ABOUT THE REGULARIING PROPERTIES OF THE NON CUT{OFF KAC EQUATION Laurent Desvillettes ECOLE NORMALE SUPERIEURE 45, Rue d'ulm 7530 Paris Cedex 05 April 19, 00 Abstract We prove in this work that under suitable assumptions, the solution of the spatially homogeneous non cut{o Kac equation (or of the spatially homogeneous non cut{o D Boltzmann equation with Maxwellian molecules in the radial case) becomes very regular with respect to the velocity variable as soon as the time is strictly positive. 1 Introduction In the upper atmosphere, a gas is described by the nonnegative density f(t; x; v) of particles which at time t and point x, move with velocity v. Such a density satises the Boltzmann equation (Cf. [Ce], [Ch, Co], [Tr, Mu]): @f @t + v r xf = Q(f); (1:1) where Q is a quadratic collision kernel acting only on the variable v and taking in account any collisions preserving momentum and kinetic energy: Q(f)(v) = ff(v 0 )f(v) 0 3 =0 =0 with f(v)f(v )g B(jv v j; ) sin dddv ; (1:) v 0 = v + v + jv v j ; (1:3) 1
v 0 = v + v jv v j ; (1:4) v v cos = jv v j ; (1:5) and B is a nonnegative cross section. When the collisions in the gas come out of an inverse power law interaction in r 1 (with s ), the cross section s writes B(x; ) = x s 5 s 1 b(); (1:6) where b L 1 (]0; loc ]) and sin b() K(s) s+1 s 1 (1:7) for some K(s) > 0 when! 0. Most of the mathematical work about the Boltzmann equation is made under the assumption of angular cut{o of Grad (Cf. [Gr]), which means that b in eq. (1.6) is supposed to satisfy sin b() L 1 ([0; ]). Note that for inverse power laws in 1 r with s, this assumption never holds (because s of the singularity appearing in eq. (1.7)). For example, the existence of a global renormalized solution to the full Boltzmann equation (1.1) is known under this assumption (Cf. [DP, L]), but it is also the case with most of the works concerning the spatially homogeneous Boltzmann equation (Cf. [A 1], [A 3], [El], [De 1]): @f (t; v) = Q(f)(t; v); (1:8) @t with the noticeable exception of [A ], where existence is proved for the non cut{o equation (1.) { (1.8) when s > 3. We shall now concentrate on this spatially homogeneous equation (1.8). When the cut{o assumption is made, it is possible to write where Q + (f)(v) = 3 =0 Q(f) = Q + (f) f Lf; (1:9) =0 f(v 0 ) f(v 0 ) B(jv v j; ) sin dddv ; (1:10) and Lf = A f; (1:11)
with A(x) = B(jxj; ) sin d: (1:1) =0 Then, the solution f(t; v) of eq. (1.8) can be written under the form f(t; v) = f(0; v) e R t t Lf (;v)d 0 + 0 Q + (f)(s; v) e R t s Lf (;v) d ds: (1:13) But the operator Q + is known to be regularizing with respect to the variable v (at least when f L (IR 3 v), and when B satises some properties) (Cf. [L 1]). Therefore, if f(0; v) is not regular (for example if it belongs to L (IR 3 v) but not to H 1 (IR 3 v)), the solution f(t; v) of eq. (1.8) will at best keep the regularity of f(0; v) when t > 0. In particular, no regularizing eect is expected for the solution of the cut{o homogeneous Boltzmann equation (1.8). On the other hand, one can hope some regularizing properties for the solution of the non cut{o homogeneous Boltzmann equation (1.8), (1.) (when (1.6), (1.7) holds). One of the reasons of assuming such a conjecture is that an asymptotics of the Boltzmann equation when the cross section is concentrating on the grazing collisions (these collisions are those that are neglected when the cut{o assumption is made) leads to the Fokker{Planck{Landau equation (Cf. [De ], [Dg, Lu]), which is known to induce regularizing eects (or at least compactness properties, even in the spatially inhomogeneous case (Cf. [L ])). This article is devoted to the proof of such a conjecture in the simpler case of the spatially homogeneous Kac equation. We recall that the original Kac model is used to describe a one{dimensional spatially homogeneous gas in which the collisions preserve the mass and the energy, but not the momentum (Cf. [K], [MK]). Note also that the theorems of sections, 3 and 4 hold for the spatially homogeneous non cut{o D radially symmetric solutions of the Boltzmann equation with Maxwellian molecules, as is shown in Appendix C. In the Kac model, the nonnegative density f(t; v) satises @f (t; v) = K(f)(t; v); (1:14) @t 3
where K(f)(t; v) = and ff(v 00 ) f(v 00 ) f(v) f(v )g d dv ; (1:15) v 00 = v cos v sin ; (1:16) v 00 = v sin + v cos : (1:17) The analysis leading to eq. (1.13) still holds in this case. Therefore, one can at best hope that the regularity of f(0; v) is conserved for the solution f(t; v) of eq. (1.14) when t > 0. This armation is indeed easily proved when f(0; v) L 1 ((1 + jvj )dv) (Cf. theorem A.1 of Appendix A), but also in the more dicult case when f(0; v) lies in some Holder spaces (Cf. [G]) (Note also the results in the same spirit for the Boltzmann equation of [We]). We will therefore concentrate in this work on the equation where K (f)(t; v) = with @f @t (t; v) = K (f)(t; v); (1:18) ff(v 00 ) f(v 00 ) f(v) f(v )g (jj) ddv ; (1:19) (x) x (1:0) when x! 0 + and ]1; 3]. This kernel is obtained by analogy with the non cut{o kernel (1.), (1.6), (1.7) of Boltzmann equation. However, the analysis in the case when = 3 (corresponding to the Coulombian interaction in the case of the Boltzmann equation (Cf. [Dg, Lu])) is very dierent from the analysis when ]1; 3[. Therefore, we will only consider in the sequel the latter case. We begin in section by proving that the existence of a solution holds for eq. (1.18) { (1.0). We prove then in section 3 our main theorem. Namely, if f(0; v) L 1 (IR v ; (1 + jvj )dv) for all > 0, the solution f(t; v) of eq. (1.18) { (1.0) lies in C 1 (IR v ) for all t > 0. Finally, in section 4, we consider the case when only a nite number of moments are known to be initially bounded for f. The reader will also nd for the sake of completeness some classical results used throughout this work in appendix A and B at the end of the paper, appendix C being devoted to the extension of the results to the D radially symmetric Boltzmann equation with Maxwellian molecules. 4
Existence for the non cut{o Kac equation We prove in this section the following theorem: Theorem.1: Let f 0 0 be an initial datum such that f 0 (v) (1 + jvj + j log f 0 (v)j)dv < +1; (:1) and let 0 be a cross section satisfying the following property: 9 0 ; 1 > 0; ]1; 3[; 8x ]0; ]; 0 jxj (x) 1 jxj : (:) Then, there exists a nonnegative solution f(t; v) L 1 ([0; +1[ t ; L 1 (IR v ; (1+ jvj )dv)) to eq. (1.18), (1.19), (.) with initial datum f 0 in the following sense: For all function W ;1 (IR v ), we have @ @t where f(t; v) (v) dv = K (v; v ) = The conservation of mass f(t; v) dv = K (v; v ) f(t; v) f(t; v ) dv dv; (:3) f(v 00 ) (v)g (jj) d: (:4) f 0 (v) dv (:5) holds for these solutions, but the energy may decrease. Moreover, if for some p IN, there exists C :1 > 0 such that f 0 (v) (1 + jvj p ) dv C :1 ; (:6) one can nd C : > 0 such that for all t 0, f(t; v) (1 + jvj p ) dv C : : (:7) Finally, if assumption (.6) holds for some p, the conservation of energy f(t; v) jvj dv = f 0 (v) jvj dv (:8) 5
holds. Remark: The analogue of this theorem is proved in [A ] for the Boltzmann equation (with s > 3). The proof given here is very similar to that of [A ]. The sense to give to the right term of denition (.4) will become clear in the sequel. Note however that because of the singularity of, this term is not dened if is not regular (W ;1 ). Proof of theorem.1: We introduce for all n IN the truncated sequence n = ^ n: (:9) Note that because of assumption (.), there exists for all > 1 a strictly positive C :3 () such that for all n IN, (j1 cos j = + j sin j ) n (jj) d C :3 (): (:10) It is also clear that (j1 cos j = + j sin j ) j(jj) n (jj)j d! 0: (:11) n!+1 Then, we consider the (unique) nonnegative solution f n (t; v) of the classical Kac equation @f n @t (t; v) = K n (f n )(t; v) (:1) with initial datum f 0 (for the existence and uniqueness of such a solution, Cf. theorem A.1 of Appendix A). This solution is known to satisfy the conservation of mass and energy, and the entropy inequality (Cf. theorem A.1 and A. of Appendix A): f n (t; v) dv = f 0 (v) dv; (:13) f n (t; v) jvj dv = f n (t; v) log f n (t; v) dv f 0 (v) jvj dv; (:14) f 0 (v) log f 0 (v) dv: (:15) 6
It is now classical (Cf. [De 3] for example) that eq. (.13) { (.15) ensure the existence of a constant C :4 such that f n (t; v) (1 + jvj + j log f n (t; v)j)dv C :4 : (:16) Because of Dunford{Pettis theorem (Cf. [B]) and of estimate (.16), one can extract from (f n ) nin a subsequence still denoted by (f n ) nin and converging to a function f in L 1 ([0; +1[ t ; L 1 (IR v )) weak *. Moreover, for all q L 1 ([0; +1[) and all L 1 loc ([0; +1[ tir v ) such that we have +1 q(t) 0 lim sup jvj!+1 tir f n (t; v) (t; v) dvdt! n!+1 j (t; v)j jvj = 0; (:17) +1 0 q(t) f(t; v) (t; v) dvdt: (:18) Denoting for all W ;1 (IR v ) Kn (v; v ) = f(v 00 ) (v)g n (jj) d; (:19) it is clear that (using the change of variables (v; v ; )! (v 00 ; v 00; )), @ f n (t; v) (v) dv = K @t n(v; v ) f n (t; v) f n (t; v ) dv dv: (:0) We shall now prove that when W ;1 (IR v ), it is possible to pass to the limit in eq. (.0) and to obtain eq. (.3). We begin by the Lemma 1: There exists a constant C :5 > 0 (depending on ) and a sequence C :5 (n) converging to 0 such that the following estimates hold: 1. for all W ;1 (IR v ), jk n(v; v )j C :5 (1 + jvj +5 4 + jv j +5 4 ) jjjjw ;1 (IRv); (:1). for all W ;1 (IR v ), jk (v; v ) K n(v; v )j C :5 (n) (1 + jvj +5 4 + jv j +5 4 ) jjjjw ;1 (IRv): (:) 7
Proof of lemma 1: Note that u=0 (v 00 ) (v) = (v cos v sin ) (v) = (v(cos 1) v sin ) 0 (v) + (v(cos 1) v sin ) 1 (1 u) v 00 + u (v(cos 1) v sin ) du: (:3) Therefore, for all ]0; 1[, j(v 00 ) (v) + v sin 0 (v)j j(v 00 ) (v) + v sin 0 (v)j 1 v (cos 1) 0 (v) + (v(cos 1) v sin ) 1 u=0 (1 u) v 00 + u (v(cos 1) v sin ) du 8 (1 + jv j 1 ) (3 jjjj W 1;1 (IRv)) 1 (j cos 1j + j sin j ) (1 + jvj + jv j ) jjjj W ;1 (IR v) C :6 jjjj W ;1 (IRv)(j cos 1j + j sin j ) (1 + jvj 1+ + jv j 1+ ) (:4) for some strictly positive constant C :6. But! sin is odd and therefore jkn(v; v )j = j f(v 00 ) (v)g n (jj) dj = j f(v 00 ) (v) + v sin 0 (v)g n (jj) dj C :6 jjjj W ;1 (IRv) We now use eq. (.10) with = 1+, and obtain 4 (j cos 1j + j sin j ) n (jj) d (1 + jvj 1+ + jv j 1+ ): (:5) jkn(v; v )j C :3 ( 1 + ) C :6 (1 + jvj +5 4 + jv j +5 4 ) jjjjw ;1 (IRv); (:6) which clearly implies estimate (.1). In order to get estimate (.), we use exactly the same proof, except that eq. (.10) is replaced by eq. (.11). 8
Lemma : There exists a constant C :7 > 0 (depending on ) such that when W ;1 (IR v ) satises jjjjjj = sup one has the following estimate: j 0 (v)j < +1; (:7) 1 + jvj jk n(v; v )j C :7 (jj 00 jj L 1 (IRv) + jjjjjj)(1 + jvj + jv j ): (:8) Proof of lemma : According to eq. (.3), j(v 00 ) (v) + v sin 0 (v)j = v (cos 1) 0 (v) + (v(cos 1) v sin ) 1 u=0 (1 u) 00 (v + u (v(cos 1) v sin )) du j cos 1j jvj j 0 (v)j + 4 (j cos 1j + j sin j ) (jvj + jv j ) jj 00 jj L 1 (IRv) C :8 (j cos 1j + j sin j ) (1 + jvj + jv j ) (jj 00 jj L 1 (IRv) + jjjjjj) (:9) for some constant C :8 > 0. Using now the oddity of! sin as in lemma 1, and estimate (.10) for =, we get estimate (.8). We now come back to the proof of theorem.1. Suppose that W ;1 (IR v ), q L 1 ([0; +1[), and v IR. because of lemma 1, +1 Kn(v; v ) f n (t; v ) dv q(t) dt t=0 +1 K (v; v ) f(t; v ) dv q(t) dt +j +1 t=0 +1 t=0 t=0 jkn(v; v ) K (v; v )j f n (t; v ) dv q(t) dt C :5 (n) jjqjj L 1 ([0;+1[ t) jjjj W ;1 (IRv) K (v; v ) ff n (t; v ) f(t; v )g dv q(t) dtj 9 Then, (1 + jvj +5 4 + jv j +5 4 )fn (t; v )dv
+1 + j K (v; v ) ff n (t; v ) f(t; v )g dv q(t) dtj: (:30) t=0 But the rst term of eq. (.30) tends to 0 because of estimate (.16). Moreover, because of lemma 1, we have for all v IR, jk (v; v lim )j = 0: (:31) jvj!+1 jv j Therefore, estimate (.18) ensures that the second term of eq. (.30) tends to 0. Finally, we obtain for all W ;1 (IR v ) and v IR, the convergence in L 1 ([0; +1[ t ) weak * of L n(t; v) = Kn(v; v ) f n (t; v ) dv (:3) towards L (t; v) = K (v; v ) f(t; v ) dv : (:33) We now observe that for all W ;1 (IR v ) and v IR, the sequence is bounded in L 1 (IR v ). More precisely, Moreover, 1 n(v; v ) = @ Kn (v; v ) (:34) @v j @ Kn (v; v )j = j @v sin 00 (v 00 ) n (jj) dj C :3 () jj 00 jj L 1 (IRv): (:35) j @K n (v; v )j = j sin 0 (v 00 ) n (jj) dj @v = j sin f 0 (v 00 ) 0 (v)g n (jj) dj j sin j jvj j cos 1j + jv j j sin j v 00 + u(v(cos 1) v sin ) du n (jj) d u=0 C :3 () jj 00 jj L 1 (1 + jvj + jv j) (:36) 10
because of estimate (1.10). Therefore, using lemma, for all W ;1 (IR v ) and v IR, j @L n @t (t; v)j = j @ Kn @t (v; v ) f n (t; v ) dv j = j wir wir C :7 K K n (v;) n (w; v ) f n (t; w) f n (t; v ) dv dwj jj @ Kn @v (v; )jj L 1 (IRv ) + jjjkn(v; )jjj (1 + jwj + jv j ) f n (t; w) f n (t; v ) dv dw C :4 C :7 fc :3 ()jj 00 jj L 1 (IRv) + C :3 () C :5 (1 + jvj) jj 00 jj L 1 (IRv)g: (:37) It is also clear that jl n (t; v)j j C :5 (1 + jvj +5 4 + jv j +5 4 ) f n (t; v ) dv j jjjj W ;1 (IRv) C :4 C :5 (1 + jvj +5 4 ) jjjjw ;1 (IRv): (:38) Therefore, for all W ;1 (IR v ) and v IR, the sequence L n(; v) is bounded in W 1;1 ([0; +1[ t ). Using now the weak convergence (.33) and Rellich theorem (Cf. [B]), it is clear that for all W ;1 (IR v ), and a.e. (t; v) [0; +1[ t IR v, the sequence L n tends to L. Therefore, for all q L 1 ([0; +1[) and all T > 0 such that Supp q [0; T ], = +1 t=0 +1 t=0 f sup t[0;t ] + +1 t=0 f f Kn (v; v ) f n (t; v) f n (t; v ) dvdv K (v; v ) f(t; v) f(t; v ) dvdv g q(t)dt L n(t; v) f n (t; v) dv jl n(t; v) L (t; v)j f n (t; v) dv L (t; v) f(t; v) dvgq(t) dt jjqjj L 1 ([0;+1[ t) L (t; v) (f n (t; v) f(t; v)) dvg q(t) dt : (:39) 11
But according to estimate (.38), lim jvj!+1 sup t[0;+1[ sup jl n (t; v )j = 0: (:40) nin jv j Therefore, estimate (.18) ensures that the second term of (.39) tends to 0. We nally use Egorov's theorem, estimate (.38), the equiintegrability of the sequence f n (obtained by estimate (.16)) and the convergence a.e. of L n to L, in order to obtain the convergence of the rst term of (.39) to 0. As announced before, we can now pass to the limit in eq. (.0) and obtain the rst part of theorem.1. In order to prove the second part of theorem.1, we observe that if assumption (.6) holds, then theorem A. (Cf. Appendix A) ensures the existence of C A:3 > 0 such that f n (t; v) (1 + jvj p ) dv C A:3 (:41) (note that C A:3 does not depend on n). But estimates (.18), (.0) and (.1) imply for a.e. t 0 the convergence of R f n(t; v)(v)dv to R f(t; v)(v)dv when C c (IR v ). Therefore, for all R > 0, t > 0, f(t; v) (1 + jvj p ) dv C A:3 : (:4) jvjr Then, estimate (.7) holds because of Fatou's lemma. Finally, we prove the conservation of mass (.5). We observe that for some function R C (IR v ) such that Supp ( R ) [ R 1; R + 1], j f(t; v) dv f 0 (v) dvj 1 ff n (t; v) + f(t; v)g jvj dv R jvjr + j R (v) ff n (t; v) f(t; v)g dvj (:43) for any R > 0. But according to the properties used in the proof of estimate (.4), estimate (.43) ensures that the conservation of mass (.5) holds. In the same way, we can see that under assumption (.6) with p, the conservation of energy (.8) holds. 1
3 Regularization properties when all polynomial moments are initially bounded This section is devoted to the proof of the following theorem: Theorem 3.1: Let f 0 0 be an initial datum such that for all p IN, there exists C 3:1 (p) > 0 satisfying f 0 (v) (1 + jvj p + j log f 0 (v)j) dv C 3:1 (p); (3:1) and let 0 be a cross section satisfying estimate (.). Then, if f(t; v) is a nonnegative solution of eq. (1.18), (1.19), (.) in the sense of eq. (.3) with initial datum f 0, we have for all t > 0 and all q IN: f(t; v) L 1 ([ t; +1[ t ; C q (IR v )); (3:) or in abridged form, f(t; v) L 1 (]0; +1[ t ; C 1 (IR v )): (3:3) Proof of theorem 3.1: According to theorem.1, we know that for all p IN, there exists C 3: (p) > 0 satisfying 8t [0; +1[; f(t; v) (1 + jvj p ) dv C 3: (p): (3:4) Therefore, the Fourier transform ^f(t; ) = of f is such that for all p IN, e iv f(t; v) dv (3:5) @p ^f j @ p (t; )j C 3:(p): (3:6) But v! e iv lies in W ;1 (IR v ), and therefore it is possible to use eq. (.3). Then, a simple calculation leads to the following equation for the Fourier transform of f: @ ^f @t (t; ) = f ^f(t; cos ) ^f (t; sin ) ^f(t; 0) ^f(t; )g (jj) d: (3:7) 13
Note that this equation is used in [G], and that it also appeared in [De 1], though for the Laplace transform of f. We rewrite it under the form @ ^f @t (t; ) = 1 + We now use the notations a(t; ) = 1 and b(t; ) = Therefore, and But f ^f(t; sin ) + ^f (t; sin ) ^f(t; 0)g (jj) d ^f(t; ) f ^f(t; cos ) ^f (t; )g ^f(t; sin ) (jj) d: (3:8) f ^f(t; sin ) + ^f(t; sin ) ^f(t; 0)g (jj) d; (3:9) f ^f(t; cos ) ^f(t; )g ^f(t; sin ) (jj) d: (3:10) @ ^f @t (t; ) = a(t; ) ^f(t; ) + b(t; ); (3:11) ^f(t; ) = ^f(0; ) e R t 0 a(;)d + t 0 b(s; ) e R t s a(;)d ds: (3:1) ^f(t; sin ) + ^f (t; sin ) ^f(t; 0) 0; (3:13) because f 0. Therefore, a(t; ) is real and a(t; ) 1 f ^f(t; 0) ^f(t; sin ) ^f(t; sin )g (jj) d: (3:14) Then, we make the change of variables We get u = jj sin : (3:15) a(t; ) 1 jj ujj f ^f(t; 0) ^f(t; u) ^f(t; u)g ( arcsin j u j) 14 du jj q 1 j u j : (3:16)
But for any x [0; ], and therefore (x) 0 jxj ; (3:17) a(t; ) 0 4 jj ujj f ^f(t; 0) ^f(t; u) ^f (t; u)g j u j du jj And since jj 0 4 jj 1 f ^f(t; 0) ^f(t; u) ^f(t; u)g juj du: (3:18) ujj ^f(t; 0) ^f(t; u) ^f(t; u) = juj 1 r1 @ ^f (1 jrj) (t; r u) dr @ we get = juj 1 r1 (1 jrj) Re( @ ^f (t; ru)) dr; (3:19) @ But a(t; ) jj 1 0 4 jj 1 juj (1 jrj) Re( ujj r1 and estimate (.8) ensures that @ ^f (t; r u)) dr du: @ (3:0) @ ^f @ (t; 0) = f(t; v) jvj dv; (3:1) @ ^f @ (t; 0) = f 0 (v) jvj dv: (3:) Moreover, if we denote E = f 0 (v) jvj dv; (3:3) we get (thanks to estimate (3.6)), that for any such that jj E C 3: (3) ; (3:4) 15
the estimate @ ^f Re( @ (t; )) E holds. But estimates (3.0) and (3.4) ensure that (3:5) a(t; ) 0 16 E jj 1 inf(jj; E C 3: (3) ) juj du E u inf(jj; C 3: (3) ) 0 16(3 ) E jj 1 3 E 3 inf (jj; C 3: (3) ) : (3:6) Therefore, there exists C 3:3 > 0, C 3:4 > 0, such that when jj C 3:3, t 0, a(t; ) C 3:4 jj 1 : (3:7) We will now use eq. (3.1) and estimate (3.7) to prove theorem 3.1 by induction. Lemma 3: We make the assumptions of theorem 3.1. We suppose moreover that there exists 0 such that for all t 1 > 0, 1 > 0, we can nd C 3:5 ( 1 ; t 1 ) > 0 satisfying 8 IR; sup j ^f(s; )j C 3:5( 1 ; t 1 ) st 1 1 + jj : (3:8) 1 Then, for all t 1 > 0, 1 > 0, we can nd C 3:6 ( 1 ; t 1 ) > 0 satisfying 8 IR; sup j ^f(s; )j C 3:6( 1 ; t 1 ) 1 st + 1 1 + jj 1 : (3:9) Proof of lemma 3: We x t 1 > 0; 1 > 0. According to eq. (3.1), for any t t 1, ^f(t; ) = ^f(0; ) e R t a(;)d 0 + t 1 0 b(s; ) e R t t a(;)d s ds + b(s; ) e R t a(;)d s ds: (3:30) t 1 Therefore, estimate (3.7) ensures that for any t t 1, jj C 3:3, j ^f(t; )j j ^f(0; )j e C 3:4t jj 1 16
t + sup 1 s[0; t 1 jb(s; )j t 1 t e C 3:4 jj 1 + sup jb(s; )j e (t s) C 3:4 jj 1 ds: ] s t t 1 1 (3:31) But for all s [0; +1[; IR, jb(s; )j = f ^f(s; cos ) ^f(s; )g ^f(s; sin ) (jj) d = 1 @ (cos 1) ^f u=0 @ (s; + u (cos 1))du ^f(s; sin ) (jj) d C :3 () C 3: (0) C 3: (1) jj: (3:3) Therefore, estimate (3.31) implies that for any t t 1, jj C 3:3, j ^f(t; )j C 3: (0) e C 3:4t jj 1 + C :3 () C 3: (0) C 3: (1) t 1 jj t 1 e C 3:4 jj 1 + sup s t 1 According to assumption (3.8), we have for all > 0, 8 IR; sup s t 1 jb(s; )j : (3:33) C 3:4 jj 1 j ^f(s; )j C3:5(; t1 : (3:34) 1 + jj Therefore, using corollary B.3 of Appendix B and assumption (3.1), there exists for all > 0 a strictly positive constant C 3:7 (; t 1 ) such that ) 8 IR; sup j @ ^f s t 1 @ (s; )j C 3:7(; t 1 ) : (3:35) 1 + jj We now compute (for all jj C 3:3, > 0), sup jb(s; )j = sup f ^f(s; cos ) ^f(s; )g ^f (s; sin ) (jj) d s t 1 s t 1 4 sup j ^f(s; cos ) ^f(s; 1 1 ( )j +) jj 1 + j cos 1j 1 + s t 1 4 1 @ ^f 1 (s; + u (cos 1)) du + @ j ^f(s; sin )j (jj) d u=0 17
+ sup + sup s t 1 s t 1 j j 4 j ^f(s; cos ) ^f(s; )j j ^f(s; sin )j (jj) d 3 4 jj j ^f(s; cos ) ^f(s; )j j ^f(s; sin )j (jj) d: (3:36) We now use estimates (3.34), (3.35) and resume the computation (for all jj C 3:3 ; > 0). sup jb(s; )j C :3 ( 1 + ) jj 1 + C 3: (0) s t 1 ( C 3:5(; t 1 ) 1 + j p j )1 ( 1 +) ( C 3:7(; t 1 ) 1 + j p ) 1 + j + 1 ( C 3:5 (; t 1 4 ) C 3: (0) p 1 + j + 1( 3 C 3:5 (; t 1 j 4 ) C3: (0) 1 + j ) ) p j C :3 () jj 1 + C 3: (0) (C3:5 (; t 1 1 ))1 ( +) (C 3:7 (; t 1 )) 1 + + 4 1 ( 4 ) C 3: (0) jj + C 3:5 (; t 1 ) C 3:8 (; t 1 ) jj 1 + (3:37) for some strictly positive constant C 3:8 (; t 1 ). We now use estimate (3.37) to precise estimate (3.33). We get for all t t 1, jj C 3:3, > 0, j ^f(t; )j C 3: (0) e C 3:4t 1 jj 1 + C :3 () C 3: (0) C 3: (1) t 1 jj t 1 e C 3:4 jj 1 + C 3:8(; t 1 ) C 3:4 jj 1 + : (3:38) Taking = 1, we get some strictly positive constant C 3:9( 1 ; t 1 ) such that when t t 1, jj is large enough, j ^f(t; )j C 3:9 ( 1 ; t 1 ) jj 1 + 1 : (3:39) Finally, using estimate (3.6) for p = 0, we obtain estimate (3.9). We now come back to the proof of theorem 3.1. We already know (because of estimate (3.6) when p = 0) that assumption (3.8) holds when 18
= 0. Then, lemma 3 clearly implies by induction that for any t > 0; q 0, there exists a strictly positive constant C 3:10 ( t; q) such that 8 IR; sup j ^f(t; )j C 3:10( t; q) tt 1 + jj q : (3:40) Using now the Sobolev inequalities (or more simply the fact that H 1 (IR) = C 1 (IR)), we get theorem 3.1. 4 Regularization properties when some polynomial moments are initially bounded We extend in this section the results of section 3 when assumption (3.1) does not hold any more. Theorem 4.1: Let f 0 0 be an initial datum such that 9r IN; r ; C 4:1 > 0; f 0 (v) (1 + jvj r + j log f 0 (v)j) dv C 4:1 ; (4:1) and let 0 satisfy (.). Then, if f(t; v) is a nonnegative solution of eq. (1.18), (1.19), (.) in the sense of eq. (.3) with initial datum f 0, we have for all t > 0 and all > 0: f(t; v) L 1 ([ t; +1[ t ; H r 1 (IR v )): (4:) Corollary 4.: In particular, under the assumptions of theorem 4.1, we have for all t > 0 and all > 0: f(t; v) L 1 ([ t; +1[ t ; C r ;1 (IR v )): (4:3) Proof of theorem 4.1: Corollary 4. is a straightforward consequence of theorem 4.1 and of classical Sobolev inequalities (Cf. [B]). We now prove theorem 4.1. We use the same strategy as in theorem 3.1. Estimates (3.4) and (3.6) still hold, but only for p r. Moreover, eq. (3.9), (3.10), (3.1) also hold, and lead to estimate (3.7) as in theorem 3.1. 19
However, lemma 3 is changed in the following way: Lemma 4: We make the assumptions of theorem 4.1. We suppose moreover that there exists 0 such that for all t 1 > 0, 1 > 0, we can nd C 4: ( 1 ; t 1 ) satisfying 8 IR; sup j ^f(s; )j C 4:( 1 ; t 1 ) st 1 1 + jj : (4:4) 1 Then, for all t 1 > 0, 1 > 0, we can nd C 4:3 ( 1 ; t 1 ) satisfying 8 IR; sup st 1 j ^f(s; )j C 4:3 ( 1 ; t 1 ) 1 + jj f1 1 r 1 1 g+ 1 : (4:5) Proof of lemma 4: We x t 1 > 0; 1 > 0. It is clear that estimate (3.33) still holds. However, using theorem B. of Appendix B, we only get for all > 0 a strictly positive constant C 4:4 (; t 1 ) such that 8 IR; sup j @ ^f s t 1 @ (s; )j C 4:4 (; t 1 ) 1 + jj (1 : (4:6) 1 ) r Then, we note that estimate (3.36) still holds, but estimate (3.37) becomes (for all jj C 3:3 ; > 0), sup jb(s; )j C :3 ( 1 + ) jj 1 + C 3: (0) s t 1 ( C 4:(; t 1 ) p 1 + j j )1 ( 1 +) ( C 4:4 (; t 1 ) 1 p ) 1 + j j(1 1 ) r + + 1 ( C 4: (; t 1 4 ) C 3: (0) p 1 + j + 1 ( 3 C 4: (; t 1 j 4 ) C3: (0) 1 + j ) ) p j C 4:5 (; t 1 ) jj 1 1 1 +f 1+ r g+(+ r ) (4:7) for some strictly positive constant C 4:5 (; t 1 ). Then, estimate (3.38) becomes for all jj C 3:3 ; > 0, j ^f(t; )j C 3: (0) e C 3:4t 1 jj 1 + C :3 () C 3: (0) C 3: (1) t 1 jj t 1 e C 3:4 jj 1 0
+ C 4:5(; t 1 ) C 3:4 jj 1 f1 1 1 r g+(+ r ) : (4:8) Taking = 1 ( + r ) 1, we get some strictly positive constant C 4:6 (; t 1 ) such that when t t 1, jj is large enough, j ^f(t; )j C 4:6 (; t 1 ) jj 1 1 1 f1 r g+ 1 : (4:9) Then, lemma 4 is obtained exactly as lemma 3. We now come back to the proof of theorem 4.1. We already know that assumption (4.4) of lemma 4 holds when = 0. Moreover, using lemma 4 by induction, we can see that for all t 1 > 0; 1 > 0; n IN, there exists a strictly positive constant C 4:7 ( 1 ; t 1 ; n) such that 8 IR; sup j ^f(t; )j C 4:7( 1 ; t 1 ; n) tt 1 1 + jj n ; (4:10) 1 where ( n ) nin is the sequence dened by 0 = 0; (4:11) n+1 = n f1 1 1 r g + 1 : (4:1) But this sequence is strictly increasing and converges to r. Therefore for all > 0; t > 0, there exists C 4:8 (; t) such that 8 IR; sup j ^f(s; )j C 4:8(; t ) st 1 + jj r : (4:13) Finally, estimate (4.13) ensures that and theorem 4.1 is proved. ^f(t; ) L 1 ([ t; +1[ t ; H r 1 (IR v )); (4:14) Appendix A: Standard properties of the classical Kac equation We prove in this appendix some classical facts about the spatially homogeneous Kac equation, and present some others that can easily be deduced from the theory of the Boltzmann equation (Cf. [A 1]). 1
Theorem A.1: Let f 0 0 be an initial datum such that f 0 (v) (1 + jvj ) dv < +1: (A:1) Then, there exists a unique nonnegative solution f(t; v) of eq. (1.18), (1.19) in L 1 ([0; +1[ t ; L 1 (IR v ; (1 + jvj )dv)) with initial datum f 0 as soon as the cross section in (1.19) belongs to L 1 ([0; ]). This solution satises the conservation of mass and energy for all t 0: f(t; v) dv = f 0 (v) dv; (A:) f(t; v) jvj dv = f 0 (v) jvj dv: (A:3) Proof of theorem A.1: We introduce the sequence (f n (t; v)) nin, de- ned by f 0 (t; v) = f 0 (v); (A:4) t f n+1 (t; v) = f 0 (v) + ff n (s; v 00 ) f n (s; v 00) s=0 f n+1 (s; v) f n+1 (s; v )g (jj) ddv ds; (A:5) and present a proof of existence in the Cauchy{Lipschitz style. Note that it is easy to obtain (by induction) the conservation of mass and energy for f n : f n (t; v) dv = f 0 (v) dv; (A:6) f n (t; v) jvj dv = f 0 (v) jvj dv: (A:7) Therefore, it is possible to write explicitly f n+1 as a function of f n, and the sequence (A.4), (A.5) is well dened. It is also clear that f n 0. Then, we dene for all n IN, u n (t) = jf n (t; v) f n 1 (t; v)j (1 + jvj ) dv: (A:8) We get u n+1 (t) t s=0 f n (s; v ) jf n (s; v) f n 1 (s; v)j
+ t + s=0 t s=0 (1 + jvj cos + jv j sin ) (jj) d dvdv ds f n 1 (s; v) jf n (s; v ) f n 1 (s; v )j (1 + jvj cos + jv j sin ) (jj) d dvdv ds f n+1 (s; v ) jf n+1 (s; v) f n (s; v)j (1 + jvj ) t + s=0 (1 + jvj ) C A:1 t s=0 (jj) d dvdv ds f n (s; v) jf n+1 (s; v ) f n (s; v )j (jj) d dvdv ds fu n (s) + u n+1 (s)g ds; (A:9) for some strictly positive constant C A:1. Moreover, we can prove in the same way that for all t 0, u 1 (t) C A: t; (A:10) where C A: > 0. But estimate (A.9) ensures that (when t [0; T ], n 1), t u n+1 (t) (C A:1 + C A:1 T eca:1t ) u n (s) ds: s=0 (A:11) Therefore, for all T > 0, s [0; T ]; n 1, u n (s) (C A:1 + C A:1 T ec A:1T ) n 1 s n n! C A:: (A:1) This estimate ensures that the sequence (f n ) nin satises the Cauchy property in L 1 loc ([0; +1[ t; L 1 (IR v ; (1 + jvj )dv)). Its limit f clearly satises eq. (1.18), (1.19). Moreover, f 0 and the conservation of mass and energy (A.3), (A.4) holds. The last property also ensures that f L 1 ([0; +1[ t ; L 1 (IR v ; (1 + jvj )dv)). The uniqueness of such a solution is then directly obtained by a Cauchy{ Lipschitz type argument. 3
We now consider the polynomial moments of the solution of the spatially homogeneous Kac equation. Theorem A.: Let f 0 0 be an initial datum such that f 0 (v) (1 + jvj + j log f 0 (v)j)dv < +1: (A:13) Then, for all t s 0, the unique nonnegative solution f(t; v) of eq. (1.18), (1.19) with initial datum f 0 (when the cross section in (1.19) belongs to L 1 ([0; ])) satises f(t; v) log f(t; v) dv f(s; v) log f(s; v) dv f 0 (v) log f 0 (v) dv < +1: (A:14) Moreover, if 9r IN; f 0 (v) (1 + jvj r ) dv < +1; (A:15) there exists C A:3 > 0 (independent of ) such that for all t 0: f(t; v) (1 + jvj r ) dv C A:3 : (A:16) Proof of theorem A.: For estimate (A.14), we refer to [A 1], where it is proved for the Boltzmann equation (for example for Maxwellian molecules with an angular cut{o). We now prove estimate (A.16) in the case when r =. We can write = @ @t f(t; v)jvj 4 dv = (jv cos v sin j 4 jvj 4 ) f(t; v) f(t; v ) (jj) ddv dv f cos 4 + sin 4 1g (jj) d f(t; v) jvj 4 dv + 6 cos sin (jj) d ( f(t; v) jvj dv) f(t; v) dv 4
= cos sin (jj) d f(t; v) jvj 4 dv f(t; v) dv + 6( f(t; v) jvj dv) : (A:17) Therefore, a simple application of the maximum principle yields f(t; v)jvj 4 dv sup f 0 (v) jvj 4 dv; 3 ( R f(t; v) jvj dv) R : f(t; v) dv (A:18) Finally, when r >, the same kind of computation can be done. Note that a rigorous proof is given in the case of the Boltzmann equation with Maxwellian molecules in [Tr]. Appendix B: Interpolation between derivatives We give here for the sake of completeness the proof of some classical results used in sections, 3 and 4. Theorem B.1: Let f lie in C (IR) and satisfy 1. There exists C B:1 > 0; > 0, such that. There exists C B: > 0, such that Then, 8x IR; jf(x)j C B:1 1 + jxj : (B:1) 8x IR; jf 00 (x)j C B: : (B:) 8x IR jf 0 (x)j s 8 CB:1 C B: 1 + jxj : (B:3) Proof of theorem B.1: Suppose that jf 0 (x 0 )j > s 8 CB:1 C B: 1 + jx 0 j : (B:4) 5
Then, because of estimate (B.), for all t [0; 1], s s x f 0 C B:1 0 + t sgn (x 0 ) C B: (1 + jx 0 j f 0 (x 0 ) ) CB:1 C B: 1 + jx 0 j : (B:5) But estimate (B.5) ensures that s x f 0 C B:1 0 + t sgn (x 0 ) C B: (1 + jx 0 j ) s CB:1 C B: 1 + jx 0 j : (B:6) Therefore, But s I = x f C B:1 0 + sgn (x 0 ) C B: (1 + jx 0 j f(x 0 ) ) C B:1 1 + jx 0 j : (B:7) s I x f C B:1 0 + sgn (x 0 ) C B: (1 + jx 0 j + jf(x 0 )j ) < C B:1 1 + jx 0 j : (B:8) Thus, we get a contradiction and conclude that theorem B.1 holds. Theorem B.: Let p IN, p, and f lie in C p (IR). If f satises the following properties: 1. There exists C B:3 > 0; > 0, such that. There exists C B:4 > 0, such that 8x IR; jf(x)j C B:3 1 + jxj : (B:9) 8q [1; p]; 8x IR; jf (q) (x)j C B:4 : (B:10) Then, for all > 0, there exists C B:5 () > 0, such that 8x IR; jf 0 (x)j 6 C B:5 () 1 + jxj (1 : (B:11) 1 )+ p
Proof of theorem B.: We use theorem B.1 and give a proof by induction. We know that if there exists C B:6 > 0 and a nite sequence (u q ) q[0;p] such that for all q [0; p], 8x IR; jf (q) (x)j C B:6 ; (B:1) uq 1 + jxj then there exists C B:7 > 0 such that for all q [0; p], where 8x IR; jf (q) (x)j C B:7 ; (B:13) vq 1 + jxj v 0 = ; 8i [1; p 1]; v i = 1 (u i 1 + u i+1 ); v p = 0: (B:14) Therefore, we dene by induction the sequence and r 0 (0) = ; 8i [1; p]; r i (0) = 0; (B:15) r 0 (n+1) = ; 8i [1; p 1]; r i (n+1) = 1 (r i 1(n)+r i+1 (n)); r p (n+1) = 0: (B:16) It is clear that for all n IN, there exists C B:8 (n) > 0, such that for all q [0; p], 8x IR; jf (q) (x)j C B:8(n) : (B:17) rq(n) 1 + jxj But for all i [0; p], the sequence (r i (n)) nin tends towards r i, where r 0 = ; 8i [1; p 1]; r i = 1 (r i 1 + r i+1 ); r p = 0: (B:18) Therefore, which yields theorem B.. r 1 = (1 1 p ) ; (B:19) Finally, using theorem B. by induction, we get the Corollary B.3: Let f lie in C 1 (IR) and satisfy: 7
1. There exists C B:9 > 0; > 0, such that 8x IR; jf(x)j C B:9 1 + jxj : (B:0). For all q IN, there exists C B:10 (q) > 0, such that 8x IR; jf (q) (x)j C B:10 (q): (B:1) Then, for all > 0, there exists C B:11 () > 0, such that 8x IR jf 0 (x)j C B:5() : (B:) 1 + jxj Appendix C: The case of the radially symmetric D Boltzmann equation with Maxwellian molecules We consider now the radially symmetric solutions of the D spatially homogeneous Boltzmann equation with Maxwellian molecules (Note that one can prove the existence of such solutions exactly as in section, provided of course that the initial datum is radially symmetric). The corresponding Boltzmann kernel can be written Q(f)(v) = ff(v 0 )f(v) 0 f(v)f(v )g b(jj) sin ddv ; (C:1) with v 0 = v + v + R ( v v ); (C:) v 0 = v + v R ( v v ): (C:3) We suppose moreover that b satises sin b(jj) K jj (C:4) for some K > 0 when! 0 (this is the non cut{o case) and that b is regular outside 0. Using the fact that f is radially symmetric, one can recast the kernel Q under the form: Q(f)(v) = f( v + R ( v ) + v + R ( v )) 8
f( v + R ( v ) + v + R ( v )) f(v)f(v ) b(jj) sin ddv = f(r (v) cos ( ) + R (v ) sin ( )) f(r (v) sin ( ) + R (v ) cos ( )) f(v)f(v ) b(jj) sin ddv = f(v cos ( ) v sin ( ))f(v sin ( ) + v cos ( )) f(v)f(v ) b(jj) sin ddv : (C:5) Thus, we can see that the equation is very similar to the Kac equation. The main dierence is simply that now v is in IR instead of IR. It is then possible to prove all the theorems of the previous sections with exactly the same proof. Note however that for the 3D radially symmetric solutions of the spatially homogeneous Boltzmann equation with Maxwellian molecules, the analogy with the Kac equation is not so clear. This case shall be discussed in a future work. Aknowledgment: I would like to thank Professor Golse for his valuable remarks during the preparation of this work. 9
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