Regularity Criteria in Weak L³ for 3D Incompressible Navier-Stokes Equations

抄録

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 R³ with radius r. We will show that if p ∈ L^m(0,1; L¹(B₂)), m > 2, and if u is sufficiently small in L^∞(0,1; L^3,∞(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.