Critical thresholds in a nonlocal Euler system with relaxation

TL;DR

This paper introduces a nonlocal Euler system with relaxation, proving local existence and uniqueness, deriving a critical threshold in 1D, and showing that nonlocality prevents finite-time blow-up for many initial conditions.

Contribution

It provides the first rigorous analysis of a nonlocal Euler system with relaxation, including existence, uniqueness, and critical threshold results.

Findings

Global smooth solutions exist for many initial data.

Nonlocal velocity regularizes finite-time breakdown.

Critical threshold identified in one-dimensional case.

Abstract

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.