On the Regularity of Navier-Stokes Equations in Critical Space

TL;DR

This paper establishes regularity criteria for Navier-Stokes solutions in certain critical scaling-invariant spaces, showing that solutions in these spaces are smooth and do not blow up at the initial time.

Contribution

It proves that solutions in specific critical Lebesgue spaces are necessarily smooth and regular, extending understanding of Navier-Stokes regularity criteria.

Findings

Solutions in the specified spaces are smooth in the domain.

No finite-time blow-up occurs for solutions in these spaces.

Regularity holds up to the initial time t=0.

Abstract

This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $ u(x,t) $ and $ p(x,t) $ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$ is in the scaling invariant spaces $L_t^{\infty}L_{x_3}^{p_1}L_{x_h}^{p_2}(Q_T)$ , where $ \frac{1}{p_1}+\frac{2}{p_2}=1 $ , $p_1\geq 2$ and $ x_h = (x_1, x_2) $ , then $ u $ is a smooth solution in $ Q_T $ and doesn't blow up at $ t = 0 $. In particular, if $ u(x,t) \in L_t^{\infty}L_{x_3}^{\infty}L_{x_h}^{2}(Q_T)$, then $u(x,t)$ is a smooth solution in $ Q_T $ and regular up to $ t = 0 $.