The energy conservation and regularity for the Navier-Stokes equations

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 $||u||_{L^{2,\infty}([0,T];BMO(\Omega))}<\infty$ for arbitrary $\Omega\subseteq\mathbb{R}^3$ then the local energy equality is valid on $[0,T]\times\Omega$. 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 $||u_h||_{L^{2,\infty}([0,T];BMO(\mathbb{R}^3))}\leq c_M$ then $u$ is regular on $(0,T]\times\mathbb{R}^3$.