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

Abstract

In this paper, we investigate the initial value problem for the Euler–Riesz system, where the interaction forcing is given by ∇(−Δ)sρ 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.