Rigorous justification of the hydrostatic-incompressible approximation for weakly stratified isothermal flow
arXiv:2607.27477
The paper rigorously derives the hydrostatic-incompressible limit of the isothermal compressible Navier‑Stokes equations under small Mach number, small aspect ratio, and weak stratification, proving uniform existence and convergence to the primitive equations.
Abstract
We consider the limit of small Mach number and small vertical-to-horizontal aspect ratio for the isothermal compressible Navier-Stokes system. In addition, we consider the scale in which the stratification is weak. Owing to the anisotropic nature of the problem, the dynamics exhibit a three-wave separation phenomenon, consistent of a slow wave, a fast horizontal acoustic wave, and an even faster vertical acoustic wave. These three waves, unfortunately, are not mutually orthogonal, and the corresponding projections are parametrized by the small parameter, which significantly complicates the nonlinear analysis. Without any restriction on the size of the initial waves, we establish the uniform existence and uniqueness of solutions to the compressible Navier-Stokes system for any fixed small parameter, by carefully analyzing the evolutions of both the energy and the acoustic waves. Moreover, we prove that, as the small parameter tends to zero, the solutions converge to that of the incompressible primitive equations governing atmospheric and oceanic flows.
30 pages