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Incompressible and vanishing vertical viscosity limit for the compressible Navier-Stokes system with Dirichlet boundary conditions

Nader Masmoudi, Changzhen Sun, Chao Wang, Zhifei Zhang

TL;DR

The paper justifies the incompressible and vanishing vertical viscosity limits for strong solutions of the scaled compressible Navier–Stokes system with anisotropic horizontal/vertical dissipation and Dirichlet boundary conditions. It develops a detailed multi-scale approximate solution that combines interior oscillations with two boundary-layer types and introduces a damping mechanism to control fast acoustic waves. A rigorous conormal-energy framework is then used to prove uniform-in-$\epsilon$ and $\nu$ bounds for the remainder, establishing convergence to the incompressible system with horizontal dissipation as $\epsilon,\nu\to 0$ under a non-resonant spectral condition. The results are significant for understanding low-Mach-number, highly anisotropic viscous flows in bounded domains and provide a precise description of how boundary layers and fast waves influence the limiting dynamics. The work also extends the theory to ill-prepared data in the presence of Dirichlet boundaries, highlighting the role of horizontal dissipation in enabling Sobolev-regular well-posedness without requiring analytic regularity.

Abstract

In this paper, we show the incompressible and vanishing vertical viscosity limits for the strong solutions to the isentropic compressible Navier-Stokes system with anistropic dissipation, in a domain with Dirichlet boundary conditions in the general setting of ill-prepared initial data. We establish the uniform regularity estimates with respect to the Mach number $ε$ and the vertical viscosity $ν$ so that the solution exists on a uniform time interval $[0,T_0]$ independent of these parameters. The key steps toward this goal are the careful construction of the approximate solution in the presence of both fast oscillations and two kinds of boundary layers together with the stability analysis of the remainder. In the process, it is also shown that the solutions of the compressible systems converge to those of the incompressible system with only horizontal dissipation, after removing the fast waves whose horizontal derivative is bounded in $L_{T_0}^2L_x^2$ by $\min\{1, (ε/ν)^{\frac14}\}.$

Incompressible and vanishing vertical viscosity limit for the compressible Navier-Stokes system with Dirichlet boundary conditions

TL;DR

The paper justifies the incompressible and vanishing vertical viscosity limits for strong solutions of the scaled compressible Navier–Stokes system with anisotropic horizontal/vertical dissipation and Dirichlet boundary conditions. It develops a detailed multi-scale approximate solution that combines interior oscillations with two boundary-layer types and introduces a damping mechanism to control fast acoustic waves. A rigorous conormal-energy framework is then used to prove uniform-in- and bounds for the remainder, establishing convergence to the incompressible system with horizontal dissipation as under a non-resonant spectral condition. The results are significant for understanding low-Mach-number, highly anisotropic viscous flows in bounded domains and provide a precise description of how boundary layers and fast waves influence the limiting dynamics. The work also extends the theory to ill-prepared data in the presence of Dirichlet boundaries, highlighting the role of horizontal dissipation in enabling Sobolev-regular well-posedness without requiring analytic regularity.

Abstract

In this paper, we show the incompressible and vanishing vertical viscosity limits for the strong solutions to the isentropic compressible Navier-Stokes system with anistropic dissipation, in a domain with Dirichlet boundary conditions in the general setting of ill-prepared initial data. We establish the uniform regularity estimates with respect to the Mach number and the vertical viscosity so that the solution exists on a uniform time interval independent of these parameters. The key steps toward this goal are the careful construction of the approximate solution in the presence of both fast oscillations and two kinds of boundary layers together with the stability analysis of the remainder. In the process, it is also shown that the solutions of the compressible systems converge to those of the incompressible system with only horizontal dissipation, after removing the fast waves whose horizontal derivative is bounded in by
Paper Structure (41 sections, 28 theorems, 386 equations)

This paper contains 41 sections, 28 theorems, 386 equations.

Table of Contents

  1. Introduction and backgrounds
  2. Approximate solution
  3. Main results
  4. Construction of the approximate solution
  5. First two interior equations
  6. Corrections for the fast variable
  7. The first boundary layers
  8. Determine the damping term $\overline{S}$
  9. Corrections for the mean flow--slow variable
  10. Prandtl layer correction
  11. Second interior profile.
  12. Further corrections for the boundary conditions
  13. The approximate solution
  14. Verification of \ref{['esR6a']}
  15. Energy functions and main strategy
  16. ...and 26 more sections

Key Result

Theorem 1.1

Suppose that $a_1, a_2,a_3$ is such that the small divisor estimate no-resonantasp holds true for some $r_0.$ There exist ${T}_1>0$ which is independent of $\epsilon,\nu,$ and an approximate solution $U^a$ (defined by app sol) such that $(\epsilon\partial^{\alpha}_t)^iU^a\in L_{T_1}^{\infty}H_{co}^{ where the error satisfies Moreover, we have for any $2\leq p<+\infty$ Here $v^{INS}$ solves the in

Theorems & Definitions (49)

  • Theorem 1.1: Existence of the approximate solution
  • Theorem 1.2: Stability of the remainder
  • Remark 1.3
  • Remark 1.4
  • Theorem 3.1: Uniform estimates
  • Proposition 3.2
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 39 more