ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR HARD POTENTIALS

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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 the spatially homogeneous Landau equation with hard potentials in terms of the entropy itself. We deduce from this explicit estimates on the speed of convergence towards equilibrium for the solution of this equation. In the case of so-called overmaxwellian potentials, the convergence is exponential. We also compute a lower bound for 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 dissipation : second method 13 4. The trend towards equilibrium : overmaxwellian case 16 5. Improved results 18 6. The trend towards equilibrium : the case of true hard potentials 21 7. Poincare-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]), (1) ∂f∂t (t, v) = Q(f, f)(t, v), v ? R N , t ≥ 0, 1

• landau equation


• course maxwellian molecules


• dissipation


• spatially homogeneous


• boltzmann equa- tion


• cross section


• maxwellian molecules


• entropy dissipation

Sujets

• Dissipation
• Cross section

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 172
• Langue: English

Extrait

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 Maxwelli