Existence theorems for the Cauchy problem of 2D nonhomogeneous incompressible non-resistive MHD equations with vacuum

TL;DR

This paper proves the existence and uniqueness of local strong solutions for the 2D nonhomogeneous incompressible non-resistive MHD equations with vacuum, under certain decay and regularity conditions.

Contribution

It establishes the existence and uniqueness of local strong solutions for the 2D MHD equations with vacuum, extending previous results to nonhomogeneous and non-resistive cases.

Findings

Unique local strong solutions under decay conditions

Classical solutions with additional regularity

Solutions exist with vacuum as far field density

Abstract

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