Uniqueness criterion of weak solutions for the 3D Navier-Stokes equations

TL;DR

This paper proves new uniqueness results for weak solutions of the 3D Navier-Stokes equations under specific integrability conditions, contributing to the understanding of solution behavior at singular times.

Contribution

It establishes conditions under which weak solutions to the 3D Navier-Stokes equations are unique, especially relating to singular times and specific function spaces.

Findings

Weak solutions are unique in $L^{5/2}(0,T;V)$ if they share the same initial data.

In $L^{3}(0,T;V)$, the difference of solutions is continuous in time and solutions coincide when initial data are equal.

The results provide criteria for solution uniqueness at singular times.

Abstract

In this paper we establish a new uniqueness result of weak solutions for the 3D Navier-Stokes equations. Under assumption that there is not uniqueness of weak solution in singular time, we prove that if two weak solutions $u$ and $v$ of 3D Navier-Stokes equations belong to $L^{\frac{5}{2}}\left( 0,T;V\right) $ with the same initial datum, then we get $u=v$. In the class $L^{3}\left( 0,T;V\right) $, we prove that $u-v\in$ $C^{0}\left( 0,T;H\right) $ and $u=v\ $when $u_{0}=v_{0}$.