Existence of Rotating Stars with Variable Entropy

We model a rotating star as a compressible fluid subject to gravitational forces. In most of 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 $s$ and with an equation of state $p=Ke^s\rho^\gamma$. Our analysis reduces to a hyperbolic equation for the entropy coupled to an elliptic equation for the density, together with a mass constraint.