Cullen, Mike; Gangbo, Wilfrid A variational approach for the 2-dimensional semi-geostrophic shallow water equations. (English) Zbl 0985.76008 Arch. Ration. Mech. Anal. 156, No. 3, 241-273 (2001). 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. Cited in 3 ReviewsCited in 20 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M30 Variational methods applied to problems in fluid mechanics 86A05 Hydrology, hydrography, oceanography Keywords:existence of weak solutions; semi-geostrophic equations; Monge transport theory; free surface boundary condition; time regularity Citations:Zbl 0915.35024 PDFBibTeX XMLCite \textit{M. Cullen} and \textit{W. Gangbo}, Arch. Ration. Mech. Anal. 156, No. 3, 241--273 (2001; Zbl 0985.76008) Full Text: DOI