Existence of Rotating Stars with Variable Entropy

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 $ω$ and small entropy variation $s$ and with an equation of state $p=Ke^sρ^γ$. 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.

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