On Endpoint Regularity Criterion Of The 3d Navier-Stokes Equations

Let (u, pi) with u = (u(1), u(2), u(3)) be a suitable weak solution of the three dimensional Navier-Stokes equations in R-3 x (0,T). Denote by (B) over dot(infinity,infinity)(-1) the closure of C-0(infinity) in (B) over dot(infinity,infinity)(-1). We prove that if u is an element of L-infinity(0,T; (B) over dot(infinity,infinity)(-1)), u(x, T) is an element of (B) over dot(infinity,infinity)(-1)), and u(3) is an element of L-infinity(0, T; L-3,L-infinity) or u(3) is an element of L-infinity(0,T; B (over dot)(p,q)(-1+3/p)) with 3 < p, q < infinity, then u is smooth in R-3 x (0,T]. Our result improves a previous result established by Wang and Zhang [Sci. China Math. 60, 637-650 (2017)].