Small inertia regularization of an anisotropic aggregation model

TL;DR

This paper rigorously analyzes the zero-inertia limit of an anisotropic aggregation model, revealing how second-order relaxation systems approximate discontinuous velocity solutions through fast transitions within a specific time scale.

Contribution

It provides a rigorous analysis of the zero-inertia limit for anisotropic aggregation models, including the characterization of fast transition layers near velocity discontinuities.

Findings

Solutions perform fast transitions of size O(ε^{2/3}) near discontinuities.

Numerical simulations confirm the theoretical transition scale.

The relaxation model effectively captures discontinuous solutions in the zero-inertia limit.

Abstract

We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter $\varepsilon$, which can be interpreted as inertia or response time. We examine rigorously the limit $\varepsilon \to 0$ of solutions to the relaxation system. Of major interest is how discontinuous (in velocities) solutions to the first-order model are captured in the zero-inertia limit. We find that near such discontinuities, solutions to the second-order model perform fast transitions within a time layer of size $\mathcal{O}(\varepsilon^{2/3})$. We validate this scale with numerical simulations.