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Exrait
ON THE SPATIALLY HOMOGENEOUS LANDAU
EQUATION FOR HARD POTENTIALS
PART I : EXISTENCE, UNIQUENESS AND
SMOOTHNESS
L. DESVILLETTES AND C. VILLANI
Abstract. We study the Cauchy problem for the homogeneous
Landau equation of kinetic theory, in the case of hard potentials.
Weprovethat fora large class of initial data, there exists a unique
weaksolutiontothisproblem,whichbecomesimmediatelysmooth
and rapidly decaying at inﬁnity.
Contents
1. Introduction 1
2. Preliminaries and main results 4
2.1. Notations 4
2.2. Main deﬁnitions 6
2.3. Main results 11
3. Appearance and propagation of moments 19
4. Ellipticity of the diﬀusion matrix 28
5. Approximated problems 31
5.1. The approximated nonlinear equation 32
5.2. H older estimates for a and b 35ij i
5.3. Uniqueness for a linear parabolic equation 38
6. Smoothing eﬀects 41
7. Initial data with inﬁnite entropy 50
8. Uniqueness by Gronwall’s lemma 53
9. in a wider class 59
10. Maxwellian lower bound 62
References 65
1. Introduction
The spatially homogeneous Landau equation (also called Fokker–
Planck–Landau) is a common model in kinetic theory (Cf. [5, 24]). It
reads
12 L. DESVILLETTES AND C. VILLANI
@f
(1) =Q(f;f);
@t
+where f(t;v)‚0 is the density of particles which at time t2R have
Nvelocity v 2R (N ‚ 2). The kernel Q(f;f) is a quadratic nonlocal
operator acting only on the v variable and modelling the eﬀect of the
(grazing) collisions between particles. It is deﬁned by the formula
(2) ‰ ? ¶ Z
@ @f @f
Q(f;f)(v)= dv a (v¡v ) f(v ) (v)¡f(v) (v ) :⁄ ij ⁄ ⁄ ⁄
@v N @v @vi j j
Here as well as in the sequel, we use the convention of Einstein for
repeated indices.
The nonnegative symmetric matrix (a ) is given by the formulaij i;j
? ¶
zzi j
(3) a (z)= – ¡ Ψ(jzj);ij ij 2jzj
where the nonnegative function Ψ only depends on the interaction be-
tween particles.
ThisequationisobtainedasalimitoftheBoltzmannequationwhen
grazing collisions prevail. See [35] for instance for a detailed study of
the limiting process, and further references on the subject.
Itishomogeneousinthatoneassumesthatthedistributionfunction
does not depend on the position of the particles, but only on their
velocities. We mention that very little is known in the inhomogeneous
case for large data (Cf. [25, 33]), while the spectral properties of the
linearized equation have been addressed in [9].
We are only concerned here with so–called hard potentials, which
means that
?+2(4) 9Λ>0;? 2(0;1]; Ψ(jzj)=Λjzj :
This case corresponds to interactions with inverse s¡power forces for
s>2N¡1.
Infact,mostofourstudycanbeextendedtothecasewhenΨ(jzj)=
2jzj ´(jzj) for some continuous function ´ such that ´ is smooth for
jzj>0 and ´(jzj)!+1 asjzj! +1. In particular, the assumption
that 0 < ? • 1 in (4) can easily be relaxed to 0 < ? < 2, and even to
? > 0if slightchangesin the assumptions on the initial data are made.
However, weshallkeeptheexpressiongivenby (4)inthesequelforthe
sake of simplicity.
On the other hand, the very particular case of Maxwellian molecules
? =0 is quite diﬀerent. It is studied in detail in [34].
RSPATIALLY HOMOGENEOUS LANDAU EQUATION 3
Finally, just as for the Boltzmann equation, little is known for soft
potentials, i.e. ? < 0 (Cf. [2, 12, 20, 35]), and even less for very
soft potentials, i.e. ? < ¡2 (Cf. [35]). These appear as a challenge
for future research, especially the very interesting and diﬃcult case
? =¡3, corresponding to the Coulomb interaction.
In this paper, we give a detailed discussion of the Cauchy problem
for equation (1) – (4), and we precise the qualitative properties of the
solutions. In particular, we are interested in smoothing eﬀects. Our
results can be summarized in the following way : under rather weak
hypotheses on the initial data, there is a unique (weak) solution f to
eq. (1) – (4). Moreover for all time t> 0, f(t;¢) belongs to Schwartz’s
space of rapidly decreasing smooth functions, and is bounded from
below by a Maxwellian distribution. Precise statements are presented
in section 2.
The organization of the paper is as follows. First of all, the decay
when jvj ! +1 of the solutions of (1) – (4) is studied in section 3.
1We prove there that all moments (in L ) of the solutions immediately
become ﬁnite. Then, a lemma of ellipticity used throughout the paper
is given in section 4. In section 5, we design convenient approximated
equations : they will be useful to rigorously justify many of the for-
mal manipulations that will be performed on solutions of the Landau
equation. The smoothing eﬀects are studied in sections 6 and 7. In
sections 8 and 9, the problem of uniqueness is addressed. Finally, in
section 10, we investigate the properties of positivity of the solution
of (1) – (4).
The Cauchy problem for the homogeneous Landau equation has al-
readybeenstudiedbyArsen’evandBuryak(Cf. [3])inthecasewhenΨ
is smooth and bounded, and when initial data are smooth and rapidly
decreasing. Even though the framework of their paper is very diﬀerent
from ours, we shall retain some of their ideas here (in particular in
sections 5, 7 and 9).
The Boltzmann equation for hard potentials has been extensively
studied under the hypothesis of angular cutoﬀ of Grad (Cf. [8], [21]),
that is, when the eﬀect of grazing collisions is neglected (Cf. [1, 12,
22, 26, 28, 36]). It is known that in this context there is a pointwise
Maxwellian lower bound, while the smoothing property does not hold,
and apparently has to be replaced by the much weaker statement that
1all the moments (in L ) of f immediately become ﬁnite, and that the
smoothness is propagated. Very little is known when one does not4 L. DESVILLETTES AND C. VILLANI
make the cutoﬀ assumption [2, 13, 16]. Our work supports the general
conjecture that smoothing eﬀects are associated to grazing collisions.
Thisconjectureisinfactprovenincertainparticularcases(Cf. [13,14,
15, 27]). We mention that the proof given below is much less technical
than the ones given in the aforementioned works, and essentially does
notdependonthedimension. Infact,itseemstobeageneralrulethat
the Landau equation is simpler to study than the Boltzmann equation,
or at least than the Boltzmann equation without angular cutoﬀ, in the
same way as derivatives are usually simpler to handle than fractional
derivatives.
Inafollowingcompanionpaper,weshallstudythelong–timebehav-
ior of the solution to (1) – (4), and give precise estimates for the speed
of convergence towards equilibrium. Here again, we shall obtain much
betterandsimplerresultsthanwhatisknownfortheBoltzmannequa-
tion. We mention that these results can actually help for the study of
thetrendtowardsequilibriumfortheBoltzmannequation(Cf. [7,32]).
Acknowledgement : The authors thank S. Mischler and H. Zaag
for several fruitful discussions during the preparation of this work.
2. Preliminaries and main results
2.1. Notations. In all the sequel, we shall assume for simplicity that
N =3, which is the physically realistic case. For s‚0;p‚1, we set
Z
2 s=2kfk 1 = jf(v)j(1+jvj ) dv =M (f);L ss
3
Z
p p 2 s=2kfk p = jf(v)j (1+jvj ) dv;
Ls
3
ZX
2 2 2 s=2kfk = j@ f(v)j (1+jvj ) dv;k ﬁHs
3
0•jﬁj•k
3where ﬁ =(i ;i ;i )2N ,jﬁj=i +i +i , and1 2 3 1 2 3
i i i1 2 3@ f =@ @ @ f:ﬁ 1 2 3
1 3˙We shall also use homogeneous spaces like H (R ), and their normss
deﬁned by
Z
2 2 2 s=2kfk = jrf(v)j (1+jvj ) dv:˙ 1Hs 3
RRRRSPATIALLY HOMOGENEOUS LANDAU EQUATION 5
T
k 3 3 1We recall that H (R ) is Schwartz’s spaceS(R ) of C func-sk‚0;s‚0
tions whose derivatives of any order decrease at inﬁnity more rapidly
¡1than any power ofjvj .
For a given initial datum f , we shall use the notationsin
Z Z
1
2M = f (v)dv; E = f (v)jvj dv;in in in in
3 2 3
Z
H = f (v)logf (v)dv;in in in
3
for the initial mass, energy and entropy.
It is classical that if f ‚ 0 and M ;E ;H are ﬁnite, then fin in in in in
belongs to
‰ Z ?
¡ ¢
3 1 3LlogL(R )= f 2L (R ); jf(v)jjlog jf(v)j jdv < +1 :
3
We shall use the standard notation f =f(v ) (and ` =`(v ), etc...).⁄ ⁄ ⁄ ⁄
Moreover,
zzi j
Π (z)=– ¡ (z =0)ij ij 2jzj
?will denote the orthogonal projection upon z (the plane which is or-
thogonal to z). By rescaling time if necessary, we shall consider in the
sequel only the case Λ=1 in (4), so that
?+2a (z)=jzj Π (z); (? 2(0;1]):ij ij
2 3We note that a belongs to C (R ), and thatij
?+2tr (a )(z)=a (z)=2jzj :ij ii
Next, we deﬁne
?(5) b (z)=@ a (z)=¡2jzj z;i j ij i
?(6) c(z)=@ a (z)=¡2(? +3)jzj ;ij ij
and when no confusion can occur,
a =a ⁄f; b =b ⁄f; c=c⁄f:ij ij i i
ff fSometimes we shall write a , b , c instead of a , b and c to recall theij iij i
dependence upon f.
RRRR66 L. DESVILLETTES AND C. VILLANI
2.2. Main deﬁnitions. With these notations, the Landau equation
can be written alternatively under the form
¡ ¢
(7) @ f =r¢ arf¡bf ;t
or
(8) @ f =a @ f¡cf:t ij ij
At the formal level, one can see that the solutions of eq. (1) – (4)
satisfy the conservation of mass, momentum and energy, that is,
Z Z
(9) M(f(t;¢))· f(t;v)dv = f (v)dv =M ;in in
3 3
Z Z
(10) f(t;v)vdv = f (v)vdv;in
3 3
Z Z
2 2jvj jvj
(11) E(f(t;¢))· f(t;v) dv = f (v) dv =E ;in in
3 2 3 2
and the entropy dissipation identity (i.e. the H-theorem)
(12) Z Z
d d
H(f(t;¢))· f(t;v) logf(t;v)dv = Q(f;f)(t;¢) logf(t;¢)
dt dt 3 3
Z Z ? ¶
1 @f @fi i
=¡ a (v¡v )ff (v)¡ (v )ij ⁄ ⁄ ⁄
2 3 3 f f£
? ¶
@ f @ fj j
(v)¡ (v ) dvdv •0:⁄ ⁄
f f
Let us now recall a deﬁnition from [35] (see also [20]).
1 3Deﬁnition 1. Let f 2L (R ) and f ·f(t;v) be a nonnegative func-in 2
+ + +1 1 3 1 1 3 0 3tionbelongingtoL (R ;L (R ))\L (R ;L (R ))\C(R ;D(R )),t 2 v loc t 2+? v t v
and such that E(f(t;¢))• E . Such a function f is called a weak so-in
lution of the Landau equation (1)–(4) with initial datum f if ’ ·in
+ 3’(t;v)2D(R £R ),t v
Z Z Z Z Z+1 +1
(13) ¡ f ’(0)¡ dt f@ ’= dt Q(f;f)’;in t
0 0
RRRRRRRRRRSPATIALLY HOMOGENEOUS LANDAU EQUATION 7
where the last integral is deﬁned by
(14)Z Z Z
Q(f;f)’= a f@ ’+2 bf@’ij ij i i
Z Z ‡ ·1
= dvdv ff a (v¡v ) @ ’+(@ ’)⁄ ⁄ ij ⁄ ij ij ⁄
2
Z Z ‡ ·
+ dvdv ff b (v¡v ) @’¡(@’) :⁄ ⁄ i ⁄ i i ⁄
Note that under our assumptions on f, each term of (14) is well–
deﬁned. Indeed, we have the estimates
ﬂ ﬂ‡ ·ﬂ ﬂ
?+2 ?+2ﬂ ﬂa (v¡v ) @ ’+(@ ’) •C(1+jvj +jv j );ij ⁄ ij ij ⁄ ⁄ﬂ ﬂ
ﬂ ﬂ‡ ·ﬂ ﬂ
?+2 ?+2ﬂ ﬂb (v¡v ) @’¡(@’) •C(1+jvj +jv j );i ⁄ i i ⁄ ⁄ﬂ ﬂ
and the integrals in the deﬁnition are well-deﬁned in view of the in-
equality
Z ZT
?+2 ?+2 2dt dvdv ff (1+jvj +jv j )•Tkfk +⁄ ⁄ ⁄ 11 3L ( ;L ( ))t 2 v
0
+2kfk + kfk 1 1 3 :1 1 3 L ([0;T];L ( ))L ( ;L ( )) 2+? vt v
In fact, by a straightforward density argument, it suﬃces that ’2
+ +2 3 1 3 2 ¡1C (R ;C (R ))\C (R ;C(R )) and @ ’;(1+jvj ) @ ’ be boundedc ij tt v c t v
+ 3onR £R .t v
The formulation of deﬁnition 1 seems to be the weakest available
+ 0 3one. It should be noted that the assumption f 2 C(R ;D(R )) is int v
fact a consequence of the other assumptions.
Wealsomentionanotherweakformulationwhichisvalidwhenmore
1regularity is available (say, f in a suitable weighted H -type space) :
Z Z Z
(15) Q(f;f)’=¡ arfr’+ fb¢r’;
where(asweshalloftendointhesequel)weusethenotationarfr’=
a @f@ ’:ij i j
Then, we recall the Boltzmann equation (in dimension 3 and for
inverse power forces for the sake of simplicity),
(16)
Z Z Z2… …
0 0@ f = dv d` d K (jv¡v j)‡(?)(f f ¡ff )·Q (f;f);t ⁄ ⁄ ⁄ B⁄
3 0 0
RRRRRR8 L. DESVILLETTES AND C. VILLANI
0 0 0 0where f =f(v), f =f(v ),⁄ ⁄
8
v+v jv¡v j⁄ ⁄> 0v = + ?;>< 2 2
(17)
> v+v jv¡v j> ⁄ ⁄0:v = ¡ ?;⁄ 2 2
and ? is the unit vector whose coordinates are (?;`) in a spherical
system centered at (v+v )=2 and with axis v¡v .⁄ ⁄
The following assumption on Q (“hard potentials”) will systemat-B
ically be made in the sequel.
Assumption A: The “kinetic” cross section K is of the form
?(18) K(jzj)=jzj ; ? 2(0;1];
and the “angular” cross section ‡ is a nonnegative function, locally
bounded on (0;…] with possibly one singularity at ?=0, such that
2 1(19) ?7¡!? ‡(?)2L (0;…):
Note that in the “physical” cases,
(?¡3)=2(20) ‡(?)»C? as ?!0;
forsomeC >0,sothatthesingularityisnonintegrable,butassumption
A is still satisﬁed.
Remark.Itisclearthat,changing‡ ifnecessary,onecanallowK(jzj)=
?bjzj for any b>0.
The following deﬁnition is also taken from [35].
1 3Deﬁnition 2. Let f 2L (R ) and f ·f(t;v) be a nonnegative func-in 2
+ + +1 1 3 1 1 3 0 3tionbelongingtoL (R ;L (R ))\L (R ;L (R ))\C(R ;D(R )),t 2 v loc t 2+? v t v
and such that E(f(t;¢))•E . Then, f is called a weak solution of thein
Boltzmann equation (16) – (17) under assumption A and with initial
+ 3datum f if for all ’·’(t;v)2D(R £R ),in t v
Z Z Z Z Z+1 +1
¡ f ’(0)¡ dt f@ ’= dt Q(f;f)’;in t
0 0
where the last integral is deﬁned by
Z
Q(f;f)’=
Z Z Z Z
1 ? 0 0¡ dvdv ff jv¡v j d`d ‡ (?)(’ +’ ¡’¡’ ):⁄ ⁄ ⁄ ⁄⁄4SPATIALLY HOMOGENEOUS LANDAU EQUATION 9
Note that once again, one can enlarge the space of admissible ’
0thanks to a density argument, and the assumption of continuity in D
is an automatic consequence of the other assumptions. Let us mention
also that if one assumes condition (20) instead of (19), then one can
+1 1 3dispend with the condition that f 2L (R ;L (R )) to give a senseloc t 2+? v
to the solutions.
We now can precisely state the links between the Boltzmann and
Landauequations(inthecaseofhardpotentials). Wegivethefollowing
deﬁnition (Cf. also [35]).
1Deﬁnition 3. Let f 2 L , and let (‡ ) be a family of “angular”in " ">02
cross sections satisfying (19). We shall say that (‡ ) is “concentrat-" ">0
ing on grazing collisions” if
for all ? >0; ‡ (?)¡¡!0 uniformly on [? ;…);0 " 0
"!0
and for some real number Λ>0,
Z …… ?2d sin ‡ (?)¡¡!Λ:"
"!02 20
We recall that this last quantity is related to the total cross section
for momentum transfer (Cf. [29]).
Deﬁnition 4. Let (‡ ) be a family of “angular” cross sections con-" ">0
?centrating on grazing collisions, and let K (jzj) = K(jzj) =jzj , (? 2"
(0;1]); be a ﬁxed “kinetic” cross section. Let us denote by Q theB"
corresponding Boltzmann collision operator. We shall deﬁne a family
of asymptotically grazing solutions of the Boltzmann equation with ini-
tial datum f as a family (f ) of weak solutions of the Boltzmannin " ">0
equation (
@ f =Q (f ;f );t " B " ""
f (0)=f" in
in the sense of deﬁnition 2.
The following result can be found in [35]. It gives a ﬁrst proof of
existence for equation (1) – (4).
Theorem 1. Let ? 2 (0;1], (‡ ) be a sequence of “angular” cross" ">0
sections concentrating on grazing collisions, and K satisfy (18). Let
1 3f 2L \LlogL(R ) for some – >0. Thenin 2+–
(i) There exists a family (f ) of asymptotically grazing solutions" ">0
of the Boltzmann equation with initial datum f .in
(ii) One canextract fromthe family (f ) a subsequenceconverging" ">0
p + 1 3weakly in L (R ;L (R )) for all 1 • p < +1 to some function ft vloc10 L. DESVILLETTES AND C. VILLANI
which is a weak solution of the Landau equation (1) – (4) with initial
datum f . Moreover, for all time t‚0, f satisﬁes the conservation ofin
mass and momentum (9), (10), and the decay of energy and entropy
Z Z
2 2jvj jvj
(21) E(f(t;¢))· f(t;v) dv• f (v) dv =E ;in in
2 2
(22) Z Z
H(f(t;¢))· f(t;v)logf(t;v)dv• f (v)logf (v)dv =H :in in in
Remarks.
1(1) Theassumptionthatf beinL forsome– > 0maypossiblyin 2+–
be dispended with, but we shall not try to do so. In fact, in
view of recent computation by X. Lu, it seems natural to con-
jecture that the optimal condition for existence is the ﬁnitenessR
2 2of f (v)(1+jvj )log(1+jvj )dv. Indeed, for the Boltzmannin
1 1equation, this condition is equivalent to f 2L (L ).t ?+2
(2) For ? > 2, the corresponding assumption would be that f 2in
1L for some – > 0.
?+–
Of course, it is also possible to give a direct proof of existence for
the Landau equation, thanks to a convenient approximated problem.
For example, we give the
Deﬁnition5. Afamily(Ψ ) iscalledafamilyofapproximatedcross" ">0
1 +sections for eq. (1) – (4) if Ψ is a bounded C function onR which"
¡1coincides with Ψ for 0 < " <jzj < " and satisﬁes the estimates (for
jzj>0)
?+2jzj 2+?<Ψ (jzj)•1+jzj ;"
2
Ψ (jzj)"0Ψ (jzj)•(2+?) :" jzj
We denote ? ¶
zzi j"a (z)= – ¡ Ψ (jzj);ij "ij 2jzj
" " " "and b =@ a , c =@b .j ii ij i
"Then, a family (f ) is called a family of approximated initial data">0in
1 3 " 1 3of f (2L \LlogL(R ) for some – > 0) if f 2C (R ), satisﬁesin 2+– in
2 2jvj jvj00 ¡– " ¡–"" 2 2C e ‚f (v)‚C e"" in
0 0 " 1for some C ;C ;– ;– > 0, and if f converges (strongly) in L \" "" " in 2+–
3LlogL(R ) for some – > 0 towards f .in
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