Dissipative measure-valued solutions and weak-strong uniqueness for the Euler alignment system

TL;DR

This paper introduces dissipative measure-valued solutions for the Euler alignment system, establishing a weak-strong uniqueness principle that ensures solutions coincide when both exist from the same initial data.

Contribution

It develops a novel framework for measure-valued solutions using tensor Young measures and proves a weak-strong uniqueness result for the Euler alignment system.

Findings

Dissipative measure-valued solutions are well-defined for the Euler alignment system.

A weak-strong uniqueness principle is established for these solutions.

The approach ensures solutions coincide when classical solutions exist from the same initial data.

Abstract

We introduce the concept of a dissipative measure-valued solution to the Euler alignment system. This approach incorporates a modified total energy balance, utilizing a binary tensor Young measure. The central finding is a weak (measure-valued)--strong uniqueness principle: if both a dissipative measure-valued solution and a classical smooth solution originate from the same initial data, they will be identical as long as the classical solution exists.