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


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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
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