Blowup criterion for Navier-Stokes equation in critical Besov space with spatial dimensions d ≥ 4

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions d >= 4. By establishing an epsilon regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in (B) over dot(p,q)(-1+d/p)(R-d), d < p, q < infinity becomes singular at a finite time T-*, then lim sup(t -> T*) parallel to u(t) parallel to ((B) over dotp,q-1+d/p(Rd)) = infinity. The corresponding result in 3D case has been obtained in [24]. As a by-product, we also prove a regularity criterion for the Leray-Hopf solution in the critical Besov space, which generalizes the results in [17], where blowup criterion in critical Lebesgue space L-d (R-d) is addressed. (C) 2019 Elsevier Masson SAS. All rights reserved.