Global well-posedness of the compressible bipolar Euler-Maxwell system in R^3

TL;DR

This paper proves the global well-posedness of the compressible bipolar Euler-Maxwell system in three dimensions for small initial data in H^3, and establishes decay rates for solutions and their derivatives under additional conditions.

Contribution

It constructs the first global unique solutions with large higher derivatives for small initial data and derives decay rates without smallness assumptions on L^p norms.

Findings

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

Decay rates of solutions are established under additional initial data conditions.

Decay rate for density difference reaches (1+t)^{-13/4} in L^2 norm.

Abstract

We first construct the global unique solution by assuming that the initial data is small in the H^3 norm but its 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. As an immediate byproduct, the L^p-L^2 (1\le p\le 2) type of the decay rates follow without requiring the smallness for L^p norm of initial data. In particular, the decay rate for the difference of densities could reach to (1+t)^{-13/4} in L^2 norm.