Lower bounds on blowing-up solutions of the 3D Navier--Stokes equations in H^3/2, H^5/2, and B^5/2_2,1

2015/03/14 by David S. McCormick, Eric J. Olson, McCormick, David S. +9
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1503.04323

arxiv created 2015/03/14 · arxiv updated 2015/03/17

Abstract

If u is a smooth solution of the Navier--Stokes equations on \mathbb R³ with first blowup time T, we prove lower bounds for u in the Sobolev spaces H^3/2, H^5/2, and the Besov space B^5/2_2,1, with optimal rates of blowup: we prove the strong lower bounds ‖u(t)‖ _H^3/2≥ c(T-t)^-1/2 and ‖u(t)‖_ B^5/2_2,1≥ c(T-t)^-1, but in H^5/2 we only obtain the weaker result \limsup_{t→ T^-}(T-t)‖u(t)‖ _H^5/2≥ c. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of u.

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