Relaxation to fractional porous medium equation from Euler--Riesz system

TL;DR

This paper analyzes the high-force limit of the Euler--Riesz system and rigorously derives the fractional porous medium equation as its relaxation limit, using energy and Wasserstein distance methods.

Contribution

It provides a unified asymptotic analysis approach for the Euler--Riesz system, establishing a quantified relaxation limit to the fractional porous medium equation.

Findings

Established the relaxation limit from Euler--Riesz to fractional porous medium equation.

Developed a unified approach using modulated energy and Wasserstein distances.

Provided quantitative convergence results in high-force regimes.

Abstract

We perform asymptotic analysis for the Euler--Riesz system posed in either $\mathbb{T}^d$ or $\mathbb{R}^d$ in the high-force regime and establish a quantified relaxation limit result from the Euler--Riesz system to the fractional porous medium equation. We provide a unified approach for asymptotic analysis regardless of the presence of pressure, based on the modulated energy estimates, the Wasserstein distance of order $2$, and the bounded Lipschitz distance.