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

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; L(B1)), m > 2, and if u is sufficiently small in L∞(0, 1; L(B1)), without any assumption on its gradient, then u is bounded in B1−τ × (τ, 1) for any τ > 0. It is a borderline case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. arXiv:1310.8307 This is a joint work with Yuwen Luo.