ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES

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ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES C. VILLANI Abstract. We establish a simplified form for the Landau equa- tion with Maxwellian-type molecules. We study in detail the Cauchy problem associated to this equation, and some qualitative features of the solution. Explicit solutions are also given. Contents 1. Introduction 1 2. Simplified Expression 3 3. The isotropic case : Fokker-Planck equation 5 4. The collision operator and the Cauchy problem 8 5. Final decomposition of the Landau collision operator 14 6. Weak formulations and applications : time-evolution of the moments, Maxwellian tails, energy estimates 15 7. Positivity 21 8. Long-time behaviour 22 9. Self-similar solutions 26 Appendix A. Inverse Fourier Transform of e?|?|2/2|?|? 27 Appendix B. The Nikolskii Transform 28 References 29 1. Introduction The Landau equation (also called sometimes Fokker-Planck) is a common kinetic model in plasma physics. It is a nonlinear partial differential equation where the unknown function, f , is the density of a “gas” in the phase space of all positions and velocities of “particles”. We shall assume that these vary in RN , N ≥ 2. In the case of a gas composed of a single species and if we assume that the density function is spatially homogeneous, ie does not depend on the position but only 1

  • landau equation

  • ?ij ?

  • bi ?

  • landau collision

  • spatially homogeneous

  • ∆f ? ∑

  • landau equa- tion only


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ON THE SPATIALLY HOMOGENEOUS LANDAU
EQUATION FOR MAXWELLIAN MOLECULES
C. VILLANI
Abstract. We establish a simplified form for the Landau equa-
tionwithMaxwellian-typemolecules. WestudyindetailtheCauchy
problem associated to this equation, and some qualitative features
of the solution. Explicit solutions are also given.
Contents
1. Introduction 1
2. Simplified Expression 3
3. The isotropic case : Fokker-Planck equation 5
4. The collision operator and the Cauchy problem 8
5. Final decomposition of the Landau collision operator 14
6. Weak formulations and applications : time-evolution of the
moments, Maxwellian tails, energy estimates 15
7. Positivity 21
8. Long-time behaviour 22
9. Self-similar solutions 26
2¡j»j =2 ‚Appendix A. Inverse Fourier Transform of e j»j 27
Appendix B. The Nikolskii T 28
References 29
1. Introduction
The Landau equation (also called sometimes Fokker-Planck) is a
common kinetic model in plasma physics. It is a nonlinear partial
differential equation where the unknown function, f, is the density of
a “gas” in the phase space of all positions and velocities of “particles”.
NWe shall assume that these vary inR , N ‚ 2. In the case of a gas
composedofasinglespeciesandifweassumethatthedensityfunction
is spatially homogeneous, ie does not depend on the position but only
12 C. VILLANI
on the velocity, the Landau equation takes the following form
(1) ( " #)Z
@f @ @f(v) @f(v )⁄
=Q(f;f)= dv a ( v¡v ) f(v ) ¡f(v) ;⁄ ij ⁄ ⁄
@t @v N @v @vi j ⁄j
Nfor t‚0, v2R , and the unknown function is assumed to be nonneg-
ative, integrable together with its moments up to order 2. Here and
below, we shall always use the convention of implicit summation over
repeated indices. The matrix (a (z)) is nonnegative symmetric, andij
depends on the interaction between particles. If we assume that any
stwo particles at distance r interact with a force proportional to 1=r ,
then
?+2 2(2) a (z)=Λjzj (– ¡zz =jzj );ij ij i j
with ? = (s¡5)=(s¡1) for N = 3, and Λ is some positive constant
that we shall normalize to be 1.
ThisequationisobtainedasalimitoftheBoltzmannequation,when
allthecollisionsbecomegrazing. See[21]forbackgroundandreferences
on the subject. To avoid confusions with other kinetic equations, we
shall never refer to it as the Fokker-Planck equation.
The study of the spatially homogeneous equation, besides its intrisic
interest, is important for numerical applications : indeed, simulation
algorithms generally make a “splitting” and consider separately the
variation of f due to the gas inhomogeneities, and the one due to
collisions.
Aparticularlysimplecaseoccurswhens=5inthethree-dimensional
case, or more generally s = 2N¡1 : such molecules are the so-called
“Maxwellianmolecules”. Althoughthesehavebeenintensivelystudied
for the Boltzmann equation (see for example [3]), we are aware of very
little work in that direction in the frame of Landau equation. Note
that Maxwellian molecules are commonly used in numerical simula-
tions [5, 14]. The purpose of our study is to investigate this simple
case as completely as possible, in the hope to have a better under-
standing of the Landau equation in general. In fact, from the study of
Maxwellian molecules, one can also deduce general results (see [9, 19]).
A wide class of potentials is studied in other works [8, 9].
It is also of interest to compare the results obtained in the frame
of the Landau equation, to those already known for the Boltzmann
equation. Indeed, the Landau equation takes into account only “graz-
ing collisions”, which often entail great mathematical difficulties in the
studyoftheBoltzmann equation–thoughthesecan generallybe over-
come in the homogeneous Maxwellian case [3].
RLANDAU EQUATION FOR MAXWELLIAN MOLECULES 3
In section 2 below, we shall obtain a simplified expression for the
Landau equation with Maxwellian molecules, formula (9); after a brief
discussion of the special case of isotropic distributions, where explicit
solutionsareeasilyavailable,weshallstudyinsection4theformofthe
collisionoperatorinthegeneralcase, thenturntotheCauchyproblem
associated to the equation, and insist on its regularizing properties.
This will lead us to rewrite the Landau collision operator as the sum
of several operators. For the convenience of the reader, this alternative
form is recast in section 5, which is entirely self-contained. Sections 6
to 8 are devoted to some qualitative features of the solutions, as the
decaytoequilibrium,thetemperaturetails,andthepositivity. Finally,
in section 9, we exhibit a family of particular self-similar solutions.
2. Simplified Expression
We can rewrite the Landau equation as
? ¶
@f @ @f
(3) = a ¡bf ;ij i
@t @v @vi j
where a = a ⁄f, b = b ⁄f, b = @ a . It is easily checked that, atij ij i i i j ij
least formally, the following quantities are conserved
Z Z Z
1 2(4) f(v)dv; f(v)vdv; f(v)jvj dv;
2
corresponding respectively to the mass, momentum and kinetic energy
of the whole gas. We set
Z Z Z
2(5) f =M; fv =MV; fjvj =2E;
Nwhere M >0, V 2R , E >0. The equation can also be rewritten, at
least formally,
(6) @ f =a @ f¡cft ij ij
where c=c⁄f, c=@b . It is in this form that we shall study it. Now,i i
in the Maxwellian case,
8 ? ¶
vv> i j2 2> a (v)=jvj – ¡ =jvj – ¡vv ;ij ij ij i j> 2> jvj><
b (v)=¡(N¡1)v;> i i>>: c(v)=¡N(N¡1):4 C. VILLANI
Writing f =f(v ), we find⁄ ⁄
Z Z
2a = jv¡v j – f dv ¡ (v¡v ) (v¡v ) f dvij ⁄ ij ⁄ ⁄ ⁄ i ⁄ j ⁄ ⁄
Z
¡ ¢
2=– Mjvj +2E¡2M(V ¢v) +M(Vv +V v ¡vv )¡ fvv ;ij i j j i i j i j
Z
b =¡(N¡1) (v¡v )f dv =¡(N¡1)M(v ¡V );i ⁄ i ⁄ ⁄ i i
c=¡N(N¡1)M:
Thus, we are led to compute all second moments of f. Setting
Z
(7) B = fvv ;ij i j
we have (at least formally)
Z
¡ ¢d
B =¡ a @ f¡bf @ (v v )fifl ij j i i fi fl
dt
which, after elementary computation, integrating systematically by
parts, yields the ordinary differential equation
¡ ¢d 2 2B =8– ME +4M NV V ¡– jVj ¡4NMB ;fifl fifl fi fl fifl fifl
dt
the solution of which is
‡ ·
¡4NMtB (t)=B (1)¡ B (1)¡B (0) e :fifl fifl fifl fifl
For example, if V = 0, B (1) = (2E=N)– . We note that thefifl fifl
evolutionofthemomentsoforder2isentirelydeterminedbythe“gross
conditions” M, V, B , of f . This is also true for the Boltzmannij 0
equation [11]. Thus, in turn, the coefficients a ;b;c, depend only onij i
these quantities.
Now, for the sake of simplicity, we assume without loss of generality
˜that V = 0, and we change the unknown function f to f such that
¡N 2˜f =Mfi f(:=fi), fi =2E=(MN). Then, calling the new function f,
we have
Z Z Z
2(8) f =1; fv =0; fjvj =N;LANDAU EQUATION FOR MAXWELLIAN MOLECULES 5
andthecorrespondingequilibriumdistributionissimplytheMaxwellian
with temperature 1,
2¡jvj =2e
M(v)= ;
N=2(2…)
˜while B (1) = – . Finally we rescale time : t = (N ¡ 1)t. Thus,ij ij
equation (1) simply reduces to
8
@ f =a @ f +Nf> t ij ij>>>> ¡ ¢1< 2 ¡fita =– + jvj – ¡vv ¡e Dij ij ij i j ij(9) N¡1
>> ?Z ¶> 4N 1>: fi = ; D = f vv ¡– :ij 0 i j ij
N¡1 N¡1
Thisequationislinear. ItisequivalenttothenonlinearLandauequa-
tion only if we consider those solutions that have mass 1, bulk velocity
zero, and energy N=2. We mention that this expression was obtained
independently (and simultaneously) by Lemou [14] (for N =3).
Before going further, we shall study the simple isotropic case.
3. The isotropic case : Fokker-Planck equation
Assuming that f is radially symmetric, and satisfies the normaliza-
tion conditions (8), we have
Z
fvv =– ; D =0:i j ij ij
2We set f =’(x), x=jvj =2. Thus
2@f @ f0 00 0=v’(x); =vv ’ (x)+– ’(x);i i j ij
@v @v@vi i j
00 0Δf =2x’ (x)+N’(x);
2X @ f 2 00 0vv =4x ’ (x)+2x’(x);i j
@v@vi ji;j
X @f 0v =2x’(x)i
@vi
i
and we notice that
ˆ !
2X X@ f @f2 0jvj Δf¡ vv =2(N¡1)x’(x)=(N¡1) v :i j j
@v@v @vi j j
i;j j6 C. VILLANI
In that case the Landau equation can be rewritten
@f
=Lf =Δf +v¢r f +Nf =r¢(rf +fv):v
@t
This is the so-called linear Fokker-Planck (or Ornstein-Uhlenbeck)
equation, well-known in kinetic theory as well as in probability, and
associated with the so-called Ornstein-Uhlenbeck adjoint semigroup
(its adjoint is Δ ¡v¢r , associated to the standard Dirichlet formv v
for the Gaussian measure). It can be used to describe the relaxation
of Brownian molecules in a gas. Its appearance in topics related to
the Boltzmann equation was already noticed by Bobylev [2, 3] : in
particular, this is satisfied by Bobylev’s explicit solutions.
(We point out that in these papers the Fokker-Planck operator does
not appear as a special case of the Landau equation, but is related to
the invariance properties of the Boltzmann equation.)
The occurrence of the Fokker-Planck equation is not surprising, in
thatitisoneofthemostsimplelinearequationsforwhichconservation
of nonnegativity, mass, energy and decrease of entropy hold (this lastp
one due for example to the Heisenberg inequality applied to f).
The explicit solution of this equation is well-known and can be ob-
tained readily by Fourier-transform techniques :
8 2jvj
¡> 2–e Nt t< f(v)= ⁄e f (e¢);0N=2(2…–)
>:
¡2t– =1¡e :
From this expression the asymptotic behaviour is clear : for t!1
the first term of the convolution product converges to M, while the
second term converges to a Dirac mass. We note that the Ornstein-
Uhlenbeckadjointsemigroupiscommonlyusedtoobtainsmoothinter-
polations between an arbitrary density and its associated Maxwellian
(see [6] for an application in the field of kinetic theory).
In [14] Lemou obtained a different expression of radial solutions of
the spatially homogeneous Landau equation, in the form of a conver-
gent series : it is a tedious but simple matter to check that the two
expressions are equivalent provided that the convergence of all expres-
sions is ensured. In particular, taking as initial data, for N =3,
? ¶
4 2jvj jvj 1
f (v)=M(v) 1+ ¡ + ;0
120 12 8
we recover the simplest of Lemou’s solutions,LANDAU EQUATION FOR MAXWELLIAN MOLECULES 7
? ? ¶¶
4 2jvj jvj 1¡4th(t;v)=M(v) 1+e ¡ + :
120 12 8
The expressions obtained by Lemou are well adapted to treat initial
dataofthetype(Maxwellian£polynomial),whiletheonegivenabove
is more convenient for arbitrary initial data, which is compensated
by a much greater cost in computation (the expression is only semi-
explicit). We indicate briefly a systematic method to change from
one expression to the other. It suffices for that purpose to compute
convolution products of the formM ⁄P(v)M , P being a polynomial1 2
in v and M ;M Maxwellian distributions. These computations are1 2
most easily done in Fourier representation, where they take the form
1 2 1 2M P(D)M ,whereD standsforthederivationoperator,M andM
being Maxwellian distributions. Expanding this expression, we obtain
sums of terms of the form (Maxwellian£ polynomial). Then, we come
back in the usual representation, having only to compute derivatives
of Maxwellians.
Remark 1. We note that Bobylev’s family of invariant solutions [2]
consists precisely in those obtained by applying the adjoint Ornstein-
2Uhlenbeck semigroup to initial data of the form Mjvj (up to a nor-
malization). They satisfy the Boltzmann equation with Maxwellian
potential,
+(10) @ f =Q(f;f)=Q (f;f)¡flf;t
Z
+ 0 0Q (f;f)= d!dv b(fi)f(v)f(v );⁄ ⁄
N¡1 NS £
0 0 v¡v⁄with v =(v¡(v¡v ;!!), v =(v +(v¡v ;!)!), cosfi =( ;!),⁄ ⁄ ⁄⁄ jv¡v j⁄R
and b is a nonnegative function (or measure) such that d!b(fi) = fl.
Using a spherical change of variables with axis v¡v as in [18], we⁄
write b(fi)d! =‡(?)d d`; with fi =…¡2?. Then
Z
2„= ‡(?)sin ?d
is, up to a universal constant, the first eigenvalue of the linearized
collision operator, and the only parameter upon which these solutions
depend. Therefore,theyareinvariantundertheasymptoticsofgrazing
collisions [21] if one chooses a sequence of cross-sections
¡ ¢? ¶ 2 ?sin1 ? "‡ (?)= ‡ 1 :" ?•"…=22" " sin ?
It can be easily checked that (apart from the equilibrium distribution)
they are the only solutions of the Fokker-Planck equation taking the
R8 C. VILLANI
form M(v)P(v), where M is a Maxwellian distribution and P a colli-
sional invariant. We think it likely that they constitue the only simple
class of solutions invariant by the asymptotics of grazing collisions.
Finally, we note that Bobylev has exhibited a countable family of
radial semi-explicit solutions (in the form of a convergent series) that
checkbothFokker-PlanckandBoltzmannequationforsomeparticular
values of the parameter „.
Remark 2. In one dimension of velocity-space, the Landau-type
model corresponding to the Kac equation is simply
@ f =@ f +x@ f +f =@ (@ f +xf);t xx x x x
R R
2together with the conditions f =1= fx .
Afterthisbriefdiscussionwegobacktothegeneralanisotropiccase.
4. The collision operator and the Cauchy problem
The matrix a is clearly nonnegative in view of its definition. Weij
analyse its positivity. We can write
' “ ' “1¡fit 2(11) a = – ¡e D + jvj – ¡vv ·A +Bij ij ij ij i j ij ij
N¡1
The matrix (B ) is obviously nonnegative in view of the Cauchy-ij
Schwarz inequality; we note that it is simply (up to a constant factor)
2jvj times the orthogonal projection on the space orthogonal to v. At
each non-zero velocity v it has only one degenerate direction, namely
that of v (so that when applied to radial functions, this operator is
totally degenerate and reduces to a first-order operator, as we have
seen in the preceding section).
As for the first part :
¡fit ¡fit(N¡1)A =N– ¡e B ¡(1¡e )–ij ij ij ij
¡fit ¡fit=(N¡1)(1¡e )– +e (N– ¡B ):ij ij ij
To prove that the matrix (A ) is nonnegative, we just have to proveij
that
(B )•N(– )ij ij
in the sense of matrices. This is an easy matter in view of
Z Z ZX X
2 2 2 2B »» = f(v)vv »» = f(v¢») •j»j fjvj =Nj»j :ij i j i j i j
ij ij
ThedegeneracycanonlyhappenifthereisalwaysequalityinCauchy-
Schwarz inequality, that is, if f is concentrated on some »-axis.LANDAU EQUATION FOR MAXWELLIAN MOLECULES 9
Thus we have the following
P
Proposition 1. The second-order operator a @ is the sum of twoij ij
elliptic operators ;
(i) the first one has coefficients depending (smoothly) only on t and is
always strictly elliptic unless t = 0 and f is concentrated on a single0
line, whose direction is the only degeneracy direction;
(ii) the second one has coefficients depending only on v, reduces to 0
for v = 0 and elsewhere has always one unique degenerate direction,
namely that of v.
1As we shall always assume f 2L , we shall never be concerned with
the degeneracy indicated in part 1 of this proposition. However, it
wouldbepossibletoassumeonlythat f isanonnegativemeasurewith
bounded mass and energy.
We can now rewrite the equation as
(12) @ f =Lf =L f +L f;t 1 2
where 8 X X
> L f = A @ f + v@f +Nf; 1 ij ij i i>< ij i
X X>> L f = B @ f¡ v@f: 2 ij ij i i:
ij i
This decomposition provides an easy interpretation of the equation.
Indeed, suppose for simplicity that (D ) = 0 : then L is the usualij 1
Fokker-Planck operator (that we shall denote by L). As for L , we2
can make its structure more clear by a change to polar coordinates.
To obtain practical explicit expressions, we shall first illustrate this
in the cases N = 2 and N = 3. Writing f(v ;v ) = f(r;?), where1 2
r ‚ 0, v = rcos?, v = rsin?, we obtain, after some computation,1 2
the following simple expression.
Proposition 2. For N =2, in polar coordinates,
L f =@ f:2 ??
Moreover, if (D )=0, thenij
? ¶
1 1
L f =@ f + @ f + +r @ f +2f1 rr ?? r2r r
ThusweseethatthesemigroupassociatedtoLconsistsinaFokker-
Planck diffusion process, plus an additional diffusion on each centered
sphere. This second diffusion is very important for large r, where it10 C. VILLANI
strongly prevails over the spherical diffusion induced by the Laplace
operator.
Wealsoremarkfromtheirexpressioninpolarcoordinatesthatthese
two operators commute. But we know explicit formulas for the associ-
ated semigroups; indeed we have
+1X
2tL ¡n t in 2(13) e f = c (f ;r)e e0 n 0
n=¡1
where Z
1 ¡in c (f;r)= f(r;?)e d :n
2…
Thus, denoting by M the centered Maxwellian with mass 1 and–
temperature –,
£¡ ¢ ⁄
tL tL tL 2t tL t1 2 2e f =e e f =M ⁄e e f (e¢) =0 0 – 0
Z +1X1 2 2 2¡r =2– ¡r =– rr cos(?¡? )=2– 2t t ¡n t in ⁄ ⁄ ⁄⁄r dr d e e e e c (f ;e r )e e⁄ ⁄ ⁄ n 0 ⁄
(2…–)
n=¡1
The convergence of the series in (13) is absolute for almost all r and
allt>0,andmustbetakenindistributionalsensefort=0. Tojustify
the expression above it suffices to note thatL preserves the positivity2
and the integrability of its solution.
From the qualitative point of view, we notice that the solution be-
1comes immediately very regular (C ) and that it has a tendency to
become radial very quickly, due to the important spherical diffusion.
For N = 3, we use the usual spherical coordinates, that is, v =1
rsin?cos`, v =rsin?sin`, v =rcos?. We recall that2 3
1
Δ=Δ + Δ ;r ?;`2r
where
1 2Δ f = @ (r @ f);r r r2r
1 1
Δ f = @ (sin?@ f)+ @ f;?;` ? ? ``2sin? sin ?
that is, Δ is the Laplace-Beltrami operator.?;`
It is possible to check
Proposition 3. If N =3, in spherical coordinates,
1
L f = Δ f:2 ?;`
2