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Non blow-up criterion for the 3-D magneto-hydrodynamics equations in the limiting case. (English) Zbl 1372.35050
Summary: In this paper, we prove that suitable weak solution \((u,b)\) of the 3-D MHD equations can be extended beyond \(T\) if \(u \in L^{\infty}(0,T;L^3(\mathbb{R}^3))\) and the horizontal components \(b_h\) of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.
MSC:
35B44 Blow-up in context of PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
35D30 Weak solutions to PDEs
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