ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES

Fourier Transform

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Niveau: Supérieur, Doctorat, Bac+8
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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• Transform

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• Publié par: mijec
• Nombre de lectures: 74
• Langue: English

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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 , assoc