Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces

TL;DR

This paper rigorously analyzes the hydrodynamic limits of Vlasov-type equations with alignment and nonlocal forces, deriving Euler-type equations under various regimes and providing explicit estimates using the relative entropy method.

Contribution

It introduces a comprehensive analysis of the hydrodynamic limits for kinetic models with nonlocal interactions, including new estimates and methods for both diffusive and non-diffusive cases.

Findings

Derivation of isothermal Euler equations with nonlocal forces under strong alignment and diffusion.

Derivation of pressureless Euler-type equations without diffusion.

Explicit estimates on the convergence towards hydrodynamic equations.

Abstract

In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker--Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulombian interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. For the sake of completeness, the existence of weak and strong solutions to the kinetic and fluid equations are also established.