Dispersive effects of weakly compressible and fast rotating inviscid fluids

Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data

“…In the same spirit, [25,43,44] are devoted to the rotating fluid system, [24] focusses on the non-viscous version of System (P E ε ), [48] studies the stratified system, and for the periodic case we refer to [46,47]. Let us also mention [6] using similar dispersive methods for the 2Dquasigeostrophic system.…”

Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime

“…An important step as long as concerns singular perturbation problems is to deduce formally a limit system to whom (PBS ε ) converges. Several works on geophysical fluids such as [17,8,22] or [33] suggest that the solutions of (PBS ε ) converge (in a sense which we do not specify at the moment) to an element belonging to the nonoscillatory space CE 0 .…”

“…where P ±,ε are the projections onto the eigenspaces generated by E ε ± defined in (33), and the convolution kernels K ±,r,R are defined in (59). Considering the dispersive estimate (60) given in Lemma 5.5 we can apply what is known as T T argument (see [5,Chapter 8]) in the same way as it is done in [17], [15], [14], [8], [10] to deduce that…”

“…where the projectors P i,ε defined in (33), are the projections onto the eigendirections E ε i , i = 0, ± defined in (31) and (32). The initial data is localized onto the very high and low frequencies along the eigendirections of the eigenvectors E ± defined in (32).…”

Fast rotation limit for the 2-D non-homogeneous incompressible Euler equations

“…Similar problems to the one presented in (1.1) have been studied by several authors, who in different ways inspected the well-posedness issues and the asymptotic analysis of models for geophysical flows. For example, in the context of compressible fluids, we refer to [7], [29], [32] for the first works on 2-D viscous shallow water models (see also [17], [19], [20] for the inviscid case), to [22], [23], [26], [27] for the barotropic Navier-Stokes system and to [37] for weakly compressible and inviscid fluids (see also [25] for other singular limits in thermodynamics of viscous fluids). In the compressible case, the fact that the pressure is a given function of the density implies a double advantage in the analysis: on the one hand, one can recover good uniform bounds for the oscillations (from the reference state) of the density; on the other hand, at the limit, one disposes of a stream-function relation between the densities and the velocities.…”