Periodic Solutions of 2D Isothermal Euler-Poisson Equations with Possible Applications to Spiral and Disk-like Galaxies

TL;DR

This paper constructs periodic solutions to 2D isothermal Euler-Poisson equations, providing insights into galaxy formation and evolution, and highlighting the stabilizing effect of rotation against blowup phenomena.

Contribution

It introduces new periodic solutions with rotation for 2D isothermal Euler-Poisson equations, extending previous non-rotational solutions and suggesting 3D applications.

Findings

Periodic solutions prevent blowup due to rotation.

Applications to spiral and disk-like galaxy formation.

Extension of previous non-rotational solutions.

Abstract

Compressible Euler-Poisson equations are the standard self-gravitating models for stellar dynamics in classical astrophysics. In this article, we construct periodic solutions to the isothermal ($\gamma=1$) Euler-Poisson equations in $R^{2}$ with possible applications to the formation of plate, spiral galaxies and the evolution of gas-rich, disk-like galaxies. The results complement Yuen's solutions without rotation (M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341(2008), 445--456.). Here, the periodic rotation prevents the blowup phenomena that occur in solutions without rotation. Based on our results, the corresponding $3$D rotational results for Goldreich and Weber's solutions are conjectured.