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

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.