On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum

TL;DR

This paper proves the existence and uniqueness of local strong solutions for the 2D density-dependent incompressible MHD equations with vacuum, under certain decay conditions on initial data.

Contribution

It establishes the local well-posedness of the 2D nonhomogeneous incompressible MHD equations with vacuum, a problem not previously fully addressed.

Findings

Unique local strong solutions exist under decay conditions.

Initial density can have compact support.

Solutions are unique and depend continuously on initial data.

Abstract

This paper concerns the Cauchy problem of the nonhomogeneous incompressible magnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) space with vacuum as far field density. In particular, the initial density can have compact support. We prove that the 2D Cauchy problem of the nonhomogeneous incompressible MHD equations admits a unique local strong solution provided the initial density and the initial magnetic decay not too slow at infinity.