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

TL;DR

This paper establishes a blowup criterion for the Navier-Stokes equations in higher dimensions ($d \, \geq \, 4$) within critical Besov spaces, extending known results from 3D and Lebesgue spaces.

Contribution

It introduces an $\, \epsilon$ regularity criterion for higher-dimensional Navier-Stokes solutions in critical Besov spaces, generalizing previous 3D and Lebesgue space results.

Findings

Blowup in critical Besov norm implies finite-time singularity.

Established regularity criterion for Leray-Hopf solutions in critical Besov space.

Extended blowup criteria to dimensions $d \geq 4$.

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