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

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.