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

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 $\varphi (x,y)$, and $m=m(n,\alpha )$ is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator \[ \mathcal{L}_ϕf(x) = \int_{\mathbb{T}^n} ϕ(x,y) (f(y) - f(x) ) dy, \] 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.

33 pages