Global Strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system

TL;DR

This paper proves the existence of unique global strong solutions with vacuum for the 2D nonhomogeneous incompressible MHD system, extending previous results by allowing initial vacuum states using advanced Sobolev inequalities.

Contribution

It establishes the first global strong solution with vacuum for the 2D nonhomogeneous incompressible MHD system, improving prior results that required positive initial density.

Findings

Global strong solution with vacuum established for 2D nonhomogeneous incompressible MHD.

Extension of results to Navier-Stokes equations as a corollary.

Use of critical Sobolev inequality of logarithmic type in the proof.

Abstract

In this paper, we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible MHD system, as long as the initial data satisfies some compatibility condition. As a corollary, the global existence of strong solution with vacuum to the 2D nonhomogeneous incompressible Navier-Stokes equations is also established. Our main result improves all the previous results where the initial density need to be strictly positive. The key idea is to use some critical Sobolev inequality of logarithmic type, which is originally due to Brezis-Wainger.