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 ε L2/1-α (0, T; Ḟ_∞∞^-α) for 0 < α < 1, where Ḟ_{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 Ḟ_{p1, q1}^-α ∩ Ḟ_{p2, q2}^s+α ⊂ Ḟ _{p, q}^s, 1/p = 1/p1 + I/p2, 1/q = 1/q1 + 1/q2 as the Hölder type inequality plays an important role for our results.