Regularity and integrability of rotating shallow-water equations

Abstract

We consider classical shallow-water equations for a rapidly rotating fluid layer. The Poincaré/Kelvin linear propagator describes fast oscillating waves for the linearized system. We show that solutions of the full nonlinear shallow-water equations can be decomposed as U(t,x 1,x 2) + Ũ(t,x 1,x 2) + W’(t,x 1,x 2) + r, where Ũ is a solution of the quasigeostrophic (QG) equation. Here r is a remainder, which is uniformly estimated from above by a majorant of order 1/f 0. The vector field W’(t,x 1,x 2) describes the rapidly oscillating ageostrophic (AG) component. This component is exactly solved in terms of Poincaré/Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations. The mathematically rigorous control of the error r, based on estimates of small divisors, is used to prove the existence, on a long time interval T *, of regular solutions to classical shallow-water equations with general initial data ( T * → +∞, as 1/f 0 → 0).