# On uniqueness of dissipative solutions to the isentropic Euler system

## TL;DR

This paper investigates the conditions under which dissipative solutions to the isentropic Euler system are unique and coincide with weak solutions, emphasizing the role of Besov regularity and Lipschitz bounds.

## Contribution

It establishes the equivalence of dissipative and weak solutions under specific regularity and Lipschitz conditions, clarifying the uniqueness criteria.

## Key findings

- Dissipative solutions match weak solutions with Besov regularity.
- One-sided Lipschitz bounds ensure solution uniqueness.
- Provides conditions for the equivalence of solution concepts.

## Abstract

The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure--valued solution of the problem. We show that dissipative solutions coincide with weak solutions starting from the same initial data on condition that: {\bf (i)} the weak solution enjoys certain Besov regularity; {\bf (ii)} the symmetric velocity gradient of the weak solution satisfies a one--sided Lipschitz bound.