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Global Strong Solutions to the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Initial Data and Vacuum

Qinghao Lei

TL;DR

The paper proves global existence of axisymmetric strong and weak solutions to the three-dimensional compressible Navier–Stokes equations in a cylindrical domain excluding the axis with slip boundary, allowing vacuum and large initial data, under a density-dependent bulk viscosity with $\lambda(\rho)=\rho^{\beta}$ and $\beta>\frac{4}{3}$, $\gamma>1$. The authors develop a novel density upper bound by estimating the effective viscous flux $G$ through axisymmetric reductions, conformal mapping, and Green's function techniques, leveraging slip boundary cancellation. Building on this bound, they derive higher-order energy and regularity estimates, including Beale–Kato–Majda type bounds and exponential decay, to extend local solutions globally in time. The results extend existing two-dimensional and periodic-domain theories to a three-dimensional axisymmetric setting with vacuum, using a combination of axisymmetric analysis and conformal-analytic methods. Overall, the work provides a rigorous framework for global solvability and long-time behavior of compressible flows with vacuum under slip conditions in axisymmetric geometries.

Abstract

This paper investigates the three-dimensional axisymmetric compressible Navier-Stokes equations under slip boundary conditions in a cylindrical domain excluding the axis. For initial density allowed to vanish, we establish the global existence and large time asymptotic behavior of strong and weak solutions, provided the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds. It should be noted that these results are obtained without any restrictions on the size of initial data. The key idea is to derive a pointwise estimate of the effective viscous flux by exploiting the axisymmetry of the solutions, along with the conformal mapping and the pull back Green's function, and then to cancel out the singularity using the slip boundary conditions.

Global Strong Solutions to the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Initial Data and Vacuum

TL;DR

The paper proves global existence of axisymmetric strong and weak solutions to the three-dimensional compressible Navier–Stokes equations in a cylindrical domain excluding the axis with slip boundary, allowing vacuum and large initial data, under a density-dependent bulk viscosity with and , . The authors develop a novel density upper bound by estimating the effective viscous flux through axisymmetric reductions, conformal mapping, and Green's function techniques, leveraging slip boundary cancellation. Building on this bound, they derive higher-order energy and regularity estimates, including Beale–Kato–Majda type bounds and exponential decay, to extend local solutions globally in time. The results extend existing two-dimensional and periodic-domain theories to a three-dimensional axisymmetric setting with vacuum, using a combination of axisymmetric analysis and conformal-analytic methods. Overall, the work provides a rigorous framework for global solvability and long-time behavior of compressible flows with vacuum under slip conditions in axisymmetric geometries.

Abstract

This paper investigates the three-dimensional axisymmetric compressible Navier-Stokes equations under slip boundary conditions in a cylindrical domain excluding the axis. For initial density allowed to vanish, we establish the global existence and large time asymptotic behavior of strong and weak solutions, provided the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds. It should be noted that these results are obtained without any restrictions on the size of initial data. The key idea is to derive a pointwise estimate of the effective viscous flux by exploiting the axisymmetry of the solutions, along with the conformal mapping and the pull back Green's function, and then to cancel out the singularity using the slip boundary conditions.

Paper Structure

This paper contains 5 sections, 29 theorems, 280 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. A Priori Estimates (I): Upper Bound of the density
  4. A Priori Estimates (II): Higher Order Estimates
  5. Proofs of Theorems \ref{['th0']}--\ref{['th3']}

Key Result

Theorem 1.1

Assume that and that $K$ is a smooth, symmetric, positive semi-definite $3 \times 3$ axisymmetric matrix-valued function satisfying $K + 2D(n)$ is positive definite on some subset $\Sigma \subset \partial \Omega$ with $|\Sigma|>0$. Suppose that the initial data $(\rho_0,\mathbf{m}_0)$ satisfy for some $q>3$, and $\rho_0$, $\mathbf{u}_0$ are axisymmetric and periodic in $x_3$ with period $1$. The

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • ...and 38 more