Topological and Purely Topological Alignment Dynamics

Abstract: We study the Euler Alignment system of collective behavior, equipped with topological' interaction protocols, which were introduced to the mathematical literature by Shvydkoy and Tadmor. Interactions subject to these protocols may depend on both the Euclidean distance between agents and on the mass distribution between them -- thetopological' component. When the interaction protocol is regular, we prove sufficient conditions for the existence of global-in-time classical solutions, related to the initial nonnegativity of a conserved quantity of the system. The remainder of our results explore the case where the interactions are `purely' topological and the interactions do not depend on the Euclidean distance. We show that in this case, the system decouples into an autonomous velocity equation in mass coordinates together with a scalar conservation law with time-dependent flux determined by the velocity. We analyze the long-time behavior for the dynamics associated to both regular and singular protocols.