The combined incompressible quasineutral limit of the stochastic Navier-Stokes-Poisson system

TL;DR

This paper establishes the convergence of stochastic compressible Navier-Stokes-Poisson solutions to stochastic incompressible Navier-Stokes solutions under the quasineutral limit, introducing new dispersive estimates for stochastic PDEs.

Contribution

It develops novel dispersive estimates for stochastic PDEs to analyze acoustic waves, enabling the proof of the quasineutral limit in a stochastic setting.

Findings

Convergence in law to stochastic incompressible Navier-Stokes solutions.

Dispersive estimates for stochastic acoustic wave equations.

Application to stochastic plasma models in zero-electron-mass limit.

Abstract

This paper deals with the combined incompressible quasineutral limit of the weak martingale solution of the compressible Navier-Stokes-Poisson system perturbed by a stochastic forcing term in the whole space. In the framework of ill-prepared initial data, we show the convergence in law to a weak martingale solution of a stochastic incompressible Navier-Stokes system. The result holds true for any arbitrary nonlinear forcing term with suitable growth. The proof is based on the analysis of acoustic waves but since we are dealing with a stochastic partial differential equation, the existing deterministic tools for treating this second-order equation breakdown. Although this might seem as a minor modification, to handle the acoustic waves in the stochastic compressible Navier-Stokes system, we produce suitable dispersive estimate for first-order system of equations, which are an added value to the existing theory. As a by-product of this dispersive estimate analysis, we are also able to prove a convergence result in the case of the zero-electron-mass limit for a stochastic fluid dynamical plasma model.