The well-posedness of the compressible non-isentropic Euler-Maxwell system in R^3

TL;DR

This paper proves the global well-posedness and decay rates of solutions for the compressible non-isentropic Euler-Maxwell system in three dimensions, even with large higher derivatives, under small initial data assumptions.

Contribution

It establishes the existence, uniqueness, and decay rates of solutions for the Euler-Maxwell system with minimal regularity assumptions on initial data.

Findings

Global unique solutions exist for small initial data in H^3.

Decay rates for density and temperature reach (1+t)^{-13/4} in L^2 norm.

Higher order derivatives can be large while solutions remain well-posed.

Abstract

We first construct the global unique solution by assuming that the initial data is small in the $H^3$ norm but the higher order derivatives could be large. If further the initial data belongs to $\Dot{H}^{-s}$ ($0\le s<3/2$) or $\dot{B}_{2,\infty}^{-s}$ ($0< s\le3/2$), we obtain the various decay rates of the solution and its higher order derivatives. In particular, the decay rates of the density and temperature of electron could reach to $(1+t)^{-13/4}$ in $L^2$ norm.