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

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.