Gravitational Navier-Stokes-Poisson Equations

• Gravitational NSP equations are a coupled system modeling compressible, viscous, self-gravitating fluids, crucial for understanding gaseous stars and accretion disks.
• They integrate mass conservation, momentum balance with viscosity and pressure, and Poisson’s equation to capture fluid inertia, thermal effects, and gravitational forces.
• Rigorous analyses demonstrate the existence of global weak solutions, decay properties, and instability thresholds, offering insights into free-boundary behavior and stellar dynamics.

The gravitational Navier–Stokes–Poisson (NSP) equations govern the evolution of compressible, viscous, self-gravitating fluids. They are foundational in the mathematical modeling of gaseous stars, accretion disks, and related astrophysical systems where fluid inertia, thermal pressure, viscosity, and Newtonian gravity are coupled. The system typically consists of continuity (mass balance), momentum conservation incorporating viscosity and gravitational force via Poisson's equation for the potential, and, in some settings, additional energy or entropy transport equations. Analytical, numerical, and physical studies of the NSP system address existence, regularity, decay, stability, free-boundary behavior, and connections to compressible Euler and Stokes models.

1. Fundamental Equations and Constitutive Laws

The gravitational NSP system for a compressible, viscous, self-gravitating fluid consists of:

• Continuity equation:

$$\partial_t \rho + \operatorname{div}(\rho \mathbf{u}) = 0,$$

with $\rho \ge 0$ the mass density, $\mathbf{u}\in\mathbb{R}^3$ the velocity.

• Momentum equation:

$$\partial_t(\rho \mathbf{u}) + \operatorname{div}(\rho \mathbf{u}\otimes\mathbf{u}) + \nabla P(\rho) - \operatorname{div}\left[\mu(\rho) D(\mathbf{u}) + \lambda(\rho)(\operatorname{div}\mathbf{u}) \mathbf{I}\right] + \rho \mathbf{u} + \rho \nabla \Phi = 0,$$

where $P(\rho)$ is the pressure law, $$D(\mathbf{u}) = \frac{1}{2}(\nabla\mathbf{u} + \nabla \mathbf{u}^\top)$$ the deformation tensor, $\mu(\rho)$ and $\lambda(\rho)$ viscosity coefficients, and $\Phi$ the gravitational potential.

• Poisson equation for gravity:

$-\Delta \Phi = \rho - \overline{\rho},$

with normalization $\overline{\rho}$ (domain average).

Density-dependent viscosity and non-monotone pressure laws,

$$\mu(\rho) = \rho, \quad \lambda(\rho)=0, \quad P \in C^1([0,\infty)), \quad P(0)=0, \quad -b \le P'(s) - a s^{\gamma-1} \le b, \quad \gamma>1,$$

are essential for capturing physical degeneracy and the Bresch–Desjardins (BD) entropy structure. Weak formulations and Faedo–Galerkin approximations underpin modern existence proofs (Ye, 2015).

2. Existence, Regularity, and Compactness

Global-in-time weak solutions exist under broad initial data and for adiabatic exponents $\gamma > 4/3$, including density-dependent viscosity and non-monotone pressure (Ye, 2015). The weak solution $(\rho, \mathbf{u}, \Phi)$ satisfies:

• Integrability and regularity: $$\rho \in L^\infty(0,T; L^{\gamma}) \cap L^2(0,T; H^1)$$, $\sqrt{\rho} \mathbf{u} \in L^\infty(0,T; L^2)$, $\mathbf{u} \in L^2(0,T; H^1)$, $\Phi \in L^\infty(0,T; H^1)$.
• Total energy and BD entropy bounds:

$$E(t) + \int_0^t \int \left[ \mu(\rho) |D(u)|^2 + \rho |u|^2 \right] \le E(0),$$

with $E(t)$ the instantaneous energy.

Compactness (Aubin–Lions lemma, Div–Curl) and strong convergence for $\rho$, $\rho \mathbf{u}$, with pressure handled via Egoroff and dominated convergence, underpin the limit transitions.

3. Free-Boundary, Spherical Symmetry, and Expanding Solutions

In star-like settings, the NSP system admits spherically symmetric solutions with a free boundary. The reduced equations on $[0, a(t)]$,

$$\begin{cases} \partial_t \rho + \frac{1}{r^2} \partial_r(r^2 \rho u) = 0, \ \rho(\partial_t u + u \partial_r u) + \partial_r \rho^\gamma = \partial_r \left( \left(\eta + \frac{4}{3} \varepsilon\right) \frac{1}{r^2} \partial_r (r^2 u)\right) - \frac{4\pi \rho}{r^2} \int_0^r \rho(s) s^2 ds, \end{cases}$$

with $p(\rho) = K \rho^\gamma$, capture the evolution of gaseous stars (Cao, 23 Nov 2025, Kong et al., 2018).

Global weak solutions exist for $\gamma \in (6/5, 4/3]$ under mass/invariant-set conditions. For strong solutions, support expands algebraically:

$$a(t) \sim \left( \frac{t}{\eta + \frac{4}{3}\varepsilon} \right)^{1/3} \quad \text{for} \ \gamma \in (6/5,4/3),$$

or $a(t) \gtrsim (1+t)^{1/4}$ for mass-critical $\gamma=4/3$ (Cao, 23 Nov 2025). The Lane–Emden steady states, modeling nonrotating stars, are strongly unstable for $\gamma < 4/3$, indicated by support expansion.

4. Stability, Instability, and Critical Exponents

Lane–Emden stationary solutions exist for polytropic exponents $6/5<\gamma \leq 4/3$; finite-mass solutions are possible for $\gamma>6/5$ (Jang et al., 2011). However, both linear and nonlinear instability persist in the regime $6/5<\gamma<4/3$, regardless of viscosity magnitude.

• Linear instability: For each $6/5<\gamma<4/3$, there exists a mode growing $\propto e^{\lambda t}$.
• Nonlinear instability: Small initial perturbations yield energy growth to $O(1)$ size in finite time, with no stabilizing effect from viscosity.

Critical threshold exponents ($\gamma=4/3$ for mass stability, $\gamma=6/5$ for energy criticality) mark the transition between stable and unstable regimes for gaseous stars.

5. Decay, Long-Time Behavior, and Stationary Solutions

Solutions to the gravity-coupled NSP system exhibit enhanced time-decay rates for perturbations. For initial data in $H^n$ (small $H^3$ norm), solutions decay as

$$\|\nabla^\ell (\rho-1, \mathbf{u}, \nabla \Phi)(t)\|_{L^2} \leq C(1+t)^{-\frac{\ell+s}{2}},$$

with an extra half-power for the density,

$$\|\nabla^\ell (\rho-1)(t)\|_{L^2} \leq C(1+t)^{-\frac{\ell+s+1}{2}}.$$

Dispersion from Poisson coupling and viscous dissipation combine to accelerate decay over classical compressible Navier–Stokes flows (Wang, 2011).

Stationary solutions in unbounded domains must be trivial under suitable integrability:

$$\|\rho\|_{L^\infty} + \|\Phi\|_{L^\infty} + \|\nabla v\|_{L^2} + \|v\|_{L^{N/(N-1)}} < \infty \implies v \equiv 0,$$

so only constant density or density-Poisson-coupled equilibria are possible (Chae, 2011).

6. Thermodynamic Extensions, Moving Domains, and Radiation

The compressible NSP system generalizes to account for heat conduction and radiative transfer, as in viscous star or accretion disk contexts (Bhandari et al., 2023, Ducomet et al., 2017). The extended Navier–Stokes–Fourier–Poisson system incorporates temperature, entropy, and radiative intensity, with boundary movement and slip or stress-free conditions:

• Additional transport equations for temperature ($\theta$)
• Ballistic energy inequality
• Penalization, compactness, and weak convergence methods for time-dependent domains

In accretion disk models, the thin-layer limit rigorously reduces a 3D rotating radiation-NSP system to a 2D model with self-gravitating and radiative coupling, including Coriolis and angular momentum transport.

7. General Pressure Laws and Inviscid Limits

Analysis extends to general pressure laws admitting non-standard asymptotics for physical models such as white dwarf stars:

$$P(\rho) \sim \kappa_1 \rho^{\gamma_1} \ \text{as} \ \rho \to 0, \quad P(\rho) \sim \kappa_2 \rho^{\gamma_2} \ \text{as} \ \rho \to \infty,$$

with $2P'(\rho) + \rho P''(\rho)>0$ for thermodynamic stability. Finite-energy weak solutions are constructed as vanishing-viscosity limits of NSP solutions, employing entropy pair analysis, $L^p$ compactness, and BD-entropy methods. Strong convergence and the reduction of Young measure solutions to Dirac masses are achieved through compensated compactness tailored to weak entropy dissipation (Chen et al., 2023).

---

Editor’s Table: Summary of Core Results

Regime / Extension	Existence/Decay	Stability / Expansion
Standard $(\gamma>4/3)$	Global weak solutions (Ye, 2015), enhanced decay (Wang, 2011)	Stable Lane–Emden stars
Instability $(6/5<\gamma<4/3)$	Weak/strong solution expansion (Cao, 23 Nov 2025, Kong et al., 2018)	Lane–Emden strongly unstable (Jang et al., 2011)
Moving domain, heat/radiation	Global weak solutions with slip/Dirichlet (Bhandari et al., 2023), 2D disk reduction (Ducomet et al., 2017)	Ballistic energy bounds, compactness
General pressure law	Global finite-energy solution, inviscid limit (Chen et al., 2023)	Entropy pair analysis, $L^p$ compactness

A plausible implication is that dissipative and dispersive mechanisms in the NSP system facilitate both strong time decay and intricate instability thresholds, deeply influenced by pressure law, dimension, viscosity structure, and boundary or symmetry constraints. The interplay between gravitational potential, entropy methods, and compensated compactness is central in extending existence theory to realistic astrophysical settings. Open questions include lowering critical exponents, uniqueness, regularity, and steady-state selection (Ye, 2015, Cao, 23 Nov 2025).