Global well posedness for a two-fluid model

TL;DR

This paper establishes the global well-posedness of a two-fluid charged fluid model with electromagnetic interactions in two dimensions and constructs weak solutions in three dimensions, extending and improving previous results.

Contribution

It proves global well-posedness in 2D for all initial data and constructs weak solutions in 3D, requiring less regularity than prior work.

Findings

Global well-posedness in 2D for any initial data.

Existence of global weak solutions in 3D.

Local well-posedness of Kato-type solutions in 3D.

Abstract

We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posdness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions \`a la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984) ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.