In this lecture, we will show first that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of strong solutions to the Navier-Stokes equations. Our result will be applied to the criterion on uniqueness and regularity of weak solutions in the marginal class of Serrin's. For the proof, we will establish various bilinear estimates in the BMO norm by means of the symbol calculus given by Coifman-Meyer. Then we will next deal with the similar blow-up phenomena to the Euler equations. To this end, we will introduce a critical Sobolev inequality of logarithmic type in the BMO and the Besov spaces which may be regarded as generalization of Brezis-Wainger's.
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