Large time decay for the magnetohydrodynamics equations in Sobolev–Gevrey spaces

Our paper shows that global solutions (u,b)∈C([0,∞);Ha,σs(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,b)\in C([0,\infty );H_{a,\sigma }^s({\mathbb {R}}^3))$$\end{document} of the Magnetohydrodynamics equations present the following asymptotic behavior: limt→∞ts2‖(u,b)(t)‖H˙a,σs(R3)2=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{t\rightarrow \infty }t^{\frac{s}{2}}\Vert (u,b)(t)\Vert ^2_{{\dot{H}}_{a,\sigma }^{s}({\mathbb {R}}^3)}=0, \end{aligned}$$\end{document}where a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document}, σ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma > 1$$\end{document}, s>1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>1/2$$\end{document} and s≠3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\ne 3/2$$\end{document}. It is important to point out that the assumption related to existence of global solutions for this same system can be made since the existence and uniqueness of local solutions were recently established; more precisely, it has been proved that there is a time T>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T > 0$$\end{document} such that (u,b)∈C([0,T];Ha,σs(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u,b)\in C([0,T];H_{a,\sigma }^s({\mathbb {R}}^3))$$\end{document}.