Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations

We shall show that every strong solution u(t) of the Navier-Stokes equations on (0, T) can be continued beyond t > T provided u is an element of L (2)/(1-alpha) (0, T; F-infinity,infinity(-alpha)) for 0 < alpha < 1, where F-p,q(s) denotes the homogeneous Triebel-Lizorkin space. As a byproduct of our continuation theorem, we shall generalize a well-known criterion due to Serrin on regularity of weak solutions. Such a bilinear estimate F-p1,q1(-alpha) boolean AND F-p2,q2(s+alpha) 1/P = 1/p1 + 1/p(2), 1/q = 1/q, + 1/q(2) as the Hblder type inequality plays an important role for our results. (C) 2004 WILEY-VCH Vertag GmbH & Co. KGaA, Weinheim.