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We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-KatoMajda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations in R n and the gradient flow to the harmonic map to the sphere are discussed. Namely the Serrin-Ohyama type regularity criteria are improved in the terms of the Besov spaces. Subject Classification (1991): 35Q05, 35L60, 75C05. Mathematics

We prove that the BM O norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class.

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