Liouville-Type Theorems for the 3D Stationary MHD Equations

In this paper, we consider the Liouville-type theorems for the 3D stationary incompressible MHD equations. Using the Caccioppoli type estimate, we proved the smooth solutions (u, b) are identically equal to zero when (u,b)is an element of Lp(R3),p is an element of(32,3).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,b)\in L<^>{p}({\mathbb {R}}<^>{3}),\ p\in (\frac{3}{2},3).$$\end{document} In addition, under an additional assumption in the setting of the Sobolev space of negative order H-center dot-1(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}<^>{-1}({\mathbb {R}}<^>{3}),$$\end{document} we can extend the index p is an element of(3,+infinity).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (3,+\infty ).$$\end{document} In fact, our results combine with the result of Yuan and Xiao (J Math Anal Appl 491(2):124343, 2020) that p is an element of[2,92],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [2,\frac{9}{2}],$$\end{document} which implies a very intriguing and novel result for the 3D stationary MHD equations with p is an element of(32,+infinity).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p\in (\frac{3}{2},+\infty ).$$\end{document}