2020/08/03 by Daniel Lear, Trevor M. Leslie, Lear, Daniel +5 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2008.01049
openalex publication_date 2020/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel ϕ and unidirectional velocity u = (u, 0, …, 0). We demonstrate a striking correspondence between the entropy function e0 = ∂1 u0 + ϕ*ρ0 and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of C1 hypersurfaces: the image of the zero set of e0 under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of e0. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.