A REMARK ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION

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1 A REMARK ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU Abstract. We consier the non-linear spatially homogeneous Lanau equation with Maxwellian molecules in a close-to-equilibrium framework an show that the Cauchy problem for the fluctuation aroun the Maxwellian equilibrium istribution enjoys a Gelfan-Shilov regularizing effect in the class S / / (R ), implying the ultra-analyticity of both the fluctuation an its Fourier transform, for any positive time.. Introuction In the work [3], we consier the spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules in a close-to-equilibrium framework an stuy the smoothing properties of the Cauchy problem for the fluctuation aroun the Maxwellian equilibrium istribution. The Boltzmann equation escribes the behavior of a ilute gas when the only interactions taken into account are binary collisions [5]. In the spatially homogeneous case with Maxwellian molecules, it reas as the equation t f = Q(f, f), (.) f t=0 = f 0, for the ensity istribution of the particles f = f(t, v) 0, t 0, v R, with, where the non-linear term ( v v ) (.) Q(f, f) = v v σ (f f f f)σv, R S b stans for the Boltzmann collision operator whose cross section is a non-negative function satisfying to the assumption (.3) (sin θ) b(cos θ) θ 0 + θ s, for some 0 < s <. The notation a b means that a/b is boune from above an below by fixe positive constants. The term (.3) is not integrable in zero π 0 (sin θ) b(cos θ)θ = +. This non-integrability plays a major role regaring the qualitative behaviour of the solutions to the Boltzmann equation an this feature is essential for the smoothing effect to be present, see the iscussion in [3] an all the references herein. In [3], we consier the spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules (.) in the raially symmetric case with initial ensity istributions (.4) f 0 = µ + µ g 0, g 0 L (R ) raial, g 0 L, Date: January 3, Mathematics Subject Classification. 35B65. Key wors an phrases. Lanau equation, Gelfan-Shilov regularity, Ultra-analyticity, Smoothing effect.

2 Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU close to the Maxwellian equilibrium istribution (.5) µ (v) = (π) e v, v R, Q(µ, µ ) = 0, where is the Eucliean norm on R, in the physical 3-imensional case = 3. The main result in [3] shows that the Cauchy problem for the fluctuation f = µ 3 + µ 3 g, aroun the Maxwellian equilibrium istribution (.6) t g + L g = µ / 3 Q( µ 3 g, µ 3 g), g t=0 = g 0 L (R 3 ), where L g = µ / 3 Q(µ 3, µ / 3 g) µ / 3 Q(µ / 3 g, µ 3 ), enjoys the same Gelfan-Shilov regularizing effect as the Cauchy problem efine by the evolution equation associate to the fractional harmonic oscillator t g + H s g = 0, (.7) g t=0 = g 0 L (R 3 H = v + v ), 4, where 0 < s < is the positive parameter appearing in the assumption (.3). More specifically, we prove that uner the assumption (.4), the Cauchy problem (.6) amits a unique global raial solution g L (R + t, L (R 3 v)), which belongs to the Gelfan-Shilov class S /s /s (R3 ) for any positive time (.8) t > 0, g(t) S /s /s (R3 ). The efinition of the Gelfan-Shilov regularity is recalle in appenix (Section 4.). In the present work, we stuy the spatially homogeneous Lanau equation with Maxwellian molecules t f = Q L (f, f), (.9) f t=0 = f 0. The Lanau collision operator Q L (f, f) is unerstoo as the limiting Boltzmann operator in the grazing collision limit asymptotic [,, 6, 7, 8], when s tens to in the singularity assumption (.3). In the physical 3-imensional case, the linearize non-cutoff Boltzmann operator with Maxwellian molecules was actually showe to be equal to the fractional linearize Lanau operator with Maxwellian molecules [] (Theorem.3), L = a(h, S )L s L, up to a positive boune isomorphism on L (R 3 ), c > 0, f L (R 3 ), c f L (a(h, S )f, f) L c f L, commuting with the harmonic oscillator H = v + v 4 an the Laplace-Beltrami operator S = (v j k v k j ), j,k 3 j k on the unit sphere S. In view of this link between the linearize Boltzmann an Lanau operators, an in analogy with the Gelfan-Shilov smoothing result proven in [3] for the spatially homogeneous non-cutoff Boltzmann equation, we may therefore expect that the spatially homogeneous Lanau equation also enjoys specific Gelfan-Shilov smoothing properties. The purpose of this note is to confirm this insight an to check that the Cauchy

3 ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 3 problem for the fluctuation aroun the Maxwellian equilibrium istribution associate to the spatially homogeneous Lanau equation with Maxwellian molecules actually enjoys a Gelfan-Shilov regularizing effect in the class S / / (R ), implying the ultra-analyticity of both the fluctuation an its Fourier transform, for any positive time.. The Lanau equation The Lanau equation written by Lanau in 936 [] is the equation t f + v x f = Q L (f, f), (.) f t=0 = f 0, for the ensity istribution of the particles f = f(t, x, v) 0 at time t, having position x R an velocity v R, with. The term Q L (f, f) is the Lanau collision operator associate to the Lanau bilinear operator ( Q L (g, f) = v a(v v ) ( g(t, x, v )( v f)(t, x, v) ( v g)(t, x, v )f(t, x, v) ) ) v, R where a = (a i,j ) i,j stans for the non-negative symmetric matrix (.) a(v) = v γ ( v I v v) M (R), < γ < +. In this work, we stuy the spatially homogeneous case when the ensity istribution of the particles oes not epen on the position variable t f = Q L (f, f), (.3) f t=0 = f 0, for Maxwellian molecules, that is, when the parameter γ = 0 in the assumption (.). At least formally, it is easily checke that the mass, the momentum an the kinetic energy are conserve quantities by this evolution equation (.4) f(t, v)v = M, f(t, v)vv = MV, f(t, v) v v = E, t 0, R R R with M > 0, V R, E > 0. The Cauchy problem (.3) associate to the spatially homogeneous Lanau equation with Maxwellian molecules an some quantitative features of the solutions were thoroughly stuie by Villani [7]. The propositions 4 an 6 of the work [7] show that, for each non-negative measurable initial ensity istribution f 0 having finite mass an finite energy (.5) f 0 0, 0 < f 0 (v)v = M < +, 0 < f 0 (v) v v = E < +, R R the Cauchy problem (.3) amits a unique global classical solution f(t, v) efine for all t 0. Furthermore, this solution is showe to be a non-negative boune smooth function f(t) 0, f(t) L (R v) C (R v), for any positive time t > 0. In this work, we stuy a close-to-equilibrium framework. To that en, we consier the linearization of the spatially homogeneous Lanau equation f = µ + µ g, aroun the Maxwellian equilibrium istribution (.6) µ (v) = (π) e v, v R. By using that Q L (µ, µ ) = 0, an setting (.7) L L g = µ / Q(µ, µ / g) µ / Q(µ / g, µ ),

4 4 Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU the original spatially homogeneous Lanau equation (.3) is reuce to the Cauchy problem for the fluctuation (.8) t g + L L g = µ / Q L ( µ g, µ g), g t=0 = g 0. An explicit computation [] (Proposition.) shows that the linearize Lanau operator with Maxwellian molecules acting on the Schwartz space is equal to ( L L = ( ) H ) [ ( S + S ( ) H )] P [ ( + S ( ) H )] P, where H = v + v 4 is the harmonic oscillator, S = (v j k v k j ), j,k j k stans for the Laplace-Beltrami operator on the unit sphere S an P k are the orthogonal projections onto the Hermite basis efine in Section 4.. The linearize Lanau operator is a non-negative operator (L L g, g) L (R v ) 0, satisfying (.9) (L L g, g) L (R ) = 0 g = Pg, where Pg = (a + b v + c v )µ /, with a, c R, b R, stans for the L -orthogonal projection onto the space of collisional invariants (.0) N = Span µ /, v µ /,..., v µ /, v µ / }. By elaborating on the solutions constructe by Villani [7], the purpose of this note is to stuy the Gelfan-Shilov regularizing properties of the Cauchy problem (.8) for the fluctuation aroun the Maxwellian equilibrium istribution. For the sake of simplicity, we may assume without loss of generality that the ensity istribution satisfies (.4) with V = 0. Furthermore, by changing the unknown function f to f as (.) f = M α f ( ) E, α = α M, we may reuce our stuy to the case when (.) f(t, v)v =, f(t, v)vv = 0, f(t, v) v v =, t 0. R R R Let f 0 = µ + µ g 0 0, with g 0 L (R v) L (R v), be a non-negative initial ensity istribution having finite mass an finite energy such that (.3) f 0 (v)v =, f 0 (v)vv = 0, f 0 (v) v v =. R R R Such an initial ensity istribution is rapily ecreasing with a finite temperature tail T (f 0 ) = sup β 0 : f 0 (v)e β v v < + }, R since (.4) f 0 (v)e v ( 4 v = µ (v) + g R (π) 0 (v) ) v < +, 4 R

5 ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 5 when g 0 L (R v). The analysis of the evolution of the temperature tail le in [7] (Section 6, p ) shows that f 0 (v)e v 4 v < + t > 0, f(t, v)e v 4 v < +. R R This implies that the fluctuation f = µ + µ g 0, aroun the Maxwellian equilibrium istribution efine by (.5) g(t) = µ / (f(t) µ ) L (R v) C (R v) S (R v), t > 0, belongs to L (R v) an therefore remains a tempere istribution for all t > 0. following statement is the main result containe in this note: The Theorem.. Let f 0 = µ + µ g 0 0, with g 0 L (R v) L (R v), be a non-negative measurable function having finite mass an finite energy such that (.6) f 0 (v)v =, f 0 (v)vv = 0, f 0 (v) v v =. R R R Let f(t) = µ + µ g(t), with g(t) L (R v) C (R v) when t > 0, be the unique global classical solution of the Cauchy problem associate to the spatially homogeneous Lanau equation with Maxwellian molecules tf = QL(f, f), f t=0 = f 0, constructe by Villani [7]. Then, there exists a positive constant δ > 0 such that ( ) / C > 0, t 0, e tδh g(t) L = e δ(k+)t P k g(t) L Ce ( )t ( g 0 L + ), k 0 with H = v + v 4, where L stans for the L (R v)-norm an P k are the orthogonal projections onto the Hermite basis efine in Section 4.. In particular, this implies that the fluctuation belongs to the Gelfan-Shilov space S / / (R ) for any positive time t > 0, g(t) S / / (R ). Remark. The orthogonal projection P k : S (R v) S (R v) is well-efine on tempere istributions since the Hermite functions are Schwartz functions. This result shows that the Cauchy problem (.8) enjoys an ultra-analytic regularizing effect in the Gevrey class G / (R ) both for the fluctuation an its Fourier transform in the velocity variable for any positive time g(t), ĝ(t) G / (R ), t > 0. Let us recall that the existence, uniqueness, the Sobolev regularity an the polynomial ecay of the weak solutions to the Cauchy problem (.3) have been stuie by Desvillettes an Villani for har potentials [8] (Theorem 6), that is, when the parameter satisfies 0 < γ in the assumption (.). Uner rather weak assumptions on the initial atum, e.g. f 0 L +δ, with δ > 0, they prove that there exists a weak solution to the Cauchy problem such that f C ([t 0, + [, S (R v)), for all t 0 > 0, an for all t 0 > 0, s > 0, m N, sup f(t, ) H m s < +. t t 0

6 6 Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU The Gevrey regularity f(t, ) G σ, for any σ >, for all positive time t > 0 of the solution to the Cauchy problem (.3) with an initial atum f 0 with finite mass, energy an entropy satisfying t 0 > 0, m 0, sup t t 0 f(t, ) H m γ < +, was later establishe by Chen, Li an Xu for the har potential case an the Maxwellian molecules case [3]. Uner the same assumptions on the solution, this result was later extene to analytic regularity [4]: t 0 > 0, c 0, C > 0, t t 0, e c 0( v) / f(t, ) L C(t + ), in the har potential case an the Maxwellian molecules case. Regaring specifically the Maxwellian molecules case γ = 0, Morimoto an Xu establishe in the ultra-analyticity [4] (Theorem.), 0 < t < T, f(t, ) G / (R ), 0 < T 0 < T, c 0 > 0, 0 < t T 0, e c 0t v f(t, ) L e t f 0 L, of any positive weak solution f(t, x) > 0 to the Cauchy problem (.3) satisfying f L (]0, T [, L (R ) L (R )), with 0 < T +, with an initial atum satisfying f 0 L (R ) L (R ). The result of Theorem. allows to specify further the property of ultraanalytic smoothing proven by Morimoto an Xu in the close-to-equilibrium framework [4]. This result points out the specific ecay of the fluctuation both in the velocity an its ual Fourier variable. As for the Boltzmann equation, the Gelfan-Shilov regularity seems relevant to escribe the regularizing properties of the Lanau equation in the close-toequilibrium framework. 3. Proof of Theorem. The proof of Theorem. is elementary an relies only on spectral arguments following the results establishe by Villani [7]. Let f 0 = µ + µ g 0 0, with g 0 L (R v) L (R v), be a non-negative measurable function having finite mass an finite energy such that (3.) f 0 (v)v =, R f 0 (v)vv = 0, R f 0 (v) v v =. R Following [7] (p. 966), we may choose an orthonormal basis of R iagonalizing the nonnegative symmetric quaratic form q(x) = f 0 (v)(x v) v = R j,k= x j x k R f 0 (v)v j v k v 0, where x v = x jv j, x = (x,..., x ), v = (v,..., v ), stans for the stanar ot prouct in R. In this orthonormal basis of R v, the unique solution to the Cauchy problem associate to the spatially homogeneous Lanau equation with Maxwellian molecules (.3) is showe to satisfy [7] (Section 5), (3.) t f = with ( T j (t)) j f + ( ) (vf) + S f, T j (t) = f(t, v)vj v = + (T j (0) )e 4t, R

7 ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 7 an (3.3) R f(t, v)v =, R f(t, v)v j v = 0, f(t, v) v v = T j (t) =, R j k f(t, v)v j v k = 0, R when t 0. These conitions imply that the fluctuation satisfies g(t) L (R v) N, that is, µ (v)g(t, v)v = 0, v j µ (v)g(t, v)v = 0, v µ (v)g(t, v)v = 0, R R R together with j k v j v k µ (v)g(t, v) = 0, R when t 0. The equation (3.) may be rewritten for the fluctuation as t g = µ / ( α j e 4t ) j (µ + µ g)+( )µ / (vµ +v µ g)+ S g, with (3.4) α j = R v j µ (v)g 0 (v)v, α j = 0. It follows that [ ( t g = ( ) v + v 4 ) S By using that α j = 0, we notice that [ ( t g = ( ) v + v 4 ) S ]g e 4t [ α j j + v j 4 v j j ] g e 4t ]g e 4t where A +,j is the creation operator efine in Section 4.. We consier S n = n P k, k=0 α j (v j )µ /. α j [(A +,j ) g + v j µ / ], the orthogonal projection onto the n + lowest energy levels of the harmonic oscillator, where P k stans for the orthogonal projection onto the Hermite basis efine in Section 4.. As mentione above, the orthogonal projection S n is well-efine on tempere istributions since the Hermite functions are Schwartz functions. This gives a sense for the orthogonal projection of the fluctuation S n g(t) S (R v) as a Schwartz function. Then, a irect

8 8 Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU computation shows that for all t 0, δ > 0, n, t( e tδh S n g L ) δ(h(e tδh S n g), e tδh S n g) L = Re( t S n g, e δth S n g) L = ( )(H(e tδh S n g), e tδh S n g) L (( S )(e tδh S n g), e tδh S n g) L + ( ) etδh S n g L e 4t e 4t α j (e tδh S n (A +,j ) g, e tδh S n g) L, α j (e tδh (v j µ / ), e tδh S n g) L since the harmonic oscillator an the Laplace-Beltrami operator on S are commuting selfajoint operators. We euce from (4.5), (4.6) an (4.7) that an (e tδh S n (A +,j ) g, e tδh S n g) L = e δt ((A +,j ) e tδh S n g, e tδh S n g) L = e δt (A +,j e tδh S n g, A,j e tδh S n g) L e tδh (v j µ / ) = e tδh ((A +,j + A,j ) Ψ 0 ) = e tδh (A +,j + A,j + A +,j A,j + A,j A +,j )Ψ 0 = (e δt A +,j + e δt A,j + A +,j A,j + A,j A +,j )e δt Ψ 0 = e (+ )δt Ψ ej + e δt Ψ 0. It follows that t( e tδh S n g L ) + ( δ)(h(e tδh S n g), e tδh S n g) L ( ) etδh S n g L + e (4 δ)t By using that H = α j A +,j e tδh S n g L A,j e tδh S n g L + e (4 δ )t ( ) e 4δt + α j e tδh S n g L. (A +,j A,j + A,j A +,j ), we notice that A +,j u L A,j u L By using that we obtain that ( A +,j u L + A,j u L ) = (Hu, u) L. A +,j e tδh S n g L A +,j e tδh S n g L, t( e tδh S n g L ) + ( δ)(h(e tδh S n g), e tδh S n g) L ( ) etδh S n g L + e (4 δ)t( ) sup α j (H(e tδh S n g), e tδh S n g) L j + e (4 δ )t ( ) e 4δt + α j e tδh S n g L.

9 ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 9 We notice that 0 < + α j = vj (µ + µ g 0 )v, R since the initial ensity istribution f 0 = µ + µ g 0 0 satisfies R f 0 v =. On the other han, we euce from (3.4) that ( + α j ) =. This implies that < α j <, because. We may choose the positive constant 0 < δ such that sup α j δ. j It follows that t( e tδh S n g L ) ( ) etδh S n g L +e (4 δ )t e 4δt + ( ) α j e tδh S n g L ( ) etδh S n g L + 3( ) e tδh S n g L ( ) e tδh S n g L + 9 ( ). We obtain that for all t 0, n, which implies that for all t 0, e tδh S n g(t) L e ( )t g 0 L + 9 (e( )t ), e tδh g(t) L e ( )t g 0 L + 9 (e( )t ). It follows that there exists a positive constant C > 0 such that e tδh g(t) L (R v) Ce ( )t ( g 0 L + ), t 0, an we euce from (4.8) that for any positive time This ens the proof of Theorem.. g(t) S / / (R ), t > Appenix 4.. The harmonic oscillator. The stanar Hermite functions (φ n ) n 0 are efine for x R, ( ) n (4.) φ n (x) = n n! π e x n ( x n (e x ) = n n! x ) n(e x a n ) = +φ 0, π x n! where a + is the creation operator a + = ( x ). x The family (φ n ) n 0 is an orthonormal basis of L (R). We set for n 0, α = (α j ) j N, x R, v R, (4.) ψ n (x) = /4 φ n ( / x), ψ n = ( x n! ) nψ0, x (4.3) Ψ α (v) = ψ αj (v j ), E k = SpanΨ α } α N, α =k,

10 0 Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU with α = α + +α. The family (Ψ α ) α N is an orthonormal basis of L (R ) compose by the eigenfunctions of the -imensional harmonic oscillator (4.4) H = v + v 4 = ( ) + k P k, I = P k, k 0 k 0 where P k is the orthogonal projection onto E k whose imension is ( ) k+. The eigenvalue / is simple in all imensions an E 0 is generate by the function (4.5) Ψ 0 (v) = (π) 4 e v 4 = µ / (v), where µ is the Maxwellian istribution efine in (.5). Setting we have Ψ α = A ±,j = v j v j, j, α!...α! Aα +,...Aα +, Ψ 0, α = (α,..., α ) N, (4.6) A +,j Ψ α = α j + Ψ α+ej, A,j Ψ α = α j Ψ α ej (= 0 if α j = 0), where (e,..., e ) stans for the canonical basis of R. In particular, we reaily notice that for all t 0, δ > 0, (4.7) e tδh A +,j = e δt A +,j e tδh, e tδh A,j = e δt A,j e tδh. 4.. Gelfan-Shilov regularity. We refer the reaer to the works [9, 0, 5, 6] an the references herein for extensive expositions of the Gelfan-Shilov regularity. The Gelfan- Shilov spaces S µ ν (R ), with µ, ν > 0, µ + ν, are efine as the spaces of smooth functions f C (R ) satisfying to the estimates or, equivalently A, C > 0, α v f(v) CA α (α!) µ e A v /ν, v R, α N, A, C > 0, sup v R v β α v f(v) CA α + β (α!) µ (β!) ν, α, β N. These Gelfan-Shilov spaces S µ ν (R ) may also be characterize as the spaces of Schwartz functions f S (R ) satisfying to the estimates C > 0, ε > 0, f(v) Ce ε v /ν, v R, f(ξ) Ce ε ξ /µ, ξ R. In particular, we notice that Hermite functions belong to the symmetric Gelfan-Shilov space S / / (R ). More generally, the symmetric Gelfan-Shilov spaces S µ(r µ ), with µ /, can be nicely characterize through the ecomposition into the Hermite basis (Ψ α ) α N, see e.g. [6] (Proposition.), (4.8) f S µ µ(r ) f L (R ), t 0 > 0, ( (f, Ψ α ) L exp(t 0 α µ ) ) α N l (N ) < + f L (R ), t 0 > 0, e t 0H /µ f L < +, where (Ψ α ) α N stans for the Hermite basis efine in Section 4., an where H = v + v 4,

11 ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION is the -imensional harmonic oscillator. The Cauchy problem efine by the evolution equation associate to the harmonic oscillator t f + Hf = 0, (4.9) f t=0 = f 0 L (R ), enjoys nice regularizing properties. The smoothing effect for the solutions to this Cauchy problem is naturally escribe in term of the Gelfan-Shilov regularity. The characterization (4.8) proves that there is a regularizing effect for the solutions to the Cauchy problem (4.9) in the symmetric Gelfan-Shilov space S / / (R ) for any positive time, whereas the smoothing effect for the solutions to the Cauchy problem efine by the evolution equation associate to the fractional harmonic oscillator (4.0) t f + H s f = 0, f t=0 = f 0 L (R ), with 0 < s <, occurs for any positive time in the symmetric Gelfan-Shilov space S /s /s (R ). Acknowlegements. The research of the first author was supporte by the Grant-in- Ai for Scientific Research No , Japan Society for the Promotion of Science. The research of the secon author was supporte by the CNRS chair of excellence at Cergy-Pontoise University. The research of the last author was supporte partially by The Funamental Research Funs for Central Universities an the National Science Founation of China No. 76. References [] R. Alexanre, C. Villani, On the Lanau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire (004), no., 6-95 [] A.A. Arsen ev, O.E. Buryak, On a connection between the solution of the Boltzmann equation an the solution of the Lanau-Fokker-Planck equation, Math. USSR Sbornik 69 (99), no., [3] H. Chen, W.-X. Li, C.-J. Xu, Gevrey regularity for solution of the spatially homogeneous Lanau equation, Acta Math. Sci. Ser. B Engl. E. 9 (009), no. 3, [4] H. Chen, W.-X. Li, C.-J. Xu, Analytic smoothness effect of solutions for spatially homogeneous Lanau equation, J. Differential Equations, 48 (00), no., [5] C. Cercignani, The Boltzmann Equation an its Applications, Applie Mathematical Sciences, vol. 67 (988), Springer-Verlag, New York [6] P. Degon, B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Moels Methos Appl. Sci. (99), no., 67-8 [7] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys. (99), no. 3, [8] L. Desvillettes, C. Villani, On the spatially homogeneous Lanau equation for har potentials. I. Existence, uniqueness an smoothness, Comm. Partial Differential Equations, 5 (000), no. -, [9] I.M. Gelfan, G.E. Shilov, Generalize Functions II, Acaemic Press, New York (968) [0] T. Gramchev, S. Pilipović, L. Roino, Classes of egenerate elliptic operators in Gelfan-Shilov spaces, New evelopments in pseuo-ifferential operators, 5-3, Oper. Theory Av. Appl. 89, Birkhäuser, Basel (009) [] L.D. Lanau, Die kinetische Gleichung für en Fall Coulombscher Wechselwirkung, Phys. Z. Sowjet. 0 (936) 54, translation: The transport equation in the case of Coulomb interactions, D. ter Haar (E.), Collecte papers of L.D. Lanau, Pergamon Press, Oxfor, 98, [] N. Lerner, Y. Morimoto, K. Prava-Starov, C.-J. Xu, Phase space analysis an functional calculus for the linearize Lanau an Boltzmann operators, preprint (0) [3] N. Lerner, Y. Morimoto, K. Prava-Starov, C.-J. Xu, Gelfan-Shilov smoothing properties of the raially symmetric spatially homogeneous Boltzmann equation without angular cutoff, preprint (0) [4] Y. Morimoto, C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differential Equations, 47 (009)

12 Y. MORIMOTO, K. PRAVDA-STAROV & C.-J. XU [5] F. Nicola, L. Roino, Global pseuo-ifferential calculus on Eucliean spaces, Pseuo-Differential Operators, Theory an Applications, 4, Birkhäuser Verlag, Basel (00) [6] J. Toft, A. Khrennikov, B. Nilsson, S. Norebo, Decompositions of Gelfan-Shilov kernels into kernels of similar class, J. Math. Anal. Appl. 396 (0), no., 35-3 [7] C. Villani, On the spatially homogeneous Lanau equation for Maxwellian molecules, Math. Moels Methos Appl. Sci. 8 (998), no. 6, [8] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann an Lanau equations, Arch. Ration. Mech. Anal. 43 (998), no. 3, Y. Morimoto, Grauate School of Human an Environmental Stuies, Kyoto University, Kyoto , Japan aress: morimoto@math.h.kyoto-u.ac.jp K. Prava-Starov, Université e Cergy-Pontoise, Département e Mathématiques, CNRS UMR 8088, Cergy-Pontoise, France aress: karel.prava-starov@u-cergy.fr C.-J. Xu, School of Mathematics, Wuhan university 43007, Wuhan, P.R. China, an, Université e Rouen, CNRS UMR 6085, Département e Mathématiques, 7680 Saint-Etienne u Rouvray, France aress: Chao-Jiang.Xu@univ-rouen.fr