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.
Abstract. At the equator, the Coriolis force from rotation vanishes identically so that multiple time scale dynamics for the equatorial shallow water equation naturally leads to singular limits of symmetric hyperbolic systems with fast variable coefficients. The classical strategy of using energy estimates for higher spatial derivatives has a fundamental difficulty since formally the commutator terms explode in the limit. Here this fundamental difficulty is circumvented by exploiting the special structure of the equatorial shallow water equations in suitable new variables involving the raising and lowering operators for the quantum harmonic oscillator, and obtaining uniform higher derivative estimates in a new function space based on the Hermite operator. The result is a completely new theorem characterizing the singular limit of the equatorial shallow water equations in the long wave regime, even with general unbalanced initial data, as a solution of the equatorial long wave equation. The results presented below point the way for rigorous PDE analysis of both the equatorial shallow water equations and the equatorial primitive equations in other physically relevant singular limit regimes.
The equatorial shallow-water equations at low Froude number form a symmetric hyperbolic system with large variable-coefficient terms. Although such systems are not covered by the classical Klainerman-Majda theory of singular limits, the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long-wave equations as the height and Froude number tend to 0. Their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory.
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