Global solutions to multi-dimensional topological Euler alignment systems

TL;DR

This paper develops a systematic regularity theory for multi-dimensional Euler alignment systems with topological diffusion, identifying classes of global smooth solutions and establishing an improved continuation criterion based on Lipschitz control.

Contribution

It introduces a new continuation criterion for global solutions, applicable to multi-dimensional topological Euler alignment systems, using fractional parabolic theory and quartic paraproduct estimates.

Findings

Identified classes of global smooth solutions: parallel shear flocks and nearly aligned flocks.

Established an improved continuation criterion based on Lipschitz norm control.

Enhanced the theoretical understanding of topological effects in multi-dimensional flocking models.

Abstract

We present a systematic approach to regularity theory of the multi-dimensional Euler alignment systems with topological diffusion introduced in \cite{STtopo}. While these systems exhibit flocking behavior emerging from purely local communication, bearing direct relevance to empirical field studies, global and even local well-posedness has proved to be a major challenge in multi-dimensional settings due to the presence of topological effects. In this paper we reveal two important classes of global smooth solutions -- parallel shear flocks with incompressible velocity and stationary density profile, and nearly aligned flocks with close to constant velocity field but arbitrary density distribution. Existence of such classes is established via an efficient continuation criterion requiring control only on the Lipschitz norm of state quantities, which makes it accessible to the applications of fractional parabolic theory. The criterion presents a major improvement over the existing result of \cite{RS2020}, and is proved with the use of quartic paraproduct estimates.