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

Abstract: Whether the 3D compressible Navier-Stokes-Poisson equations admit global classical solutions for general large initial data has long been a challenging open problem. In this paper, we provide an affirmative answer to this question under spherical symmetry on solid balls . Specifically, we consider the initial-boundary value problem for the multi-dimensional compressible equations with density-dependent viscosity coefficients satisfying the BD-type entropy equality, namely, assuming $μ=ρ^α,\ λ=(α-1)ρ<sup>α$ with $N=2, α\in (\frac{1}{2},1]$ and $N=3, α\in (\frac{5}{6},1]$, we establish the global existence of spherically symmetric classical solutions to the compressible Navier-Stokes-Poisson equations for both gaseous stars and plasmas with arbitrarily large initial data on solid balls. Our key observation lies in successfully handling the singularity at the center of the ball. By controlling the growth orders of the density and the gravitational potential at the central singularity, leveraging the structural advantages of the BD entropy and spherical symmetry, and fully exploiting the coupling between the effective velocity and the velocity, we establish $L<sup>\infty$ estimates for the key quantities, which in turn yield upper and lower bound estimates for the density. This can be regarded as the first result on the existence of global classical solutions for arbitrarily large initial data to the compressible Navier-Stokes-Poisson equations in a truly multi-dimensional domain with high-dimensional features.

• The paper proves global-in-time existence and uniqueness of classical solutions for the compressible Navier-Stokes-Poisson equations under arbitrarily large spherically symmetric initial data.
• It employs innovative weighted a priori estimates and BD entropy-based methods to manage the central singularity and handle degeneracy in density-dependent viscosities.
• The study establishes uniform bounds and high-order regularity, offering a robust framework for astrophysical and plasma models characterized by strong nonlinearity.

Global Existence of Classical Solutions for Multi-Dimensional Compressible Navier-Stokes-Poisson Equations with Spherically Symmetric Large Data

Introduction and Problem Setting

This paper establishes the global-in-time existence and uniqueness of spherically symmetric, classical (smooth) solutions to the compressible Navier-Stokes-Poisson (NSP) equations with arbitrarily large initial data in the physically relevant dimensions $N = 2, 3$, posed on solid balls. Unlike prior works, which typically impose strong smallness or near-equilibrium data restrictions, the present analysis handles the full range of large, spherically symmetric data away from vacuum. The significant challenge is managing the geometric singularity at the origin and the nonlinear coupling between density, velocity, and gravitational or electrostatic potential under general density-dependent viscosities.

The governing equations are:

$$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$

subject to boundary and symmetry conditions, with density-dependent shear and bulk viscosities satisfying the Bresch–Desjardins (BD) entropy structure, i.e., $\mu(\rho) = \rho^\alpha$, $\lambda(\rho) = (\alpha-1)\rho^\alpha$, for a specified range of $\alpha$.

Main Results

Global Classical Solutions for Arbitrarily Large Spherically Symmetric Data

Existence and Uniqueness:

• For $N=2$, for $\alpha \in (\frac{1}{2},1)$, $\gamma > 2 - \alpha$, any initial data $(\rho_0,\mathbf{u}_0)$ in $H^3$ bounded away from vacuum, there exists a unique global spherically symmetric classical solution with density uniformly bounded above and below for all $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$0.
• For $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$1, for $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$2, $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$3, the same conclusion holds.
• For $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$4, the admissible range is $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$5 in 2D and $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$6 in 3D, again yielding global classical solutions for arbitrarily large data.

Regularity: The constructed solutions possess strong regularity in space and time, with density, velocity, and potential fields in high-order Sobolev spaces as specified in the paper.

Uniform Bounds:

$$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$7

for all $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$8, for a continuous function $$\begin{cases} \rho_t + \mathrm{div} (\rho \mathbf{u}) = 0, \ \rho(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u}) + \nabla P(\rho) = -\rho \nabla \Phi + \mathrm{div}( \mu(\rho) \mathcal{D} \mathbf{u}) + \nabla(\lambda(\rho) \mathrm{div} \mathbf{u} ), \ \Delta \Phi = \kappa (\rho - \bar{\rho}), \end{cases}$$9 depending on the initial data and $\mu(\rho) = \rho^\alpha$0.

Parameters and Admissible Ranges: The results hold for all large spherically symmetric data, within the ranges for $\mu(\rho) = \rho^\alpha$1 and $\mu(\rho) = \rho^\alpha$2 determined by the dissipative and structural properties of the equations and the strength of the singularity at the origin. The sharp ranges for admissibility exploit the fine structure of the BD entropy and the explicit Sobolev embeddings under spherical symmetry.

Analytical Approach

Key Innovations

1. Handling Central Singularity via Weighted A Priori Estimates

The spherical geometry introduces pronounced singularities in the NSP system, especially at $\mu(\rho) = \rho^\alpha$3, due to terms like $\mu(\rho) = \rho^\alpha$4 and $\mu(\rho) = \rho^\alpha$5. The core of the analysis lies in establishing weighted a priori bounds that allow control over both the solution and its derivatives, despite these geometric singularities.

2. BD Entropy and Effective Velocity

A central structural tool is the BD entropy, which yields additional dissipation and regularity beyond classical energy methods, even for degenerate viscosity coefficients. By introducing the effective velocity $\mu(\rho) = \rho^\alpha$6, one can derive energy–entropy functionals whose dissipation mechanisms eliminate the most severe terms.

3. Hierarchies of Integrability and Sobolev Embedding

A fundamentally new aspect is the derivation of precise, dimension-dependent weighted $\mu(\rho) = \rho^\alpha$7 estimates for the density and velocity, which are then interpolated and embedded into pointwise control via sharp one-dimensional Sobolev inequalities. The construction exploits the specific mapping between Lagrangian and Eulerian variables to transfer singularities into integrable weights.

4. Closing Estimates for Arbitrary Large Data

Rather than linearization or small-data perturbation, every nonlinear feedback term is handled directly. The entropy structure and high integrability at key exponents permit closing estimates on the density, velocity, and their derivatives, uniformly in time. The approach flexibly distinguishes the physically distinct cases of self-gravity ($\mu(\rho) = \rho^\alpha$8) and plasma ($\mu(\rho) = \rho^\alpha$9).

5. Higher-Order Regularity and Bootstrapping

Once lower-order estimates are propagated, the analysis is bootstrapped to higher-order derivatives using intricate elliptic estimates, energy-type inequalities for time derivatives, and commutator bounds, thereby establishing classical regularity.

Numerical and Theoretical Implications

• Strong Nonlinearity and Degeneracy: The admissible range for $\lambda(\rho) = (\alpha-1)\rho^\alpha$0 thus enables treatment of highly degenerate viscosities, going well beyond the constant viscosity regime. Important models of astrophysical systems (e.g., polytropic gaseous stars, degenerate plasmas) are thereby included.
• Uniform-in-Time Control: Uniform (in time) upper and lower bounds for the density and the absence of finite time blow-up are obtained for arbitrarily large smooth data sets, under symmetry.
• Vacuum Exclusion: The analysis crucially excludes initial vacuum for classical solutions, aligning with the regime of well-posedness for strong solutions.

Technical Highlights and Contrasts with Prior Work

• This work provides the first existence theorem for arbitrarily large data yielding global classical solutions to the compressible multi-dimensional NSP system with degenerate, density-dependent viscosity in a fully multi-dimensional domain, extending previous results for small data, weak solutions, or planar symmetry.
• The critical innovations are the geometric weighted functionals and the effective use of the BD entropy structure, which together yield $\lambda(\rho) = (\alpha-1)\rho^\alpha$1 control over density and velocity gradients, overcoming the singularity at $\lambda(\rho) = (\alpha-1)\rho^\alpha$2 without vacuum or smallness constraints.

Outlook and Future Directions

• Extension to Non-Symmetric Flows: While spherical symmetry is pivotal in the removal of the singularity at the center, the methods here refine the tools for analyzing blow-up and regularity criteria in the absence of symmetry and may inform global existence results in more general domains.
• Transitions to Vacuum: Extending these results to accommodate initial vacuum, or near-vacuum data, especially under degenerate viscosities, remains an important and challenging direction.
• Numerics and Physical Modeling: The explicit bounds and functional framework provide a basis for the rigorous justification of numerical schemes in astrophysics and plasma physics, where large data and strong degeneracy are generic.

Conclusion

The paper provides a comprehensive analytical foundation for the global-in-time dynamics of the multi-dimensional compressible Navier-Stokes-Poisson system with density-dependent viscosity and arbitrary spherically symmetric large data. By leveraging advanced entropy methods and highly technical weighted energy hierarchies adapted to spherical geometry, the singularities inherent at the origin are controlled, yielding existence, uniqueness, and uniform regularity for classical solutions across the physically significant parameter regimes.

Citation: "Global existence of classical solutions for the multi-dimensional compressible Navier-Stokes-Poisson equations on solid balls for arbitrary spherically symmetric large initial data" (2604.08946).