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Anelastic approximation for the degenerate compressible Navier--Stokes equations revisited

Nilasis Chaudhuri, Francesco Fanelli, Yang Li, Ewelina Zatorska

TL;DR

The paper addresses the rigorous singular limit from degenerate compressible Navier–Stokes equations with density-dependent viscosity to a generalized anelastic system under joint low-Mach/low-Froude scaling. It advances the κ-entropy weak solutions framework, employing a two-velocity formulation and relative entropy to prove weak-to-strong convergence for well-prepared data in 𝕋^3 and ill-prepared data in ℝ^3, the latter aided by dispersive estimates for acoustic waves. A key novelty is deriving convergence without auxiliary regularization terms such as drag, capillarity, or cold pressure, relying instead on a refined energetic- dispersive analysis and meticulous control of remainder terms. The results have potential implications for understanding stratified viscous flows with spatially varying density profiles in three dimensions, under minimal regularization assumptions.

Abstract

In this paper, we revisit the joint low-Mach and low-Frode number limit for the compressible Navier-Stokes equations with degenerate, density-dependent viscosity. Employing the relative entropy framework based on the concept of $κ$-entropy, we rigorously justify the convergence of weak solutions toward the generalized anelastic system in a three-dimensional periodic domain for well-prepared initial data. For general ill-prepared initial data, we establish a similar convergence result in the whole space, relying essentially on dispersive estimates for acoustic waves. Compared with the work of Fanelli and Zatorska [Commun. Math. Phys., 400 (2023), pp. 1463-1506], our analysis is conducted for the standard isentropic pressure law, thereby eliminating the need for the cold pressure term that played a crucial role in the previous approach. To the best of our knowledge, this is the first rigorous singular limit result for the compressible Navier-Stokes equations with degenerate viscosity that requires no additional regularization of the system.

Anelastic approximation for the degenerate compressible Navier--Stokes equations revisited

TL;DR

The paper addresses the rigorous singular limit from degenerate compressible Navier–Stokes equations with density-dependent viscosity to a generalized anelastic system under joint low-Mach/low-Froude scaling. It advances the κ-entropy weak solutions framework, employing a two-velocity formulation and relative entropy to prove weak-to-strong convergence for well-prepared data in 𝕋^3 and ill-prepared data in ℝ^3, the latter aided by dispersive estimates for acoustic waves. A key novelty is deriving convergence without auxiliary regularization terms such as drag, capillarity, or cold pressure, relying instead on a refined energetic- dispersive analysis and meticulous control of remainder terms. The results have potential implications for understanding stratified viscous flows with spatially varying density profiles in three dimensions, under minimal regularization assumptions.

Abstract

In this paper, we revisit the joint low-Mach and low-Frode number limit for the compressible Navier-Stokes equations with degenerate, density-dependent viscosity. Employing the relative entropy framework based on the concept of -entropy, we rigorously justify the convergence of weak solutions toward the generalized anelastic system in a three-dimensional periodic domain for well-prepared initial data. For general ill-prepared initial data, we establish a similar convergence result in the whole space, relying essentially on dispersive estimates for acoustic waves. Compared with the work of Fanelli and Zatorska [Commun. Math. Phys., 400 (2023), pp. 1463-1506], our analysis is conducted for the standard isentropic pressure law, thereby eliminating the need for the cold pressure term that played a crucial role in the previous approach. To the best of our knowledge, this is the first rigorous singular limit result for the compressible Navier-Stokes equations with degenerate viscosity that requires no additional regularization of the system.

Paper Structure

This paper contains 35 sections, 5 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. The generalized anelastic system
  3. Derivation of the anelastic approximation: previous results
  4. Main contributions of this work
  5. Formulation of the main results
  6. Weak solutions to the primitive system
  7. Strong solutions to the target system
  8. Statement of the main results
  9. Relative $\kappa$-entropy inequality
  10. Derivation for general test functions
  11. Reduction of the relative $\kappa$-entropy inequality
  12. Uniform bounds based on the relative -entropy inequality
  13. Coercivity of the relative -entropy functional
  14. Uniform estimates
  15. Singular limit for well-prepared initial data
  16. ...and 20 more sections

Key Result

Theorem 2.9

Let $\Omega=\mathbb{T}^3$ or $\hbox{R}^3$ and $m\geq 3$ an integer. Let $b\in C^m(\Omega)$ satisfy const-b-2 and $\mathbf{U}_0\in H^{m}(\Omega;\mathbb{R}^3)$ with ${\rm div} (b \mathbf{U}_0)=0$. Then, there exists $T_{\ast}>0$ such that ane-app admits a unique local strong solution $(\mathbf{U},\nab for any $0<T<T_{\ast}$, and for a positive constant $C = C(T,b,\mathbf{U}_0)$ depending only on the

Theorems & Definitions (14)

  • Definition 2.1: $\kappa$-entropy solution to one velocity system
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: $\kappa$-entropy weak solution of the two-velocity system
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Theorem 2.10: Well-prepared data
  • ...and 4 more