Global unique solution for the 3-D full compressible MHD equations in space of lower regularity

TL;DR

This paper proves the global existence and uniqueness of solutions to the 3-D full compressible MHD equations with low regularity initial data by establishing new gradient estimates using advanced decomposition techniques.

Contribution

It introduces novel $L^p$ gradient estimates and employs the div-curl decomposition to achieve global well-posedness for solutions with lower regularity initial data.

Findings

Established global well-posedness for the 3-D full compressible MHD system.

Developed new $L^p$ gradient estimates for solutions.

Utilized div-curl decomposition and modified flux techniques.

Abstract

In this paper, we establish new $L^p$ gradient estimates of the solutions in order to discuss Cauchy problem for the full compressible magnetohydrodynamic(MHD) systems in $\mathrm{R}^3$. We use the "$\rm{div}-\rm{curl}$" decomposition technique (see \cite{{HJR},{MR}}) and new modified effective viscous flux and vorticity to calculate "$\Vert\nabla \mathbf{u}\Vert_{L^3}$" and "$\Vert\nabla \mathbb{H}\Vert_{L^3}$".As a result, we obtain global well-posedness for the solution with the initial data being in a class of space with lower regularity, while the energy of which should be suitably small.