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Expanding Solutions to Free Boundary 3D Spherically Symmetric Compressible Navier-Stokes-Poisson Equations near the Lane-Emden Stars

Han Cao

TL;DR

This work analyzes the free boundary 3D spherically symmetric compressible Navier-Stokes-Poisson system with a polytropic EOS $P(\rho)=\rho^{\gamma}$, $\gamma\in(\tfrac{6}{5},\tfrac{4}{3}]$, proving global existence of weak solutions under constant viscosity and near-Lane-Emden data, as well as an invariant data framework that extends prior results. The authors develop a variational approach to control the negative gravitational energy and construct approximate problems, deriving comprehensive a priori estimates in both Eulerian and Lagrangian coordinates to obtain compactness and convergence to a global weak solution. They also establish algebraic expanding rates for the vacuum free boundary, showing that the support grows like $a(t) \sim t^{1/3}$ in many regimes and depending on viscosity with density-dependent generalizations, thereby illustrating strong instability of Lane-Emden configurations within NSP dynamics. The results advance understanding of gaseous star models with self-gravity, vacuum boundaries, and variable viscosity, and provide a robust framework for global weak solutions near critical Lane-Emden states.

Abstract

We consider the gravitational Navier-Stokes-Poisson equations with the equation of state $P(ρ)=Kρ^γ$, where $γ\in(\frac{6}{5},\frac{4}{3}]$, which models the viscous polytropic gaseous stars. We prove the existence of global weak solutions to the equations with constant viscosity and radially symmetric initial data. For $γ=\frac{4}{3}$, we require the initial data having mass less than the mass of the Lane-Emden stars; for $γ\in(\frac{6}{5},\frac{4}{3})$, we require that the initial data belong to an invariant set where initial initial data can be taken near the Lane-Emden stars. For $γ\in(\frac{6}{5},\frac{4}{3})$, we show that the invariant set contains some initial data that are not allowed in previous literature. We also prove the support of any strong solution expands to infinity for the Navier-Stokes-Poisson equations with constant viscosity and a class of density-dependent viscosity, which indicates the strong instability of Lane-Emden solutions for the Navier-Stokes-Poisson equations.

Expanding Solutions to Free Boundary 3D Spherically Symmetric Compressible Navier-Stokes-Poisson Equations near the Lane-Emden Stars

TL;DR

This work analyzes the free boundary 3D spherically symmetric compressible Navier-Stokes-Poisson system with a polytropic EOS , , proving global existence of weak solutions under constant viscosity and near-Lane-Emden data, as well as an invariant data framework that extends prior results. The authors develop a variational approach to control the negative gravitational energy and construct approximate problems, deriving comprehensive a priori estimates in both Eulerian and Lagrangian coordinates to obtain compactness and convergence to a global weak solution. They also establish algebraic expanding rates for the vacuum free boundary, showing that the support grows like in many regimes and depending on viscosity with density-dependent generalizations, thereby illustrating strong instability of Lane-Emden configurations within NSP dynamics. The results advance understanding of gaseous star models with self-gravity, vacuum boundaries, and variable viscosity, and provide a robust framework for global weak solutions near critical Lane-Emden states.

Abstract

We consider the gravitational Navier-Stokes-Poisson equations with the equation of state , where , which models the viscous polytropic gaseous stars. We prove the existence of global weak solutions to the equations with constant viscosity and radially symmetric initial data. For , we require the initial data having mass less than the mass of the Lane-Emden stars; for , we require that the initial data belong to an invariant set where initial initial data can be taken near the Lane-Emden stars. For , we show that the invariant set contains some initial data that are not allowed in previous literature. We also prove the support of any strong solution expands to infinity for the Navier-Stokes-Poisson equations with constant viscosity and a class of density-dependent viscosity, which indicates the strong instability of Lane-Emden solutions for the Navier-Stokes-Poisson equations.

Paper Structure

This paper contains 18 sections, 23 theorems, 358 equations.

Table of Contents

  1. Introduction
  2. Formulation and main results
  3. Variational methods
  4. Existence of global weak solutions
  5. A priori estimates
  6. Basic energy estimate
  7. Further estimates in Eulerian coordinates
  8. Further estimates in Lagrangian coordinates
  9. Compactness arguments
  10. Convergence near the free boundary
  11. Convergence near the origin
  12. Verification of weak solutions
  13. Proof of Theroem \ref{['Thm1: ExistenceOfGlobalWkSoln']} and Theorem \ref{['Thm2']}
  14. Expanding rate of the free boundary
  15. Proof of Theorem \ref{['Thm3']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $T>0$ and $\gamma\in(\frac{6}{5},\frac{4}{3}]$. Suppose $(\rho_0(r),u_0(r))$ are initial data satisfying and Assumption cond11. Then there exists a global weak solution $(\rho(r,t),u(r,t),a(t))$ to (nsp3)-(boundary conditions) for $t\in[0,T]$, which satisfies $\rho(r,t)\ge0\ a.e.$ and

Theorems & Definitions (51)

  • Definition 2.1: global weak solutions
  • Remark 2.1
  • Theorem 2.1: Global existence
  • Theorem 2.2: Higher order regularities
  • Theorem 2.3: Algebraic expanding rate of the free boundary
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Corollary 3.1
  • Lemma 4.1
  • ...and 41 more