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In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions. [ABSTRACT FROM AUTHOR]

A dispersive decay estimate is proven for the linear propagator of the three-scale rotating stratified Boussinesq equations in the stratification-dominant regime. Because the phase function of that propagator is both singular and degenerate, obtaining the estimate requires novel techniques involving cutting phase-space shells into several pieces and using different methods in each region. Using the decay estimate, we prove the long-time existence of solutions to the initial-value problem for the nonlinear equations and obtain a rate of convergence to the limit system and intermediate asymptotics. [ABSTRACT FROM AUTHOR]

Singular limits of the rotating stratified Boussinesq equations are considered as the Froude number and Rossby number tend to zero at different rates. The limiting dynamics are validated for the rotation-dominant and the stratification-dominant cases, respectively. Furthermore, the convergence rate estimates are obtained. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]