12
pages

- phase space
- limn fn
- xk ?xj
- initial condition
- then fn ?
- fn
- coulomb interaction
- vlasov equation
- take many

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P.E. Jabin

Equipe Tosca, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis

Laboratoire Dieudonn´e, Univ. de Nice, Parc Valrose, 06108 Nice cedex 02

Antoine Gerschenfeld

´Ecole Normale Sup´erieure, ParisIntroduction

0.1 Introduction

The validity of kinetic models as a limit of systems of many interacting particles is

still an important open issue. The number of particles to take into account is so large

in most applications (plasma physics, galaxies formation...) that the use of continuous

models is absolutely required.

The same issues directly arise for the use of particle methods. Those methods rely

ontheassumptionthatalarge(butnottoolarge)numberof“meta-particles”correctly

represents the dynamics of a much larger number of real particles. This assumption

would be directly implied by the convergence of the system to the unique solution to

some equation.

Thescalingunderconsiderationhereleadstoso-calledmeanﬁeldlimits.Thoselim-

its were classically establishedunderstrong regularityassumptions forthe interaction,

which are not satisﬁed in many physical situations of interest. We aim at describing

those classical approaches but also to present the new ideas recently developed for the

singular cases.

We consider N identical particles with positions/velocities (X ,V ) in the phasei i

space, interacting through the 2-body interaction kernel K(x), which leads to the

evolution equations

dX =V ,i idt P (0.1)d 1V = K(X −X ).i i jjdt N

The 1/N factor in the second equation is a scaling term so that positions, velocities

and accelerations are now of order 1.

The kernel K may take many diﬀerent forms depending on the physical setting.

The guiding example and the one with the most important physical applicationsK(x)

is Coulomb interaction, which reads in dimension d

α

K(x) =−∇φ(x) , φ(x) = + (regular terms),

d−2|x|

where α> 0 (resp. α< 0) corresponds to the repulsive (resp. attractive) case.

dIn what follows, the dynamics will be considered on the torus X ∈ Π , d ≥ 2,i

dmainly to simplify the exposition. Note that even then the velocities are still inR .

0.2 Well-posedness of the microscopic dynamics

TheCauchy-Lipschitztheoremappliesto(0.1)ifK(x)isLipschitz,inwhichcasethere

exists an unique solution for any initial condition. In the repulsive Coulomb case, it is

still possible to apply it by remarking that the energy conservation,

X X1 α 12E(t) = |V| + =E(0),i 2N N |X −X |i j

i i=j

1implies that the |X −X | admit a time-independent lower bound in : one mayi j 2N

therefore consider K as Lipschitz on its attainable domain for a given set of initial

conditions.Oneshouldnote,however,thattheformofthisestimatemakesitimproper

to use in the N →∞ limit.

v6It is possible to assume less regularity on K by restricting the set of acceptable

initial conditions. In particular, results by (DiPerna and Lions, 1989), (Ambrosio,

2004) and (Hauray, 2005) apply to almost-every initial condition.

0.3 Existence of the macroscopic limit

N0 N0 N0 N0 N0Given a sequence of initial conditions Z = (X ,..,X ,V ,..,V ) with corre-1 N 1 N

Nsponding solutions Z (t), one expects the empirical density on phase space,

X1 N Nf (t,x,v) = δ(x−X (t))⊗δ(v−V (t)),N i iN

i

to converge, in some sense, as N →∞, to a limit f satisfying an evolution equation,

0the ”limiting dynamics”, with initial conditions f = lim f (0,·).N N

N NIf K is continuous or if X (t) =X (t) for all t and i =j, then, posing K(0) = 0,i j

one can write the N-body evolution in the form of a Vlasov equation :

∂ f +v·∇ f +(K? ρ )·∇ f = 0t N x N x N v NR (0.2)

ρ (t,x) = dvf (t,x,v) .N N

1 d dThenf →f in weak-? topology (for the space of Radon measuresM (Π ×R )),N

and f solves (0.2) for the initial conditions lim f (0,·).N N

Equation (0.2) cannot be obtained from (0.1) with such an immediate method for

any kind of singular interaction K 6∈C . However, even for a Coulomb potential, Eq.0

(0.2) is well posed provided some assumptions on the initial conditions are made, such

1 ∞as f(0,·)∈ L ∩L and with compact support in velocities; See (Horst, 1981; Lions

andPerthame,1991;Pfaﬀelmoser,1992;Schaeﬀer,1991).However,thenon-linearterm

(K? ρ )·∇ f makes the f →f limit highly nontrivial for non-continuous K.x N v N N

0.4 Physical space models

The above question is easier to solve in the case of hydrodynamics-related models,

which evolve according to a ﬁrst-order equation of the form

Xd 1

X = μ μ K(X −X ). (0.3)i i j i j

dt N

j=i

P

Using ρ (t,x) = μ δ(x−X (t)), it can be rewritten asN i ii

∂ ρ+∇ ((K?ρ)ρ) = 0 . (0.4)t x

For instance, in dimension 2, the above yields the incompressible Euler equation for

2μ =±1,K(x) =x /|x |.i ⊥

As a rule of thumb, the N →∞ limit is easier to take in this case than in (0.1).

A crucial ingredient to the study is a bound on d (t) = inf |X (t)− X (t)|.min i=j i j

This oﬀers direct control over the right-hand term in (0.3),which becomes regular if

N −1/dd ∼N for singular force terms K (up to a coulombian singularity).min

vi6666Macroscopic limit in the regular case

More precisely, assume that, up to time t and for x ∼ x , there exists a locallyi

bounded F such that

X 1 dmin K(x−X (t)) ≤F .j 1/dN N j=i 1,∞W

Let (k,l) be the particles such that d (t) = |X −X|. If one also assumes thatmin k l

μ =μ , thenk l

X