Examples of dispersive effects in non-viscous rotating fluids

Abstract

Strichartz estimates for rotating fluids have already been used to show that the velocity fields converge, as the Rossby number goes to zero, to a solution of a nearly two-dimensional Navier-Stokes system. Using a similar method, it is possible to get results of convergence also in the non-viscous case-to solutions of a nearly two-dimensional Euler system. The initial data do not need to be well prepared, and the limit can be as singular as a vortex patch or a Yudovich solution. Résumé Des estimations de Strichartz pour les fluides tournants ont déjàété utilisées pour montrer que les champs de vitesses convergent, lorsque le nombre de Rossby tend vers zéro, vers une solution d'un système de Navier-Stokes quasi bidimensionnel. En utilisant une méthode analogue, il est possible d'obtenir des résultats de convergence aussi dans le cas non visqueux-vers des solutions d'un système d'Euler quasi bidimensionnel. Il n'est pas nécessaire que les données initiales soient bien préparées, et la limite peut etre aussi singulière qu'une poche de tourbillon ou une solution de Yudovich.

The incompressible limit of solutions of the two‐dimensional compressible Euler system with degenerating initial data

“…Similar results have also been obtained about rapidly rotating fluids [11] and about solutions of the Boussinesq system and their quasi-geostrophic limit [10].…”

“…Similar results to those presented here have also been obtained about rapidly rotating fluids [9] and about solutions of the Boussinesq system and their quasigeostrophic limit [12]. These two examples are more complicated, because of the necessity of proving new dispersive estimates-instead of simply using the well-known estimates on solutions of the wave equation [15], as we do below.…”

The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data

“…Similar results have also been obtained about rapidly rotating fluids [9] and about solutions of the Boussinesq system and their quasigeostrophic limit [8]. These two examples are more complicated, because special dispersive estimates have to be proved, while the classical Strichartz estimates on solutions of the wave equation [15] are here perfectly suitable.…”

“…In the next section, we reformulate the system one last time, mainly for the convenience of notations, but also to stress its similarity with the other systems for which analogous results have been obtained [8,9].…”

Dispersive Effect of the Coriolis Force and the Local Well-Posedness for the Navier-Stokes Equations in the Rotational Framework

“…We remark that similar dispersive estimates were obtained by Dutrifoy [8] in the context of non-viscous rotating fluids. Our estimates in Lemma 2.3 gives an improvement of [8] …”