Analysis of Oscillations and Defect Measures for the Quasineutral Limit in Plasma Physics

Abstract

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier–Stokes–Poisson system in three dimensions. We show that as λ → 0 the velocity field u λ strongly converges towards an incompressible velocity vector field u and the density fluctuation ρ λ −1 weakly converges to zero. In general, the limit velocity field cannot be expected to satisfy the incompressible Navier–Stokes equation; indeed, the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self-interacting wave packets. We provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis. Moreover, we were able to identify an explicit pseudo-parabolic PDE satisfied by the leading correctors terms. Our results include all the previous results in the literature; in particular, we show that the formal limit holds rigorously in the case of well prepared data.