Abstract
Any weak solution u to the Navier-Stokes equations is showed to be regular under the assumption that \( \left\| u \right\|_{L_w^2 (0,T;L^\infty (\mathbb{R}^3 ))} \) is sufficiently small, which is a limiting case of the regularity criteria derived by Kim and Kozono. Our result gives a positive answer to the question proposed by Kim and Kozono. For the incompressible magnetohydrodynamic equations, we also show the regularity of weak solution only under the assumption that \( \left\| u \right\|_{L_w^2 (0,T;L^\infty (\mathbb{R}^3 ))} \) is sufficiently small.
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He, C., Wang, Y. Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations. Sci. China Math. 53, 1767–1774 (2010). https://doi.org/10.1007/s11425-010-3135-3
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DOI: https://doi.org/10.1007/s11425-010-3135-3