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

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, we show that if the mild solution $u$ with initial data in ${\dot{B}}_{p,q}^{-1+d\mathrm{/}p}({\mathbb{R}}^d)$, $d<p,q<\mathrm{\infty }$ becomes singular at a finite time $T_{\ast }$, then $$\underset{t\to T_{\ast }}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{{\dot{B}}_{p,q}^{-1+d\mathrm{/}p}({\mathbb{R}}^d)}=\mathrm{\infty }\mathrm{.}$$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. 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~\cite{DoDu09}, where blowup criterion in critical Lebesgue space $L^d({\mathbb{R}}^d)$ is obtained.