Weak and Strong Solutions to the Forced Fractional Euler Alignment System

TL;DR

This paper studies the existence, uniqueness, and properties of weak and strong solutions for a forced fractional Euler alignment system with singular interaction kernels, extending classical results to less regular solutions and analyzing their long-term behavior.

Contribution

It introduces a framework for weak and strong solutions in larger function spaces and extends classical properties and energy laws to these less regular solutions.

Findings

Existence and uniqueness of weak and strong solutions are established.

Onsager-type criteria for energy law validity are derived.

Fast alignment and flocking behaviors are confirmed in the forceless case.

Abstract

We consider a hydrodynamic model of self-organized evolution of agents, with singular interaction kernel $\phi_\alpha(x)=1/|x|^{1+\alpha}$ ($0<\alpha<2$), in the presence of an additional external force. Well-posedness results are already available for the unforced system in classical regularity spaces. We define a notion of solution in larger function spaces, in particular in $L^\infty$ ("weak solutions") and in $W^{1,\infty}$ ("strong solutions"), and we discuss existence and uniqueness of these solutions. Furthermore, we show that several important properties of classical solutions carry over to these less regular ones. In particular, we give Onsager-type criteria for the validity of the natural energy law for weak solutions of the system, and we show that fast alignment (weak and strong solutions) and flocking (strong solutions) still occur in the forceless case.