Littlewood-Paley Decomposition and regularity issues in Boltzmann homogeneous equations I. Non cutoff and Maxwell cases

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1 Littlewood-Paley Decomposition and regularity issues in Boltzmann homogeneous equations I. Non cutoff and Maxwell cases Radjesarane ALEXANDRE and Mouhamad EL SAFADI MAPMO UMR 668 Uniersité d Orléans, BP ORLEANS Cedex, FRANCE February 004 Abstract We use Littlewood Paley decompositions and related arguments to study some regularity questions in Boltzmann equation. This is done in the framework of homogeneous solutions, with a scattering cross section of Maxwell type, without assuming the usual Grad s cutoff assumption. Although the recent results of Desillettes and Wennberg include larger assumptions on the cross sections than those dealt with here, our method is extremely simple in comparison. Keywords- Boltzmann equation, Littlewood Paley decomposition, non cutoff cross sections. MSC- 35D05, 35D0, 35K70, 8B40. Introduction Consider Boltzmann homogeneous equation, see the classical books of Cercignani, Chapman and Cowling [5, 6] alexandr@labomath.uni-orleans.fr safadi@labomath.uni-orleans.fr t f(t, ) = Q(f, f)(t, ) t 0, R n. (.)

2 where f is a positie function depending only (homogeneous framework) upon the two ariables t 0 (time) and R n (elocity) with f(0, ) = f 0 (). We assume that n, and in fact the most interesting case is n = 3. We assume that the initial datum f 0 satisfies the usual entropic hypothesis, that is f 0 0, f 0 (){+ + log( + f 0 ()}d < + (.) R n Of course, f 0 is suppose not to be the null function. Boltzmann quadratic operator Q depends on as follows, Q(f, f) = d dσb(, σ)(f f ff ), σ where IR n, σ S n, f = f(), f = f( ), f = f( ), f = f( ) and = + σ are the so called post (or pre) collisional elocities. and = + + σ Herein, we asssume that the scattering cross section B(, σ) > 0 depends only on the deiation angle θ (Maxwell case): B(, σ) = b(cosθ), cos θ = <, σ>, 0 θ π. (.3) The aboe range of alues of θ means more precisely that b is supported on this set. This is really unimportant, as can be checked from the rest of the paper, but that simplifies somehow some points, say keeping away from the non singular or cutoff case. In this first part of our work, we are interested in taking a collision section B associated to an intermolecular potential, under the non cutoff assumption, that is rs σ B(, σ) L (S n ). More precisely, we assume b(cosθ) has a singularity around θ 0 of the form sinθb(cosθ) κ when θ 0 and γ = θ+γ s The aboe framework is by now standard, and for more details, the reader may consult the works of Alexandre, Desillettes, Villani and Wennberg [3, 7] We assume that a weak solution to Boltzmann (.) has already been constructed and that it satisfies the usual entropic estimates, to say, for a fixed T > 0, sup t [0,T ] R n f(t, )( + + log( + f(t, ))d <. (.4)

3 These follow from the formal conseration laws: mass conersation : momenty conseration : energy conseration : entropy estimate : t flogf + 4 f(t, )d = f(t, )d = f(t, )d = σ f 0 ()d; f 0 ()d; f 0 ()d; b( σ){f f ff }log{ f f ff } = 0. We refer for instance to the papers of Arkeryd, Goudon and Villani [4, 0, 6] for more details. We shall also assume that our weak solutions satisfy the mass conseration and the entropy dissipation rate estimate to say T 0 σ b( σ){f f ff }log{ f f ff } < +. (.5) In fact, this last assumption is not necessary, and may be omitted, as will be clear from the rest of the paper. Howeer, only for physical reasons, we do assume these ones. Such weak solutions hae been constructed within the general non cutoff case, see for instance the works of Arkeryd, Goudon, Villani [4, 0, 6], and also the probabilistic counterpart, see for instance the paper of Fournier, Méléard and the references therein [9]. In this part of our work, we are interested in regularity issues associated to such solutions. That is, is this weak solution more regular than the initial datum, and if so, can we hae estimates on this regularity? Such questions, for the non cutoff case, started with the works of Desillettes [7], see also the references in the reiew of Villani [7], and hae been the subject of recent works, see Alexandre, Fournier, Méléard [,, 9]. The up to date recent results about this regularity question are due to Desillettes and Wennberg [8]. So, what the need for another paper around this question? There a few important reasons for doing that; the first one is the following point: is any weak solution regular? the aboe 3

4 mentionned works show only that one can construct a weak solution which will be regular. The second one is to proide other methods, which could be of some help in the non homogeneous case, a question the first author started in Alexandre []. On these few points, we improe the issues, but actually assuming the Maxwell case on the cross section. Remoing this hypothesis is the subject of our second part in progress. Our arguments use Littlewood Paley decomposition, as can be found in the standard texts of Runst, Sickel, Stein, Triebel [, 3, 4], see also the Lecture Notes of Tao [5], and in fact only the point that working on small annuli instead of the full frequency space is the important point of simplification. Apart from this particular point, and it is already quite surprising, we only use elementary analytical arguments. The result we want to proe here is gien by Theorem. Let f be any weak homogeneous solution of Boltzman quation. Let B be a Maxwell collision kernel without angular cuttoff, as aboe, and f 0 be an initial datum with finite mass, energy and entropy R n f 0 ()( + + log( + f 0 ())d <. Then, for s R +, t > 0, one has f H s (R n ). In fact, we are also able to gie an estimate on the Sobole norm of f, wrt the norm gien by series as known by Harmonic Analysis people, s Pde people. Also, it is now clear how to obtain estimates in other functional spaces, such as Beso or Triebel spaces. The paper is organized as follows: Section is deoted to the proof of this result, diided in a few steps. We recall therein the basic tools needed for applying Littlewood Paley theory. A final Section contains a small Appendix related to basic facts used during the proof. Proof of the main result We diide it into three parts. In the first one, for reader s clarity, we recall some basic elements from Littlewood Paley theory. Second part is deoted to the obtention of a simple differential inequality inoling Fourier quantities related to f at say fixed frequency. Finally, in the third part, we conclude the proof and get in particuliar estimates on the Sobole norm of our solution. 4

5 . Background on Littlewood-paley theory and Sobole spaces. Let us recall the Littlewood Paley decomposition, which is a ery basic (but surprisingly ery powerful) way to cut into small pieces functions defined on the phase plane. After describing its basic properties, we shall end up by recalling the link with Sobole spaces. Readers already familiar with this theory can skip this sub-section. More details may be found in the books of Runst, Sickel, Stein and Triebel [, 3, 4]. See also the ery beautiful lecture notes by Tao [5]. We fix once for all in this paper a collection {ψ k = ψ k ()} k N of smooth functions such that supp ψ 0 { IR n, }, supp ψ k { IR n, k k+ } for all k, for eery multi-index α, there exists a positie number c α such that k α D α ψ k () c α for all k = 0,,,... and all IR n and ψ k () = for all IR n. k=0 To simplify also some computations, all functions ψ k, for k, are constructed from a single one ψ 0, ie we are gien ψ such that supp ψ {, } and ψ > 0 if and ψ k () ψ( k ), for all k and IRn. Then, we define the Littlewood Paley projection operators p k, for k 0, by p k f() = ψ k () ˆf(). For k, p k is a kind of frequency projection on the annulus { k } because it has (frequency) support on the annulus { R n ; k k+ }. By telescoping the series, we thus hae the Littlewood Paley decomposition f = p k f for all f S. k N This decomposition takes a single function and writes it as a superposition of a countably infinite family of functions p k f, each one haing a frequency of magnitude k, for k. 5

6 Let us gie some comments for the unfamiliar reader. The Littlewood Paley pieces p k f of f react ery well with deriaties. For instance, roughly speaking, we hae p k f k p k f. More precisely, one can show the following facts. Let k be an integer, k, and let f be a function with Fourier support in the annulus { k k+ }. Then we hae f k f for all k. Inequalities of this type are known as Berstein, Polya, Nikoolsky inequalities and turn out to be ery useful, see [, 3, 4]. In particular, we hae p k f k p k f for arbitrary f. On the other hand, by the triangle inequality, we hae one Littlewood Paley type inequality sup k p k f f p k f. k Usual Sobole spaces can be described by the following important result, see Runst, Sickel, Stein and Triebel [, 3, 4] for details Lemma. For all s 0, f H s k N ks p k f. Although the aboe result gies only an equialence of norms between the rhs and the left one, we shall use the right one as our definition of Sobole spaces. After all, for a non integer s, the usual definition of Sobole spaces by Fourier transform is not so explicit. Furthermore, this point is actually central in our arguments below.. Obtaining a differential inequality For clarity, we diide this section in fie steps. In the first step, we apply Fourier transform on equation (.), in the form used in [3], and we use a few simple arguments to obtain a simple equality. Then, in the next three steps, we estimate each term appearing in this equality, getting an useful inequality for the Fourier transform of our solution f. 6

7 Finally, in step 5, by an iteration process, we deduce the H s norm of our solution, for t > 0. Step : Application of the Fourier transformation. Applying Fourier transform on Boltzmann equation (.) in a standard way, see for instance [3], yields the equation where t ˆf = Q(f, f) = [ ˆf( + ) ˆf( ) ˆf() ˆf(0)] b(.σ) dσ, (.6) S n + = + σ and = σ. Note that this application of Fourier transform works with our assumptions on f, see [3]. In the following, we let k be an integer such that k 3. The reason for choosing this minimal alue of k will be clear below. Multiplying (.6) by ψ( ), we get k t pk f = [ ˆf( + ) ˆf( )ψ( S n k ) ˆf() ˆf(0)ψ( )] b( k.σ) dσ, where p k f() = ψ( k ) ˆf() is the component of order k of f in the Littlewood Paley decomposition, see the preious section. Letting the term ˆf( + ) ˆf( )ψ( + ) appear, the aboe equation is transformed into k t pk f = [ ˆf( ) p k f( + ) p k f() ˆf(0)] b( S n.σ)dσ Sn + ˆf( + ) ˆf( )[ψ( ) ψ(+ )] b( k k.σ)dσ. Multiplying by the conjugated term p k f, we obtain ( t pk f) p k f = [ ˆf( ) p k f( + ) p k f() p k f() p k f() ˆf(0)] b( S n.σ)dσ Sn + ˆf( + ) ˆf( ) p k f()[ψ( ) ψ(+ )] b( k k.σ)dσ. In the same way, conjugation and multiplication by p k f yields ( t pk f) p k f = [ ˆf( ) p k f( + ) p k f() p k f() p k f() ˆf(0)] b( S n.σ)dσ+ Sn + ˆf( + ) ˆf( ) p k f()[ψ( ) ψ(+ )] b( k k.σ)dσ. 7

8 Since t p k f = ( t pk f) p k f + p k f( t pk f), we get finally t p k f() = [ ˆf( ) p k f( + ) p k f() + ˆf( + ) p k f( + ) p k f() p k f() p k f()] ˆf(0)b( S n.σ)dσ Sn + [ ˆf( + ) ˆf( ) p k f() + ˆf( + ) ˆf( ) p k f()][ψ( ) ψ(+ k k )]b(.σ)dσ. (.7) To proceed further on, we shall use the following classical result (Bochner s type inequality) about the link between Fourier transform and positie definite functions, see [] for instance. Lemma. Let d N. Let two sets of numbers (x i ) i d (resp (λ j ) j d ) in R 3 (resp in Cl ). Then for any positie function g in L (R 3 ), one has i,j d λ i λ j ĝ(x i x j ) 0. We apply this lemma, taking d =, x =, x = +, λ = p k f(), λ = p k f( + ) and our solution f, and we notice that p k f() ˆf(0) pk f() p k f( + ) ˆf( ) p k f( + ) p k f() ˆf( ) + p k f( + ) ˆf(0) 0. (.8) Letting the term (.8) appear in our equation (.7), integrating wrt to, we get the equality t p k f d + S n [ p k f() ˆf(0) pk f() p k f( + ) ˆf( ) p k f( + ) p k f() ˆf( )+ p k f( + ) ˆf(0)]b(.σ) Sn = [ p k f() ˆf( + ) ˆf( ) + p k f() ˆf( + ) ˆf( )][ψ( (.9) ) ψ(+ k k )]b(.σ)dσd (.0) + [ p k f( + ) p k f() ] ˆf(0)b(.σ)dσd. (.) S n Our next task will be to analyse each term appearing in the aboe equality, leaing the time deriatie. Step : Lower bound on the term (.9). For the integral (.9), one has, using [3], S n [ p k f() ˆf(0) pk f() p k f( + ) ˆf( ) p k f( + ) p k f() ˆf( )+ p k f( + ) ˆf(0)]b(.σ) 8

9 Note that we hae = (π) n f (p k f p k f) b( σ.σ)dd dσ p k f() { [ ˆf(0) ˆf( ) ]b( S n.σ)dσ}d. S n [ ˆf(0) ˆf( ) ]b(.σ)dσ C f 0 γ, where C f0 is a (strictly) non negatie constant, depending on n, f 0 L, f 0 L, f 0 LLogL and b. This result can be found in [3]. Consequently, we obtain the following inequality [ p k f() ˆf(0) pk f() p k f( + ) ˆf( ) p k f( + ) p k f() ˆf( )+ p k f( + ) ˆf(0)]b( S n.σ) (π) n C f 0 p k f γ d. (.) Step 3: Another expression for the term (.). For this one, we consider the following truncated term, where ε > 0 is a fixed non negatie constant, to be sent to 0 at the end S ɛ =.σ >ɛ[ p k f( + ) p k f() ] ˆf(0)b( S n.σ)dσd. S ɛ is well defined, and it writes also as S ɛ = G ɛ.σ >ɛ p k f() ˆf(0)b(.σ 0.σ)dσd where G ɛ =.σ >ɛ p k f( + ) ˆf(0)b( S n.σ)dσd. By the change of ariables used in [3], see also [7], +, we get G ɛ =.σ >ɛ p k f() ˆf(0) n where Then it follows that S ɛ = ˆf(0).σ S n ψ σ () = (.σ)..σ >ɛ p k f() [ n (.σ) b(ψ σ())dσd ( b(ψ σ()) b(.σ).σ)]dσd. 9

10 Since p k f is bounded, with compact support, applying Lebesgue dominated conergence theorem, we hae where S = ˆf(0) which is the integral (.). S ɛ S for ɛ 0 S n p k f() [ n ( b(ψ σ()) b(.σ).σ)]dσd The inner spherical integral is well defined, in the non cutoff case. Indeed, since n.σ 7 ( b(( 8.σ).σ) ) dσ = = π 4 π Arccos 7 8 Arccos 7 8 sin n θ sin n θ n cos n (θ) cos n (θ/) it follows that the spherical inner integral becomes π 0 = sin n θ [ π 0 n/ π sin n θ [ which is well defined, een in the non cutoff case. 0 b(cosθ) dθ b(cosθ) dθ, cos n b(cosθ) b(cosθ) ] dθ (θ/) cos n ] b(cosθ) dθ (θ/) sin n θ ( cos n (θ/)) b(cosθ) dθ Now, since we are in the Maxwell case, the quantity n [ S n ( b(ψ σ()) b(.σ).σ)]dσ is a constant depending solely on the parameter γ, denoted C(γ). It follows that and finally, the integral (.) becomes S = ˆf(0)C(γ) p k f() d S n [ p k f( + ) p k f() ] ˆf(0)b(.σ)dσd = ˆf(0)C(γ) p k f L. (.3) 0

11 Step 4: Upper bound for the term (.0). This integral is nothing else than Re pk f() ˆf( + ) ˆf( )A k b( S n.σ)dσd where A k = ψ( k ) ψ(+ k ). Thus (.0) is bounded from aboe by f 0 (0) p k f() ˆf( + ) A k b(.σ)dσd. (.4) S n Let us notice that for the alues of the ariable, ie k k+, appearing in this integral, it follows that k + k+, see the Appendix if necessary. Thus ˆf( + ) = p k f( + ) + p k f( + ) + p k f( + ) + p k+ f( + ) Plugging this expression of ˆf( + ) in equation (.4), the term (.0) is bounded from aboe by f 0 (0) + f 0 (0) + f 0 (0) + f 0 (0) k k+ k k+ k k+ k k+ We can also easily estimate A k as follows where C = sup η ψ(η). p k f() p k f( + ) A k b(.σ)dσd S n (.5) p k f() p k f( + ) A k b(.σ)dσd S n (.6) p k f() p k f( + ) A k b(.σ)dσd (.7) S n p k f() p k+ f( + ) A k b(.σ)dσd. (.8) S n A k ψ( k ) ψ(+ k ) C k + k C k Next, we shall estimate the last four integrals, so that the term p k (f) will appear, using Cauchy Schwartz s inequality, and being careful on the integral oer S n, due to the singularity around θ 0. For the first integral (.5), it writes also as f 0 (0) p k f() { p k f( ) A k b(.σ)dσ } d k k+ S n

12 by making the change of ariables σ σ. Because of the arious support conditions in the aboe expression, we can as well assume that k k+ and k k+. In particuliar, since = sin θ, it follows that sin θ k k k+ 4 which means in particuliar, that for each supp ψ k, we are, as regards the spherical integral, away from the singularity point θ 0. Writing this spherical integral as p k f( ) A k b( S n.σ)dσ = p k f( ) b ( S n.σ) Ak b (.σ)dσ and applying Cauchy Schwartz inequality wrt the ariable σ, we find that it is bounded from aboe by ( p k f( ) b( S n.σ)dσ) ( A k b( S n.σ)dσ) (.9) Second term of the integral (.9) is bounded by a constant, denoted by Γ. Indeed, we know that so that = sin θ k+ sin θ A k C k 4C sin θ Thus A k b( π.σ)dσ 4C sin θ b(cosθ)dθ < + S n 0 Therefore, the integral (.9) is also bounded from aboe by Γ ( S n p k f( ) b(.σ)dσ). Recall that this last spherical integral is well defined in the non cutoff case, as we notice aboe (range of alues of ). Thus, the integral (.5) is bounded from aboe by f 0 (0)Γ p k f() ( p k f( ) b( S n.σ)dσ). Applying now Cauchy Schwartz inequality wrt ariable, it bounded by f 0 (0)Γ ( p k f() d) ( p k f( ) b( S n.σ)dσd).

13 Note that the second integral oer ariable is a shorthand notation for in fact integration oer k k+. Getting back to the second term, making the change of ariables, we hae where S n p k f( ) b(.σ)dσd = G(γ) =.σ 7 8 p k f() = p k f() { = G(γ)( p k f() d),.σ 7 8 n (.σ 7 8 (.σ) b(ψ (σ))dσ. n.σ) b(ψ (σ))dσd N (.σ) b(ψ (σ))dσ} d Since θ 0, it follows that this integral is well defined in the non cutoff case (we simply work as in the cutoff case). Indeed =.σ 7 8 π Arccos 7 8 n (.σ) b(ψ (σ))dσd = sin n θ which is a finite integral. cos n (θ/) π 4 Arccos 7 8 b(cosθ) dθ n/ sin n θ π Arccos 7 8 Consequently, the integral (.5) will be bounded from aboe by n cos (θ) b(cosθ) dθ sin n θ b(cosθ) dθ f 0 (0) G(γ) Γ pk (f) L. (.0) Concerning the third term (.7), we shall deal with it, as for (.5), the only difference being due to the term p k f( + ) instead of p k f( + ). Therefore, we need to check that the integral oer S n is well defined, again in the non cutoff case. Making the change of ariables, σ σ, it is the same as f 0 (0) p k f() p k f( ) A k b( k k+ S n.σ)dσ. Again, we note that the inner spherical integral is well defined. Indeed, from the support condition, one has k k, and sin θ k k k+ 8 and in particuliar, we are away from the singularity θ 0. 3

14 Therefore, the integral (.7) becomes bounded from aboe as f 0 (0) G(γ) Γ pk (f) L p k (f) L (.) Finally, for the third term (.8), making the change of ariables σ σ, we can proceed as aboe by noticing that the inner spherical integral is well defined: since k k+, we hae sin θ k k+ and thus again, we are away from the singularity θ 0. Finally, the same process works also for the second term (.6). Consequently, using estimates (.0) and (.) and similar ones for the other integrals, integral (.0) is bounded from aboe by f 0 (0) G(γ) Γ [ pk (f) L + + p k (f) L p k (f) L + p k (f) L p k (f) L + p k (f) L p k+ (f) L ]. Step 5: The differential inequality From (.), (.3) and (.), we obtain the following estimate t p k f L + (π) n C f 0 p k f γ d ˆf(0)C(γ) p k f L + f 0 (0)G(γ)Γ [ pk (f) L + p k (f) L p k (f) L + p k (f) L p k (f) L + p k (f) L p k+ (f) L ] (.) where Λ p k f L + Λ [ p k (f) L + p k (f) L p k (f) L + p k (f) L p k (f) L + p k (f) L p k+ (f) L ] Λ = ˆf(0)C(γ) + f 0 (0)G(γ)Γ Λ = f 0 (0)G(γ)Γ. Since k k+, we obtain the inequality t p k f L + 4(π) n C f 0 kγ p k f Λ p k f L + Λ p k (f) L p k (f) L +Λ p k (f) L p k (f) L + Λ p k (f) L p k+ (f) L. (.3) 4

15 Using simple inequalities such as p k (f) L p k (f) L [ p k (f) L + p k (f) L ] inequality (.3) becomes { t p k f L + [ 4(π) n C f 0 kγ Λ 3Λ / ] p k f L (.4) Λ / p k f L + Λ / p k f L + Λ / p k+ f L and diiding (.4) by kn, we get t p k f L kn + [ 4(π) n C f 0 kγ Λ 3Λ / ] p k f L kn Λ / p k f L (k )n + Λ / p k f L (k )n + n Λ / p k+ f L (k+)n. Therefore, we hae obtained a chain of inequalities of the form, for k 3 where U k = p k f L kn, C k =.3 Conclusion and regularity t U k + C k U k βu k + βu k + β U k+ (.5) 4(π) n C f 0 kγ Λ 3Λ /, β = Λ / and β = n β. From Bersnstein inequality (also called Polya Plancherel Nikoolski inequality), see [, 3, 4], we first note that p k f L n(k+) p k f L. Since p k f L f L = f 0 L, we hae obtained U k M with M = n f 0 L. Starting from this estimate, we can then apply a simple result stated in Lemma 3. of the Appendix to obtain the following estimate on U k : for all p 3, there exist constants γ p and α p such that for all k p. U k Mγ p e C k (p )t + Mα p (C k (p ) ) p Since p k f L = kn U k, one has by Lemma. f H s = k=0 ks p k (f) L C + k=3 ks kn U k = C + k=3 k(s+n) U k. 5

16 In iew of the aboe estimate on U k, we notice then, that for each s 0, the series k.s U k is, for each fixed t > 0, a conergent series, for it is enough to take this iterated estimate for a large number of iterations p, with respect to s (in fact wrt s + n). This ends the proof. 3 Appendix Lemma 3. With the notations of Littlewood Paley theory, we hae i) If supp ψ k then ψ k () + ψ k () + ψ k+ () =, for all k, and also for k = 0 if we take the conention that ψ 0. ii) If supp ψ k then k + k+. Proof: i) This point is well known, see for instance Triebel [4], and follows easily by noticing that if suppψ k, then / supp ψ j, for either j k or j k +. ii) Let us assume that supp ψ k i.e k k+ ( i.e k ). Notice that the modulus of + writes + = ( + cosθ) with cosθ = <, σ >, 0 cosθ. Since + cos θ, it follows that +cos θ, and thus we hae k + k+. Moreoer, as + k+, we get the result. Lemma 3. Let β, M two non negatie numbers. Let be gien a sequence (U k ) k (resp (c k ) k ) of positie numbers depending continuously on time (resp of non negatie numbers) satisfying (say for k 3) t U k + C k U k βu k + βu k + β U k+, U k M k, t 0. Then, for any p iterations, there exist constants γ p, α p such that, for all k p, one has U k Mγ p e C k (p )t + Mα p (C k (p ) ) p 6

17 Proof: It is done by iteration. First start from the fact that U k M, and using this in the right hand side of the differential inequality, leads to the aboe conclusion, for p =. Then, start from the aboe first iteration, and replace again in the rhs of the aboe inequality, and just repeat the process. References [] Alexandre.R Around 3D Boltzmann non linear operator without cutoff. A new formulation. MAN, Vol 34, N3, (000). [] Alexandre.R Solutions of Boltzmann equation without cutoff and for small initial data. J. Stat. Physics 04, no. -, (00). [3] Alexandre.R, Desillettes.L, Villani.C, Wennberg.B Entropy dissipation and long range interactions. Arch. Rat. Mech. Anal., 5-4, (000). [4] Arkeryd.L Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rat. Mech. Anal., 77, - (98). [5] Cercignani.C The Boltzmann equation and its applications. Springer-Verlag, New- York (988). [6] Chapman.S, Cowling.T The mathematical theory of non uniform gases. Cambridge Uniersity Press (95). [7] Desillettes.L Regularization properties of the -dimensional non radially symmetric non cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules. Transport Theory Stat Phys. 6-3, (997). [8] Desllettes.L, Wennberg.B Regularity in Boltzmann equation without cutoff. Preprint, see (003). [9] Fournier.N, Méléard.S Existence results for D homogeneous Boltzmann equation without cutoff and for non Maxwell molecules by use of Malliain calculus. Preprint 6, Uniersité Paris VI (000). [0] Goudon.T On Boltzmann equation and Fokker-Planck asymptotics: influence of grazing collisions. J. Stat. Phys , (997). 7

18 [] Runst.T, Sickel. W Sobole spaces of fractional order, Nemytskij operators, and Non linear PdE. De Gruyter, New York (996). [] Schwartz.L Théorie des distributions. Hermann, Paris (966). [3] Stein.E-M Harmonic Analysis. Real ariable methods, orthogonality, and oscillatory integrals. Princeton Uni. Press, Princeton (993). [4] Triebel.H Theory of function spaces. Birkhauser Verlag, Basel and al. (983). [5] Tao.T Littlewood-paley decomposition. Lecture Notes For 54A. See web page [6] Villani.C On a new class of solutions for the spatially homogeneous, Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 3-3, (998). [7] Villani.C A reiew of mathematical topics in collisional kinetic theory. in Handbook of Fluid Mechanics, Ed. S. Friedlander, D.Serre (00). 8