The Regularity of Very Weak Solutions to Magneto-Hydrodynamics Equations

In this paper, employing the duality technique, we prove that the very weak solution of Magneto-Hydrodynamics equations is regular in R3×(0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3\times (0, T]$$\end{document} if it belongs to the Banach space Lp(h,T;Lq(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}(h,T;L^{q}(\mathbb {R}^{3}))$$\end{document} with 2p+3q=1,q∈(3,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{2}{p}+\frac{3}{q}=1,\ \ q\in (3,\infty )$$\end{document} for any small h>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h>0$$\end{document}. Secondly, we further prove the integrability condition imposed on the magnetic field can be removed by using the energy method and the regularity theory of the heat operator, which is of independent interest.