We study the regularity of a distributional solution (
u,
p) of the 3D incompressible evolution Navier-Stokes equations. Let
Br denote concentric balls in
R3 with radius
r. We will show that if
p ∈
Lm(0,1;
L1(
B2)),
m > 2, and if
u is sufficiently small in
L∞(0,1;
L3,∞(
B2)), without any assumption on its gradient, then
u is bounded in
B1 × (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.
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