On a Model Boltzmann Equation without Angular Cuto1 by L. DESVILLETTES(1) & F. GOLSE(2) April 1, 1999 (1) CMLA, Ecole Normale Superieure de Cachan, 94235 Cachan Cedex, FRANCE, [email protected] (2) Universite Paris 7 and Ecole Normale Superieure, DMI, 45, rue d'Ulm, 75230 Paris Cedex 05, FRANCE, [email protected]

Abstract

A model Boltzmann equation (see formulas (1.1.6) { (1.1.9) below) without Grad's angular cuto assumption is considered. One proves 1. the instantaneous smoothing in both position and velocity variables by the evolution semigroup associated to the Cauchy problem for this model; 2. the derivation of the analogue of the Landau-Fokker-Planck equation in the limit when grazing collisions prevail.

1 Introduction and Main Result. 1.1

Introduction

Let us rst recall the Boltzmann equation of rare ed gases (see [Ce] for example), @tf + v
rxf = Q~ B (f ) : (1:1:1) In this equation, the unknown f  f (t; x; v) is the density of particles which at time t  0 and position x 2 R have velocity v 2 R , and Q~ B is a 3

1

AMS classi cation: 45K05, 76P05

1

3

quadratic operator acting only on the v-dependence of f which takes into account only the binary collisions in the gas. It reads Q~B (f )(v) = =

Z



Z

v 2R3 !2S 2

f (v + (!
(v v)) !) f (v (!
(v v)) !) f (v) f (v)

B (jv vj; j jvv

v v j




!j) d!dv ;

(1:1:2) where B is a cross section depending on the type of interactions between the particles of the gas. In the case where interparticle forces are proportional to r s (where r is the distance between the two particles under consideration) one has s

B (x; y) = jxj s s(y); and

5 1

s+1

(1:1:3)

(1:1:4) s(y) y! jyj s ; so that Q~ B is de ned only as a singular integral operator. The singularity of s, together with the dissipative character of Q~ B make it plausible that this operator behaves as some nonlinear diusion operator, i.e. a nonlinear analogue (of some fractional power) of the Laplacian
acting on the velocity space. On the contrary, when the singularity in s is removed, that is, when Grad's \angular cuto assumption" is made (cf. [Gr]), (in other words, when B 2 Lloc(R  S )), the collision integral behaves roughly as a bounded operator on functions of the velocity variable. This fundamental dierence can be seen on the evolution semigroup associated to the Cauchy problem for the space homogeneous Boltzmann equation, that is @tf = Q~ B (f ): (1:1:5) When s is of the form (1.1.4), one expects that the associated semigroup should be a smoothing operator for any positive value of the time variable. This has been established on various models or particular cases: see [De 1], [De 3], [De 4], [Pr], for the 2D Boltzmann equation, radially symmetric or not. 0

1

3

2

2

1

On the contrary, in the cuto case, it is known (see [L 1], [We], [B, De]) that this semigroup does not regularize the initial data for any positive value of the time variable. It is therefore expected that (when s has the form (1.1.4)), the space inhomogeneous Boltzmann operator Q~ B v
rx should be the nonlinear analogue of a hypoelliptic operator, by analogy with the linear Fokker-Planck operator
v v
rx (see for example [H]). An indication of this could be that the evolution semigroup of the Boltzmann equation regularizes the initial data in both the position and velocity variable. This is a widely open question at the moment, even for the most elementary traditional simpli ed models of the Boltzmann equation. This is due to the side-eects of many additional diculties (among which the absence of maximum principle and the eect of large velocities seem to be the most important). This paper is aimed at introducing the simplest possible nonlinear model of (inhomogeneous) Boltzmann type equation (with a singularity as in (1.1.4)), and at trying to prove in this context the expected smoothing properties. This model is somehow a reduction of Kac's collision integral [K] to velocities of norm 1; it also is reminiscent of a model proposed in [C, S]. It reads @tf (t; x; v) + cos(2v) @xf (t; x; v) = Qb(f )(t; x; v) ; (1:1:6) where the unknown is the number density f  f (t; x; v). Here, t  0 is the time variable, the position variable is x 2 T , and v 2 T parametrizes the velocity cos(2v) of the particles. The collision operator is given (at the formal level) by 1

Qb()(v) =

=

Z 1 2 Z

=

1 2

1

0 ) (v ) (v 0)] b() ddv 0 ; [  ( v +  )  ( v T 1

(1:1:7)

where b is an even function on [ 1=2; 1=2] with positive values. As usual, the notation Qb(f (t; x; v)) designates the function of the variable v de ned by v 7! Qb(f (t; x;
))(v), the variables t and x being parameters in the collision integral. In order to mimick the behaviour of s in (1.1.4), we postulate that (for all  2 [ 1=2; 1=2]), C jj  b()  C jj ; (1:1:8) where C ; C > 0 and 1 < < 3. Hence the collision integral (1.1.7) is a nonlinear singular integral. 0

0

1

1

3

Finally, we introduce the initial data

f (0; x; v) = f (x; v) ; (x; v) 2 T  T : 1

0

(1:1:9)

1

In order to establish the smoothing eect of the evolution of (1.1.6) in all variables t, x and v, one must depart from the method used on the space homogeneous Boltzmann equation. Indeed, this method is based on applying the Fourier transform in the velocity variable: see [De1, 3, 4], [Pr] and [De, G]. While the collision integral is well behaved under this transformation, the advection operator is not. The present paper proposes a dierent strategy, where steps 2 and 3 are reminiscent of [L 2]: 1. use the entropy production to control (fractional) derivatives in the velocity variable of the number density; 2. write (1.1.6) in the form

@tf (t; x; v) + cos(2v) @xf (t; x; v) = h(t; x; v) ;

(1:1:10)

where h is a singular integral with respect to the velocity variable, and apply the Velocity Averaging Method to obtain some smoothness on quantities of the form Z

f (t; x; v0) (v; v0) dv; T where  2 C 1(T  T ); nally, keep track of the dependence on  of the norm of this average (in some Besov or Sobolev space): 3. replacing  by suitable approximations of the identity in step 2, try to get some regularity on f in all variables t, x and v, by using the results in step 1. 1

1

1

There are obvious shortcomings in this method: it is not completely clear how to iterate in order to obtain that f 2 C 1(R  T  T ) (whether this is true is not yet known, albeit very likely). As for the analogy with the Boltzmann equation, note that two of the diculties quoted above are not treated: the set of velocities is bounded in our model, and, more important, the natural functional spaces for the true Boltzmann equation are L or, at best, +

1

1

1

4

L log L spaces and not L1 as in our model). In either L or L log L spaces, the gain of smoothness by Velocity Averaging is marginal, and the meaning of the collision operator unclear. Thus, applying the method above to models more realistics than (1.1.6-7), and in particular to the Boltzmann equation, would certainly require new ideas and lead to tremendous technicalities. Notice however that step 1 was recently achieved in the case of the Boltzmann equation (without angular cuto) by P.-L. Lions: see [L3]. An optimal regularity estimate, also based on the entropy production term, but best suited to space homogeneous problems has also been obtained recently by Villani [V 2]. Even on the simpli ed model (1.1.6) considered here, the regularity of the solution obtained by our method is very likely not optimal. It could be that some bootstrap procedure leads to better regularity; yet this would certainly lead to rather technical developments. There are equally obvious advantages: the method seems robust in the sense that it rests on physically intrinsic arguments, like the entropy inequality. Even cavitation (i.e. the local vanishing of the density) does not wholly ruin the argument in step 1, be it in the case of our model (1.1.6-7) or in that of the Boltzmann equation without cuto (see [L3]). Notice however that cavitation can be dealt with very easily in the case of our model (1.1.67) | although better regularity can be obtained in the case where cavitation does no occur: see Proposition 4 below | while it gives rise to noticeable diculties on more physical models, as can be seen on the example of the Landau-Fokker-Planck equation (see [L2]). 1

1.2

Main results

We begin with a precise functional de nition of Qb.

Proposition 1 Let b satisfy (1.1.8). Then , the operator Qb de ned by (1.1.7) is a continuous (nonlinear) operator from C (T ) to C (T ) which extends as a continuous operator from L (T ) to D0 (T ). 2

1

1

1

0

1

1

As a corollary, it is now possible to de ne a solution f of (1.1.6) { (1.1.9) in the sense of distributions, as soon as 0  f 2 L1(T  T ) and f  0 2 L1(R  T  T ) \ C (R ; D0(T  T )). A slightly stronger notion of solutions will be needed in the sequel, that of entropic solutions, de ned below: +

1

1

0

+

1

1

5

0

1

1

De nition 1 Let b satisfy (1.1.8) and f  0 2 L1(T  T ). An entropic solution of (1.1.6) { (1.1.9) is a function f  0 2 L1 (R  T  T ) \ C (R ; D0(T  T )) satisfying (1.1.6) { (1.1.9) in the sense of distributions 1

0

+

1

1

+

1

1

1

as well as the following entropy relation: for all T > 0, ZZ

1 2

+

Z 1 2

TZ

T

1

0

f (t; x)

T T 1

ZZ

1

jf (T; x; v)j dxdv 2



jf (t; x; v + ) f (t; x; v)j b() ddv dxdt T T (1:2:1)  T T jf (x; v)j dxdv : 2

1

1

ZZ

1 2

1

1

2

0

The main result in this paper is the Theorem A Let b satisfy (1.1.8) and f  0 2 L1(T  T ). The Cauchy s  ( R  T  problem (1.1.6) { (1.1.9) admits an entropic solution f 2 Hloc T ) for all  > 0 with (1:2:2) s( ) = 2 ( + 1) (1 + 3) : If f  R a.e. for some R > 0, the value in the right hand side of (1.2.2) can be replaced by the better regularity index s( ) = 2 (

+ 11) : (1:2:3) For example, the typical case = 2 leads to respectively f 2 H = and f 2 H = : hence the smoothing eect is rather weak. The problem (1.1.6) { (1.1.9) also admits an analogue of the LandauFokker-Planck asymptotics (see [Li, Pi] as well as [De 2], [D, Lu], [V 1], [Ar, Bu]) in the limit when grazing collisions prevail | in the present case, when b is concentrated near  = 0. In the case of the true Boltzmann equation, this limit has been proved only in particular situations: the case of the linearized inhomogeneous equation is considered in [De 2], while the spatially homogeneous equation is considered in [Ar, Bu] and [V 1]. The good features of our model allow to state a very general convergence theorem. Observe the speci c scaling (1.2.4) below, aimed at zooming on the grazing collisions in the model collision integral (1.1.7): it is the analogue in the case of our model (1.1.6-7) of the scaling appearing in [De 2] and [V 1]. 1

0

1

( )

+

1

0

0

0

2

1 30

1 18

6

1

Theorem B Let 0  f 2 L1 (T  T ), and f " be an entropic solution of 1

0

1

eq. (1.1.6) { (1.1.9) with b replaced by b" de ned by

 " b"() = " 0b(  )if ifjj jj"; ; 

3

(1:2:4)

2

2

and where b satis es (1.1.8). Then there exists a subsequence (still denoted by f " ) converging in L1 (R+  T1  T1) weak-* to a solution f of

@tf (t; x; v) + cos(2v)@xf (t; x; v) = Cf (t; x)@v f (t; x; v) 2

(1:2:5)

in the sense of distributions, where

C=

=

Z 1 2

=

1 2

 b()d ;

f (t; x) =

2

Z

v2T1

f (t; x; v) dv:

(1:2:6)

The outline of the paper is as follows: section 2 reviews the precise de nition of Qb as a singular integral, thus leading in particular to proposition 1. Dierent useful expressions of Qb are also presented. Section 3 deals with the entropy relation, thereby leading to the existence of entropic solutions, and to step 1 in the method above. Section 4 establishes various results from the theory of Velocity Averaging which are crucial in proving step 2. The end of the proof of Theorem A (that is, step 3) belongs to section 5, while the Landau-Fokker-Planck approximation (Theorem B) is established in section 6.

2 The Collision Integral To the function b satisfying (1.1.8) is associated a distribution of order 2 on T , denoted by PV (b) (the principal value of b) and de ned for all g 2 C ([ 1=2; 1=2]) by 1

2

< PV (b); g >=

=

Z 1 2

=

1 2

 1 2



g() + g( ) 2 g(0) b() d :

(2:1)

With this de nition at hand, proposition 1 can be proved. In fact, we prove the slightly more precise 7

Lemma and De nition 1 For all  2 C (T ) and all v 2 T , consider 2

1

1

the functions Fv [] de ned by Fv [](; v0) = (v + ) (v0 ) (v) (v0) ; and Gv [] de ned by Gv []() = (v + ) (v) : 1. Then, < PV (b) 1; Fv [] >=  < PV (b); Gv [] > ; with Z  = T (v) dv : 2. The formula Qb()(v) =< PV (b) 1; Fv [] > ; 8v 2 T1 de nes a continuous operator Qb : C 2(T1) ! C 0(T1 ). 3. The continuous operator Qb : C 2(T1) ! C 0(T1 ) extends as tinuous operator Qb : L1 (T1 ) ! D0 (T1 ) de ned as follows:  2 L1(T1) and  2 C 2(T1), 1

Z

< Qb();  >=  T (v) < PV (b); Gv [] > dv : 1

(2:2) (2:3) (2:4) (2:5) (2:6) a confor all

(2:7)

Proof. Let (Xn )n 2 RN be an increasing sequence converging to +1, and let bn() = inf(b(); Xn ). Now, for all n 2 N, bn 2 L ([ 1=2; 1=2]) is even (because b itself is even) and for all  2 L (T ) one has, by symmetrizing 0

+

1

1

1

the integrand of (1.1.7) in the variable :

Qbn ()(v) =

=

Z 1 2 

Z 1 2

T

=

1

1 2

(v +  ) (v 0  ) + (v  ) (v 0 +  ) 

2 (v) (v0) bn() ddv0 : (2:8) Also, one can apply Fubini's theorem and integrate rst in the variable v0 in (2.8). This gives

Qbn ()(v) =  1 2

=

Z 1 2 

=

1 2



(v + ) + (v ) 2 (v) bn () d : 8

(2:9)

In other words, for all n 2 N, one has

Qbn ()(v) =< PV (bn) 1; Fv [] >=  < PV (bn); Gv [] > :

(2:10)

Now, for all g 2 C ([ 1=2; 1=2]), 2

g() + g( ) 2 g(0) = O (jj ) ; 0

so that



2



 7! g() + g( ) 2 g(0) b() 2 L (T ) : Taking  2 C (T ), one sees, by applying (2.11) to g() = (v + ) (v0 ) ; and to g() = (v + ); 1

2

1

(2:11)

1

that each side of (2.4) is well-de ned. Letting n ! +1 in (2.10) establishes, by dominated convergence, equality (2.4) as well as the relation

Qb()(v) = n!lim1 Qbn ()(v) =  < PV (b); Gv [] > : +

(2:12)

As for point 2, observe that the continuity is made obvious by formula (2.12). Now for point 3: if  2 C (T ), one obviously has 2

1

Z

T

1

=

Z

Z 1 2

(v) Qbn ()(v) dv =





 T = (v) (v ) + (v + ) 2 (v) bn() ddv : (2:13) Letting n ! +1 and applying the rst equality in (2.12) results in (2.7). // An immediate consequence of (2.7) is that the collision integral (1.2) conserves the total number of particles: 1

1 2

1 2

Corollary 1 For all  2 L (T ), one has 1

1

< Qb(); 1 >= 0 :

Proof. Apply (2.7) with  = 1. //

(2:14)

No other quantity (such as momentum, energy, etc..) is conserved in this model. This will become clear in section 3, where we prove that Qb() = 0 only when  is a constant. 9

Next we proceed to another way of de ning Qb, as the second derivative (in the velocity variable) of a nonsingular integral operator. The following Proposition can be viewed as yet another de nition of the collision integral (1.1.7); we shall not use it speci cally until section 6. Notice that it is also possibe to write Boltzmann's collision integral as a divergence (with respect to the v variable); this idea can be found in x41 of [Li, Pi] and is a possible starting point of the derivation of Landau's collision operator from Boltzmann's collision integral. The computations in [Li, Pi] have recently been put on more mathematical footing by Villani [V 2].

Proposition 2 For all r 2 R and z 2 T , consider the expression A(r; z) = (r jzj) (2:15) (where, for all z 2 T , jzj 2 [0; 1=2] designates the geodesic distance to 0). Let b satisfy (1.1.8) and consider the function Bb : R ! [0; +1]de ned by 1

+

1 2

+

1

Bb ( z ) =

=

Z 1 2

A(jj; z) b() d ; 8z 2 T :

(2:16)

1

=

1 2

Then, 1. for all z 2 T1 , one has

0  Bb(z)  D(1 + jzj )

(2:17)

2

for some D > 0 (depending only on ; C0; C1), 2. for all  2 L1(T1 ), one has (compare with [Li, Pi])

Qb() = @v Sb() ;

Sb () =    Bb :

where

2

(2:18)

Proof. One has A(jj; z)  jj 1jzjjj so that 1 2

=

Z 1 2

=

1 2

bn() A(jj; z) d 

=

Z 1 2

 1jzj b() d  D + jzD j : 2

0

(2:19)

This proves 1. Now, integrating by parts twice shows that, for all  2 [ 1=2; 1=2] and all  2 C (T ): 2

 1 2

1



(v )+(v+) 2 (v) =



Z 1 2

T

1

10

1v [

jj;v] (w)



1 v;v jj (w) @w (w) dw [

+

]

Z

= T A(jj; v w) @w (w) dw: (2:20) Notice that, for all v 2 T and all  2 R, expressions like v  are to be understood as the image of v under the translation by  (which obviously induces a map on T viewed as R=Z). On the other hand, intervals like [v ; v] are to be understood as arcs on the circle of length 1 identi ed to T. Applying (2.10) and (2.20) shows that, for all n 2 N and all  2 L (T ): 2

1

1

1

1

1

< Qbn ();  >= 

ZZ

(v) Bbn (v w) @w (w) dwdv 2

T T 1

1

1

Z

=  @w (w) (Bbn  )(w) dw : (2:21) T Here, the symbol  denotes the convolution is the sense of T . Since 1 < < 3, C dw < +1 : (2:22) T dist (v; w) Next take the limit as n ! +1 in (2.21): applying the dominated convergence theorem with (2.17) and (2.22) leads to 2

1

1

Z

0

2

1

< Qb();  >= 

Z

T

@v (v) (Bb  )(v) dv ; 2

1

(2:23)

which proves (2.18). //

3 The Entropy Relation This section is devoted to an analogue of Boltzmann's H-theorem for the model (1.1.6) { (1.1.9). The following simple lemma re ects once again the fact that the collision cross section b is even:

Lemma and De nition 2 For all  and 2 C (T ), 2

Z

T

1

< PV (b); Gv [] > (v)dv =

ZZ

= ; = T 1

[ 1 2 1 2]

11

1

Gv []()Gv[ ]()b() ddv : (3:0)

This formula extends naturally the de nition of the left hand side of (3.0) to the case where  and 2 C 1 (T1) only. In particular Qb extends as a mapping from C 1 (T1) into distributions of order 1 on T1 and veri es, for all , 2 C 1 (T1 ):

< Qb(); >=



ZZ

(v)] b() ddv : (3:1) Specializing (3.1) to the case where  = leads to the following extended de nition: for all  2 L (T ), the notation  < Qb();  > designates the following element of [0; +1]: 1 2

= ; = T 1

[ 1 2 1 2]

2

1

1

ZZ 1 2

[(v +) (v)][ (v +)

T T 1

1

j(v + ) (v)j b() ddv : 2

Proof. Formula (3.0) is recovered from (3.1) and (2.4-6) if  > 0 everyhere, which can be ensured by considering  + C in the place of . It suces to prove (3.1) in the case where  = 2 C (T ), which corresponds to the classical H Theorem. The general case follows by density and polarization. Since b is even, for all  2 C (T ) and all n  0, formula (2.9) shows that 2

1

Qbn ()(v) = 

1

1

=

Z 1 2

=

1 2

[(v + ) (v)] bn() d :

(3:2)

Hence Z

Z

(v) Qbn ()(v) dv =  T T 1

1

=  = 

Z

=

Z

1 2

T

= 1=2

1

Z

=

1 2

Z

T

=

1

1 2



1 2 Z 1

Now, since  2 C (T ), 2

=

1 2

[(v + ) (v)] (v) bn() ddv

[(v) (v )] (v ) bn() ddv [(v) (v + )] (v + ) bn() ddv =

Z 1 2

T

=

Z 1 2

=

1 2

[(v + ) (v)] bn() ddv : 2

1

j(v + ) (v)j = O ( ) 2

12

0

2

(3:3)

so that

 7! j(v + ) (v)j b() 2 L (T ) thanks to assumption (1.1.8) on b. By dominated convergence, 2

Z

T

1

(v) Qb()(v) dv = n!lim1

1

Z

+

= 

=

Z

1 2

Z 1 2

T

=

1

1 2

S1

1

(v) Qbn ()(v) dv

[(v + ) (v)] b() ddv ; 2

which proves (3.1). // A well-known consequence of the H-theorem in the case of the classical Boltzmann equation is that the only nonnegative integrable number densities for which the collision integral vanishes are local Maxwellian distributions. The analogous result for (1.1.6) { (1.1.9) is the following

Corollary 2 Let  2 L (T ) be such that Qb() = 0. Then  is equal to a 1

1

constant a.e..

Proof. If  = 0, then  = 0 a.e. and the theorem is proved. If  6= 0,  Qb() = 0. Let Z 2 C 1(R) be a nonnegative even function supported in [ 1; 1] and denote, for all  2]0; 1=2[, 1

0

 (v) = 1 Z v + k : k 2Z !

X

By (2.7), for all  2]0; 1=2[,

 Qb()   = 0 =  Qb(  ) : 1

1

(3:4)

Since    2 C 1(T ), (3.4) shows that for all  2]0; 1=2[, 1

   = C ; where C is a constant. But

Z

Z

C =   (v) dv = (v) dv; T T which shows that C is in fact independent of : hence, for all  2]0; 1=2[, 1

1

   = C 13

(3:5)

where C is a constant. As  ! 0, the left side of (3.5) converges vaguely to . Therefore  = C as a measure on T , that is to say a.e. // Note that as announced in section 2, this proves that the only conserved quantity is the mass. We do not know whether all L1 solutions of (1.1.6) { (1.1.9) in the sense of distributions necessarily satisfy an entropy inequality. However, by truncating the collision cross section b, we prove that there exist entropic solutions to (1.1.6) { (1.1.9) for any bounded nonnegative initial data. Proposition 3 Let 0  f 2 L1(T  T ), and b satisfy (1.1.8). Then, there exists an entropic solution of (1.1.6) { (1.1.9) such that (3:6) 0  f (t; x; v)  kf kL1 ; a.e. on R  T  T . If moreover f  R a.e. for some R > 0, then f (t; x; v)  R for a.e. (t; x; v) 2 R  T  T . 1

1

0

1

0

1

+

0

1

+

0

1

0

1

0

Proof. Consider rst the model equation (1.1.6) with b replaced by its truncation bn as in the proof of Lemma and de nition 1. To begin with, (2.9) holds for all  2 L (T ) (by the density of C (T ) in L (T )). This 1

1

can be recast as:

Qbn ()(v) = 

1

=

Z 1 2

1

1

(v + ) bn () d kbnkL  (v) 1

=

1 2

1

(3:7)

for all  2 L (T ). In addition, we have the Lemma 1 Let 0  f 2 L1 (T  T ). For all n 2 N, there exists a solution f n 2 L1(R  T  T ) \ C (R ; L (T  T )) to the problem (@tf n + cos(2v) @xf n )(t; x; v) + kbnkL f n (t; x) f n(t; x; v) 1

1

1

+

0

1

1

1

+

1

1

1

1

= f n (t; x)

=

Z 1 2

=

1 2

f n (t; x; v + ) bn () d ;

(3:8)

f n (0; x; v) = f (x; v) ; (x; v) 2 T  T : 1

0

It satis es

(3:9)

1

(3:10) 0  f n(t; x; v)  kf kL1 ; a.e. on R  T  T . Moreover, if f  R a.e for some R > 0, then f n (t;
)  R a.e. for all t > 0. 0

0

0

+

0

14

1

1

0

The proof of Lemma 1 is classical and deferred until after that of Proposition 3. The standard Velocity Averaging lemmas (Cf. [DP, L] for example) together with estimates (2.17), (2.18) imply that the sequence f n converges (possibly after extraction of a subsequence) to f in L1 (R  T  T ) weak * while f n converges to f a.e.. Then, for all  2 C (R  T  T ), the quantity < PV (bn); Gv () > converges a.e. (in t; x; v) to < PV (b); Gv () >, so that Qbn (f n ) converges to Qb(f ) in the sense of distributions, and f is a solution in the sense of distributions of (1.1.6). By the uniform estimates in L1 (R  T  T ) on f n , (3.6) and the last armation of proposition 3 are clear. It remains to prove the entropy condition (1.2.1). Let 0   2 C 1(R) and denote (t) =  (t=). The solution f n is extended to negative values of t by the value 0 Multiplying (3.8) by (   )  f n (where  is de ned in the proof of corollary 2), integrating in all variables and letting  ! 0 leads, for all T > 0, to +

2

1

+

1

1

1

+

1

1

0

1

ZZ 1 2

+

TZ

Z 1 2

+

T

1

0

Z 1 2

T

1

f n (t; x)

1

ZZ

2

1

1

jf n (t; x; v + )

T T



TZ

0

f n (t; x)

jf n (T; x; v)j dxdv

T T 1

ZZ

1 2

ZZ

T T 1

1



dxdt

jf n (T; x; v)j dxdv 2

T T 1

f n (t; x; v)j2 bm() ddv

1

jf n (t; x; v + )

ZZ

f n (t; x; v)j2 bn() ddv



dxdt

jf (x; v)j dxdv ;

(3:11) for all m  n 2 N according to the proof of Lemma 3.1. Since f n converges a.e. towards f , we get, by convexity and weak convergence, (the integer m being xed while n ! +1) that for all T > 0, =

1 2

T T 1

1

ZZ 1 2

+

Z 1 2

TZ

0

 (t; x) T f 1

T T 1

ZZ

1

0

2

jf (T; x; v)j dxdv 2



T T 1

1

jf (t; x; v + ) f (t; x; v)j bm() ddv dxdt 2

15



ZZ 1 2

jf (x; v)j dxdv : T T 1

(3:12)

2

0

1

Letting then m ! +1 gives the entropy inequality (1.2.1). // Proof of Lemma 1. The quickest route to this result is to generate for each n 2 N a sequence (fmn )m2N by the following iteration procedure: f n = 0; ( or f n = R if f  R > 0 a:e:) (3:13) and, for all m  1: @tfmn (t; x; v) + cos(2v) @xfmn (t; x; v) + fmn (t; x)fmn (t; x; v) 0

0

0

0

0

1

=

Z 1 2

fmn (t; x; v + ) bn() d ; (3:14) = fmn (0; x; v) = f (x; v) ; (x; v) 2 T  T : (3:15) Now (3.14) can be solved explicitly and it is easy to show that (fmn )m2N converges pointwise to a limit denoted by f n . The convergence also holds in L (by dominated convergence). Taking the limit as m ! +1 in (3.14) { (3.15) leads to the announced result. An easy in duction argument shows that, if f  R a.e., then fmn  R a.e. for all m  0.// The entropy inequality (1.2.1) provides a regularity estimate in the v variable for all entropic solutions to (1.1.6) { (1.1.9). The main diculty is to take into account cavitation, i.e. zones where the density might vanish. In the case of our model (1.1.6-7), we know from Proposition 3 that cavitation cannot occur unless it is already present in the initial data. This property is not shared by the true Boltzmann equation, and [L3] nds a way around this p by proving Besov regularity in the v variable on f instead of f itself. In the case of our model (1.1.6-7), cavitation is treated by the following simple argument: Proposition 4 Let 0  f 2 L1(T  T ), b satisfy (1.1.8) and f be an entropic solution of (1.1.6) { (1.1.9). For all T > 0;  > 0, there exists a constant C (depending on T; jjf jjL1 ; ; C ; C ; ) such that = fmn (t; x) 1

1

1 2

1

0

1

1

0

0

0

1

0

0

0

Z

TZ

Z

Z

=

1 2

1

1

jf (t; x; v + ) f (t; x; v)j jj 2

T T 1=2 If moreover f0  R0 a.e for some R0 > 0, then 1

0

Z

1

TZ

=

Z

Z 1 2

1

1 2

1

1

ddxdvdt  C :

jf (t; x; v + ) f (t; x; v)j jj ddxdvdt  C ; 2

T T 1=2 where C depends on the previous parameters and R0 . 0



+2

16

Proof. The case when f  R a.e. for some R > 0 is a simple consequence 0

0

0

of the entropy inequality (1.2.1). In the general case, one has to isolate the points where cavitation can appear. Estimate (1.2.1) gives TZ

Z

=

Z

Z 1 2

T T 1

0

2

=

1

jf (t; x; v + ) f (t; x; v)j jj

1 2

=

Z 1 2 Z Z

Z

 T

=

1

1 2

+



2

ddxdvdt 

(t;x)jj

1 2

1

1 2

+



2

jf (t; x; v + ) f (t; x; v)j dxdt 2

j j +

Z

Z

T

=

Z Z

1 2

=

1

1 2

+

 +

TZ

T

=

1

1 2

=

TZ 0

Z 1 2

T T 1

0 Z

=

Z

Z 1 2 1

=

1 2

0



2

ddv 

2

1

1 2

+



2

ddv

(t; x) jf (t; x; v + ) f (t; x; v)j jj ddxdvdt 2

jjf jjL1 1

f(t;x)jj

0

+

2 jf (t; x; v + )j + 2 jf (t; x; v)j dxdt



2

1 2

 +

g

2

 C jjf jjL T T + 4 jjf jj 1

1 2



2

j j Z



(t;x)jj

1 2

1

2

2(

1

4 (t; x) jj

L1 T

1)

=

Z 1 2

=

1 2

j j

1

1 2

+



2

ddxdt

 d;

1+

whence the desired result follows. // Proposition 4 provides the regularity estimate in the v-variable which is precisely step 1 in the method described in subsection 1.1.

4 Velocity Averaging. In this section, we return to the classical estimates of the Velocity Averaging method rst introduced in [A], [G, P, S], [G, L, P, S]. The goal is to keep track of the dependence of the estimates (in Sobolev spaces) for velocity averages like f (t; x; v0) (v; v0) dv0 (4:0) T on the norms of derivatives of the smooth function . This can be done with the original methods of proof in the references quoted above. Z

1

17

In order to deal with equations of the type (1.1.10) (speci cally, to be able to treat fractional derivatives in v in the right{hand side), we use the method of [DP, L] and [G, Po], and adapt the computations to our case. In the sequel, we say that f (z) 0 (independant of z) such that f (z)  C g(z). Let us rst establish the following technical result.

Lemma 2 For all x and y 2 R, 1.

Z

Ix;y = T 1jx

2.

v)j1 dv

y

+ cos(2

1

Z

Jx;y = T 1jx

v)j>1

y

+ cos(2

1

 (; )j2 i h 0, i( + cos(2v) )f^(; ; v) =< PV ( ); Gv [^g(; ;
)] > ; (4:12) so that, by formula (3.0) and some obvious density argument

< f^ > (; ) = 1   + cos(2v)   T "

Z

!#

1

+

=

Z

Z 1 2

T

=

1

1 2



f^(; ; v)(v)dv

+cos(2

)

5

4

Therefore

Z

Z

j < f^ > (; )j  2 kkL1 T jf^(; ; v)j dv T 1j + 2 C (kg^(; ;
)k ;) 2

2

2

1

1

1

Z

Z 1 2 1 1 2

=

T

=

3

(v)    v  Gv [^g(; ;
)]()Gv v 7! i( + cos(2v) ) ()ddv : 2



+cos(2 ( + ))



v) j dv

2

2

v   (v +  )   i( + cos(2(v + )) )

+cos(2



 2

(v)    v  i( + cos(2v) ) +cos(2

)

 2 kkL1 kf^(; ;
)kL I  ;  2

2

ddv j j  (4:14) 1+

2





   v  ddv j  ( v +  )  ( v ) j + 2 C (kg^(; ;
)k ;) i( + cos(2v) ) jj  = T + 2 C (kg^(; ;
)k ;) kkL1 v   =    v      T = i( + cos(2(v + )) ) i( + cos(2v) ) jddv j  (4:15)  2 kkL1 kf^(; ;
)kL I  ;  + 2 C (kg^(; ;
)k ;) (kk1; ;) 2 J  ;  + 2 C (kg^(; ;
)k ;) kkL1 1 T ; (4:16) 1

Z 1 2

2

2

2

=

Z

1

1 2 1

Z

Z 1 2 1 1 2 2



+cos(2 ( + ))

2

2

2

2





+cos(2

21

2

)

2

2

2

)

2

1+

2

1

1

+cos(2

2

2



2

1+

2

2

2

2

2

where

S  + cos(2(v + )) 

Z 1 2 1 1 2

Z

=

T= T

=

v)  S  + cos(2 

!

! 2

with the notation S (x) = (x)=x for all x 2 R. Then Z

S  + cos(2 (v + )) 

Z 1 2 1 1 2

=

T T

=

v)  S  + cos(2 

!

ddv j j  (4:17)

! 2

1+

 

 + cos(2(v + l))  sin(2(v + l)) dl   ddv (4:18)  j j    = S  + cos(2 (v + ))   (; )j  2 kkL1 kf^(; ;
)kL I  ; 

+

2

2

2

2

2

2

+ 4 C (kg^(; ;
)k ;) (kk1; ;) 1 J  ;  1

1+

2

2

22

2

2

2

2

2

2

(4:22)

+ 2 C () (kg^(; ;
)k ;) kkL1 1  Now it suces to appeal to Lemma 2, 2

2

2

j < f^ > (; )j  (; )j 0,  2]0; 2[, 1, 2 2 R+ and nonnegative functions h1 , h2 2 L1(R2) such that, for all  2 C 1(T1), 1

2

j < f^ > (; )j

2

23

h

i

 C kkL1 ( +  ) h (; ) + (kk1; ;) ( +  ) h (; ) : (5:2) 2

2

2

1

2

+ )

1

2

2

2

2

2

2

Then, one has Z

RT T 1

1

1

 1 ( 

2+ 2

  ; 2 ++2 ) jf^(; ; v )j2 dddv

2 inf( 1 +2

< +1 : (5:3)

Proof. One has: Z

Z

Z

2

jf^(; ; v)j dv 0) 1

1

1 2 (1+ + )

1

1+2 ( 1 ) 2 (1+ + )

2

1

2

0

2

0

0

Z

RT T 1

1

1

 1

2+ 2



1 2 2 ( +1) ( +3)

( +  ) 2



jf^(; ; v)j dddv < +1 : 2

In addition, Proposition 4 shows that

f 2 L (R  T ; H 2

1

+

2

 (T1)) ;

1

which establishes the rst part of Theorem A. The corresponding estimates if f  R a.e. for some R > 0 are Z

RT T 1

and

1

1

1

0





1 2 2 ( +1)2

  ( +  )

2+ 2

0

2

0

jf^(; ; v)j dddv < +1 2

f 2 L (R  T ; H (T )) ; and the second part of theorem A is proven. // 2

+

1

26

1

1

6 The Landau-Fokker-Planck Approximation. In this last section, we give a short proof of Theorem B. Proof of Theorem B The L1 estimate on f " shows that, up to extraction of a subsequence, f " converges to f in L1(R  T  T ) weak-*. Using the following variant of formulation (2.18), @tf " (t; x; v) + cos(2v)@xf " (t; x; v) = "f (t; x) 1

+

 @v

Z

2

and observing that

=

1 2 Z 1

=

1

1 2

=

Z 1 2 Z 1 1 1 2

=

1

juj) f " (v + " u ) du jj2 b() d

2 (1





juj) f "(v + " u ) du jj2 b() d

2 (1



=

Z 1 2

 b() d  jjf "jjL1 R T T =  jjf jjL1 T T 2 C (1=2) ; (

0

+

( 1

1

1)

1)

2

1 2

3

1

we appeal to the standard averaging lemmas (see [DP, L]) to show that the quantity f " converges a.e. on R  T (again up to extraction of a subsequence) towards f . In order to pass to the limit in the nonlinear collision term, we only need to show the following: for all smooth function ' of the variable v, the quantity +

1

  "=2 " f (t; x; v) '(v + ) + '(v ) 2 '(v) " 3 b(  ) ddv 1 T "=2  Z Z 1=2 Z 1 00 2 " " = f (t; x) T1 f (t; x; v) 1=2 1 2 (1 juj) ' (v +  u ) du jj b() d dv converges (in L1 (R  T1) weak-*) towards

"f (t; x)

Z

Z

+

f (t; x)

Z

T

1

f (t; x; v) '00(v) dv

=

Z 1 2

=

1 2

 b() d : 2

But this follows at once from the convergence a.e. of f " . // Acknowledgements. We express our thanks to Aline Bonami for her valuable advice during the preparation of this paper. Both authors were supported by the TMR \Asymptotic Methods in Kinetic Theory", ERB FMRX CT97 0157. 27

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[Pr] Proutiere, A., New results of regularization for weak solutions of Boltzmann equation, Preprint. [V 1] Villani, C., On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Preprint. [V 2] Villani, C., Conservative forms of Boltzmann's collision operator: Landau revisited, Preprint. [We] Wennberg, B., Regularity in the Boltzmann equation and the Radon transform, Comm. Part. Di. Eq., 19, no. 11-12, (1994), 2057{2074. L.D.: CMLA, Ecole Normale Superieure de Cachan, 94235 Cachan Cedex, FRANCE F.G.: Universite Paris VII & Ecole Normale Superieure, D.M.I., 45 rue d'Ulm, 75230 Paris Cedex 05 FRANCE E-mail: [email protected] & [email protected]

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