The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions

This work establishes local existence and uniqueness as well as blow-up criteria for solutions \begin{document}$ (u,b)(x,t) $\end{document} of the Magneto–Hydrodynamic equations in Sobolev–Gevrey spaces \begin{document}$ \dot{H}^s_{a,\sigma}(\mathbb{R}^3) $\end{document} . More precisely, we prove that there is a time \begin{document}$ T>0 $\end{document} such that \begin{document}$ (u,b)\in C([0,T];\dot{H}_{a,\sigma}^s(\mathbb{R}^3)) $\end{document} for \begin{document}$ a>0, \sigma\geq1 $\end{document} and \begin{document}$ \frac{1}{2} . If the maximal time interval of existence is finite, \begin{document}$ 0\leq t , then the blow–up inequality \begin{document}$ \frac{C_1\exp\{C_2(T^*-t)^{-\frac{1}{3\sigma}}\}\;\;\;\;\;\;\;}{\;\;\;\;(T^*-t)^{q}\;\;\;\;} \;\;\;\;\;\;\;\;\;\;\leq \|(u,b)(t)\|_{\dot{H}_{a,\sigma}^s(\mathbb{R}^3)} \quad \mbox{with}\,\, q = {\frac{2(s\sigma+\sigma_0)+1}{6\sigma}} $\end{document} holds for \begin{document}$ 0\leq t , \begin{document}$ a>0 $\end{document} , \begin{document}$ \sigma> 1 $\end{document} ( \begin{document}$ 2\sigma_0 $\end{document} is the integer part of \begin{document}$ 2\sigma $\end{document} ).