Abstract
We define Lagrangian solutions in physical space for 3-d incompressible semigeostrophic system with free upper boundary under various conditions for initial data, then prove their existence via the minimization with respect to a geostrophic functional, generalizing the results of Cullen and Feldman (J Math Anal 37(5): 1371–1395, 2006) and Feldman and Tudorascu (Arch Ration Mech Anal 218(1): 527–551 2015) to the situation of free upper boundary. As a byproduct of our proof, we obtain the existence of measure-valued dual space solutions when the initial measure \(\nu _0\in \mathcal {P}_2(\mathbb {R}^3)\) and is supported on \(\{-\frac{1}{\delta }\le y_3\le -\delta \}\).
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References
Ambrosio, L.: Transport Equation and Cauchy Problem for BV Vector Fields. Invent. Math 158, 227–260 (2004)
Ambrosio, L., Gangbo, W.: Hamitonian ODE in the Wasserstein spaces of probability measures. Commun. Pure Appl. Math. 61(1), 18–53 (2008)
Ambrosio, L., Gigli, N., Savare, G.: Gradient flows in metric spaces and the Wasserstein spaces of probability measures. Lectures in Mathematics. Birkhauser, ETH Zurich (2005)
Benamou, J.-D., Brenier, Y.: Weak existence for the Semi-Geostrophic equations formulated as a coupled Monge-Ampere/transport problem. SIAM J. Appl. Math 58(5), 1450–1461 (1998)
Cullen, M., Feldman, M.: Lagrangian Solutions of Semigeostrophic Equations in Physical Space. SIAM J. Math. Anal. 37(5), 1371–1395 (2006)
Cullen, M., Gangbo, W.: A Variational Approach for the 2-Dimensional Semi-Geostrphic Shallow Water Equations. Arch. Ration. Mech. Anal. 156(3), 241–273 (2001)
Cullen, M.J.P., Gilbert, D.K., Kuna, T., Pelloni, B.: Free Upper Boundary Value Problems for the Semi-Geostrophic Equations. arXiv:1409.8560 (preprint)
Mike-Cullen, H.M.: The Fully Compressible Semi-Geostrophic System from Meteorology. Arch. Ration. Mech. Anal. 167(4), 309–336 (2003)
Cullen, M., Sedjro, M.: Model of Forced Axisymmtric Flows. SIAM J. Math. Anal. 46(6), 3983–4013 (2014)
Ambrosio, L., Colombo, M., De Philippis, G., Figalli, A.: Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case. Commun. Partial Differ. Equations 37(12), 2209–2227 (2012)
Ambrosio, L., Colombo, M., De Philippis, G., Figalli, A.: A global existence result for the semigeostrophic equations in three dimensional convex domains. Discrete Contin. Dyn. Syst 34(4), 1251–1268 (2014)
Feldman, M., Tudorascu, A.: On the Semi-Geostrophic System in Physical Space with General Initial Data. Arch. Ration. Mech. Anal 218(1), 527–551 (2015)
Feldman, M., Tudorascu, A.: Lagrangian solutions for the Semi-geostrophic shallow water system in physical space with general initial data. St. Petersburg Mathematical Journal, vol 27, no. 3 (2015)
Acknowledgements
The work of the author was supported in part by the National Science Foundation under Grant DMS-1401490. The author would also like to thank Mike Cullen for suggesting me this problem, and his advisor Mikhail Feldman for helpful discussions and suggestions. Thanks also go to the anonymous referee whose suggestions helped clarify certain ambiguities in the paper.
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Communicated by L. Ambrosio.
