Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations

Abstract

We study the regularity of a distributional solution $(u,p)$ of the 3D incompressible evolution Navier-Stokes equations. Let $B_r$ denote concentric balls in ${\mathbb{R}}^3$ with radius $r$. We will show that if $p\in L^m(0,1;L^1(B_2))$, $m>2$, and if $u$ is sufficiently small in $L^{\infty }(0,1;L^{3,\infty }(B_2))$, without any assumption on its gradient, then $u$ is bounded in $B_1\times (\frac{1}{10},1)$. It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.

16 pages

Cited by in corpus (5)

• On global solutions to the Navier-Stokes system with large $L^{3,\infty }$ initial data
• The Navier-Stokes equations in nonendpoint borderline Lorentz spaces
• On blowup of nonendpoint borderline Lorentz norms for the Navier-Stokes equations
• Time-dependent singularities in the Navier-Stokes system
• Local boundary regularity for the Navier-Stokes equations in nonendpoint borderline Lorentz spaces