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

✍ Scribed by Hideo Kozono; Yukihiro Shimada

Publisher
John Wiley and Sons
Year
2004
Tongue
English
Weight
197 KB
Volume
276
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

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 ∈ $L^{{{2} \over {1 - \alpha}}}$ (0, T; $\dot F^{- \alpha}_{\infty ,\infty}$ for 0 < α < 1, where $\dot F^{s}_{p,q}$ 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 $\dot F^{- \alpha}_{p_1 , q_1} \cap \dot F^{s + \alpha}_{p_2 , q_2} \subset \dot F^{s}_{p, q}$, 1/p = 1/p~1~ + 1/p~2~, 1/q = 1/q~1~ + 1/q~2~ as the Hölder type inequality plays an important role for our results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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