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Expanding solutions near unstable Lane-Emden stars

Ming Cheng, Xing Cheng, Zhiwu Lin

TL;DR

The paper addresses the nonlinear dynamics of self-gravitating gaseous stars governed by the Euler–Poisson system with polytropic EoS $P(\rho)=K\rho^\gamma$, establishing global weak solutions and expanding supports near Lane–Emden stars for $\gamma=\tfrac{4}{3}$ and $\gamma\in(\tfrac{6}{5},\tfrac{4}{3})$ under mass constraints. It leverages a variational framework, connecting Lane–Emden masses to a Hardy–Littlewood type inequality, and uses Navier–Stokes–Poisson regularization and BD-type entropy to pass to the Euler–Poisson limit, yielding invariant expanding sets and strong instability of Lane–Emden states. The results also prove non-collapse of white dwarfs below the Chandrasekhar limit, with the Chandrasekhar mass identified as the Lane–Emden mass for $\gamma=\tfrac{4}{3}$, and extend the analysis to higher dimensions, where analogous expanding behavior and blow-up phenomena can occur. Overall, the work provides a variational and analytical foundation for nonlinear instability and expansion near equilibrium stellar configurations, with explicit mass thresholds and energy controls.

Abstract

We consider the compressible Euler-Poisson equations for polytropes $P(ρ)=Kρ^γ$ with $γ\in \left(\frac{6}{5},\frac{4}{3} \right]$ and the white dwarf stars. For $γ=\frac{4}{3},$ we establish the existence of a global weak solution for the spherically symmetric initial data with mass less than the mass of the Lane-Emden stars (i.e. non-rotating polytropes). For $γ\in \left(\frac{6}{5},\frac{4}{3} \right)$, we show the existence of global weak solution for spherical symmetric initial data in an invariant set containing a neighborhood of Lane-Emden stars. Moreover, the support of these solution expands to infinity. As a corollary, this proves the strong instability of the Lane-Emden stars for $γ\in \left( \frac{6}{5},\frac{4}{3}\right] $. For $γ\in \left(\frac{6}{5},\frac{4}{3} \right),$ our results provide the first example of expanding solutions near the Lane-Emden stars. For white dwarf stars, we prove that the solution cannot collapse if the mass of initial data is less than the Chandrasekhar limit mass, which is the supremum of the mass of the non-rotating white dwarf stars. Our proof strongly uses the variational characterization of the Lane-Emden stars. First, we relate the best constant of a Hardy-Littlewood type inequality with the mass of the Lane-Emden stars with $γ=\frac{4}{3}$, which is further shown to equal the Chandrasekhar limit mass. For $γ\in\left( \frac{6}{5},\frac{4}{3}\right) $, we show that the Lane-Emden stars are minimizers of an energy-mass functional subject to a Pohozaev type constraint. This is crucial in the construction of the invariant set of expanding solutions.

Expanding solutions near unstable Lane-Emden stars

TL;DR

The paper addresses the nonlinear dynamics of self-gravitating gaseous stars governed by the Euler–Poisson system with polytropic EoS , establishing global weak solutions and expanding supports near Lane–Emden stars for and under mass constraints. It leverages a variational framework, connecting Lane–Emden masses to a Hardy–Littlewood type inequality, and uses Navier–Stokes–Poisson regularization and BD-type entropy to pass to the Euler–Poisson limit, yielding invariant expanding sets and strong instability of Lane–Emden states. The results also prove non-collapse of white dwarfs below the Chandrasekhar limit, with the Chandrasekhar mass identified as the Lane–Emden mass for , and extend the analysis to higher dimensions, where analogous expanding behavior and blow-up phenomena can occur. Overall, the work provides a variational and analytical foundation for nonlinear instability and expansion near equilibrium stellar configurations, with explicit mass thresholds and energy controls.

Abstract

We consider the compressible Euler-Poisson equations for polytropes with and the white dwarf stars. For we establish the existence of a global weak solution for the spherically symmetric initial data with mass less than the mass of the Lane-Emden stars (i.e. non-rotating polytropes). For , we show the existence of global weak solution for spherical symmetric initial data in an invariant set containing a neighborhood of Lane-Emden stars. Moreover, the support of these solution expands to infinity. As a corollary, this proves the strong instability of the Lane-Emden stars for . For our results provide the first example of expanding solutions near the Lane-Emden stars. For white dwarf stars, we prove that the solution cannot collapse if the mass of initial data is less than the Chandrasekhar limit mass, which is the supremum of the mass of the non-rotating white dwarf stars. Our proof strongly uses the variational characterization of the Lane-Emden stars. First, we relate the best constant of a Hardy-Littlewood type inequality with the mass of the Lane-Emden stars with , which is further shown to equal the Chandrasekhar limit mass. For , we show that the Lane-Emden stars are minimizers of an energy-mass functional subject to a Pohozaev type constraint. This is crucial in the construction of the invariant set of expanding solutions.
Paper Structure (14 sections, 20 theorems, 203 equations)

This paper contains 14 sections, 20 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Expanding solutions for polytropes with $\gamma = \frac{4}{3}$
  4. The best constant of Hardy-Littlewood inequality
  5. Existence of spherically symmetric global weak solutions when $\gamma = \frac{4}{3}$
  6. Spherically symmetric finite energy weak solutions
  7. Existence of spherically symmetric global weak solutions when $\gamma = \frac{4}{3}$
  8. Expanding strong solutions
  9. Expanding solutions for polytropes with $\gamma \in \left( \frac{6}{5}, \frac{4}{3} \right)$
  10. A constrained variational problem
  11. Existence of spherically symmetric global weak solutions when $\gamma\in \left(\frac{6}{5}, \frac{4}{3} \right)$
  12. Expanding strong solutions
  13. Non-collapsing of white dwarf stars
  14. High dimensional case of the Euler-Poisson equations

Key Result

Theorem 1.1

For $\gamma = \frac{4}{3}$, consider the problem 1.1-eq1.5v34 with spherically symmetric initial data $(\rho_0,u_0)$ which has compact support and finite energy. Then, there exists a spherically symmetric global weak solution $(\rho,u)$ of 1.1-eq1.5v34 when $M(\rho_0)< M_{K}$. Here, $M_{K}$ is the m

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.1
  • proof
  • ...and 38 more