We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system.

中文翻译：

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浅水界限的三层流量

我们制定并讨论了分层流的浅水极限动力学，该分层流具有三层不同密度的不混溶流体，其上下两面由水平壁界定。我们获得了由四个方程组成的结果系统，在非Bousinesq情况下，它们可能不是局部的。我们提供了一种系统的方法来传递到Boussinesq极限，然后更仔细地研究这些方程，它们是混合类型的一阶PDE。我们表明，在对称情况下，解仍保留在不变的表面上，并使用简单的波来说明非对称情况并非如此。还提出了由两个方程组组成的简化模型，并将其与整个系统进行比较。