We give a lower bound of the entropy dissipation rate of KaY, Boltzmann and Fokker-Planck-Landau equations. We apply this estimate to the problem of the speed of convergence to equilibrium in large time for the Boltzmann equation. 8 t f +v. v x f = Q ' ( f , f ) ,

Rarefied gas dynamics is usually described by the Boltzmann equation
a J + v ' V x f = Q ( f , f ) ,
(

A simpler monodimensional model has been introduced by Ka~ in [9]: where Q is defined in [9] or [11].

The asymptotics of the Boltzmann equation when the grazing collisions become
predominant formally leads to the Fokker-Planck-Landau equation
~ t f + v S x f = Q.(f, f ) ,
(

According to Boltzmann's H-theorem, the entropy dissipation rate v~ 3

Q( f, f )(v) log f (v)dv is nonpositive. Moreover, it is equal to 0 if and only if f is a Maxwellian (cf. [t 2]). The same property holds for the collision term Q', and for Q with the additional prescription that the Maxwellian is of bulk velocity 0.

Therefore, in order to get a better understanding o f the phenomena appearing when the entropy dissipation term tends to 0, it is useful to obtain a lower bound of it in terms o f some distance from f to the space o f Maxwellians.

Such an estimate measures the speed of convergence to equilibrium when the
entropy dissipation tends to 0. This situation occurs for example when the time t
tends to infinity in Eq. (

Otf + v . Vxf = ~-Q(f, f ) ?

Accordingly, in Sect. 2, we give a lower bound of the entropy dissipation term
for the Kae equation in terms o f a distance to the Maxwellian states o f the type
i n f ~ Llogf ( v ) - m ( v ) l d v ,
m ~ F
(

We extend this result in Sect. 3 to the case o f the Boltzmann collision kernel Q, and in Sect. 4 to the case o f the Fokker-Planck-Landau collision kernel Q'.

We explain in Sect. 5 how these results give classical estimates when applied to the linearised kernels of Q, Q and Q'.

In Sect. 6, we apply the previous estimates to investigate the long time behaviour of the Boltzmann equation.

O J + v O x f = Q ( f , f ) , models a one-dimensional gas in which all collisions conserving the energy are equiprobable (cf. [9] or [11 ]). Therefore, its collision kernel Q is the following: where 2~

dO { f (v') f (v'O- f (v)f (v~)} ~

dv~ V l ~ 0 = 0 v' = v cos 0 + va sin 0 , v~ = - v sin 0 + vl cos 0 .

The theorem below is a partial answer to a question of M c K e a n (cf. [1t], p. 365; 13b]).

We shall denote by U the set o f the convex, continuous and even function from to IR such that for all x in ]R,

O < ¢ ( x ) ~ x ( e ~ - t )

Moreover, if B is a function from N * to N * , we introduce the set

Lf~= { f s LV/for all v such that [vl< R, f (v)> BR} , and P is the space of loga'-ithms of Maxwellians with zero bulk velocity

Theorem 1. Let Q be the Kag collision kernel, and R be strictly positive. Then there exists a strictly positive KR such that for all f in L~ and all ¢ in U, - ~ Q ( f , f ) ( v ) l o g f ( v ) d v ~ ¼ B ~ r c R 2 ¢ ( ~ - ~ ve~. i n f ~ m~F Ivl<R [ l o g f ( v ) - m ( v ) l d v ) .

Proof of Theorem 1. Boltzmann's H-theorem ensures that - ~ Q_(f,f)(v)logf(v)dv =~

~ ~ v6N v ~ .f { f ( v ' ) f ( v ' ~ ) - f ( v ) f ( v O } {log ( f ( v ' ) f ( v ~ ) ) - l o g ( f ( v ) f ( v O ) } 0=0

z~ i ~ 2 ( l ° g f ( v ' ) + l ° g f ( v ' x ) - l ° g f ( v ) - t ° g f ( v O ) V2+Vl~R2 0=0 dO ~

dvldV " Therefore - ~ Q ( f , f ) ( v ) l o g f ( v ) d v

v~N = 4 rc

vz+v~__<R2o=o ) "2re dvl dv by Jensen's inequality.

Before going further in the proof, we need the following lemmas: L e m m a 1.

2~ i ~ I l o g f ( v ' ) + l o g f ( v ' l ) - l o g f ( v ) - l o g f ( v l ) l v2+VI <=R2 0=0 dO ~ dvldv _->

inf ~ functionsT v2+vl__<R2

[ l o g f ( v ) + l o g f ( v l ) - T ( v 2 + v 2 ) [ d v l d v .

9 (v, v1) = log f (v) + log f (v1) ,

R , = rotation of angle ~ ,
[log f ( v ' ) + tog f (v;) - log f ( v ) - log f(v~ )[
(

Ilogf(v)+logf(vi)-T(v2+v~)Idvidv ~ K R inf S tl°gf(v)-rn(v)t dr"

rne~ Ivl__<R Proof of Lemma 2. Let ~ / b e the space of functions depending only on v2+ v2, and

: t ~ L 1(Iv[__<R ) / r ~ £ t ( v , vl) = t(v) + t(vl) ~ L 1(v2 + v~< R2)/~I .

The operator L is clearly linear and one-one (cf. [3] or [12] in the more complicated case of the Boltzmann collision kernel).

functions T v2+v~<R 2

It(v)+t(vO-T(v2 +v~)ldvldv __< inf I

a, be~_ v2+v~<=RZ < 4 R inf ~ me~ tvJ_-<R

[t(v)+t(vi)-a(v2 +v~)-2bldvldv It(v)-m(v)ldv.

In order to apply the open mapping theorem, we have to prove that the image
o f L is closed. Assume that there exists a sequence t~ in LI(Iv[<=R)/F and t in
L a(V2 ~-/.)21 ~ R 2)/l~f such that tn(v) + tn(vl) tends to t(v, V1) in L i (/)2 jr V2 ~ R 2)/~r.
Then, there exist a sequence k n in Ll(lv]<R), a sequence T, depending only on
v2+vz and 9 in L i ( v Z + v ~ R z) such that tn is the natural projection of kn on
L 1(Iv]<R)/F, t is the natural projection of 9 on L i (v2+ v2~ RZ)/M, and
k, (v) + kn(va)+ T, (vz + v2)--*9 (v, vl)
(

converges in W -1'1. Taking the double partial derivative of this expression with respect to v, v1, we obtain that converges in W-a, 1. Therefore, there exists a sequence of real numbers b, such that converges in W -3'1. Then. there exists a sequence of real numbers c. such that which has the following property:

{vl 8v

Vl ~v - v

T(vz ÷ vzl) = 0 .

Ovlj (k.(vO+k.(v)+ T.(v2 +vZ))=vak/~(v)-vk[,(v 0
(

But (

k.(v) +½b.v 2+ a. converges weakly in L 1.

Therefore, t. converges weakly in L a(Iv1< R)/[" and the image o f / . is closed. Applying the theorem of the open mapping to L, we obtain a strictly positive KR such that:

inf ~ functionsT v2-k/jl~R2 [t(v)+t(vl)- T(vZ +v~)[dvldv>=Kg inf S tt(v)-m(v)[ dv

m~t~ ivl<R k ; ( v ) - k ; ( v l )

k" (v) + b. k~(v)+b.v+c.

k~(v) + b. v l~;( v ) - v k ; (vl) k~'(v)+b.

k/,(v)+b.v 0vl {v,k ' ( v ) - vl~;(vl)} Injecting t = log f in this estimate, we obtain Lemma 2.

The proof of Theorem t easily follows from Lemmas i and 2 together with
estimate (

~ S { f ( v ' ) f ( v ' l ) - f ( v ) f ( v O } B ( v , vl,o~)dcodv, vl~IR 3 ¢9~S2 with and B is a nonnegative collision cross-section, depending on the collision process.

According to Boltzmann's H-theorem, the equilibrium is obtained for
Maxwellian densities if B is strictly positive a.e. In order to obtain an estimate of the
type (

We keep in this section the notations of Sect. 2. Moreover, we introduce F, the space of logarithms of Maxwellians:

F={avZ+b "v+c/a, c e N , b ~ N 3} , we denote by IAI the Lebesgue measure of the set A, and by S N the sphere of dimension N.

Theorem 2. Let Q be the Boltzmann collision kernel with B as in (

Q ( J ~ f ) ( v ) l ° g f ( v ) d v > l B2ISSJ ISZIR6CR~ " G

KR inf , S5t IS2[R 6 m~; Ivl__<R llogf (v)-m(v),dv) . Proof" of Theorem 2. By Boltzmann's H-theorem, - ~ Q ( f , f ) ( v ) l o g f ( v ) d v

1 : ~

1 > - B2CR = 4 with

velt ~ ? o9" v l - v byJensen'sinequality L e m m a 3. dogdvldv ~ ) (because

I f Io9 ~ [ V2+V2~R20eS 2

d o g d v l d v - I S I ISS]R 6).

[l°gf(v)+l°gf(vl)-T(v+va'vZ+v2)[dvldv" Proof of Lemma 3. Let us introduce the notation: g (v, vl) -- log f(v) + log f(vl). We compute

I f l l ° g f ( v ' ) + l ° g f ( v l ) - t ° g f ( v ) - l ° g f ( v l ) l vE+vl<:R2 ~ e S 2 o9 " v~I --v dogdvldV v +v,<=R2 ~,~s2 o 9 " , - , - ,

~=s(o) with and with

V1 - - V s(~o)=2 ~ o ' ~ ) o V1 - - V

Iv~-vl

V1 --V - 1 0 ) ' ~ (v',v~) = ~ ( v , vl)

U~(V, Vl)=½(v+va + l v - v x l a , v + v x - l v - v ~ l ~ ) ?
According to (

$ $ ]g(U~(v, v O ) - a ( v , vOldadvldv

V2+V2~R2 a e S 2

I ~ [ l ° g f ( v ' ) + l ° g f ( v ' x ) - l ° g f ( v ) - l ° g f ( v l ) [ v + v l 6 R 2 toES 2 w" ~,u,-u,l-v dc~dv~dv >

inf ~ functions T v2 + vI __<R 2

inf ~ functions T vZ+v~<R2 [g(v, v O - T ( v + v l , v 2 + v ~ ) l d v l d v I l o g f ( v ) + l o g f ( v O - T ( v + v l , v 2 + v 2 1 ) l d v l d v , which concludes the p r o o f of the lemma.

Lemma 4. For all strictly positive R, there exists a strictly positive Kg such that inf ~ functions T vz + v I <R2

I l o g f ( v ) + l o g f ( v O - T ( v + v l , t ~ + v ~ ) l d v l d v _>_KR inf S I I ° g f ( v ) - m ( v ) l dv ?

m ~ r Ivl~_R Proof of Lemma 4. Let M be the space of functions depending only on (v+v 1, v2 +v2), and let L be the operator

L : t ~ L 1(Iv[<-_R)/F-~Lt(v, vi) = t(v) + t(vl) ~ L 1(v2 + v~ <=R 2)/M .

I t ( v ) + t ( v l ) - a ( v 2 + v Z ) - b ' ( v + v l ) - 2 c l d v l dv

inf i functionsT v2+ vi <R2 It(v) + t(vl) - T(v2 + v2)ldr1 dv __<

inf .l a, cE~R,b~N 3 v2+v~<R z =<16R 3 inf ~ mer IvI__<R

I l o g t ( v ) - m ( v ) l d v .

In order to apply the open mapping theorem, we have to prove that the image of L is closed. Suppose that there exist a sequence t, in L I ( [ v I < R ) / F and t in L a(vz + v~< R 2) / M such that t. (v) + t. (vi) tends to t (v, vi) in L 1(vz + v~ < R 2)/M. Then, there exist a sequence k n in Ll(lvl < R ) , a sequence T. depending only on v + v i , vZ+v 2 and g in Ll(vZq-vZK=R z) such that t. is the natural projection o f k . on Ll(lvl < R ) / F , t is the natural projection o f g on Ll(v2+vZ<=RZ)/M, and kn(v) + k. (vl) + ~ (vz + v2)--*9 (v, v1) (14) in L 1(v2 + vz < R z).

v = ( x i , x z , x 3 ) , v i = ( Y i , Y 2 , Y 3 ) ? OYl 2 (~Y2 ~--0y3)-- (Y3 -- x3) (6~1 ~3yI

VT(v + vi, vz + v~) = 0 ,

~k. (v)+ ~k.(v0

Ok. Ok. Okn (Y2- X2) ~ - (Xl' X2, X 3 ) - (Yl--Xl) ~--2- (Xl' X2' X 3 ) - ( Y 2 - X2) ~ - (Yi, Y2, Y3)

Ok, q- (Yl -- xl) ~-~ (Yl, Y2, Y3) (15) converges in W -1,1. Moreover, the same formula holds if we change the indices

Taking the double partial derivative of this expression with respect to xl, Yl, we obtain that c32kn ~ . 82kn 81 82 tX l ' X 2 ' X 3 ) - S i - ~ (Yl,Yz,Y3) converges in W -3'1. Taking the double partial derivative of (14) with respect to xl, Y2, we obtain that 82kn t~2kn 812 (Xl,x2, x3)- ~ (Yl, Y2, Y3) converges in W -3'1. Therefore, converges in W-a,1 and there exists a sequence of real numbers a. such that converges in W -3'1. Moreover, the same convergences holds with the same sequence a, when we change the indices 1, 2, and 3 by circular permutation.

Accordingly, there exist three sequences of real numbers b.I, b2, b~ such that converges in W -2'1. Injecting Y2 =xz in formula (17), Eq. (16) ensures that 82kn 81 82 82kn ~1~ ~a. 8k. +a.xi+bi" 8i t~2kn 81 g2 t~2kn t~12 k-an (16) (17) (18) (19) converges in W -2'1. Differentiating (15) with respect to x I , we obtain that t~2kn t~2kn (y2-x2) T U (xl, x2, x3)-(yl - x 0 ~ (xl, x~, x3) 8k. 8k.

+ b2- (xl' x2, x3) - ~-2 (Yl, Y2, Y3) converges in W -2'1. By the same argument, injecting Yl = x , in formula (17),

k.(v)+½a.v2 + b. "v+c. converges weakly in L 1(b. being the vector of components b~), Finally, we obtain that k. converges weakly in L~(lv[<R)/F, which ensures that the image of L is closed. Thus we can apply the open mapping theorem to L in order to obtain a strictly positive KR such that

inf 2 I functionsT v + v t ~ R 2

]t(v)+t(vO-T(v+vl'vZ+v2)ldvdv~ > K R inf ~ Jt(v)-m(v)ldv .

m ~ r Ivl<R Injecting t = log f in this estimate, we obtain Lemma 4.

The proof of Theorem 2 easily follows from Lemmas 3 and 4 together with estimate (13). 4. On the Fokker-Planck-Landau Collision Kernd The derivation of the Fokker-Planck-Landau collision kernel may be found in [4 or 10]. We denote

Q ' ( f , f ) = d i v v ~ welt 3 ~

I tv _ w[2

j ? {f(w) Vvf(v ) - f ( v ) Vwf(w)}dw , where C is a strictly positive constant depending on the physical properties of the gas, and I is the identity tensor. We keep in this section the notations of Sects. 2 and 3. Moreover, we introduce F', the space of derivatives of logarithms of

F ' = {a+bv/a~IR3,b~lR} , and H~og, the set of functions in L z such that their logarithm is in H 1.

Theorem 3. Let Q' be the Fokker-Planck-Landau collision kernel, R be strictly positive, and 0 be as in Sect. 2. Then there exist a strictly positive K R such that for all f in t 2 ( ~ n ? o g ,

~ a ( f , f ) ( v ) l o g f ( v ) d v > C - 2 - ~ K R inf ~ IV~logf(v)-m(v)12dv. velO m~r' ivl__<R

~ v ~ R s

Q ' ( f , f ) ( v ) l o g f ( v ) d v = g

f I v ~ N 3 w~ll?

f ( v ) f ( w ) { g v l ° g f ( v ) - g w l ° g f ( w ) } ? f i _ (v - wiv)-®w [(vz - w)} {Vvlogf(v)_ Vwlogf(w)}dwd v k. =>B2C4R t~l~=<, lwl__<R~{Vvlogf(v)--Vwlogf(w)} I ?{Vvlogf(v ) - Vwlogf(w)}dwdv . (v-w) ®(v-w)] ~52w-~ j

I v - w l ~ are 1 with order n - i and 0 with order 1. Moreover, the eigenvector corresponding to the eigenvalue 0 is v - w . Therefore, for all x in R3,

I - ( v - w ) ® ( v - w ) ) [v_w]2 ] x "x> ~s~infI x + 2 ( v - w ) l z .

- f

V~ H~3.

Q ( f , f ) ( v ) l o g f ( v ) d v > B 2 C inf ~ ~ I V ~ l o g f ( v ) - V w l o g f ( w ) + 2 ( v , w ) ( v - w ) l Z d w d v = 4 R functionsX(v,w) ivl<R lwl<R Before going further in the proof, we need the following lemma: Lemma 5. For all strictly positive R, there exists a strictly positive K R such that inf S ~ [V,,l°gf(v)-Vw l ° g f ( w ) + 2 ( v , w ) ( v - w ) l z d w d v functions2(v,w) Ivl<=R Iwl<R . (20) > K R inf S I V v l ° g f ( v ) - m ( v ) [ 2dv"

reel'" tvl<__R Proof o f Lemma 5. Let M ' be the space of functions of the form: 2(v, w ) ( v - w), and let L' be the following operator:

L " t ~ L2(lv[ < R; IR3)/F'~L't(v, w) = t ( v ) - t(w) E LZ(lvl < R, Iwl < R ; IR3)/M ' .

inf S ~ f t ( v ) - t ( w ) + 2 ( v , w ) ( v - w ) [ zdwdv functions 2(v, w) Ivl_-<R Iw[ < R

inf ~ ~ I t ( v ) - t ( w ) + a ( v - w ) + b - b l Z d w d v a ~ R , b ~ 3 [vl<=R Iwl<R ~ 3 2 R 3 inf j" I t ( v ) - m ( v ) I d v .

meF' [vl<R

In order to apply the open mapping theorem, we have to prove that the image of L' is closed. Suppose that there exists a sequence tn in L2(Ivl~R; I R 3 ) / F ' and t in LZ(IvI<R, lwI<R; ]R3)/M ' such that t . ( v ) - t . ( w ) tends to t(v,w) in L2(IvI<R, ]w[ < R ; N 3 ) / M '. Then, there exists a sequence k. in L2(lv[ < R , IR3), a sequence 2. of real-valued functions and 9 in L z (Iv[< R, [wI< R; IR3) such that tn is the natural projection of k. on L2(lvr<R;IR3)/F ', -i is the natural projection of 9 on

k , ( v ) - k , ( w ) + 2,,(v, w ) ( v - w ) ~ g ( v , w) in U(Ivl <R, Iwl < R , ~3).

Accordingly, if we set k. -_ ( k .1, k .2, k.3 ), v = (v1, v2, v3), w = ( w l , w 2 , w 3 ) , (k~(v) - k~ (w)) (vj - w~) - (k.~(v) - k.~(w)) (v, - w,) converges in L 2(Iv I__<R, Iw I__<R ; IR3) for all i, j in {1,2, 3}. Taking the double partial derivative of this expression with respect to v,, w~, we obtain that converges in H - 2 . Taking the double partial derivative of (22) with respect to vi, wj, where i and j are different, we obtain that converges in H -2. Therefore, (21) (2 2) (23) (24) (25) (26) (27) converges in H - 1. In the same way, injecting v~= w~in formula (25), Eq. (24) ensures that converges in H - 2 for all distinct i, j and there exists a sequence o f real numbers a. such that converges in H - 2 Accordingly, there exist three sequences of real numbers b.~, b.2, bn3 such that

k .i+ a . v i + b . i converges in H -1. Differentiating (22) with respect to vi, we obtain that ak~ ak~

Oi (v) (vj-- wj) -- (k~ (v) - k~(w)) - (vi - w'~)- ~ (v) converges in H -1. Injecting vj=wj in formula (25), Eq. (23) ensures that: ak~ ~k~ o-T (v)+--~-i (w) Ok~oi (v)+~k/ (w)

Ok~ Ok.' ~i t-a.

ak.~ Ok.' 0i 4-a, k,i, + a,,vi + b,i, converges in H - 1 . Formulas (26) and (27) ensure that converges weakly in L 2. Finally, we obtain that k, converges weakly in L 2 (Iv l ~ R)/F', which ensures that the image of L' is closed, Thus we can apply the open mapping theorem to L' in order to obtain a strictly positive KR such that inf ~ ~ functions,~(v,w) IriS/1 Iwl~R =>KR inf I m~F" Ivl<R

I t ( v ) - m ( v ) t 2 d r "

I t ( v ) - t ( w ) + 2(v, w ) ( v - w)12dwdv Injecting t = V~log f in this estimate, we obtain Lemma 5.

The p r o o f of Theorem 2 easily follows from Lemma 5 together with estimate (20). 5. On the Linearised Kinetic Equations The proofs of the theorems of Sects. 2, 3 and 4 are still valid when we deal with the linearised collision terms of the KaY, Boltzmann or Fokker-Planck-Landau equations. More precisely, we obtain the following theorems, which are classical in the case of the Boltzmann and the Ka~ linearised equations:

We shall denote by M a given Maxwellian, and L z ( M 1/z(v)dv) the Hilbert space of functions f such that f M ~/2 is in L2, together with the norm:

I[fl[L2(M~/~(v)dr)= (Sf 2(V)M (v) dv) 1/2 Theorem 4. Denote by 0 the Ka6 collision kernel. There exists a strictly positive K such that for all f in L 2( M 1/z(v) dr): - I f ( v ) O ( M , Mf)(v)dv>Ktlf-mfllb(M'J~(o)av),

v ~ N where my is the orthogonal projection o f f on F in L 2( M */z(v)dv).

Theorem 5. Denote by Q the Boltzmann collision kernel with a cross-section B such that

B ( u ' u l ' o g ) ~

Co9" v , - v

V x _ v ' where C is a strictly positive constant. There exists a strictly positive K such that for all f in L z ( M 1/2(v) dr):

~ f ( v ) Q ( M , M f ) ( v ) d v > K l ! f - m f I l ~ ( M , ~ ( v ) a ~ ) , v e ~ a where m f is the orthogonal projection o f f on f in L2(M1/2(v)dv).

Remark. The latter theorem is classical with additional assumptions on B, (cf. [3]). The author does not know whether the result given here in all its generality is new. Theorem 6. Denote by Q' the Fokker-Ptanck-Landau collision kernel. There exists a strictly positive K such that for alI f in H 1, where ~ is the function ~ = - I , M .

V
6. On the Speed of Convergence to Equilibrium in Large Time
S o m e results are already k n o w n on this subject. F o r e x a m p l e L. A r k e r y d p r o v e d in
[2] that the convergence to a Maxwellian in the case o f the spatially h o m o g e n e o u s
B o l t z m a n n e q u a t i o n holds with exponential decay. H o w e v e r , it seems t h a t for the
full B o l t z m a n n or Ka~ equations, the convergence is slower. This section is d e v o t e d
/ N
to the e x p l a n a t i o n o f how, in some sense, the convergence is in O (7--Z.)" M o r e
precisely, we are able to p r o v e the following t h e o r e m : \ V , /
Theorem 7. Let f b e a Di Perna-Lions solution of the Boltzmann equation (cf. [7]) with
a cross section B as in (

St f inf S j l o g f - m t d v d x q t - < K" t xE~.3 m~F lvl<R t 1/~- " Proof of Theorem 7. We proceed as in T h e o r e m 9 o f [5]. A c c o r d i n g to [7], +co I I t=O x~?

f vEHI.3

- Q ( f , f ) ( t , x , v ) l o g f ( t , x , v ) d v d x d t is finite and therefore 2t

~ ~ - Q ( f , f ) ( t , x , v ) l o g f ( t , x , v ) d v d x t xe~a vs~3 d t < C t - t for some c o n s t a n t C. A c c o r d i n g to T h e o r e m 2, and using the fact that the function q~ introduced in this t h e o r e m m a y be t a k e n to be equivalent to x 2w h e n x tends to 0, we o b t a i n t h a t for s o m e strictly positive KR : 2t

5 inf S t xelR' reel" IoI__<R Remark. The same kind o f t h e o r e m w o u l d hold in the case o f the F o k k e r - P l a n c k L a n d a u e q u a t i o n if we k n e w the global existence o f a w e a k solution.