Zero-electron-mass limit of Euler-Poisson equations

TL;DR

This paper investigates the zero-electron-mass limit in multidimensional Euler-Poisson equations for plasmas, establishing uniform solutions and their convergence to incompressible Euler equations under specific conditions.

Contribution

It introduces new frequency-localization Strichartz estimates and proves the uniform global existence and convergence of solutions as the electron mass tends to zero.

Findings

Uniform global existence of classical solutions for small initial data

Development of new Strichartz-type estimates for acoustics equations

Convergence of solutions to incompressible Euler equations in the zero-electron-mass limit

Abstract

This paper is concerned with multidimensional Euler-Poisson equations for plasmas. The equations take the form of Euler equations for the conservation laws of the mass density and current density for charge-carriers (electrons and ions), coupled to a Poisson equation for the electrostatic potential. We study the limit to zero of some physical parameters which arise in the scaled Euler-Poisson equations, more precisely, which is the limit of vanishing ratio of the electron mass to the ion mass. When the initial data are small in critical Besov spaces, by virtue of the ``Shizuta-Kawashima" skew-symmetry condition, we establish the uniform global existence and uniqueness of classical solutions. Then we develop new frequency-localization Strichartz-type estimates for the equation of acoustics (a modified wave equation) with the aid of the detailed analysis of the semigroup formulation generated by this modified wave operator. Finally, it is shown that the uniform classical solutions converge towards that of the incompressible Euler equations (for \textit{ill-prepared} initial data) in a refined way as the scaled electron-mass tends to zero.