ON THE SPATIALLY HOMOGENEOUS LANDAU

EQUATION FOR HARD POTENTIALS

PART II : H-THEOREM AND APPLICATIONS

L. DESVILLETTES AND C. VILLANI

Abstract. We ﬁnd a lower bound for the entropy dissipation of

thespatiallyhomogeneousLandauequationwithhardpotentialsin

terms of the entropy itself. We deduce from this explicit estimates

onthespeedofconvergencetowardsequilibriumforthesolutionof

this equation. In the case of so-called overmaxwellian potentials,

theconvergenceisexponential. Wealsocomputealowerboundfor

the spectral gap of the associated linear operator in this setting.

Contents

1. Introduction and main result 1

2. Entropy dissipation : ﬁrst method 8

3. Entropy : second method 13

4. The trend towards equilibrium : overmaxwellian case 16

5. Improved results 18

6. The trend towards : the case of true hard

potentials 21

7. Poincar´e-type inequalities and applications 24

8. Entropy dissipation and regularity estimates 26

Appendix A. Deﬁnition of the entropy dissipation 27

Appendix B. Approximation of the entropy dissipation 29

References 30

1. Introduction and main result

We recall the spatially homogeneous Landau equation (Cf. [8, 18]),

@f N(1) (t;v)=Q(f;f)(t;v); v2R ; t‚0;

@t

12 L. DESVILLETTES AND C. VILLANI

where f is a nonnegative function and Q is a nonlinear quadratic op-

erator acting on the variable v only,

(2) ‰Z ? ¶

@ @f @f

Q(f;f)(v)= dv a (v¡v ) f (v)¡f (v ) ;⁄ ij ⁄ ⁄ ⁄

@v N @v @vi j j

where f = f(v ), and the convention of Einstein for repeated indices⁄ ⁄

is (and will systematically be) used.

NHere,(a (z)) (z2R )isanonnegativesymmetricmatrixfunctionij ij

with only one degenerate direction, namely that of z. More precisely,

(3) a (z)=Π (z)Ψ(jzj);ij ij

where Ψ is a nonnegative cross section and

zzi j

(4) Π (z)=– ¡ij ij 2jzj

?is the orthogonal projection onto z =fy=y¢z =0g:

We address the reader to Part I of this work [13] for references on

the subject.

The Landau equation is obtained as a limit of the Boltzmann equa-

tion when grazing collisions prevail. The terminology concerning the

crosssectionisthereforecloselyrelatedtothatoftheBoltzmannequa-

tion.

In this paper, we shall deal with diﬀerent types of cross sections

Ψ. We recall the important particular case of Maxwellian molecules

¡(2N¡1)(coming out of an inverse power force in r ),

2(5) Ψ(jzj)=jzj :

Any cross section Ψ, such that Ψ is locally integrable and satisfying

2(6) Ψ(jzj)‚jzj

willbecalledovermaxwellian(ofcourseMaxwellianmoleculesareover-

maxwellian).

The “true” hard potentials cross section (coming out of an inverse

¡spower force in r for s> 2N¡1) is

?+2(7) Ψ(jzj)=jzj

for some ? 2 (0;1). Such a cross section is not overmaxwellian be-

cause of its behavior near z = 0. We therefore deﬁne “modiﬁed” hard

2potentials by the requirements that Ψ is of class C , overmaxwellian,

and

?+2(8) Ψ(jzj)»jzj asjzj!+1:

RON THE HOMOGENEOUS LANDAU EQUATION 3

Note that multiplication of Ψ by a given strictly positive constant

amounts to a simple rescaling of time.

For a given nonnegative initial datum f , we shall use the notationsin

Z Z

1

2M(f )= f (v)dv; E(f )= f (v)jvj dv;in in in in

N 2 N

Z

H(f )= f (v)logf (v)dv;in in in

N

for the initial mass, energy and entropy. It is classical that if f ‚ 0in

and M(f );E(f );H(f ) are ﬁnite, then f belongs toin in in in

‰ ?Z

¡ ¢

1 NLlogL= f 2L (R ); jf(v)jjlog jf(v)j jdv < +1 :

N

The solutions of the Landau equation satisfy (at least formally, thanks

to the change of variables (v;v )$ (v ;v)) the conservation of mass,⁄ ⁄

momentum and energy, that is

Z Z

(9) M(f(t;¢))· f(t;v)dv = f (v)dv =M(f );in in

N N

Z Z

(10) f(t;v)vdv = f (v)vdv;in

N N

Z Z2 2jvj jvj

(11) E(f(t;¢))· f(t;v) dv = f (v) dv =E(f ):in in

N 2 N 2

They also satisfy (at the formal level) the entropy dissipation identity

d

(12) H(f(t;¢))=¡D(f(t;¢));

dt

where H is the entropy

Z

(13) H(f)· f(v)logf(v)dv;

N

and D is the entropy dissipation functional

Z

(14) D(f)=¡ Q(f;f)(v)logf(v)dv

N

Z Z ? ¶

1 @f @fi i

= a (v¡v )ff (v)¡ (v )ij ⁄ ⁄ ⁄

2 N N f f£

? ¶

@ f @ fj j

(v)¡ (v ) ‚0:⁄

f f

RRRRRRRRRRRRRR4 L. DESVILLETTES AND C. VILLANI

Due to the singularities at points where f vanishes, this formula is

notveryconvenientforamathematicalstudy. Therefore, asin[26], we

shall rewrite the entropy dissipation for the Landau equation in a form

which makes sense under very little assumptions on f. Since, formally,

? ¶ ‡ ·p p p p prf rf

ff (v)¡ (v ) =2 f r f(v)¡ fr f(v )⁄ ⁄ ⁄ ⁄

f f

p

=2(r ¡r ) ff ;v v ⁄⁄

the entropy dissipation is

Z Z p p

2 dvdv a(v¡v )(r¡r ) ff (r¡r ) ff :⁄ ⁄ ⁄ ⁄ ⁄ ⁄

In other words,

1

2(15) D(f)= jjKjj ;2 N NL ( £ )2

where

p

1=2K(v;v )=2Π(v¡v )Ψ (jv¡v j)(r ¡r ) f(v)f(v ):⁄ ⁄ ⁄ v v ⁄⁄

We show in Appendix A that K is well–deﬁned as a distribution on

N N 1 NR £R as soon as Ψ is locally integrable and f 2 L (R ). In

particular,asnotedin[26],thisallowstocoverthephysicalcaseswhere

Ψ has a singularity at the origin. Hence, formula (15) enables us to

deﬁne D(f) as an element of [0;+1] in the most general case, and we

shall always consider it as the deﬁnition of the entropy dissipation. Of

course, with this convention, formula (14) holds only under suitable

regularity assumptions on f (and its logarithm).

TheequalityD(f)=0holds(attheformallevel, andwhen f;Ψ>0

Na.e.) only if for all v;v 2R ,⁄

r(logf)(v)¡r(logf)(v )=‚ (v¡v )⁄ v;v ⁄⁄

for some ‚ 2 R. It is easy to check that this implies that for allv;v⁄

N Nv 2 R , rf(v) = ‚v +V for some ﬁxed ‚ 2 R and V 2 R . This

ensures in turn that f is a Maxwellian function of v,

2

(v¡u)‰ ¡

2T(16) f(v)= e ·M (v);‰;u;TN=2(2…T)

Nfor some u2R , ‰;T > 0. A rigorous proof (under suitable assump-

tions on f) can be found for instance in [23]. Other proofs shall be

given in the present paper.

This theorem is the Landau version of Boltzmann’s H-theorem, in

view of which it is expected that a solution f(t;¢) of the Landau

RRON THE HOMOGENEOUS LANDAU EQUATION 5

equation converges when t ! +1 towards the Maxwellian function

fM =M f f f deﬁned by‰ ;u ;T

Z Z

f f f‰ = f(v)dv; ‰ u = f(v)vdv;

N N

and Z

2 f f 2 ff(v)jvj dv =‰ ju j +NT :

N

Thepurposeofthispaperistostudythespeedofconvergenceoff(t;¢)

ftowards M . Let us summarize brieﬂy the state of the art concern-

ing the asymptotic behavior of the solutions to the spatially homoge-

neous Boltzmann and Landau equations. The reader will ﬁnd many

references (but unfortunately not the most recent ones) in [12] on the

general problem of the behavior when t! +1 of the solutions of the

Boltzmannequationinvarioussettings, includingthefullx–dependent

equation.

In the homogeneous setting, we are aware of essentially two types of

theorems :

† The results by Arkeryd [2] and Wennberg [27] give exponential

convergence towards equilibrium for the spatially homogeneous

Boltzmann equation with hard (or Maxwellian) potentials in

pweigh– ted L norms, namely

f ¡–tkf¡M k•Ce ;

but with a rate – > 0 (depending on the initial datum), which

is obtained by a compactness argument and is therefore not

explicit. These results are based on the study of the spectral

properties of the linearized Boltzmann operator.

† On the other hand, Carlen and Carvalho obtain in [4, 5] an es-

timate which gives only at most algebraic decay for the Boltz-

mann equation (with Maxwellian molecules or hard-spheres),

but which is completely explicit (though rather complicated).

These results rely on a precise study of the entropy dissipation

D of the Boltzmann equation. A function Φ (with Φ(0) = 0)B

is computed in such a way that

‡ ·

fD (f)‚Φ H(f)¡H(M ) :B

This function Φ is strictly increasing from 0 (but very slowly).

As a consequence, it is shown in [5] how, for a given initial

RRR6 L. DESVILLETTES AND C. VILLANI

datum f and ">0, one can compute T (f )>0 such thatin " in

ft‚T (f )=)kf(t)¡M k 1 •":" in L

The results by Carlen and Carvalho have been applied successfully to

several situations, for example in the context of an hydrodynamical–

type limit, or in order to study the trend to equilibrium when initial

data have inﬁnite entropy.

We also note that the optimal rate of convergence for the Boltz-

mann equation with Maxwellian molecules was recently obtained by

Carlen, Gabetta and Toscani in [6], using a completely diﬀerent ap-

proach, which does not seem easily adaptable to other potentials.

We ﬁnally mention that a general but somewhat weaker entropy

dissipation inequality was established by the ﬁrst author of this work

in [11] (Cf. also [28]) for various collision operators including Landau

(under suitable assumptions on Ψ). This estimate would imply a local

(in velocity variable) convergence towards equilibrium, in some sense,

¡1=2which is essentially in O(t ), if the solutions were known to satisfy

certain additional technical assumptions. The results of this paper are

not used here. Some of its ideas are however retained (Cf. section 3).

Weshallnottryheretousethespectralgapofthelinearizedoperator

(Cf. [10]),butweshallfocusasin[4]and[5]ontheentropydissipation

D(f). In fact, as we shall show in section 7, from our work one can

recoveranexplicitestimateofthespectralgapofthelinearizedLandau

operator in the case of overmaxwellian molecules. Such an estimate

seems diﬃcult to obtain by classical methods using Weyl’s theorem

(i.e. thepropertythattheessentialspectrumisunchangedbycompact

perturbations).

Our study relies on the use of the Fisher information,

Z Z

2 pjrfj 2(17) I(f)· =4 jr fj ;

f

whichhasalreadybeensuccessfullyusedinrelatedproblemsbyCarlen

and Carvalho [4] and Toscani [22] (respectively for the Boltzmann and

the linear Fokker–Planck equations). In particular, our use of the log-

arithmic Sobolev inequality was inspired by this last work.

We now state our main result. For s>0, we use the notation

‰ ?Z

1 N 1 N 2 s=2L (R )= f 2L (R ); jf(v)j(1+jvj ) dv < +1 :s

1 NTheorem 1. Let f be in L (R ), and Q a be Landau operator with2

2overmaxwellian cross section (i.e. Ψ(jzj) ‚ jzj ). Then there existsON THE HOMOGENEOUS LANDAU EQUATION 7

f f 2‚>0, explicitly computable and depending only on ‰ ;u ; and the N

scalars

Z

f

P = f(v)vv dv;i jij

N

such that

‡ ·

f(18) D(f)‚‚ I(f)¡I(M ) :

Theorem1willbeproveninsection2byasimplecomputation,which

relies on explicit calculations done in [24] for Maxwellian molecules

2(that is, Ψ(jzj) =jzj ). The reader may ﬁnd this proof somewhat un-

satisfactory, in that it seems to be heavily dependent on the particular

structure of the Landau equation with Maxwellian molecules. This is

why we present in section 3 another proof, which has interest in itself,

and gives the same result (though with a smaller ‚).

Section 4 is devoted to the applications of theorem 1 to the Landau

1equationwithovermaxwelliancross sections. Exponentialdecay (in L

norm)isprovenwithanexplicitrate. Aninterestingfeedbackproperty

due to the nonlinearity of the equation is detailed. It allows one to get

better constants in the rate of decay than one could expect at ﬁrst

sight.

However, the relaxation times obtained in section 4 are still very

large if the initial datum is only assumed to have ﬁnite mass, energy

and entropy. In section 5, we show how to remove this drawback,

and get realistic relaxation times, with a rate which diﬀers from the

(optimal)relaxationrateforthelinearFokker-Planckequationonlyby

a factor 2=3 (in dimension 3). At this point, the entropy dissipation

1is used as a control of the concentration, or equivalently the weak L

compactness.

Then, in section 6, we deal with “true” hard potentials. Algebraic

decay with explicit constants is proven for the solution of the Landau

equation in this case. Again, adapting the method described above, it

is possible to ﬁnd realistic relaxation times. Thanks to the result in

1 N[13], we also prove that the convergence holds in fact in H (R ), and

that the solution is globally (in time) stable with respect to its initial

datum. As in [13], this global stability also holds with respect to small

perturbations of the cross section Ψ.

In section 7, we show how our work can be used to get estimates for

the linearized Landau kernel. Namely, inequalities such as (18) enable

ustoﬁndsimplereﬁnementsofresultsofDegondandLemou(Cf. [10]).

Finally, in section 8, we give a last application of inequality (18).

Namely, under rather weak conditions, the weak cluster points f of

R8 L. DESVILLETTES AND C. VILLANI

asymptotically grazing solutions of the Boltzmann equation (Cf. [13])

2 + 1 Nhave automatically a square root belonging to L (R ;H (R )).t vloc

Useful results concerning the deﬁnition and approximations of D(f)

are given in Appendix A and B.

To conclude this introduction, we mention that among the numer-

ous remaining open problems in this subject, the possibility of ﬁnding

estimates in the case of soft potentials (i.e. ? < 0) seems to be a

particularly interesting and diﬃcult question.

2. Entropy dissipation : first method

First proof of theorem 1. In sections 2 and 3, we shall assume that

Z Z Z

2(19) f(v)dv =1; f(v)vdv =0; f(v)jvj dv =N;

N N N

which amounts to a simple change of coordinates of the form

N(20) (t;v)¡!(at;bv+c); a;b2R;c2R :

Then, M will denote the centered Maxwellian with normalized mass

and temperature,

2

jvj

¡

2e

f(21) M(v)=M (v)= :

N=2(2…)

We recall that

I(M)=N:

Next, it is clear that the entropy dissipation depends linearly on Ψ. In

particular,ifD istheentropycorrespondingtotheLandauk

operator with cross section Ψ ,k

Ψ ‚Ψ =)D (f)‚D (f):2 1 2 1

2Therefore, we only need to prove Theorem 1 for Ψ(jzj) =jzj . More-

over, thanks to the lemma of Appendix B, it is enough to prove the-

2orem 1 when f 2 S and jlogfj is bounded by C(1+jvj ) for some

C >0. Note that as far as the evolution problem for the Landau equa-

tion with reasonable potentials is concerned, we need theorem 1 only

for such smooth functions f, thanks to our study in [13]. We shall

however need theorem 1 in its full generality when giving applications

totheregularityoftheweakclusterpointstoBoltzmannequation(Cf.

section 8).

It is shown in [24] that for a Maxwellian cross section and a normal-

ized initial datum,

‡ ·

(22) Q(f;f)=Nr¢(rf +fv)¡ P @ f +r¢(fv) +Δ f;ij ij ?`

RRRON THE HOMOGENEOUS LANDAU EQUATION 9

where Δ denotes the Laplace-Beltrami operator of spherical diﬀu-?`

sion. We recall that in the physically realistic case N = 3, using

the usual spherical coordinates (r;?;`) deﬁned by v = rsin?cos`,1

v =rsin?sin`, v =rcos?, the action of the Laplace-Beltrami oper-2 3

ator is deﬁned by

1 1

Δ f = @ (sin?@ f)+ @ f:?` ? ? ``2sin? sin ?

We ﬁrst note that the contribution of Δ to the entropy dissipation?`

is nonnegative. Indeed (supposing that N =3 for simplicity),

Z

2Δ f logfr sin?drd d`?`

Z Z

12 2= @ (sin?@ f)logfr drd d` + @ flogf r sin?drd d`? ? `` 2sin ?

Z Z

2 2j@ fj j@ fj 1? `2 2=¡ r sin?drd d` ¡ r sin?drd d`2f f sin ?

Z

2jr fj?`

(23) =¡ ;

f

where (in polar coordinates)

0 1

0

B C@ f?r f = :@ A?` 1

@ f`

sin?

A direct proof without spherical coordinates is also obtained very

easily : indeed,

X¡ ¢

2Δ f = jvj – ¡vv @ f¡(N¡1)v¢rf;?` ij i j ij

ij

whence (integrating by parts)

Z ZX ¡ ¢@f@ fi j2Δ flogf =¡ jvj – ¡vv •0:?` ij i j

f

ij

The quantity (23) vanishes only for radially symmetric functions,

and we note that it is very large when r ! +1 compared with the

entropydissipationinducedbyausualLaplaceoperator. Thissuggests

thatsolutionstotheLandauequationhaveatendencytobecomeradial

rather fast. However, we shall not use this information in the sequel.

Noting that

Z Z Z

fv¢r(logf)= v¢rf =¡N f =¡N;10 L. DESVILLETTES AND C. VILLANI

we easily see that the entropy dissipation induced by the terms in (22)

other than Δ is?`

Z Z

2jrfj @f@ fi j

(24) N ¡P ¡N(N¡1):ij

f f

As in [24], we can always assume that (P ) is diagonal. Indeed,ij

Z

Nq :~e2R 7¡! f(v)(v;~e)(v;~e)dv

deﬁnes a (nonnegative) quadratic form, so that there exists an or-

thonormal basis (e~;:::;e~ ) which is also orthogonal for q. Let us1 N

deﬁne the ”directional temperatures”

Z

2T = fv :i i

With these notations,

Z

2X (@f)i

(25) D(f)‚ (N¡T ) ¡N(N¡1):i

f

i

We recall the following elementary lemma (Cf. also [25]).

Lemma 1. Consider ﬁ ;::;ﬁ ‚0. Under the constraints1 N

Z Z Z

2f ‚0; f =1; fv =0; fv =T;i ii

one has

Z

2X X(@f) ﬁi i

(26) ﬁ ‚ :i

f Ti

i i

Remark. This lemma is obviously one version (among others) of the

Heisenberg uncertainty principle (or Cram´er-Rao inequality).

Proof. Firstnote that, bydensity, it is suﬃcient to treat the case when

f is smooth and nonvanishing. Then

Z ? ¶2

@f @gi i

(27) 0•ﬁ ¡ f;i

f g

where

2

vi¡Y 2Tie

g(v)= p ;

2…Tii

and @g =¡(v=T )g. Expanding (27), we obtaini i i

Z Z Z

2 2(@f) v vi i i0•ﬁ +2ﬁ @f +ﬁ fi i i i 2f T Ti i