The pressureless damped Euler-Riesz equations

TL;DR

This paper studies the global existence, uniqueness, and long-time decay behavior of solutions to the pressureless damped Euler-Riesz equations in both whole space and periodic domains, revealing algebraic and exponential decay rates.

Contribution

It establishes the first comprehensive analysis of the pressureless damped Euler-Riesz equations, including global solutions and detailed decay rates in different spatial settings.

Findings

Global existence and uniqueness of classical solutions.

Solutions decay to equilibrium over time.

Exponential decay rates in both whole space and periodic cases.

Abstract

In this paper, we analyze the pressureless damped Euler-Riesz equations posed in either $\mathbb{R}^d$ or $\mathbb{T}^d$. We construct the global-in-time existence and uniqueness of classical solutions for the system around a constant background state. We also establish large-time behaviors of classical solutions showing the solutions towards the equilibrium as time goes to infinity. For the whole space case, we first show the algebraic decay rate of solutions under additional assumptions on the initial data compared to the existence theory. We then refine the argument to have the exponential decay rate of convergence even in the whole space. In the case of the periodic domain, without any further regularity assumptions on the initial data, we provide the exponential convergence of solutions.