Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations

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.