Global Well-Posedness of Classical Solutions to the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data

Abstract: This paper is concerned with the global existence and uniqueness of classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in three-dimensional bounded domains or in the whole space $\mathbb{R}<sup>N$ $(N=2,3)$ with non-vacuum far-field density. Specifically, we assume that the shear viscosity coefficient $μ(ρ)=ρ<sup>α$ and the bulk viscosity coefficient $λ(ρ)=(α-1)ρ<sup>α$, which satisfy the BD entropy relation. For arbitrarily large spherically symmetric initial data, we establish the global existence and uniqueness of spherically symmetric classical solutions under the following conditions: for $N=2$, $α\in (0.54369,1)$ and $γ\in (1,\infty)$; for $N=3$ (both in bounded domains and in the whole space), $α\in (0.67661,1)$ and $γ\in (1,6α-3)$. In the two-dimensional case $\mathbb{R}<sup>2$, the restriction on $α$ can be further relaxed to $α\in (9-6\sqrt{2},1)$ provided that the initial data satisfy additional weighted integrability conditions. Moreover, we show that the solution will not exhibit vacuum in any finite time provided that no vacuum is present initially.

• The paper establishes global well-posedness of classical solutions for degenerate compressible Navier-Stokes systems using advanced BD entropy and energy methods.
• It leverages spherical symmetry and weighted integrability to handle viscosity degeneracy and secure uniform bounds on density and velocity.
• The results provide a robust analytical foundation for simulating compressible flows in astrophysics and engineering, even with arbitrarily large spherically symmetric initial data.

Global Well-Posedness of Classical Solutions to Multi-Dimensional Degenerate Compressible Navier-Stokes with Large Spherically Symmetric Data

Problem Setting and Main Innovations

The paper establishes global existence, uniqueness, and regularity of classical solutions for the barotropic compressible Navier-Stokes equations with density-dependent, possibly degenerate viscosity coefficients in $\mathbb{R}^N$ ($N=2,3$) and bounded spherical domains. The viscosity coefficients are set as $\mu(\rho) = \rho^\alpha$ and $\lambda(\rho) = (\alpha-1) \rho^\alpha$, compatible with the Bresch–Desjardins entropy structure.

The analysis targets arbitrarily large spherically symmetric initial data under non-vacuum far-field conditions; results apply both to the Cauchy problem and to Dirichlet or slip boundary conditions in bounded domains. The well-posedness results are achieved with sharp lower bounds on the viscosity exponent $\alpha$: for $N=2$, $\alpha \in (0.54369, 1)$, and for $N=3$, $\alpha \in (0.67661, 1)$, the pressure exponent $\gamma$ is constrained to $N=2,3$0 in three dimensions.

A significant technical advancement is the extension of classical solution theory to the setting of large spherically symmetric initial data with degenerate viscosities—no smallness or proximity assumptions are imposed on initial density or velocity profiles beyond some regularity and boundedness requirements. The paper further refines these results in two dimensions via weighted integrability conditions, allowing relaxation of the restriction on $N=2,3$1 to $N=2,3$2.

Technical Approach and A Priori Estimates

The analysis leverages spherical symmetry to reduce the PDE system, facilitating handling of singularities at the origin and yielding improved energy estimates. The BD entropy structure is crucial for deriving uniform bounds on density and its spatial derivatives. The main obstacles overcome are the degeneracy of viscosity near vacuum and the nonlinearity induced by density-dependence.

Key steps include:

• Energy, BD entropy, and Sobolev estimates: The standard energy method is systematically combined with BD entropy estimates and Sobolev embedding to obtain uniform bounds on density, its derivatives, and the velocity.
• Effective velocity framework: An effective velocity $N=2,3$3 is introduced, transforming the system into a transport equation with improved regularity properties thanks to the BD entropy structure.
• Local-to-global extension: A sequence of higher-order estimates ensures the propagation of regularity and prevents blow-up, enabling global extension of local-in-time classical solutions.
• Upper and lower density bounds: For both dimensions, intricate estimates exploiting spherical symmetry, mass conservation, and weighted integrability are developed to achieve global-in-time bounds for $N=2,3$4 from above and away from zero.
• Lagrangian coordinates near the origin: To control density behavior close to the singularity, the analysis shifts to Lagrangian variables, demonstrating conservation of fluid volume and bounding the inverse density.
• Weighted integrability in 2D: By imposing additional weighted integrability on the initial data, the paper relaxes constraints on $N=2,3$5, showing the technical significance of spatial weights for degenerate systems.

Main Results and Numerical Thresholds

Existence and Uniqueness:

• For $N=2,3$6, classical solutions exist globally for $N=2,3$7 and $N=2,3$8, with further relaxation to $N=2,3$9 ($\mu(\rho) = \rho^\alpha$0) under additional weighted integrability.
• For $\mu(\rho) = \rho^\alpha$1, solutions exist globally under $\mu(\rho) = \rho^\alpha$2 and $\mu(\rho) = \rho^\alpha$3.
• The density remains bounded and strictly positive at all times provided the initial profile is non-vacuum.

Regularity:

• The solutions $\mu(\rho) = \rho^\alpha$4 satisfy high regularity in Sobolev spaces ($\mu(\rho) = \rho^\alpha$5, $\mu(\rho) = \rho^\alpha$6) and temporal continuity in these norms.
• Strong bounds on time derivatives up to second order, and spatial derivatives up to fourth order, are rigorously derived.

Density Behavior:

• The density does not reach vacuum in finite time, a strong claim given arbitrary amplitude and oscillation in the initial data.

Generality:

• The paper generalizes previous works by lowering the threshold for $\mu(\rho) = \rho^\alpha$7, covering viscosity exponents close to the critical limit where degeneracy becomes severe.

Implications and Future Directions

The results significantly expand the classical theory for compressible Navier-Stokes with physically relevant, density-dependent viscosities, contributing to both mathematical theory and potential computational modeling. The guaranteed absence of vacuum formation for large spherically symmetric data is a notable strengthening over many prior results that rely on smallness or positivity assumptions.

Practically, these results offer a mathematical foundation for simulation and modeling of compressible flows in spherical domains with degenerate viscosity, applicable in astrophysics, geophysics, and engineering scenarios where density may vary widely.

Theoretically, the paper demonstrates that the BD entropy structure is robust enough to yield global regularity even in the presence of degenerate viscosity, supporting further exploration into more general boundary conditions, non-spherical geometries, or even weaker solution concepts.

Open directions include:

• Extension to general (non-spherically symmetric) initial data or domains.
• Analysis for viscosity coefficients below the current critical thresholds.
• Treatment of compressible flow models with temperature-dependent viscosity and heat conduction, incorporating the BD entropy structure.

Conclusion

This paper rigorously demonstrates global existence, uniqueness, and regularity of classical spherically symmetric solutions to multi-dimensional compressible Navier-Stokes equations with degenerate, density-dependent viscosity. The technical methods combining BD entropy, weighted integrability, and spherical symmetry yield strong upper and lower bounds on density and velocity, extending classical well-posedness theory to a broader class of physically relevant settings. The analysis sets a foundation for further studies of degenerate compressible flows and their mathematical and computational properties (2604.18306).