Existence of Rotating Stars with Variable Entropy

TL;DR

This paper proves the existence of differentially rotating stars with variable entropy, extending previous models that assumed constant entropy, by developing a new perturbative method to handle the coupled hyperbolic-elliptic system.

Contribution

It introduces a novel perturbative approach to construct rotating star solutions with variable entropy, overcoming the limitations of prior methods that required constant entropy.

Findings

Existence of rotating stars with small angular velocity and entropy variation.

Construction of solutions bifurcating from non-rotating stars.

Development of a new method to handle coupled hyperbolic-elliptic equations.

Abstract

We model a rotating star as a compressible fluid subject to gravitational forces. In almost all the mathematical literature the entropy is considered to be constant. Here we allow it to be variable. We consider a star that steadily rotates differentially around a fixed axis, say the $z$-axis. We prove the existence of a family of such stars with small angular velocity $\omega$ and small entropy variation $s$ and with an equation of state $p=Ke^s\rho^\gamma$. Our analysis reduces to a hyperbolic equation for the modified entropy coupled to an elliptic equation for the modified density, together with a mass constraint. Due to the variable entropy and the consequent loss of both regularity and variational structure, all the methods in the previous literature fail. We develop a new ad hoc perturbative strategy that allows us to construct rotating stars that bifurcate from the non-rotating ones.