On an Euler-Schr\"{o}dinger system appearing in laser-plasma interaction

TL;DR

This paper proves the global existence, uniqueness, and decay of solutions for a coupled Euler and Schrödinger system modeling laser-plasma interactions under small initial data conditions.

Contribution

It extends previous work by establishing global solutions with Sobolev regularity for a coupled Euler-Schrödinger system in the whole space.

Findings

Existence of global-in-time unique solutions.

Solutions exhibit algebraic decay over time.

Results apply under small initial density and vector potential.

Abstract

We consider the Cauchy problem for the barotropic Euler system coupled to a vector Schr\"{o}dinger equation in the whole space. Assuming that the initial density and vector potential are small enough, and that the initial velocity is close to some reference vector field $u_0$ such that the spectrum of $Du_0$ is bounded away from zero, we prove the existence of a global-in-time unique solution with (fractional) Sobolev regularity. Moreover, we obtain some algebraic time decay estimates of the solution. Our work extends the papers by D. Serre and M. Grassin [11, 13, 19] and previous works by B. Ducomet and co-authors [4, 8] dedicated to the compressible Euler-Poisson system.