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Global Well-Posedness of the Vacuum Free Boundary Problem for the Degenerate Compressible Navier-Stokes Equations With Large Data of Spherical Symmetry

Gui-Qiang G. Chen, Jiawen Zhang, Shengguo Zhu

TL;DR

The paper establishes global-in-time classical solutions for the vacuum free boundary problem of the degenerate compressible Navier–Stokes equations with density-dependent viscosity under large, spherically symmetric data in dimensions two and three. A novel region-segmentation approach couples interior BD entropy-type analysis near the origin with density- and flow-map-weighted estimates near the moving boundary, enabling uniform control of the density and the deformation gradient η_r and η/r. The authors develop new weighted energy functionals and a damped transport framework for the effective velocity V to overcome incompatibilities between BD entropy and physical vacuum, yielding global existence without small data assumptions and, in the 2-D shallow-water case (γ=2), addressing a physically important model. The results also translate into Eulerian coordinates, confirming global well-posedness for ρ and u with the free boundary propagating at finite speed, and show automatic satisfaction of the stress-free boundary condition on ∂Ω(t). The techniques, notably the interior/exterior region decomposition and flow-map weighted estimates, provide a robust framework for related nonlinear degenerate PDEs with moving boundaries.

Abstract

The study of global-in-time dynamics of vacuum is crucial for understanding viscous flows. However, the corresponding large-data problems for multidimensional spherically symmetric flows have remained open, due to the coordinate singularity at the origin and the strong degeneracy on the moving boundary. In this paper, we analyze the vacuum free boundary problem for the barotropic compressible Navier-Stokes equations with degenerate density-dependent viscosity coefficients (as in the shallow water equations) in two and three spatial dimensions. We prove that, for a general class of spherically symmetric initial density: $ρ_0^β\in H^3$ with $β\in (\frac{1}{3},γ-1]$ ($γ$: adiabatic exponent) vanishing on the moving boundary in the form of a distance function, no vacuum forms inside the fluid in finite time, and we establish the global well-posedness of classical solutions with large initial data. Particularly, when $β=γ-1$, the initial density contains a physical vacuum, but fails to satisfy the condition required for the Bresch-Desjardins (BD) entropy estimate when $γ\ge 2$. Our analysis is mainly based on a region-segmentation method: near the origin, we develop an interior BD entropy estimate, thereby obtaining flow-map-weighted estimates for the density; while, near the boundary, we construct $ρ_0$-weighted estimates for the effective velocity, which differ fundamentally from the classical BD entropy estimates and yield novel flow-map-weighted estimates for both the fluid and the effective velocities. These estimates enable us to obtain the uniform upper bound for the density and show that no cavitation occurs inside the fluid. The methodology developed here should also be useful for solving other related nonlinear equations involving similar difficulties.

Global Well-Posedness of the Vacuum Free Boundary Problem for the Degenerate Compressible Navier-Stokes Equations With Large Data of Spherical Symmetry

TL;DR

The paper establishes global-in-time classical solutions for the vacuum free boundary problem of the degenerate compressible Navier–Stokes equations with density-dependent viscosity under large, spherically symmetric data in dimensions two and three. A novel region-segmentation approach couples interior BD entropy-type analysis near the origin with density- and flow-map-weighted estimates near the moving boundary, enabling uniform control of the density and the deformation gradient η_r and η/r. The authors develop new weighted energy functionals and a damped transport framework for the effective velocity V to overcome incompatibilities between BD entropy and physical vacuum, yielding global existence without small data assumptions and, in the 2-D shallow-water case (γ=2), addressing a physically important model. The results also translate into Eulerian coordinates, confirming global well-posedness for ρ and u with the free boundary propagating at finite speed, and show automatic satisfaction of the stress-free boundary condition on ∂Ω(t). The techniques, notably the interior/exterior region decomposition and flow-map weighted estimates, provide a robust framework for related nonlinear degenerate PDEs with moving boundaries.

Abstract

The study of global-in-time dynamics of vacuum is crucial for understanding viscous flows. However, the corresponding large-data problems for multidimensional spherically symmetric flows have remained open, due to the coordinate singularity at the origin and the strong degeneracy on the moving boundary. In this paper, we analyze the vacuum free boundary problem for the barotropic compressible Navier-Stokes equations with degenerate density-dependent viscosity coefficients (as in the shallow water equations) in two and three spatial dimensions. We prove that, for a general class of spherically symmetric initial density: with (: adiabatic exponent) vanishing on the moving boundary in the form of a distance function, no vacuum forms inside the fluid in finite time, and we establish the global well-posedness of classical solutions with large initial data. Particularly, when , the initial density contains a physical vacuum, but fails to satisfy the condition required for the Bresch-Desjardins (BD) entropy estimate when . Our analysis is mainly based on a region-segmentation method: near the origin, we develop an interior BD entropy estimate, thereby obtaining flow-map-weighted estimates for the density; while, near the boundary, we construct -weighted estimates for the effective velocity, which differ fundamentally from the classical BD entropy estimates and yield novel flow-map-weighted estimates for both the fluid and the effective velocities. These estimates enable us to obtain the uniform upper bound for the density and show that no cavitation occurs inside the fluid. The methodology developed here should also be useful for solving other related nonlinear equations involving similar difficulties.
Paper Structure (68 sections, 88 theorems, 983 equations)

This paper contains 68 sections, 88 theorems, 983 equations.

Table of Contents

  1. Introduction
  2. Background of the Problem
  3. Lagrangian reformulation of VFBP in spherical coordinates
  4. Outline of the paper
  5. Main Theorems
  6. Main results in Lagrangian coordinates
  7. Applications to the 2-D shallow water equations (the viscous Saint-Venant system)
  8. Main results in Eulerian coordinates
  9. Notations and Main Strategies
  10. Notations
  11. Notations on coordinates and operators
  12. Notations on function spaces
  13. Other notations
  14. Main strategies
  15. Some new weighted energy functionals
  16. ...and 53 more sections

Key Result

Theorem 2.1

Let $n=2$ or $3$ and If $\rho_0(r)$ satisfies distance-la for some and $u_0(r)$ satisfies then, for any $T>0$, IBVPeq:VFBP-La-eta admits a unique classical solution $(U,\eta)(t,r)$ in $[0,T]\times \bar{I}$ such that where $C(T)>1$ is a constant depending only on $(n,\mu,A,\gamma,\beta,\varepsilon_0,\rho_0,u_0,\mathcal{K}_1, \mathcal{K}_2,T)$. Moreover, such a classical solution admits the fol

Theorems & Definitions (177)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.2
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 167 more