The energy conservation and regularity for the Navier-Stokes equations

TL;DR

This paper investigates energy conservation, regularity, and singularity formation of weak solutions to the Navier-Stokes equations, establishing conditions under which energy equality holds and characterizing blowup rates in endpoint function spaces.

Contribution

It introduces new criteria for energy equality and regularity of Navier-Stokes solutions using BMO and Lorentz spaces, especially at the endpoint cases.

Findings

Construction of solutions with Type II singularities in endpoint spaces.

Proof that local energy equality holds under certain BMO conditions.

Demonstration that BMO norm blow-up rate exceeds a specific threshold near singularity.

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 $||u||_{L^{2,\infty}([0,T];BMO(\Omega))}<\infty$ 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 $(0,T]\times\mathbb{R}^3$.