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Rotating fluids with self-gravitation in bounded domains. (English) Zbl 1060.76125

Summary: We study steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state \(P = e^S \rho^\gamma\). When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant \(\gamma\). In addition, we obtain some properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of non-rotating stars. For general domains and variable angular velocities, both existence result for isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.

MSC:

76U05 General theory of rotating fluids
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
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