# A sufficient condition for uniqueness of weak solutions of the incompressible Euler system

## TL;DR

This paper establishes a new sufficient condition for the uniqueness of weak solutions to the incompressible Euler equations in dimensions two and higher, under mild regularity assumptions, with a simple proof technique applicable to related fluid dynamics equations.

## Contribution

It introduces a novel regularity-based criterion ensuring uniqueness of weak solutions, contrasting with previous nonuniqueness results.

## Key findings

- Provides a new sufficient condition for solution uniqueness
- Employs a simple proof method using commutator estimates
- Applicable to other fluid equations like Euler-Boussinesq

## Abstract

We give a new sufficient criteria to prove the uniqueness of the incompressible Euler equation in dimension $N\geq2$. In their celebrated works by V. Scheffer [18], A. Shnirelman [19], C. De Lellis and L. Sz\'ekelyhidi Jr. [7] they have obtained the nonuniqeness of weak solutions of incompressible Euler equation. Here we obtain uniqueness criteria for the same equation under some mild regularity condition on weak solutions. Our proof is simple and can be employed to other equations like inhomogeneous incompressible Euler and Euler-Boussinesq equations. One of the key ingredients in our proof is commutator estimate [5, 11].