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

Abstract

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 ϵ regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in B˙p,q−1+d/p(Rd), d<p,q<∞ becomes singular at a finite time T⁎, thenlimsupt→T⁎‖u(t)‖B˙p,q−1+d/p(Rd)=∞. 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 Ld(Rd) is addressed.