Young-Pil Choi : Lagrangian formulation and Eulerian closure in alignment dynamics

Posted by Roman Shvydkoy , part of the Analysis and Applied Mathematics Seminar.

At

April 27, 2026, 4 p.m.

In

636 SEO

Abstract

We study a continuum Lagrangian alignment system for interacting agents with weak initial data. We first prove global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We then pass from the Lagrangian description to an Eulerian one, and obtain an Euler-Reynolds-alignment system involving a nonnegative Reynolds stress and, in the nonlinear velocity-coupling case, an additional defect force caused by microscopic velocity fluctuations. Under a heavy-tailed interaction assumption, we show that these defect terms vanish asymptotically, leading to mono-kinetic closure at large times. In the linear velocity-coupling case, we further prove the global existence of weak solutions to the Euler-alignment system, including a sharp critical-threshold result in one dimension and a global existence result in higher dimensions under a large-coupling condition. We also establish mean-field convergence results for the underlying particle system, including uniform-in-time convergence in the linear case.