Cahn-Hillard and Keller-Segel systems as high-friction limits of Euler-Korteweg and Euler-Poisson equations

TL;DR

This paper demonstrates that the Cahn-Hilliard and Keller-Segel systems can be derived as high-friction limits of Euler-Korteweg and Euler-Poisson equations, providing new insights into their mathematical relationships.

Contribution

It establishes the high-friction limit connection between complex fluid models and classical phase separation and chemotaxis systems, including existence of solutions and limiting behavior.

Findings

Existence of dissipative measure-valued solutions for Euler--Poisson system.

High-friction limit of Euler--Korteweg and Euler--Poisson equations to Cahn--Hilliard--Keller--Segel system.

Additional restrictions on parameters for attractive potentials.

Abstract

We consider a combined system of Euler--Korteweg and Euler--Poisson equations with friction and exponential pressure with exponent $\gamma > 1$. We show the existence of dissipative measure-valued solutions in the cases of repulsive and attractive potential in Euler--Poisson system. The latter case requires additional restriction on $\gamma$. Furthermore in case of $\gamma \geq 2$ we show that the strong solutions to the Cahn--Hillard--Keller--Segel system are a high-friction limit of the dissipative measure-valued solutions to Euler--Korteweg--Poisson equations.