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

Shuai Wang, Guochun Wu, Xin Zhong

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.

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

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.

Paper Structure

This paper contains 10 sections, 20 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Strategy of the proof
  5. Preliminaries
  6. Auxiliary results and inequalities
  7. Uniform lower-order $L^p$-estimates
  8. A priori estimates
  9. Proof of Theorem 1.1
  10. Proof of Theorem 1.2

Key Result

Lemma 1.1

Let $R_\eta$ be the rotation about the $x_3$-axis by angle $\eta\in[0,2\pi)$, i.e., Assume that the domain $\Omega$ is invariant under $R_\eta$ and that $(\rho,\mathbf{u})$ is axisymmetric. For any admissible test functions $(\phi,\boldsymbol\psi)$ in Definition d1.1, define the rotational averages Then $(\phi_\eta,\boldsymbol\psi_\eta)$ are admissible test functions and are axisymmetric, with $

Theorems & Definitions (38)

  • Definition 1.1
  • Lemma 1.1: Axisymmetric reduction of test functions
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • ...and 28 more