Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars

TL;DR

This paper constructs explicit blowup solutions for the 2D isothermal Euler-Poisson equations modeling gaseous stars, analyzes their blowup rate, and discusses implications for related models and solution lifespan.

Contribution

It introduces a family of analytical blowup solutions for the 2D isothermal Euler-Poisson equations and extends existing results on solution lifespan for compactly supported initial data.

Findings

Constructed explicit blowup solutions in R^2

Analyzed blowup rate of these solutions

Extended lifespan results to related models

Abstract

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R^2. Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case, the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.