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Remarks on regularities for the 3D MHD equations. (English) Zbl 1068.35117
Summary: We consider the regularity criteria for the solution to the 3D MHD equations \[ \partial_t u+ u\cdot\nabla u= \nu_1\Delta u-\nabla p- {1\over 2}\nabla|b|^2+ b\cdot\nabla b+ f, \] \[ \partial_t b+ u\cdot\nabla b= \nu_2\Delta b+ b\cdot\nabla u+ g, \] \[ \text{div\,}u= \text{div\,}b= 0, \] \[ u(x,0)= u_0(x),\quad b(x,0)= b_0(x). \] It is proved that if the gradient of the velocity field belongs to \(L^{\alpha,\gamma}\) with \(2/\alpha+ 3/\gamma\leq 2\) or the velocity field belongs to \(L^{\alpha,\gamma}\) with \(2/\alpha+ 3/\gamma\leq 1\) on \([0, T]\), then the solution remains smooth on \([0, T]\). The significance is that there are no restriction on the magnetic field. Moreover, the norms \(\|\nabla u\|_{L^{\alpha,\gamma}}\) and \(\| u\|_{L^{\alpha,\gamma}}\) are scaling dimension zero for \(2/\alpha+ 3/\gamma= 2\) and \(2/\alpha+ 3/\gamma= 1\), respectively.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
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