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Blowup criterion for Navier-Stokes equation in critical Besov space with spatial dimensions $d \geq 4$ (1803.04076v1)

Published 12 Mar 2018 in math.AP

Abstract: This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $\epsilon$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B{-1+d/p}_{p,q}(\mathbb{R}d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_$, then $$ \limsup_{t\to T_} |u(t)|{\dot B{-1+d/p}{p,q}(\mathbb{R}d)} = \infty. $$ 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 $Ld(\mathbb{R}d)$ is obtained.

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