Weak Solutions to an Euler Alignment System with Singular Interactions in a Bounded Domain

TL;DR

This paper investigates the existence and uniqueness of weak solutions to a complex Euler alignment system with singular interactions, including attraction, repulsion, and self-propulsion, in bounded domains, extending the model with leader agents and fractional Bessel potentials.

Contribution

It establishes the existence of infinitely many weak solutions, constructs energy-bounded solutions, and proves a weak-strong uniqueness principle for a novel Euler alignment model with singular kernels.

Findings

Existence of infinitely many weak solutions.

Construction of energy-bounded solutions.

Weak-strong uniqueness principle established.

Abstract

Euler alignment systems appear as hydrodynamic limits of interacting self-propelled particle systems such as the (generalized) Cucker-Smale model. In this work, we study weak solutions to an Euler alignment system on smooth, bounded, connected domains. This particular Euler alignment system includes singular alignment, attraction, and repulsion interaction kernels which correspond to a Yukawa potential. We also include a confinement potential and self-propulsion. We embed the problem into an abstract Euler system to conclude that infinitely many weak solutions exist. We further show that we can construct solutions satisfying bounds on an energy quantity, and that the solutions satisfy a weak-strong uniqueness principle. Finally, we present an addition of leader-agents governed by controlled ODEs, and modification of the interactions to be Bessel potentials of fractional order $s > 2$.