On regularity and singularity for L^∞(0,T;L^3,w(ℝ³)) solutions to the Navier-Stokes equations

2016/11/15 by Choe, Hi Jun, Wolf, Jörg, Yang, Minsuk
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.04725

Abstract

We study local regularity properties of a weak solution u to the Cauchy problem of the incompressible Navier-Stokes equations. We present a new regularity criterion for the weak solution u satisfying the condition L^∞(0,T;L^3,w(ℝ³)) without any smallness assumption on that scale, where L^3,w(ℝ³) denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time t. The condition that the weak Lebesgue space norm of the veclocity field u is bounded in time is encompassing type I singularity and significantly weaker than the end point case of the so-called Ladyzhenskaya-Prodi-Serrin condition proved by Escauriaza-Sergin-Šverák.

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