快速旋转可压缩流体动力学中的各向异性和分层效应
Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids
摘要：
本文的主要目标是开发鲁棒的方法来处理地球物理流建模中出现的两个普遍存在的特征：（i）粘性应力张量的各向异性，（ii）分层效应。我们着重研究了具有科里奥利力和引力的正压Navier-斯托克斯方程。两个结果是本文的主要贡献。首先，我们建立了一个局部适定性结果的有限能量的解决方案，通过极大正则性的方法。这种方法使我们能够规避使用的有效粘性通量，这起着关键作用的弱解理论的Lions-Feireisl和Hoff，但似乎仅限于各向同性粘性应力张量。此外，我们的方法是坚固的，足以考虑到非恒定的参考密度状态，这是至关重要的分层效应时，处理。其次，我们研究了当Rossby数、Mach数和Froude数为相同数量级时，前一模型解的结构。我们证明了一个误差估计的相对熵之间的实际解决方案和它们的近似的大规模准地转流补充Ekman边界层。我们的分析适用于一大类正压压力定律。
英文摘要：
The primary goal of this paper is to develop robust methods to handle two ubiquitous features appearing in the modeling of geophysical flows: (i) the anisotropy of the viscous stress tensor, (ii) stratification effects. We focus on the barotropic Navier-Stokes equations with Coriolis and gravitational forces. Two results are the main contributions of the paper. Firstly, we establish a local well-posedness result for finite-energy solutions, via a maximal regularity approach. This method allows us to circumvent the use of the effective viscous flux, which plays a key role in the weak solutions theories of Lions-Feireisl and Hoff, but seems to be restricted to isotropic viscous stress tensors. Moreover, our approach is sturdy enough to take into account non constant reference density states; this is crucial when dealing with stratification effects. Secondly, we study the structure of the solutions to the previous model in the regime when the Rossby, Mach and Froude numbers are of the same order of magnitude. We prove an error estimate on the relative entropy between actual solutions and their approximation by a large-scale quasi-geostrophic flow supplemented with Ekman boundary layers. Our analysis holds for a large class of barotropic pressure laws.
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