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Remarks on the regularity criteria for generalized MHD equations. (English) Zbl 1211.35230
The Cauchy problem to the 3D generalized MHD equations is studied in $\{x\in{\Bbb R}^3,t>0\}$. $$\aligned &\frac{\partial u}{\partial t}+(-\Delta)^\alpha u+u\cdot\nabla u -b\cdot\nabla b+\nabla p=0,\quad \text{div}\,u=0,\\ & \frac{\partial b}{\partial t}+(-\Delta)^\beta b +u\cdot\nabla b -b\cdot\nabla u=0,\quad \text{div}\,b=0,\\ &u(x,0)=u_0(x),\quad b(x,0)=b_0(x). \endaligned$$ Here $u$ is the velocity field, $b$ is the magnetic field, $p$ is the pressure, $\alpha,\beta>0$, $(-\Delta)^\alpha$ is the fractional power of Laplacian. It is proved that if $u_0,b_0\in H^1({\Bbb R}^3)$ and $(u,b)$ is a weak solution with $\nabla u$ integrable in a certain Morrey space then $(u,b)$ is smooth on $(0,T)$.

35Q35PDEs in connection with fluid mechanics
76W05Magnetohydrodynamics and electrohydrodynamics
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI
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