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 $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<infty$ and $lim_{tto T}sqrt{T-t}||u(t)||_{L^infty}=infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $uin L^{2,infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,infty}([0,T];BMO(Omega))}<infty$ for arbitrary $Omegasubseteqmathbb{R}^3$ then the local energy equality is valid on $[0,T]timesOmega$. 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 $(0,T]timesmathbb{R}^3$.