On a Local Existence Theorem for the Evolution Equation of Viscous Gaseous Stars in a Physical Vacuum

Technical Summary: Local Existence Theorem for Viscous Gaseous Stars in a Physical Vacuum

Problem Formulation
This paper addresses the local well-posedness of the free boundary problem (VFBP) for the three-dimensional compressible Navier–Stokes–Poisson (CNSP) equations describing self-gravitating viscous gaseous stars. The system is characterized by degenerate viscosities, where the shear and bulk viscosity coefficients depend on the density as power laws, $\mu(\rho) = a_1\rho^\delta$ and $\lambda(\rho) = a_2\rho^\delta$, with $\delta \in (0, 1)$.

The physical setting involves a "physical vacuum" boundary, defined by the condition that the sound speed squared, $c^2$, has a negative normal derivative at the boundary ($-\infty < \partial_n c_0^2 < 0$). This condition implies that the initial density $\rho_0$ vanishes at the boundary like a distance function ($\rho_0^{\gamma-1} \sim \text{dist}(x, \partial\Omega)$). The authors focus on spherically symmetric, barotropic motion, reducing the 3-D problem to a 1-D initial-boundary value problem (IBVP) on a moving interval $I(t) = [0, R(t))$. The primary mathematical challenge lies in the strong degeneracy of the equations near the vacuum boundary and the coordinate singularity at the origin, which complicate the propagation of regularity for the velocity field.

Methodology
The authors employ a Lagrangian reformulation of the problem to fix the spatial domain to $I = [0, 1)$. The core methodology involves a multi-step analytical framework:

1. Linearization and Approximation: The nonlinear problem is linearized around a given flow map $\bar{\eta}$. To solve the resulting linearized problem, the authors utilize a modified Galerkin scheme. They construct a Hilbert basis $\{\xi_j\}$ derived from a specific Sturm–Liouville problem tailored to the spherical symmetry and Neumann boundary conditions. This allows for the construction of global-in-time approximation solutions.
2. Uniform Energy Estimates: A significant portion of the work is dedicated to establishing uniform energy estimates for the linearized solutions. The authors define a total energy functional $E(t)$ and a dissipation functional $D(t)$, which are split into interior and exterior components.
   • Interior Estimates: Near the origin, standard Sobolev norms are used, leveraging the smoothness of the solution in the bulk.
   • Exterior Estimates: Near the vacuum boundary, the authors introduce weighted Sobolev spaces with weights involving powers of the initial density $\rho_0$. These weights are carefully chosen to handle the degeneracy and to ensure the solution remains smooth up to the moving boundary. The weights depend on the adiabatic exponent $\gamma$ and the viscosity parameter $\delta$.
3. Bootstrap and Regularity: Through a bootstrapping argument involving elliptic estimates and cross-derivative embeddings (detailed in Appendix D), the authors derive higher-order regularity for the velocity and flow map. They specifically address the "cross-derivative" terms that arise in the momentum equation, proving that the highest-order derivatives can be controlled in the weighted spaces.
4. Picard Iteration: Finally, the local well-posedness of the nonlinear problem is established via a classical Picard iteration. The uniform estimates derived for the linearized problem are used to prove the strong convergence of the iterative sequence to a unique classical solution.

Key Contributions and Results
The paper establishes the following main results:

• Local Well-Posedness: The authors prove the existence and uniqueness of classical solutions to the spherically symmetric CNSP equations with degenerate viscosities and physical vacuum boundaries. This holds for all adiabatic exponents $\gamma > 4/3$ and viscosity parameters $\delta \in (0, 1)$.
• Regularity up to the Boundary: Unlike previous results for viscous flows which often yielded strong solutions that were not smooth up to the moving boundary, this work demonstrates that the solutions obtained are smooth all the way to the moving vacuum boundary for $t > 0$. The solutions capture the specific physical vacuum behavior of the Lane-Emden star configuration.
• General Initial Data: The results hold without smallness assumptions on the initial data, provided the initial density satisfies the physical vacuum condition ($\rho_0^{\gamma-1} \sim 1-r$) and belongs to appropriate weighted Sobolev spaces ($H^3$ for density, weighted $H^6$ for velocity).
• Boundary Conditions: The paper rigorously derives the boundary conditions satisfied by the solution, showing that the stress-free condition holds automatically and that the velocity satisfies a specific relation at the vacuum boundary involving the viscosity coefficients.

Significance
The significance of this work, as presented by the authors, lies in overcoming the intrinsic difficulties posed by the strong degeneracy of the equations near the physical vacuum. Previous studies on viscous gaseous stars often resulted in solutions that lacked smoothness at the boundary or required restrictive assumptions on the initial data (e.g., higher decay rates of density near the vacuum). By developing a tailored energy framework with specific weighted norms and utilizing a modified Galerkin scheme, this paper extends the local well-posedness theory to the physically relevant case of Lane-Emden stars with degenerate viscosities. The results bridge the gap between the well-understood inviscid Euler-Poisson case and the more complex viscous case, providing a rigorous foundation for the study of the evolution of viscous gaseous stars in a physical vacuum.