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Existence and Exponential Growth of Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Bounded Domains

Guocai Cai, Jing Li

TL;DR

This work establishes global existence of weak and classical solutions to the 3D barotropic compressible Navier–Stokes equations with slip boundary conditions in smooth bounded domains for any $\gamma>1$, even when vacuum is present initially, provided the initial energy is small. The authors develop new boundary-integral estimates tailored to slip boundaries and exploit the effective viscous flux $F$ and vorticity $\mathrm{curl}\,u$, together with a Helmholtz–Weyl decomposition, to derive comprehensive a priori estimates (low- and high-order) and long-time behavior. They prove exponential decay of the energy-related quantities and, in the presence of initial vacuum, exponential growth of the density gradient in time; they also extend the results to not-necessarily-simply-connected domains using boundary-term norm equivalences. Additionally, a weak-solution theory is established under the same small-energy regime, and the paper provides a framework to handle nontrivial domain topology via generalized boundary conditions.

Abstract

We investigate the barotropic compressible Navier-Stokes equations with slip boundary conditions in a three-dimensional (3D) simply connected bounded domain, whose smooth boundary has a finite number of two-dimensional connected components. For any adiabatic exponent bigger than one, after discovering some new estimates on boundary integrals related to the slip boundary condition, we prove that both the weak and classical solutions to the initial-boundary-value problem of this system exist globally in time provided the initial energy is suitably small. Moreover, the density has large oscillations and contains vacuum states. Finally, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. This is the first result concerning the global existence of classical solutions to the compressible Navier-Stokes equations with density containing vacuum states initially for general 3D bounded smooth domains.

Existence and Exponential Growth of Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Bounded Domains

TL;DR

This work establishes global existence of weak and classical solutions to the 3D barotropic compressible Navier–Stokes equations with slip boundary conditions in smooth bounded domains for any , even when vacuum is present initially, provided the initial energy is small. The authors develop new boundary-integral estimates tailored to slip boundaries and exploit the effective viscous flux and vorticity , together with a Helmholtz–Weyl decomposition, to derive comprehensive a priori estimates (low- and high-order) and long-time behavior. They prove exponential decay of the energy-related quantities and, in the presence of initial vacuum, exponential growth of the density gradient in time; they also extend the results to not-necessarily-simply-connected domains using boundary-term norm equivalences. Additionally, a weak-solution theory is established under the same small-energy regime, and the paper provides a framework to handle nontrivial domain topology via generalized boundary conditions.

Abstract

We investigate the barotropic compressible Navier-Stokes equations with slip boundary conditions in a three-dimensional (3D) simply connected bounded domain, whose smooth boundary has a finite number of two-dimensional connected components. For any adiabatic exponent bigger than one, after discovering some new estimates on boundary integrals related to the slip boundary condition, we prove that both the weak and classical solutions to the initial-boundary-value problem of this system exist globally in time provided the initial energy is suitably small. Moreover, the density has large oscillations and contains vacuum states. Finally, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. This is the first result concerning the global existence of classical solutions to the compressible Navier-Stokes equations with density containing vacuum states initially for general 3D bounded smooth domains.

Paper Structure

This paper contains 9 sections, 31 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A priori estimates (I): lower order estimates
  4. A priori estimates (II): higher order estimates
  5. Proofs of Theorems \ref{['th1']}-\ref{['th3']}
  6. Proof of Theorem \ref{['th1']}
  7. Proof of Theorem \ref{['th2']}
  8. Proof of Theorem \ref{['th3']}
  9. Proof of Theorem \ref{['th4']}

Key Result

Theorem 1.1

Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^3$ and its smooth boundary $\partial\Omega$ has a finite number of 2-dimensional connected components. For given positive constants $M$ and $\hat{\rho},$ suppose that the $3 \times 3$ symmetric matrix $A$ in ch1 is smooth and positive and the compatibility condition for some $g\in L^2.$ Then there exists a positive constant $\varep

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.4
  • ...and 47 more