On global solutions to the 3D viscous, compressible, and heat-conducting magnetohydrodynamic flows

This paper concerns the Cauchy problem of three-dimensional viscous, compressible, and heat-conducting magnetohydrodynamic flows. Both some new Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} gradient estimates and the “div-curl” decomposition of ‖∇u‖L3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \nabla \text{ u }\Vert _{L^3}$$\end{document} are established; the existence of global solutions to the Cauchy problem with small energy and lower regularity assumed on the initial data are obtained. Furthermore, we also prove that the global solution belongs to a new class of functions in which the uniqueness can be shown to hold.