The anelastic approximation and convergence rate for the compressible Navier-Stokes equations with ill-prepared data in bounded domains
Abstract
We justify the anelastic approximation and establish a convergence rate for local strong solutions to the isentropic compressible Navier-Stokes equations in a three-dimensional bounded domain, with ill-prepared initial data and vorticity-slip boundary conditions. When the Mach and Froude numbers vanish at the same rate ε, the system converges to an anelastic system with nonconstant hydrostatic density. Compared with the incompressible limit, the balance between singular pressure and gravity generates a variable-coefficient acoustic operator and additional commutators. To overcome these difficulties, we introduce the pressure perturbation adapted to the hydrostatic balance and establish weighted acoustic and vorticity estimates, yielding uniform-in-ε estimates. We then use the weighted Helmholtz decomposition and time-averaging technique to prove the strong convergence of solenoidal component of the momentum and obtain an error bound controlled by ε and the initial error in the projected momentum. Our analysis reveals the weighted cancellation mechanism that applies to singular limits around a class of nonconstant equilibria.
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BibTeX
@misc{ou2026anelastic,
title = {{The anelastic approximation and convergence rate for the compressible Navier-Stokes equations with ill-prepared data in bounded domains}},
author = {Yaobin Ou and Ping Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02831},
eprint = {2611.02831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02831},
}
RIS
TY - UNPB
TI - The anelastic approximation and convergence rate for the compressible Navier-Stokes equations with ill-prepared data in bounded domains
AU - Yaobin Ou
AU - Ping Wang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02831
UR - https://arxiv.org/abs/2611.02831
AB - We justify the anelastic approximation and establish a convergence rate for local strong solutions to the isentropic compressible Navier-Stokes equations in a three-dimensional bounded domain, with ill-prepared initial data and vorticity-slip boundary conditions. When the Mach and Froude numbers vanish at the same rate ε, the system converges to an anelastic system with nonconstant hydrostatic density. Compared with the incompressible limit, the balance between singular pressure and gravity generates a variable-coefficient acoustic operator and additional commutators. To overcome these difficulties, we introduce the pressure perturbation adapted to the hydrostatic balance and establish weighted acoustic and vorticity estimates, yielding uniform-in-ε estimates. We then use the weighted Helmholtz decomposition and time-averaging technique to prove the strong convergence of solenoidal component of the momentum and obtain an error bound controlled by ε and the initial error in the projected momentum. Our analysis reveals the weighted cancellation mechanism that applies to singular limits around a class of nonconstant equilibria.
ER -