Relative Entropy Method for the relaxation limit of Hydrodynamic models

TL;DR

This paper develops a method using relative entropy to derive nonlinear aggregation-diffusion models from nonlocal hydrodynamic systems, providing a rigorous relaxation limit analysis.

Contribution

It introduces a general framework for obtaining aggregation-diffusion models as limits of hydrodynamic systems using the relative entropy method, including existence results.

Findings

Derivation of aggregation-diffusion models from hydrodynamic systems

Conditions on potentials for the relaxation limit to hold

Existence of weak solutions for the hydrodynamic systems

Abstract

We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.