Relaxation to Fractional Porous Medium Equation from Euler–Riesz System

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 in the case of repulsive Riesz interactions, based on the modulated energy estimates, the Wasserstein distance of order 2, and the bounded Lipschitz distance. For the attractive Riesz interaction case, we consider the periodic domain and estimate a lower bound on the modulated internal energy to handle the modulated interaction energy.