On classical solutions to the Cauchy problem of the 2D compressible non-resistive MHD equations with vacuum

TL;DR

This paper establishes local existence, uniqueness, and blowup criteria for classical solutions to the 2D compressible non-resistive MHD equations with vacuum, under specific decay and regularity conditions.

Contribution

It proves the existence of unique local strong solutions and criteria for blowup, extending understanding of the 2D compressible non-resistive MHD equations with vacuum.

Findings

Existence of unique local strong solutions under decay conditions.

Conditions for solutions to become classical with additional regularity.

A blowup criterion depending only on density and magnetic fields.

Abstract

In this paper, we investigate the Cauchy problem of the compressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density. We prove that the 2D Cauchy problem has a unique local strong solution provided the initial density and magnetic field decay not too slow at infinity. Furthermore, if the initial data satisfies some additional regularity and compatibility conditions, the strong solution becomes a classical one. Additionally, we establish a blowup criterion for the 2D compressible non-resistive MHD depending solely on the density and magnetic fields.