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Regularity criteria for the 3D MHD equations in terms of the pressure. (English) Zbl 1160.35506

Summary: We consider the regularity criteria for weak solutions to the 3D MHD equations. It is proved that under the condition \(b\) being in the Serrin’s regularity class, if the pressure \(p\) belongs to \(L^{\alpha,\gamma}\) with \(\frac{2}{\alpha}+\frac{3}{\gamma} \leqslant 2\) or the gradient field of pressure \(\nabla p\) belongs to \(L^{\alpha,\gamma}\) with \(\frac{2}{\alpha}+\frac{3}{\gamma} \leqslant 3\) on \([0,T]\), then the solution remains smooth on \([0,T]\).

MSC:

35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
35B65 Smoothness and regularity of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35D10 Regularity of generalized solutions of PDE (MSC2000)
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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