AI 中文总结
本文针对弱分层等温流动，在小马赫数和小纵横比极限下，通过分析能量与声波演化建立可压缩Navier-Stokes系统解的一致存在唯一性，证明其解收敛到不可压缩原始方程的解，严格论证了静水压不可压缩近似。
AI 中文摘要
我们考虑等温可压缩Navier-Stokes系统在小马赫数和小垂直-水平纵横比下的极限情况，同时考虑分层较弱的尺度。由于问题的各向异性，动力学表现出三波分离现象，包括慢波、快水平声波和更快的垂直声波。不幸的是，这三种波并非相互正交，对应的投影由小参数参数化，这极大地增加了非线性分析的复杂性。在不对初始波的大小施加任何限制的情况下，我们通过仔细分析能量和声波的演化，针对任意固定的小参数，建立了可压缩Navier-Stokes系统解的一致存在性和唯一性。此外，我们证明当小参数趋于零时，该系统的解收敛到描述大气和海洋流动的不可压缩原始方程的解。
英文摘要
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.
Comments30 pages
