MR3468734 Yuwen LUO and Tai-Peng TSAI Yuwen LUO and Tai-Peng TSAI Regularity Criteria in Weak $L^3$ for 3D Incompressible Navier-Stokes Equations Funkcialaj Ekvacioj. Serio Internacia 58 2015 387--404 http://fe.math.kobe-u.ac.jp/FE/FullPapers/58-3/58_387.pdf http://www.ams.org/mathscinet-getitem?mr=MR3468734 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 $B_1\times(1/10,1)$. 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. Navier-Stokes equations, Regularity criteria, Distributional solution, Weak $L^3$. 35Q30, 35B10, 35B40. 58-387 2015 Regularity Criteria in Weak $L^3$ for 3D Incompressible Navier-Stokes Equations Yuwen LUO and Tai-Peng TSAI Yuwen LUO and Tai-Peng TSAI 1 Bennet, C.; Sharpley, R. Interpolation of Operators Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA 1988 MR0928802 2 Bergh, J.; Löfström, J. Interpolation Spaces. An Introduction Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York 1976 MR0482275
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