Global Solutions of the Compressible Euler-Poisson Equations for Plasma with Doping Profile for Large Initial Data of Spherical Symmetry

TL;DR

This paper proves the global existence of solutions for the multidimensional compressible Euler-Poisson equations in plasma with doping profiles, allowing large initial data and unbounded energy, by employing density-dependent viscosity and novel electric field estimates.

Contribution

It introduces a new approach using density-dependent viscosity to establish global solutions and inviscid limits for plasma equations with large doping profiles and initial data.

Findings

Global solutions exist for large initial data with unbounded energy.

Inviscid limit of Navier-Stokes-Poisson solutions converges to Euler-Poisson solutions.

No concentration formation occurs in the inviscid limit for large doping profiles.

Abstract

We establish the global-in-time existence of solutions of finite relative-energy for the multidimensional compressible Euler-Poisson equations for plasma with doping profile for large initial data of spherical symmetry. Both the total initial energy and the initial mass are allowed to be {\it unbounded}, and the doping profile is allowed to be of large variation. This is achieved by adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the inviscid limit of global weak solutions of the Navier-Stokes-Poisson equations with the density-dependent viscosity terms to the corresponding global solutions of the Euler-Poisson equations for plasma with doping profile can be established. New difficulties arise when tackling the non-zero varied doping profile, which have been overcome by establishing some novel estimates for the electric field terms so that the neutrality assumption on the initial data is avoided. In particular, we prove that no concentration is formed in the inviscid limit for the finite relative-energy solutions of the compressible Euler-Poisson equations with large doping profiles in plasma physics.