Viscous Polytropic Gaseous Stars

• Viscous polytropic gaseous stars are self-gravitating, compressible gas spheres governed by a polytropic law (p=Kρ^γ) with viscosity influencing the equilibrium and free boundary dynamics.
• Mathematical models employ energy methods, spectral analysis, and free boundary conditions to delineate stability thresholds and characterize expanding and contracting regimes.
• Viscosity regularizes the vacuum boundary and plays a crucial role in determining the stability of static Lane-Emden equilibria versus dynamically evolving configurations.

A viscous polytropic gaseous star is a theoretical model describing a self-gravitating, compressible polytropic gas sphere subject to viscosity, governed by the compressible Navier-Stokes-Poisson (NSP) system. Physical realizations include spherically symmetric stars where pressure obeys a polytropic law $p=K\rho^{\gamma}$ with viscosity regularizing both the interior and the vacuum boundary. The dynamics incorporate both gravitational self-attraction and dissipative effects, yielding rich equilibrium and dynamical behaviors and connecting to various stability phenomena—both linear and nonlinear. The Lane-Emden equilibrium profile underpins static configurations, while expanding and contracting dynamical regimes models stellar evolution environments.

1. Mathematical Models and Governing Equations

The motion of viscous polytropic gaseous stars is formulated as follows: in three-dimensional space under spherical symmetry, the principal variables are density $\rho(r,t) \geq 0$, radial velocity $u(r,t)$, and gravitational potential $\Phi(r,t)$. The system features a free boundary $r=R(t)$ separating the interior from vacuum. The governing equations are

• Continuity:

  $$\partial_t \rho + \frac{1}{r^2}\partial_r(r^2 \rho u) = 0$$

• Momentum (Navier-Stokes):

  $$\partial_t (\rho u) + \frac{1}{r^2}\partial_r(r^2 \rho u^2) + \partial_r p(\rho) + \rho \partial_r \Phi = \frac{1}{r^2}\partial_r \left( r^2 \tau_{rr} \right)$$

  where

  $$p(\rho) = K \rho^{\gamma}, \quad \tau_{rr} = (2\mu + \lambda)\partial_r u + \frac{2\mu}{r} u$$

  with shear viscosity $\mu>0$, bulk viscosity $\lambda+\frac{2}{3}\mu\geq0$.

• Poisson (self-gravity):

  $$\frac{1}{r^2}\partial_r(r^2\partial_r \Phi) = 4\pi \rho$$

Boundary conditions at the moving interface $r=R(t)$ ensure zero density, a vanishing normal total stress, and a kinematic condition matching fluid velocity to the interface rate, encapsulating the physical vacuum requirements and regularity at the origin (Luo et al., 2015, Cheng et al., 2023). In the isentropic model, pressure is solely a function of density; in thermodynamic variants, an additional equation for temperature is included.

2. Lane-Emden Equilibrium and Static Configurations

The static equilibrium is described by the Lane-Emden solution, a compactly supported profile $\bar{\rho}(r)\geq 0$ solving

$$\frac{1}{r^2}(r^2 p'(\bar{\rho}))' = -4\pi \bar{\rho}$$

subject to $\bar{\rho}(R)=0$ and fixed total mass $M = \int_0^R 4\pi r^2 \bar{\rho}(r)\, dr$. This equilibrium yields a physical vacuum structure at the free boundary:

$$\bar{\rho}(r) \sim (R - r)^{1/(\gamma - 1)} \quad \text{as } r\to R^-.$$

For $\gamma > 6/5$, these equilibria are compactly supported, and $\gamma > 4/3$ ensures stability against gravitational collapse in the linear regime via coercivity of weighted energy forms (Luo et al., 2015, Cheng et al., 2023). The scaling relations for equilibrium radius $R$ and mass $M$ as functions of central density $\rho_c$ are given by the Lane-Emden theory.

Parameter	Lane-Emden relation (editor's term)	Physical significance
$R(\rho_c)$	$$R(\rho_c) \sim \rho_c^{\frac{2-\gamma}{2(\gamma-1)}}$$	Star size vs. central density
$M(\rho_c)$	$$M(\rho_c) \sim \rho_c^{\frac{3\gamma-4}{2(\gamma-1)}}$$	Mass scaling vs. $\rho_c$

3. Expanding Configurations and Their Stability

Two major classes of expanding homogeneous solutions have been identified for the viscous polytropic star models:

1. Self-similar isentropic expansion ($\gamma=4/3$, $\delta<0$): Solutions expand homogeneously with $a(t)$ obeying $a^2\ddot a = \delta < 0$. The density evolves as $\rho(r,t) = a(t)^{-3} \bar{\rho}(r/a(t))$, with the free boundary scaling as $R(t) = a(t) R_0$.
2. Linear expansion (either $\gamma=4/3$, $\delta=0$, or thermodynamic $3K = c_v$): The scale evolves linearly $a(t) = a_0 + a_1 t$, and similar forms hold for $\rho(r,t)$ and $R(t)$.

The stability properties of these configurations depend crucially on viscous dissipation and the equation of state:

• Self-similar isentropic expansions exhibit nonlinear instability: The viscous dissipation $D(t)$ fails to be strictly positive under these flows; total energy is not conserved, and infinitesimal perturbations can grow unboundedly, as the viscous term loses control over zero-energy perturbation modes (Liu, 2017).
• Linear expansions are nonlinearly stable: Provided the expansion rate $a_1$ is large enough (e.g., $a_1 > 1/a_0$ in the isentropic model), viscosity participates effectively in damping perturbations, and one establishes uniform-in-time boundedness and decay of energy norms, up to and including the vacuum boundary (Liu, 2017).

4. Nonlinear Asymptotic Stability and Energy Methods

For initial perturbations near the Lane-Emden equilibrium, the energy method delivers global-in-time regularity and asymptotic stability. In the Lagrangian framework, strong solution functionals such as

$$\mathcal{E}(t) = \|(r_x - 1, v)\|_{L^\infty_x}^2 + \|x^{1/2} v\|_{L^2_x}^2 + \|p^{-\frac{1}{2}} [(r^2 r_x)^{-1} (r^2 v)_x]\|_{L^2_x}^2$$

control perturbations. The fundamental energy-dissipation inequality

$E(t) + \int_0^t D(s) ds \leq E(0)$

is established, with dissipation up to the moving vacuum boundary enabled by weighted Hardy-Sobolev inequalities (Luo et al., 2015). The sound speed attains uniform $C^{1/2}$ Hölder continuity across the boundary:

$c(\rho) = \sqrt{p'(\rho)} \sim (R(t) - r)^{1/2}$

guaranteed for all $t \geq 0$ (Luo et al., 2015). Quantitative decay rates for relaxation to equilibrium are given by

$$|R(t) - R| + \|u(\cdot, t)\|_{L^\infty} + \|\rho(\cdot, t) - \bar{\rho}(\cdot)\|_{L^\infty} \leq C(\delta) (1 + t)^{-(\gamma - 1)/\gamma + \delta}$$

for any small $\delta > 0$.

5. Spectral Stability and the Turning-Point Principle

Stability transitions in viscous polytropic stars are governed by the turning-point principle: the number of unstable eigenvalues (modes) of the linearized NSP system equals that of the (inviscid) Euler-Poisson system, and stability changes only at extremal points of the mass–radius curve $M(R)$ (Cheng et al., 2023).

• Let $n_V(\rho_c)$ denote the number of unstable modes for the viscous system, and $n_E(\rho_c)$ for the inviscid system.
• For $\gamma > 6/5$ and $M'(\rho_c) \neq 0$, $n_V(\rho_c) = n_E(\rho_c)$.
• At mass extrema ($M'(\rho_c)=0$), $n_V(\rho_c)$ jumps by one as the curve bends in the $(R, M)$ plane.
• The proof leverages an infinite-dimensional Kelvin-Tait-Chetaev theorem for second-order PDEs with dissipation: the count of unstable eigenvalues coincides with the negative Morse index of the linearized operator, which tracks through the bifurcation points.

Nonlinear stability matches linear stability: if $n_{-}(L) = 0$, Lyapunov techniques yield decay and global existence; if $n_{-}(L) > 0$, finite-amplitude instabilities occur.

6. Role and Physical Interpretation of Viscosity

Viscosity acts to regularize the vacuum free boundary and provide dissipation that controls high-order norms, particularly in the presence of the physical vacuum degeneracy $c(\rho) \sim (R - r)^{1/2}$ (Luo et al., 2015). For $\gamma > 4/3$, viscosity strengthens the stabilizing effects of pressure against gravitational collapse.

However, as shown in expanding settings (Liu, 2017), the viscous stress degenerates under exact homogeneous expansion, failing to uniformly control the full dynamics in the self-similar case, yielding instability. In contrast, properly weighted energy-dissipation methods restore control (and stability) in linear expansion regimes, both for isentropic and thermodynamic models, provided dissipation is strong relative to growth and forcing.

This indicates a delicate balance: macroscopic viscosity can stabilize linear global expansion but not self-similar blow-up or rapid expansion. A plausible implication is that viscosity sets strict limits on the types of expansion feasible for massive, radiation-dominated stars.

7. Extensions, Models, and Open Problems

Research on viscous polytropic gaseous stars points to several directions:

• Generalization to non-radial, rotating, or non-spherical configurations remains open.
• More realistic microphysical viscosity laws (e.g., density-dependent viscosity) and full radiative transport systems are yet to be fully characterized.
• Nonlinear collapse branches ($\dot a(0) < a_{\rm esc}$) invite further analysis beyond small perturbations.
• The basin of nonlinear stability and global existence for large data are active subjects.
• The distinction between the stability of isentropic and thermodynamic models, especially coupling with internal energy and heat-generation, exposes complex interactions between dissipation and dynamical regimes.

These analyses employ advanced PDE methodologies—weighted energy, Lagrangian coordinates, spectral gap analysis—and underscore how dissipative mechanisms and equation of state determine the long-term evolution and macroscopic stability regimes of gaseous stellar objects (Luo et al., 2015, Cheng et al., 2023, Liu, 2017).