Local well-posedness of the topological Euler alignment models of collective behavior

TL;DR

This paper proves local well-posedness for multi-dimensional topological Euler-alignment models, establishing existence and uniqueness of classical solutions under certain regularity conditions and analyzing the operator's coercivity.

Contribution

It introduces new sharp coercivity estimates for the topological alignment operator, enabling well-posedness results for these collective behavior models.

Findings

Local existence and uniqueness of solutions in specified Sobolev spaces.

Sharp coercivity estimates for the alignment operator.

Global well-posedness in 1D with additional techniques.

Abstract

In this paper we address the problem of well-posedness of multi-dimensional topological Euler-alignment models introduced in \cite{ST-topo}. The main result demonstrates local existence and uniqueness of classical solutions in class $(\rho,u) \in H^{m+\alpha} \times H^{m+1}$ on the periodic domain $\mathbb{T}^n$, where $0<\alpha<2$ is the order of singularity of the topological communication kernel $\phi(x,y)$, and $m = m(n,\alpha)$ is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator \[ \mathcal{L}_\phi f(x) = \int_{\mathbb{T}^n} \phi(x,y) (f(y) - f(x) ) dy, \] which render proper a priori estimates and help stabilize viscous approximation of the system. In dimension 1, this result, in conjunction with the technique developed in \cite{ST-topo} gives global well-posendess in the natural space of data mentioned above.