## 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 coe±cients. 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 di±culty 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.

Original language | English (US) |
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Pages (from-to) | 375-397 |

Number of pages | 23 |

Journal | Communications in Mathematical Sciences |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - 2006 |

## Keywords

- Equatorial long wave equation
- Fast averaging
- Shallow water equations
- Singular limit

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics