We are concerned with two kinds of singular limits of the Cauchy problem to the two-layer rotating shallow water equations as the Rossby number and the Froude number tend to zero. First we construct the uniform estimates for the strong solutions to the system under the condition that the Froude number is small enough. Different from the previously studied cases, the large operator of this model is not skew-symmetric. One of the key new ideas in this paper is to obtain the uniform estimates using the special structure of the system rather than the antisymmetry of the large operator. After that the convergence of the equations with ill-prepared data to a two-layer incompressible Navier-Stokes system is proved with the help of Strichartz estimates constructed in this paper.

当Rossby数和Froude数趋于零时，我们对两层旋转浅水方程组的柯西问题有两种奇异极限。首先，在弗洛德数足够小的条件下，为系统的强解构造统一估计。与先前研究的情况不同，此模型的大算子不是倾斜对称的。本文的主要新思想之一是使用系统的特殊结构而不是大型算子的反对称性来获得统一估计。之后，借助本文构建的Strichartz估计，证明了具有不良数据的方程组到两层不可压缩Navier-Stokes系统的收敛性。