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

TL;DR

This paper rigorously justifies the convergence of solutions of degenerate compressible Navier-Stokes equations to the anelastic system in low-Mach and low-Froude regimes, using a novel entropy framework without extra regularization.

Contribution

It provides the first rigorous singular limit analysis for degenerate viscosity Navier-Stokes equations without additional regularization, extending previous work to standard pressure laws.

Findings

Convergence established for well-prepared initial data in 3D periodic domains.

Dispersive estimates used for ill-prepared initial data in the whole space.

Elimination of the cold pressure term simplifies the analysis.

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 $\kappa$-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.