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A variational approach for the 2-dimensional semi-geostrophic shallow water equations. (English) Zbl 0985.76008

Summary: Existence of weak solutions to the three-dimensional semi-geostrophic equations with rigid boundaries was proved by J. D. Benamou and Y. Brenier [SIAM J. Appl. Math. 58, No. 5, 1450-1461 (1998; Zbl 0915.35024)], using Monge transport theory. This paper extends the results to a free surface boundary condition, which is more physically appropriate. This extension is at present for the two-dimensional shallow water case only. In addition, we establish stronger time regularity than was possible in the cited work.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M30 Variational methods applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 0915.35024
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