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 by Roman and Tadmor (2018 arXiv:1806.01371). The main result demonstrates local existence and uniqueness of classical solutions in class (ρ, u) ∈ Hm+α × Hm+1 on the periodic domain , where 0 < α < 2 is the order of singularity of the topological communication kernel ϕ(x, y), and m = m(n, α) is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator which render proper a priori estimates and help stabilize viscous approximation of the system.