Singular limits of the Cauchy problem to the two-layer rotating shallow water equations

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. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper focuses on two singular limits of the Cauchy problem, transforming it into two-layer rotating shallow water equations as the Rossby number and Froude number approach zero. Unique from previous studies, the large operator in this model is not skew-symmetric. A key idea in this paper is obtaining uniform estimates using the system's specific structure rather than the antisymmetry of the large operator. Convergence of the equations with ill-prepared data to a two-layer incompressible Navier-Stokes system is then proven using Strichartz estimates constructed in this paper.