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Remarks on the equatorial shallow water system. (English) Zbl 1187.35181

Summary: This article recalls the results given by A. Dutrifoy, A. Majda and S. Schochet in [Commun. Pure Appl. Math. 62, No. 3, 322–333 (2009; Zbl 1156.76013)] in which they prove an uniform estimate of the system as well as the convergence to a global solution of the long wave equations as the Froud number tends to zero. We prove the convergence with weaker hypothesis and show that the life span of the solutions tends to infinity as the Froud number tends to zero.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
35B35 Stability in context of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs

Citations:

Zbl 1156.76013
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References:

[1] Dutrifoy (A), Majda (A.) and Schochet (S.).— A simple justification of the singular limit for equatorial shallow water dynamics, Communications in Pure and Applied Mathematics, 62, p. 305-443 (2008). · Zbl 1156.76013
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[4] Chemin (J.-Y.).— A propos d’un problème de pénalisation de type antisymétrique, Journal de Mathématiques Pures et Appliquées, 76, p. 739-755 (1997). · Zbl 0896.35103
[5] Klainerman (S.) and Majda (A.).— Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Communications in Pure and Applied Mathematics, 34, p. 481-524 (1981). · Zbl 0476.76068
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