Incompressible hydrodynamic limits for a kinetic model of waves–particles interaction

Abstract

The macroscopic dynamics of a kinetic equation involving a model wave–particle collision operator of plasma physics (see Degond and Peyrard, C. R. Acad. Sci. Paris 323, 1996) is investigated. Using relative entropy estimates about an absolute Maxwellian, it is shown, as in Bardos, Golse and Levermore (Comm. Pure Appl. Math. 46(5), 1993), that any properly scaled sequence of solutions has fluctuations that converge to a limiting density of the form $g=U(x,t).v +P_0 g(x,t)({\vert v\vert ^2/2})$ with fluid variables that satisfy the incompressibility and Boussinesq relations. For a convenient scaling of the initial fluctuations, the momentum densities globaly converge to a solution of the Stokes equation. Using an infinite number of entropies, we derive the diffusion equation for the energy distribution function of the particles. A similar discrete time version of this result holds for the Navier–Stokes equation coupled with a diffusion equation for the energy distribution function under an additional mild weak compactness assumption.