Local well-posedness to the 2D Cauchy problem of full compressible magnetohydrodynamic equations with vacuum at infinity

TL;DR

This paper establishes local existence and uniqueness of strong solutions for the 2D full compressible MHD equations with vacuum at infinity, allowing for vacuum states and various decay conditions on initial data.

Contribution

It introduces a spatial weighted energy method to prove local well-posedness for the 2D compressible MHD equations with vacuum, accommodating non-decaying temperature and vacuum at infinity.

Findings

Proves local existence and uniqueness of strong solutions.

Allows vacuum states both inside the domain and at infinity.

Initial temperature decay conditions are relaxed.

Abstract

This paper concerns the Cauchy problem of two-dimensional (2D) full compressible magnetohydrodynamic (MHD) equations in the whole plane $\mathbb{R}^2$ with zero density at infinity. By spatial weighted energy method, we derive the local existence and uniqueness of strong solutions provided that the initial density and the initial magnetic field decay not too slowly at infinity. Note that the initial temperature does not need to decay slowly at infinity. In particular, vacuum states at both the interior domain and the far field are allowed.