Global existence of classical solutions for the multi-dimensional compressible Navier-Stokes-Poisson equations on solid balls for arbitrary spherically symmetric large initial data

1. Problem Statement

The paper addresses a long-standing open problem in mathematical fluid dynamics: the global existence of classical solutions for the 3D compressible Navier-Stokes-Poisson (NSP) equations with large initial data on a solid ball (a bounded domain containing the origin).

• The System: The equations govern the motion of compressible gaseous stars ($\kappa > 0$) or plasmas ($\kappa < 0$) under self-consistent gravitational or electric fields. The system includes the continuity equation, the momentum equation with density-dependent viscosity, and the Poisson equation for the potential.
• The Viscosity: The shear viscosity $\mu(\rho)$ and bulk viscosity $\lambda(\rho)$ satisfy the Bresch-Desjardins (BD) entropy relation:
  $$\mu(\rho) = \rho^\alpha, \quad \lambda(\rho) = (\alpha - 1)\rho^\alpha$$
  where $\alpha$ is a parameter determining the viscosity behavior.
• The Challenge:
  1. Large Initial Data: Previous results often required small perturbations from a steady state. This paper handles arbitrarily large initial data (away from vacuum).
  2. Spherical Singularity: In spherical coordinates, the momentum equation contains singular terms (e.g., $\frac{u}{r}$ in 2D and $\frac{u}{r^2}$ in 3D) arising from the geometry and the Poisson term. These singularities are particularly severe at the center of the ball ($r=0$).
  3. Multi-dimensional Complexity: While 1D results exist, extending them to truly multi-dimensional domains with high-dimensional features (specifically $N=2$ and $N=3$) while maintaining classical regularity is extremely difficult.

2. Methodology

The authors employ a combination of a priori estimates, Lagrangian coordinates, and weighted energy methods to overcome the singularities and establish global existence.

A. Coordinate Transformation

The problem is transformed from Eulerian coordinates $(r, t)$ to Lagrangian coordinates $(x, \tau)$, where $x$ represents the mass coordinate. This maps the time-dependent domain $[0, R(t)]$ to a fixed domain $[0, 1]$, simplifying the handling of the free boundary (though here the domain is a solid ball with fixed boundaries, the transformation aids in handling the density gradient).

B. Effective Velocity

A crucial step is the introduction of the effective velocity $w$:
$$w = u + r^{N-1}(\rho^\alpha)_x$$
This variable decouples the viscous terms and allows for the derivation of the BD entropy estimate, which provides crucial control over the density gradient and the singularity at the origin.

C. Handling the Singularity (Key Innovation)

The core difficulty lies in the Poisson term (e.g., $\frac{x}{r^2}$ in 3D). The authors develop a sophisticated weighted estimate strategy:

1. Weighted $L^\infty$ Bounds: They establish bounds for quantities like $\|\rho^{\alpha - 1/2} r^\xi\|_{L^\infty}$ near the origin. This controls the growth of density and potential at the singularity.
2. Relationship between $x$ and $r$: By combining the basic energy estimate and the BD entropy estimate, they derive a precise relationship between the Lagrangian mass coordinate $x$ and the Eulerian radius $r$ (e.g., $x \leq C r^\beta$). This relationship is essential to eliminate the singularity caused by the Poisson term in higher-order estimates.
3. Case Differentiation:
   • 2D Case ($N=2$): The singularity is manageable with specific constraints on $\alpha$ and $\gamma$.
   • 3D Case ($N=3$): The singularity is stronger ($\sim 1/r^2$). The authors treat the cases $\kappa = 1$ (gaseous stars) and $\kappa = -1$ (plasmas) separately. For gaseous stars, they require $\gamma > 4/3$ to absorb the singular integral into the potential energy.

D. Higher Integrability and Sobolev Embedding

The authors construct a sequence of higher-order integrability estimates for the velocity $u$ and effective velocity $w$.

• They utilize the set $A_{set}$ (a dense set of integers) to choose exponents that allow for the closure of estimates.
• By establishing $L^{2k}$ bounds for $u$ and $w$, they derive $L^\infty$ bounds for the density via Sobolev embedding inequalities in one dimension (after radial reduction).
• They prove a coupled Gronwall-type inequality to control the $L^\infty$ norms of $u$ and $w$ simultaneously.

3. Key Contributions

1. First Global Classical Solutions for Large Data in Multi-D: This is the first result establishing the global existence of classical (not just weak) solutions for arbitrarily large initial data for the compressible NSP equations in a truly multi-dimensional domain ($N=2, 3$) with high-dimensional features.
2. Resolution of the Central Singularity: The paper provides a rigorous method to handle the singularity at the center of the ball for density-dependent viscosity, a problem that has hindered previous attempts.
3. Extension of BD Entropy to NSP: The authors successfully adapt the BD entropy framework (previously used for Navier-Stokes) to the NSP system, overcoming the additional difficulties posed by the non-local Poisson coupling.
4. Specific Parameter Ranges: The paper identifies precise ranges for the viscosity exponent $\alpha$ and the adiabatic exponent $\gamma$ for which global solutions exist:
   • 2D: $\alpha \in (1/2, 1]$, $\gamma > 2-\alpha$ (for $\alpha < 1$) or $\gamma > 1$ (for $\alpha = 1$).
   • 3D: $\alpha \in (5/6, 1]$, with specific constraints on $\gamma$ (e.g., $\gamma > 4/3$ for gaseous stars).

4. Main Results

The paper proves the existence of unique global spherically symmetric classical solutions $(\rho, u, \Phi)$ for the initial-boundary value problem on a solid ball $\Omega = B_R \subset \mathbb{R}^N$ ($N=2, 3$).

• Theorem 2.1 & 2.2 (Case $\alpha < 1$):

  • For $N=2$, if $1/2 < \alpha < 1$ and $\gamma > 2-\alpha$.
  • For $N=3$, if $5/6 < \alpha < 1$ and $\gamma$ satisfies specific bounds (e.g., $4/3 < \gamma < 4\alpha - 1 - \alpha^2$).
  • Result: Global existence with uniform upper and lower bounds on density: $(C(T))^{-1} \leq \rho(x,t) \leq C(T)$.

• Theorem 2.3 & 2.4 (Case $\alpha = 1$):

  • For $N=2$, if $\gamma > 1$.
  • For $N=3$, if $4/3 < \gamma < 3$.
  • Result: Global existence with uniform upper and lower bounds on density.

In all cases, the solutions satisfy high regularity conditions (e.g., $\rho \in C([0, T]; H^3)$, $u \in C([0, T]; H^3)$).

5. Significance

• Theoretical Breakthrough: This work resolves a critical gap in the theory of compressible fluids with self-gravity/electric fields. It moves beyond the "small data" regime, which is often physically unrealistic for astrophysical objects like stars.
• Astrophysical and Plasma Applications: The results are directly applicable to modeling the long-term evolution of gaseous stars and plasmas, where density-dependent viscosity and large initial perturbations are common.
• Methodological Advancement: The techniques developed for handling the central singularity via weighted estimates and the specific coupling between the effective velocity and the Poisson term provide a new toolkit for analyzing singular PDEs in multi-dimensional domains.
• Regularity: Unlike many previous works that only established weak solutions (which may develop shocks or vacuum), this paper ensures the existence of classical solutions, meaning the solution remains smooth and differentiable for all time, preventing the formation of singularities in the solution itself (though the domain geometry is singular).