Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Magnetohydrodynamic Equations with Vacuum

TL;DR

This paper proves the global existence, uniqueness, and decay rates of strong solutions for 2D density-dependent MHD equations with vacuum, allowing large initial data and vacuum states.

Contribution

It establishes the first global strong solution results for 2D nonhomogeneous incompressible MHD with vacuum and large initial data.

Findings

Global existence and uniqueness of strong solutions

Decay rates of velocity, magnetic field, and pressure gradients

Initial vacuum states and large data are allowed

Abstract

This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Magnetohydrodynamic (MHD) equations with vacuum as far field density. We establish the global existence and uniqueness of strong solutions to the 2D Cauchy problem on the whole space $\mathbb{R}^2$, provided that the initial density and the initial magnetic decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the gradients of velocity, magnetic and pressure.