Kinetic Limit of Dynamical Description of Wave-Particle Self-consistent Interaction in an Open Domain
In a closed domain Ω of space, we consider a system of N particles σ N =(x 1,v 1,…,x N ,v N ) interacting via a pair potential U. In this region, particles also interact self-consistently with a wave Z=Aexp(iϕ). We consider injection of particles in Ω, so N varies in time.
Given initial data (Z N (0),σ N (0)) and a boundary source/sink, the system evolves according to a Hamiltonian dynamics to (Z N (t),σ N (t)). In the limit of infinitely many particles (kinetic limit), this generates a Vlasov-like kinetic equation for the distribution function f(x,v,t) coupled to an envelope equation for Z(t)=Z ∞(t). The solution (Z ∞,f) exists and is unique for any initial data with finite energy, provided that Ω has smooth enough boundaries.
Further, for any finite time t, given a sequence of initial data such that σ N (0)→f(0) weakly and Z N (0)→Z(0) as N→∞, the states generated by the Hamiltonian dynamics (Z N (t),σ N (t)) are such that lim N→∞(Z N (t),σ N (t))=(Z ∞(t),f(x,v,t)).
The authors thank N. Dubuit for fruitful discussion and Marco A. Amato for initiating and collaborating in the ongoing researches.
B. Vieira Ribeiro is supported by a grant from CAPES Foundation through the PDSE program, process number: 8510/11-3.