Commutator estimates with fractional derivatives and local existence for the generalized MHD equations

This paper considers the problem of the local existence for the generalized MHD equations with fractional dissipative terms Λ2αu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^{2\alpha } u$$\end{document} for the velocity field and Λ2βb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^{2\beta } b$$\end{document} for the magnetic field, respectively. Based on some new commutator estimates, local existence for the generalized MHD equations is established, which recovers and improves previous results.