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 $d\ge 4$. By establishing an $ϵ$ regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in ${\dot{B}}_{p,q}^{-1+d/p}({\mathbb{R}}^d)$, $d<p,q<\infty$ becomes singular at a finite time $T_{⁎}$, then

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 $L^d({\mathbb{R}}^d)$ is addressed.