Localized continuation criterion, improved local existence and uniqueness for the Euler-Poisson system in a bounded domain

TL;DR

This paper improves the understanding of singularity formation in the Euler-Poisson system by refining initial conditions, removing nonphysical assumptions, and establishing a localized continuation criterion that extends previous results to the compressible case.

Contribution

It introduces a refined local existence and uniqueness theorem for the Euler-Poisson system with compact support initial data and develops a localized continuation criterion that generalizes previous incompressible results.

Findings

Removes nonphysical exterior velocity assumptions in Makino's star model

Establishes improved local existence and uniqueness for compact support data

Proves a localized continuation criterion controlling solution breakdown

Abstract

To understand the formations of singularities of the Euler-Poisson system with vacuum, we revisit Makino's star model in this article. We first remedy, to some extent, the inconveniences of Makino's star model and remove its imposed nonphysically exterior free-falling velocity field by only specifying the velocity field on a compact support of the density. Makino has coined the presence of such an exterior velocity field as a ``Cheshire cat phenomenon'' and he [33] and Rendall [45] have both emphasized the difficulty of removing this phenomenon. Moreover, we obtain an improved local existence and uniqueness theorem for initial data for the density and velocity that both have compact support. Finally, we are able to prove a localized strong continuation criterion in which the breakdown of solutions is only controlled by quantities defined on the compact support of the solution. In addition, the localized strong continuation criterion also generalizes the continuation criterion from the incompressible Euler equation in a bounded domain [16,47] to the compressible Euler-Poisson system.