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

The study of global-in-time dynamics of vacuum is crucial for understanding viscous flows. In particular, physical vacuum, characterized by a moving boundary with nontrivial finite normal acceleration, naturally arises in the motion of shallow water. The corresponding large-data problems for multidimensional spherically symmetric flows remain open, due to the combined difficulties of coordinate singularity at the origin and degeneracy on the moving boundary. In this paper, we analyze the free boundary problem for the barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients (as in the shallow water equations) in two and three spatial dimensions. For a general class of spherically symmetric initial densities: $ρ_0^β\in H^3$ with $β\in (\frac{1}{3},γ-1]$ ($γ$: adiabatic exponent), vanishing on the moving boundary in the form of a distance function, we establish the global well-posedness of classical solutions with large initial data. We note that, when $β=γ-1$, $ρ_0$ contains a physical vacuum, but fails to satisfy the condition required for the Bresch-Desjardins (BD) entropy estimate when $γ\ge 2$, precluding the use of the BD entropy estimate to handle the degeneracy of the shallow water equations ({\it i.e.}, the case $γ=2$) on the physical vacuum boundary. Our analysis relies on a region-segmentation method: near the origin, we develop an interior BD entropy estimate, leading to flow-map-weighted estimates for the density; near the boundary, to handle the physical vacuum singularity, we introduce novel $ρ_0$-weighted estimates for the effective velocity, which are fundamentally different from the classical BD entropy estimate. Together, these estimates yield the desired global regularities.

💡 Research Summary

The paper addresses the vacuum free‑boundary problem (VFBP) for the compressible Navier‑Stokes equations with density‑dependent viscosity, a model that coincides with the viscous Saint‑Venant (shallow‑water) system when the adiabatic exponent γ equals 2. The authors consider spherically symmetric flows in two and three space dimensions and allow the initial data to be arbitrarily large. The initial density ρ₀ is assumed to belong to H³ and to vanish at the moving vacuum boundary like a power of the distance to the boundary, i.e. ρ₀(x)≈(1−|x|)^{β} with β∈(1/3,γ−1]. When β=γ−1 the profile corresponds to a physical vacuum, where the sound speed c satisfies a non‑trivial finite normal acceleration condition on the free surface.

The main result is the global existence and uniqueness of classical solutions (ρ,u)∈C(