The energy conservation and regularity for the Navier-Stokes equations

Abstract

In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies ${\lim }_{t\to T}\sqrt{T-t}||u(t)|{|}_{BMO}<\infty$ and ${\lim }_{t\to T}\sqrt{T-t}||u(t)|{|}_{L^{\infty }}=\infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $u\in L^{2,\infty }(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u|{|}_{L^{2,\infty }([0,T];BMO(\hat{I}\text{c◯}))}<\infty$ for arbitrary $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^3$ then the local energy equality is valid on $[0,T]\times \hat{I}\text{c◯}$. As a corollary, we also prove $||u|{|}_{L^{2,\infty }([0,T];BMO({\mathbb{R}}^3))}<\infty$ implies the global energy equality on $[0,T]$. Thirdly, we show that as the solution $u$ approaches a finite blowup time $T$, the norm $||u(t)|{|}_{BMO}$ must blow up at a rate faster than $\frac{c}{\sqrt{T-t}}$ with some absolute constant $c>0$. Furthermore, we prove that if $||u_3|{|}_{L^{2,\infty }([0,T];BMO({\mathbb{R}}^3))}=M<\infty$ then there exists a small constant $c_M$ depended on $M$ such that if $||u_h|{|}_{L^{2,\infty }([0,T];BMO({\mathbb{R}}^3))}\le c_M$ then $u$ is regular on $(0,T]\times {\mathbb{R}}^3$.