Global axisymmetric solutions and incompressible limit for the 3D isentropic compressible Navier-Stokes equations in annular cylinders with swirl and large initial data

TL;DR

This paper proves the global existence of weak solutions for 3D compressible Navier-Stokes equations in annular cylinders with large initial data, and shows convergence to incompressible solutions as bulk viscosity increases.

Contribution

It establishes the existence of solutions with large initial data and vacuum states under high bulk viscosity, extending previous results to boundary-influenced geometries.

Findings

Solutions exist globally with large initial data and vacuum states.

Large bulk viscosity suppresses compressible effects, enabling convergence to incompressible flow.

The approach uses advanced inequalities and estimates building on prior foundational works.

Abstract

We establish the global existence of weak solutions to the isentropic compressible Navier-Stokes equations in three-dimensional annular cylinders with Navier-slip boundary conditions, allowing large axisymmetric initial data and vacuum states, provided that the bulk viscosity is sufficiently large. We identify a regime in which compressible and incompressible effects coexist. The compressible component interacts with pressure and density to produce an effective dissipation mechanism, while the divergence-free component enjoys improved regularity. This shows that large bulk viscosity strongly suppresses the compressible effect, thereby relaxing restrictions on the size of the initial data. Moreover, such solutions converge globally in time to weak solutions of the inhomogeneous incompressible Navier-Stokes system as the bulk viscosity tends to infinity. The proof relies on a Desjardins-type logarithmic interpolation inequality and Friedrichs-type commutator estimates. Our results build upon the works of Hoff (Indiana Univ. Math. J. 41 (1992), pp. 1225-1302) and Danchin-Mucha (Comm. Pure Appl. Math. 76 (2023), pp. 3437-3492), and further develop Hoff-type time-weighted estimates uniform in the bulk viscosity in the presence of boundaries.