From gas dynamics with large friction to gradient flows describing diffusion theories

TL;DR

This paper investigates how gradient flows in Wasserstein distance emerge as high friction limits of Euler flows, providing a unified framework for deriving diffusion equations from fluid dynamics models.

Contribution

It introduces a relative energy method to connect Euler flows with gradient flows, proving convergence from Euler-Poisson and Euler-Korteweg systems to diffusion equations.

Findings

Proves convergence from Euler-Poisson to Keller-Segel system.

Establishes convergence from Euler-Korteweg to Cahn-Hilliard equation.

Develops a unified approach for high friction limits in fluid models.

Abstract

We study the emergence of gradient flows in Wasserstein distance as high friction limits of an abstract Euler flow generated by an energy functional. We develop a relative energy calculation that connects the Euler flow to the gradient flow in the diffusive limit regime. We apply this approach to prove convergence from the Euler-Poisson system with friction to the Keller-Segel system in the regime that the latter has smooth solutions. The same methodology is used to establish convergence from the Euler-Korteweg theory with monotone pressure laws to the Cahn-Hilliard equation.