On uniqueness of weak solutions of the incompressible Navier-Stokes equations

TL;DR

This paper investigates the uniqueness of weak solutions to the 3D incompressible Navier-Stokes equations, introducing a new approach and establishing uniqueness in certain smooth function spaces, along with auxiliary problem analysis.

Contribution

It presents a novel method for proving uniqueness of weak solutions in smooth function spaces and explores auxiliary problems related to the Navier-Stokes equations.

Findings

Uniqueness of velocity in smooth function spaces is established.

Auxiliary problems related to the main Navier-Stokes problem are shown to have unique solutions.

A conditional result on the uniqueness of weak solutions is proved.

Abstract

In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity for the considered problem is proved for given functions from spaces that posseses some smoothness. Moreover, these spaces are dense in respective spaces of functions, under which were proved existence of the weak solutions. In addition here the solvability and uniqueness of the weak solutions of auxiliary problems associated with the main problem is investigated, and also one conditional result on uniqueness is proved.