Geodesic
Lagrangian solutions for the semi-geostrophic shallow water system in physical space with general initial data
Algebra i analiz, Tome 27 (2015) no. 3, pp. 272-300.

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In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs the said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space and ultimately enables us to achieve the desired results.
Keywords: Semi-Geostrophic Shallow Water system, flows of maps, optimal mass transport, Wasserstein metric, optimal maps, absolutely continuous curves. 
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M. Feldman; A. Tudorascu. Lagrangian solutions for the semi-geostrophic shallow water system in physical space with general initial data. Algebra i analiz, Tome 27 (2015) no. 3, pp. 272-300. http://geodesic.mathdoc.fr/item/AA_2015_27_3_a12/

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