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A simple justification of the singular limit for equatorial shallow-water dynamics. (English) Zbl 1156.76013

Summary: 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 [A. Dutrifoy and A. Majda, Commun. Partial Differ. Equations 32, No. 10, 1617–1642 (2007; Zbl 1166.35361)] 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 a 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.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 1166.35361
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References:

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