Strong Solutions for Three-dimensional Nonhomogeneous Incompressible Heat Conducting Magnetohydrodynamic Equations with Vacuum

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 3D nonhomogeneous heat conducting MHD equations with vacuum, establishing a blowup criterion based on velocity conditions independent of temperature and magnetic field.

Contribution

It introduces a weak Serrin-type blowup criterion for strong solutions and demonstrates global solutions under small initial data, even with initial vacuum.

Findings

Global existence and uniqueness of strong solutions

A blowup criterion based on velocity conditions

Applicability to initial vacuum scenarios

Abstract

This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the global existence and uniqueness of strong solution to the 3D nonhomogeneous heat conducting MHD equations under some smallness condition on the initial data. In addition, the initial vacuum is allowed.