Weak solutions for Euler systems with non-local interactions

TL;DR

This paper demonstrates the existence of infinitely many weak solutions for modified Euler systems with non-local interactions, relevant to collective behavior modeling, and establishes a weak-strong uniqueness principle.

Contribution

It adapts convex integration to prove multiple weak solutions for non-local Euler systems and identifies conditions for dissipative solutions and uniqueness.

Findings

Existence of infinitely many weak solutions for non-local Euler models.

Identification of initial data sets with multiple dissipative solutions.

Establishment of weak-strong uniqueness for these systems.

Abstract

We consider several modifications of the Euler system of fluid dynamics including its pressureless variant driven by non-local interaction repulsive-attractive and alignment forces in the space dimension $N=2,3$. These models arise in the study of self-organisation in collective behavior modeling of animals and crowds. We adapt the method of convex integration to show the existence of infinitely many global-in-time weak solutions for any bounded initial data. Then we consider the class of \emph{dissipative} solutions satisfying, in addition, the associated global energy balance (inequality). We identify a large set of initial data for which the problem admits infinitely many dissipative weak solutions. Finally, we establish a weak-strong uniqueness principle for the pressure driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction.