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Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

Jing Li, Zhouping Xin

TL;DR

The paper proves global existence of weak solutions to the barotropic compressible Navier–Stokes equations with density-dependent, potentially degenerate viscosities in 2D and 3D under periodic and Cauchy settings. The authors construct smooth approximations that satisfy energy, BD entropy, and Mellet–Vasseur estimates, then leverage compactness results to pass to the limit, accommodating vacuum states. In 2D, global weak solutions are obtained on ${ m T}^2$ or ${ m R}^2$; in 3D, solutions exist under carefully chosen α, γ ranges, including a special case with h = ρ, g = 0. The work extends the scope of Lions’ open problems by enabling global weak existence with vacuum for a broad class of density-dependent viscosities and large initial data, and it connects BD entropy methods with Mellet–Vasseur type compactness in a unified framework.

Abstract

This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensional periodic domains or whole space for large initial data. This, in particular, solved an open problem in [P. L. Lions. Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford University Press, 1998].

Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

TL;DR

The paper proves global existence of weak solutions to the barotropic compressible Navier–Stokes equations with density-dependent, potentially degenerate viscosities in 2D and 3D under periodic and Cauchy settings. The authors construct smooth approximations that satisfy energy, BD entropy, and Mellet–Vasseur estimates, then leverage compactness results to pass to the limit, accommodating vacuum states. In 2D, global weak solutions are obtained on or ; in 3D, solutions exist under carefully chosen α, γ ranges, including a special case with h = ρ, g = 0. The work extends the scope of Lions’ open problems by enabling global weak existence with vacuum for a broad class of density-dependent viscosities and large initial data, and it connects BD entropy methods with Mellet–Vasseur type compactness in a unified framework.

Abstract

This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensional periodic domains or whole space for large initial data. This, in particular, solved an open problem in [P. L. Lions. Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford University Press, 1998].

Paper Structure

This paper contains 14 sections, 31 theorems, 341 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['th2']}: ${\Omega }=\mathbb{T}^2$
  3. A priori estimates
  4. Compactness results
  5. Proof of Theorem \ref{['th2']}: ${\Omega }=\mathbb{T}^2$
  6. Proof of Theorem \ref{['vth2']}: ${\Omega }=\mathbb{T}^3$
  7. A priori estimates
  8. Compactness results
  9. Proof of Theorem \ref{['vth2']}: ${\Omega }=\mathbb{T}^3$
  10. Proof of Theorem \ref{['qvth2']}: ${\Omega }=\mathbb{T}^3$
  11. A priori estimates
  12. Compactness results
  13. Proof of Theorem \ref{['qvth2']}: ${\Omega }=\mathbb{T}^3$
  14. Proofs of Theorems \ref{['th2']}, \ref{['vth2']}, and \ref{['qvth2']}: Cauchy Problem

Key Result

Theorem 1.1

Let $\Omega=\mathbb{R}^2$ or $\mathbb{T}^2 .$ Suppose that $\alpha$ and $\gamma$ satisfy Moreover, assume that the initial data $(\rho_0,m_0)$ satisfy pini1. Then there exists a global weak solution $(\rho,u )$ to the problem ii1aa'1hgvv1en1.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • ...and 27 more