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 |