any t ], T[. Proof : Since ξj w = ifv j f, it follows from f L 1 1 x, v that ξ wt, η, ξ L η,ξ. Noting ξ tη Gδt, η, ξ, η Gδt, η, ξ L η,ξ, we see that the first two terms of 3.5 are obvious. To check the last term in 3.5, note ξj G δ t, η, ξ = c tξ j t η 1 jg δ t, η, ξ 1 + δe φt,η,ξ Nδξ j 1 + δ η + ξ G δt, η, ξ. 3.6 Then, we have ξ G δ t, η, ξ wt, η, ξ = G δ t, η, ξ ξ wt, η, ξ + ξ G δ t, η, ξ wt, η, ξ = G δ t, η, ξ ξ wt, η, ξ + 4c tξ t η δe φt,η,ξ G δt, η, ξ wt, η, ξ 4Nδξ 1 + δ η + ξ G δt, η, ξ wt, η, ξ. Since G δ t, η, ξ L x, v we have G δ t, η, ξ ξ wt, η, ξ L. Using 1 1, 1 + δe φt,η,ξ Nδξ 1 + δ η + ξ 1, 1

14 and ξ t ηg δt, η, ξ δe φt,η,ξ wt, η, ξ ξ t η G δt, η, ξ wt, η, ξ ξ tη G δ t, η, ξ wt, η, ξ + t η G δt, η, ξ wt, η, ξ L. We have proved Lemma 3. We take now G δ t, η, ξ wt, η, ξ as test function in the equation of 3.1. Then we have d dt G δt,, wt,, L + ξ tηg δ t, η, ξwt, η, ξ dηdξ = wt, η, ξ ξj ξ j tη j G δ t, η, ξ wt, η, ξ dηdξ + t G δ t,, wt,,, G δ t,, wt,,. 3.7 We prove now the following; Proposition 3.1. We have t G δ t,, w, G δ t,, w L L = c ξ tηg δ t, η, ξwt, η, ξ R dηdξ d 3 c c t η G δ t, η, ξwt, η, ξ 1 dηdξ δe φt,η,ξ Re wt, η, ξ ξj c t + c t + c 3c ξ j tη j G δ t, η, ξ wt, η, ξ dηdξ ξ tηg δ t, η, ξwt, η, ξ dηdξ + d + N δ/c G δ t,, wt,, L c c t η G δ t, η, ξwt, η, ξ 1 dηdξ δe φt,η,ξ Proof of Proposition 3.1 : The estimate 3.8 is deduced from t G δ t, η, ξ = c ξ tη 3 c t η 1 G δ t, η, ξ. 1 + δe φt,η,ξ 13

15 Since it follows from 3.6 that I = Re wt, η, ξ ξj ξ j tη j G δ t, η, ξ wt, η, ξ dηdξ = Re c t ξ j tη j ξ j t R η j G δ t, η, ξwt, η, ξ 1 dηdξ d 1 + δe φt,η,ξ we get Re Re Nδξ j ξ j tη j R 1 + δ η + ξ G δt, η, ξwt, η, ξ dηdξ d ξ j tη j ξj G δ t, η, ξwt, η, ξ G δ t, η, ξwt, η, ξdηdξ, I = c t ξ j tη j ξ j t R η j G δ t, η, ξwt, η, ξ 1 dηdξ d 1 + δe φt,η,ξ Nδξ j ξ j tη j R 1 + δ η + ξ G δt, η, ξwt, η, ξ dηdξ d + d G δt,, wt,, L = c t ξ tη G δ t, η, ξwt, η, ξ 1 dηdξ 1 + δe φt,η,ξ + c t ξ tη η G δ t, η, ξwt, η, ξ 1 dηdξ 1 + δe φt,η,ξ Nδξ j ξ j tη j R 1 + δ η + ξ G δt, η, ξwt, η, ξ dηdξ d For the last term, noting + d G δt,, wt,, L. Nδξ j ξ j tη j 1 + δ η + ξ N /c δ ξ + c ξ tη 1 + δ η + ξ N δ/c + c ξ tη we finally obtain I c t + c t 3c + c ξ tηg δ t, η, ξwt, η, ξ dηdξ + d + N δ/c G δ t,, wt,, L c c t η G δ t, η, ξwt, η, ξ 1 dηdξ. 1 + δe φt,η,ξ 14

16 Thus we have proved Proposition 3.1. End of proof of Theorem 1. : Now the equation 3.7, the estimate 3.8 and 3.9 deduce d dt G δt,, wt, L + 3c 4c t c t ξ tηg δ t, η, ξwt, η, ξ dηdξ 3c d + N δ/c G δ t,, wt,, L. Then for any < T < T choose c > depends on T small enough such that then for any < t T, which gives 3c 4c T c T 3c, d dt G δt,, wt,, L d + N δ/c G δ t,, wt,, L R, d Take δ, we have = G δ t,, wt,, L e d+n δ/c t f L. e c e c By using 3.3, we get finally R t ξ sη ds c 1t 3 η ˆft, η, ξ tη dηdξ R t ξ+t sη ds c 1t 3 η ˆft, η, ξ dηdξ e d t f L. e ct v+t3 x ft,, L e d t f L for any < t T, where c = cc >. This is the desired estimate 1.5, which implies We have thus proved Theorem 1.. ft,, A 1/. 4. Linear model of inhomogeneous Landau equations We prove now the Theorem 1.3 in this section. By the change of variables t, x, v t, x + v t, v, the Cauchy problem 1.8 is reduced to { f t = v t x āµ v t x f bµf, 4.1 f t= = g x, v, 15

17 where ft, x, v = gt, x + v t, v. Recall that and ā ij µ = a ij µ = δ ij v + 1 v i v j ; bj µ = vi a ij µ = vj ; i, j = 1,,d, i=1 ā ij µξ i ξ j ξ, for all v, ξ. i In view of this Cauchy problem, we set t Ψt, η, ξ = c ξ sη ds, for a sufficiently small c > which will be chosen later on. Then we can use the 3. with α = 1 to estimate Ψ. Set F δ t, η, ξ = e Ψ 1 + δe Ψ N for N = d + 1, < δ 1 N. If A is a first order differential operator of t, η, ξ variables then we have 1 AF δ = 1 + δe Ψ Nδ AΨF δ, 4. and δe Ψ Nδ 1. Taking F δ t, D x, D v f = F δ t f H N as a test function in the weak solution formula of 4.1, we have 1 d dt F δtf L + = v j f + j,k=1 āµ v t x F δ tf, v t x F δ tf L vj t xj F δ t f dxdv + 1 t F δ f, Fδ tf L { } ā jk µ F δ t vj t xj f F δ t ā jk µ vj t xj f We prove now the following results. vk t xk F δ tf dxdv. 16

18 Proposition 4.1. We have v t x F δ tf L āµ v t x F δ tf, v t x F δ tf. 4.3 L t F δ t f, F δ tf c v t x F δ tf L F δ tf L. 4.4 L Re v j f vj t xj F δ t f d R F δtf L d +c t v t x F δ ft L F δ ft L. 4.5 Proof : The estimate 4.3 is a direct consequence of the elliptic condition 1.7. Using the Fourier transformation and noting 4., we see that 4.4 is derived from 1 t F δ t, η, ξ = 1 + δe Ψ Nδ t ΨF δ, t Ψ = c ξ tη. For 4.5, we have firstly Re v j F δ tf R 6 vj t xj F δ tf = d F δtf L. For the commutators [v j, F δ t], using Fourier transformation, we have that for t > and ˆf = ˆft, η, ξ [F δ t, D x, D v, v j ]ft, x, v vj t xj F δ t, D x, D v ft, x, v dxdv = = = 3 F δ t, D x, D v v j ft v j F δ t, D x, D v ft vj t xj F δ t, D v ft dxdv { i ξj Fδ t, η, ξ ˆft F δ t, η, ξ i ξj ˆft } F δ t, η, ξ iξ j tη j ˆftdηdξ ξj F δ t, η, ξ ˆftξj tη j F δ t, η, ξ ˆftdηdξ c t ξ tη F δ t, η, ξ ˆft dηdξ c t v t x F δ ft L F δ ft L, where, in view of 4., we have used the fact that 1 ξj Ψt, η, ξ ξ j tη j c 3 ξ j sη j ξ sη ξ j tη j ds c t ξ tη. Thus 4.5 has been proved. 17

19 For the commutator terms, we have Proposition 4.. There exists a constant C 1 > independent of δ > such that j,k=1 { } ā jk µ F δ t vj t xj f F δ t ā jk µ vj t xj f vk t xk F δ tf C 1 { c t v t x F δ tf L + F δtf L }. 4.6 Proof : In order to prove 4.6, we introduce the polar coordinates of ξ centered at tη, that is, r = ξ tη and ω = ξ tη ξ tη Sd 1. Note again that / ξ j = ω j / r + r 1 Ω j where Ω j is a vector field on S d 1. We have again ω j Ω j =, Ω j ω j = d 1,. By means of Plancherel formula, we have j,k=1 { } ā jk µ F δ t vj t xj F δ t ā jk µ vj t xj f vk t xk F δ tf { ] = ξ k tη k [δ jk ξ ξk ξj, F δ t, η, ξ ξ j tη j ˆft } j,k=1 = J. Noting again ξ = r + d 1 r r + 1 r F δ t, η, ξ ˆftdξdη Ω l, we have with F δ t, η, r, ω = F δ t, η, r ω + tη = F δ t, η, ξ j,k=1 [ = [ { ω k δ jk r + d 1 r r + 1 r r + d 1 r l=1 l=1 Ω l { ω k r + } r 1 Ω k ω j r + r 1 Ω j ] r, Fδ 18 }, Fδ ] ω j

20 1 r Note again that [ + k=1 ω k r + r 1 ω k Ω k [ ω j Ω l, Fδ ]ω j = A 1 + A + A 3. l=1 [ A 1 + A = r + d 1 r ] r, Fδ On the other hand, we have in view of 4. A 3 = 1 r = 1 r j,l=1 j,l=1 ω j ] r + r 1 Ω j ω j, Fδ [ + r + d 1, r r F ] δ =. [ ] ω j Ω l [Ω l, Fδ ] Ω l, [Ω l, Fδ ] ω j Ωl Ψ ω j Ω l 1 + δe Ψ NδΩ lψ Ωl Ψ 1 + δe Ψ NδΩ lψ F δ Ωl Ψ + Ω l 1 + δe Ψ NδΩ lψ Fδ ω j. Putting w j = ω j Fδ w with wt, η, r, ω = ˆft, η, r ω + tη, we have J = Re J = Re r A 3 w F δ w r d 1 drdωdη η S d 1 { Ωl Ψ = Re Ω l j,l=1 η S 1 + δe Ψ NδΩ } lψ w j w j r d 1 drdωdη d 1 Ωl Ψ + j,l=1 η S 1 + δe Ψ NδΩ lψ d 1 Ωl Ψ + Ω l 1 + δe Ψ NδΩ lψ w j r d 1 drdωdη = J 1 + J. Since Ω l = Ω l + d 1ω l, the integration by parts gives Hence we obtain J = J 1 = j,l=1 η d j,l=1 η S d 1 {Ω l Ωl Ψ +d 1ω Ωl Ψ l 1+δe NδΩ lψ Ψ 1+δΨ S d 1 1+δe NδΩ lψ Ψ 1+δΨ } w j r d 1 drdωdη. { δe Ψ Nδ Ω l Ψ

21 = 1 d 1ω l 1 + δe Ψ Nδ η S d 1 { δe Ψ Nδ } Ω l Ψ w j r d 1 drdωdη Ω l Ψ 1 d δe Ψ Nδ } ω l Ω l Ψ F δ w r d 1 drdωdη. Since there exists a constant C d > such that t Ω l Ψ = c r ξ j sη j ξ sη dsω lω j c C d tr, 4.8 we have J C d {c t η l=1 l=1 r F δ w r d 1 drdωdη S d 1 } + F δ w r d 1 drdωdη, S d 1 which yields 4.6. The proof of Proposition 4. is now complete. η End of proof of Theorem 1.3 : From Propositions 4.1 and 4., there exist constants C, C 3 > independent of δ > and t > such that 1 d 1 dt F δft L + c t C v t x F δ ft L So that if 1 c t C, we have, Using the fact F δ f = 1 1+δ g, we get Take the limit δ. Then we have C 3 F δ ft L. d dt F δft L C 3 F δ ft L. 4.9 F δ ft L e C3t g L. e Ψt,η,ξ ˆft, η, ξ dηdξ e C3t g L. 4.1 On the other hand, by Lemma 3.1, there exists a c 1 > such that e Ψt,η,ξ ˆft, η, ξ R t dηdξ = e c ξ sη ds ĝt, η, ξ tη dηdξ R d R t = e c ξ+t sη ds ĝt, η, ξ dηdξ R d η ĝt, η, ξ dηdξ. e cc1t ξ +t

22 Finally, for any < T < T, choosing c > small enough such that 1 c T C, we have proved, e cc1t v1/ +t x 1/ gt, x, v dxdv e C3t g L for any < t T, which completes the proof of Theorem 1.3 with C = C 3 depending only on d. Remark 4.1. The formulas 4.7 and 4.8 show that we cannot get the ultra-analytic effect of order 1/ as in Theorem 1.. It is the same reason why we do not consider the symmetric term Qg, µ in the equation 1.8 as in [1]. Acknowledgments: Authors wish to express their hearty gratitude to Seiji Ukai who communicated Lemma 3.1. The research of the first author was supported by Grant-in- Aid for Scientific Research No , Japan Society of the Promotion of Science, and the second author would like to thank the support of Kyoto University for his visit there. References [1] R.Alexandre, Y.Morimoto, S.Ukai, C.-J.Xu and T.Yang, Uncertainty principle and kinetic equations, J. Funct. Anal [] R. Alexandre and M. Safadi, Littlewood Paley decomposition and regularity issues in Boltzmann equation homogeneous equations. I. Non cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., [3] R. Alexandre and M. Safadi, Littlewood Paley decomposition and regularity issues in Boltzmann homogeneous equations. II. Non cutoff and non Maxwell cases. Discrete Contin. Dyn. Syst. A, [4] H. Chen, W. Li and C.-J. Xu, The Gevrey Hypoellipticity for linear and non-linear Fokker-Planck equations, to appear J. Diff. Equat. [5] H. Chen, W. Li and C.-J. Xu, Gevrey class regularity for the solution of the spatially homogeneous Landau equation, Kinetic and related models 1 8 no [6] Y. Chen, L. Desvillettes and L. He, Smoothing Effects for Classical Solutions of the Full Landau Equation, to appear in Arch. Rat. Mech. and Analysis [7] Y. Chen, Smoothness of Classical Solutions to the Vlasov-Poisson-Landau System, Kinetic and Related Models 18, no. 3, [8] L. Desvillettes, G,Furiolo and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules. Trans. Amer. Math. Soc., [9] L. Desvillettes and C. Villani, On the Spatially Homogeneous Landau Equation for Hard Potentials. Part I: Existence, Uniqueness and Smoothness. Comm. Partial Differential Equations 5, no. 1-, [1] L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. Comm. Partial Differential Equations 5, no. 1-, [11] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54 1, no. 1, 1 4. [1] L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm. Partial Differential Equations, [13] Y. Guo, The Landau equation in a periodic box. Comm. Math. Phys. 31, no. 3, [14] L. Hörmander, The analysis of linear partial differential operators II, Springer-Verlag, [15] Z. H. Huo, Y.Morimoto, S.Ukai and T.Yang, Regularity of entropy solutions for spatially homogeneous Boltzmann equation without Angular cutoff, Kinetic and Related Models 18, no. 3, [16] Y. Morimoto and C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto Univ. 47 7,

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