Global Existence and Incompressible Limit for the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
Qinghao Lei
TL;DR
This work analyzes the 3D axisymmetric compressible Navier–Stokes equations with slip boundary in a cylindrical domain excluding the axis, establishing global existence and exponential decay for weak, strong, and classical solutions even with large initial data and vacuum, provided the bulk viscosity is sufficiently large. It also proves an incompressible limit: as the bulk viscosity $\nu=2\mu+\lambda$ tends to infinity, solutions converge to the axisymmetric inhomogeneous incompressible Navier–Stokes equations, with the divergence of the velocity decaying like $O(\nu^{-1/2})$. The analysis leverages axisymmetric reductions, sharp a priori estimates (lower and higher order), the effective viscous flux, div–curl and Beale–Kato–Majda type controls, and Green’s function arguments to bound density and derive exponential decay rates. Under extra regularity and compatibility, the convergence strengthens to a unique global strong solution of the incompressible system. Overall, the results provide a comprehensive global well-posedness and singular limit theory for axisymmetric compressible flows with large viscosity effects in a physically relevant geometric setting.
Abstract
In this paper, we study the three-dimensional axisymmetric compressible Navier-Stokes equations with slip boundary conditions in a cylindrical domain excluding the axis. We establish the global existence and exponential decay of weak, strong, and classical solutions with large initial data and vacuum, under the assumption that the bulk viscosity coefficient is sufficiently large. Moreover, we prove that as the bulk viscosity coefficient tends to infinity, the solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations.