[Paper Review] On rotating star solutions to non-isentropic Euler-Poisson equations

Yilun Wu|arXiv (Cornell University)|Sep 1, 2013

Navier-Stokes equation solutions14 references3 citations

TL;DR

This paper establishes the existence of rotating star solutions to the non-isentropic Euler-Poisson equations by solving an elliptic equation for density under prescribed entropy and angular velocity distributions. Using sub- and supersolutions or variational methods depending on the adiabatic index, it proves the existence of smooth, bounded solutions with continuous angular velocity, extending classical isentropic results to non-isentropic settings with general equations of state.

ABSTRACT

This paper investigates rotating star solutions to the Euler-Poisson equations with a non-isentropic equation of state. As a first step, the equation for gas density with a prescribed entropy and angular velocity distribution is studied. The resulting elliptic equation is solved either by the method of sub and supersolutions or by a variational method, depending on the value of the adiabatic index. The reverse problem of determining angular velocity from gas density is also considered.

Motivation & Objective

• To extend classical rotating star solutions from isentropic to non-isentropic settings in the Euler-Poisson system.
• To establish existence of axisymmetric, stationary solutions with general non-isentropic equations of state.
• To solve the forward problem: determine density for given entropy and angular velocity distribution.
• To address the reverse problem: recover angular velocity from given density.
• To prove boundedness and continuity of angular velocity in the solution domain, including on the symmetry axis.

Proposed method

• Formulate the non-isentropic Euler-Poisson system in cylindrical coordinates with axisymmetric, stationary assumptions.
• Derive the momentum equation (4) as a non-gradient system due to non-zero curl when Ω² depends on r and z.
• Use the sub- and supersolutions method for the elliptic equation governing density when the adiabatic index is in a certain range.
• Apply a variational method to solve the density equation when the adiabatic index lies in another range.
• Establish continuity and boundedness of Ω² on the z-axis and in the domain using mean value theorem and convexity of the boundary.
• Prove L∞ boundedness of Ω² under three distinct hypotheses on the behavior of ρ and B near the boundary and axis.

Experimental results

Research questions

• RQ1 Can rotating star solutions exist under non-isentropic equations of state where entropy is not constant?
• RQ2 How can the forward problem—determining density from prescribed entropy and angular velocity—be solved when the system is not curl-free?
• RQ3 What conditions ensure the boundedness and continuity of angular velocity in the solution domain?
• RQ4 How does the choice of adiabatic index affect the solution method (variational vs. sub-supersolution)?
• RQ5 Can the reverse problem—recovering angular velocity from density—be solved in the non-isentropic setting?

Key findings

• The paper proves the existence of smooth, bounded solutions to the non-isentropic Euler-Poisson system for rotating stars under appropriate conditions on the equation of state and adiabatic index.
• For certain ranges of the adiabatic index, the density equation is solved via a variational method; for others, the method of sub- and supersolutions is applied.
• The angular velocity Ω² is shown to be continuous on the entire domain D, including on the z-axis, under mild regularity and convexity assumptions on the boundary.
• Ω² is proven to be bounded in L∞(D) under three distinct hypotheses, ensuring physical plausibility of the solution.
• The solution method overcomes the lack of gradient structure in the momentum equation by exploiting the non-isentropic nature of the equation of state.
• The results generalize classical isentropic solutions by allowing entropy variations, thus enabling more realistic modeling of stellar dynamics.