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 Ḟp, qs 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 Ḟp1, q1-α ∩ Ḟp2, q2s+α ⊂ Ḟ p, qs, 1/p = 1/p1 + I/p2, 1/q = 1/q1 + 1/q2 as the Hölder type inequality plays an important role for our results.
- Littlewood-Paley decomposition
- Navier-Stokes equations
- Triebel-Lizorkin space
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