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

TL;DR

This paper rigorously analyzes the quasineutral limit in a viscous plasma model, showing convergence of velocity and density fluctuations, and introduces microlocal defect measures and correctors to describe the process.

Contribution

It provides a detailed mathematical framework for the quasineutral limit using microlocal defect measures and identifies a pseudo parabolic PDE for the correctors, extending previous results.

Findings

Velocity field converges strongly to incompressible limit

Density fluctuations vanish weakly as the limit is approached

Explicit PDE for leading correctors is derived

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 $3-D$. We show that as $\lambda\to 0$ the velocity field $u^{\lambda}$ strongly converges towards an incompressible velocity vector field $u$ and the density fluctuation $\rho^{\lambda}-1$ 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 shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis. Moreover we will be able to identify an explicit pseudo parabolic pde satisfied by the leading correctors terms. Our results include all the previous results in literature, in particular we show that the formal limit holds rigorously in the case of well prepared data.