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

TL;DR

This paper proves the global well-posedness of classical solutions for the vacuum free boundary problem in spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity, addressing large initial data and physical vacuum conditions.

Contribution

It introduces a novel region-segmentation method and weighted estimates to handle degeneracy and vacuum singularities, extending the understanding of multidimensional viscous flows.

Findings

Established global well-posedness for large initial data.

Developed interior BD entropy estimates near the origin.

Created new weighted estimates for effective velocity at the vacuum boundary.

Abstract

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: $\rho_0^{\beta}\in H^3$ with $\beta\in (\frac{1}{3},\gamma-1]$ ($\gamma$: 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 $\beta=\gamma-1$, $\rho_0$ contains a physical vacuum, but fails to satisfy the condition required for the Bresch-Desjardins (BD) entropy estimate when $\gamma\ge 2$, precluding the use of the BD entropy estimate to handle the degeneracy of the shallow water equations ({\it i.e.}, the case $\gamma=2$) 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 $\rho_0$-weighted estimates for the effective velocity, which are fundamentally different from the classical BD entropy estimate. Together, these estimates yield the desired global regularities.