On well-posedness and singularity formation for the Euler-Riesz system

TL;DR

This paper studies the well-posedness and finite-time blowup of solutions for the Euler-Riesz system, extending classical results to cases with nonlocal interactions and providing conditions for singularity formation.

Contribution

It develops a functional framework for local existence and uniqueness, covers the Euler-Poisson case, and establishes finite-time blowup criteria for a broad class of initial data.

Findings

Established local well-posedness for Euler-Riesz system.

Proved finite-time blowup under certain initial conditions.

Extended analysis to both attractive and repulsive interaction forces.

Abstract

In this paper, we investigate the initial value problem for the Euler-Riesz system, where the interaction forcing is given by $\nabla(-\Delta)^{s}\rho$ for some $-1<s<0$, with $s = -1$ corresponding to the classical Euler-Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler-Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler-Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler-Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler-Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.