Remarks on the 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 using two different approaches, one with smoothness assumptions and one local approach, also exploring auxiliary problem solutions.

Contribution

It introduces a new approach for proving uniqueness under smoothness conditions and provides local results without such conditions, expanding understanding of weak solution uniqueness.

Findings

Uniqueness established for smooth functions using a novel approach.

Local uniqueness results obtained without complementary conditions.

Analysis of auxiliary problems supports main results.

Abstract

This article studies the uniqueness of the weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case. Here, the investigation is provided using two different approaches. The first (the main) result is obtained for given functions possessing a certain smoothness using the new approach. The second result is without the complementary conditions but is, in some sense, the "local" result investigated by another approach. In addition, here the solvability and uniqueness of the weak solutions to auxiliary problems lead out from the main problem are investigated.