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Vanishing viscosity limit for the compressible Navier-Stokes equations with non-linear density dependent viscosities

Luca Bisconti, Matteo Caggio, Filippo Dell'Oro

TL;DR

The paper studies the vanishing viscosity limit for the 3D compressible Navier–Stokes equations with density-dependent viscosities in a bounded domain, including a drag term and no-slip boundary conditions. It develops two Kato-type boundary-layer criteria that guarantee convergence to a strong Euler solution as $\varepsilon\to0$ and $r_1(\varepsilon)\to0$, using a relative energy framework enhanced by a Kato-type fake boundary layer. The analysis extends prior work to general nonlinear viscosity laws by avoiding the augmented BDG approach and employing a direct relative-energy method, yielding convergence in density $\varrho\to\varrho^E$ in $L^\gamma$ and momentum $\varrho\mathbf{u}\to\varrho^E\mathbf{u}^E$ in $L^1$. The results provide explicit boundary-layer conditions and demonstrate how vanishing viscosity and drag parameters influence the inviscid limit, broadening applicability to nonlinear density-dependent viscosities and enriching the understanding of boundary-layer effects in compressible flows.

Abstract

In a three-dimensional bounded domain $Ω$ we consider the compressible Navier-Stokes equations for a barotropic fluid with general non-linear density dependent viscosities and no-slip boundary conditions. A nonlinear drag term is added to the momentum equation. We establish two conditional Kato-type criteria for the convergence of the weak solutions to such a system towards the strong solution of the compressible Euler system when the viscosity coefficient and the drag term parameter tend to zero.

Vanishing viscosity limit for the compressible Navier-Stokes equations with non-linear density dependent viscosities

TL;DR

The paper studies the vanishing viscosity limit for the 3D compressible Navier–Stokes equations with density-dependent viscosities in a bounded domain, including a drag term and no-slip boundary conditions. It develops two Kato-type boundary-layer criteria that guarantee convergence to a strong Euler solution as and , using a relative energy framework enhanced by a Kato-type fake boundary layer. The analysis extends prior work to general nonlinear viscosity laws by avoiding the augmented BDG approach and employing a direct relative-energy method, yielding convergence in density in and momentum in . The results provide explicit boundary-layer conditions and demonstrate how vanishing viscosity and drag parameters influence the inviscid limit, broadening applicability to nonlinear density-dependent viscosities and enriching the understanding of boundary-layer effects in compressible flows.

Abstract

In a three-dimensional bounded domain we consider the compressible Navier-Stokes equations for a barotropic fluid with general non-linear density dependent viscosities and no-slip boundary conditions. A nonlinear drag term is added to the momentum equation. We establish two conditional Kato-type criteria for the convergence of the weak solutions to such a system towards the strong solution of the compressible Euler system when the viscosity coefficient and the drag term parameter tend to zero.

Paper Structure

This paper contains 21 sections, 5 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Assumptions on viscosity coefficients $\mu$ and $\lambda$
  4. Weak solutions to the compressible Navier-Stokes system
  5. Strong solution to the compressible Euler system
  6. Statement of the main results
  7. Relative energy inequality
  8. "Fake" boundary layer and uniform bounds
  9. Uniform bounds on the weak solutions and the boundary layer
  10. Estimating the terms $\mathcal{R}_1$-$\mathcal{R}_5$
  11. Estimate of $\mathcal{R}_1$
  12. Estimate of $\mathcal{R}_2$-$\mathcal{R}_4$
  13. Estimate of $\mathcal{R}_5$
  14. Proof of Theorem \ref{['main-1']}
  15. Estimate of $\mathcal{R}_6$
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $T > 0$ be arbitrarily fixed. Let also the initial data $(\varrho_0,\mathbf{u}_0)$ for the compressible Navier-Stokes system cont-mom be such that Then, for every $\varepsilon>0$ and $r_1>0$, there exists at least a weak solution $(\varrho, \mathbf{u})$ to system cont-mom on the time interval $[0,T]$ with boundary conditions bc-bc-bis.

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Remark 2.3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.1
  • Remark 3.1
  • ...and 2 more