# American Institute of Mathematical Sciences

April  2014, 34(4): 1251-1268. doi: 10.3934/dcds.2014.34.1251

## A global existence result for the semigeostrophic equations in three dimensional convex domains

 1 Scuola Normale Superiore, Piazza Cavalieri 7, 56123 Pisa 2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy 3 Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany 4 Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin TX 78712

Received  July 2012 Revised  December 2012 Published  October 2013

Exploiting recent regularity estimates for the Monge-Ampère equation, under some suitable assumptions onthe initial datawe prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equationsin 3-dimensional convex domains.
Citation: Luigi Ambrosio, Maria Colombo, Guido De Philippis, Alessio Figalli. A global existence result for the semigeostrophic equations in three dimensional convex domains. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1251-1268. doi: 10.3934/dcds.2014.34.1251
##### References:
 [1] L. Ambrosio, M. Colombo, G. De Philippis and A. Figalli, Existence of Eulerian solutions to the semigeostrophic equations in physical space: The 2-dimensional periodic case, Comm. Partial Differential Equations, 37 (2012), 2209-2227. doi: 10.1080/03605302.2012.669443.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar [3] L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.  Google Scholar [4] L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in "Calculus of Variations and Non-Linear Partial Differential Equations'' (eds. B. Dacorogna and P. Marcellini), Lecture Notes in Mathematics, 1927, Springer, Berlin, (2008), 1-41. doi: 10.1007/978-3-540-75914-0_1.  Google Scholar [5] J.-D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equation formulated as a coupled Monge-Ampère/transport problem, SIAM J. Appl. Math., 58 (1998), 1450-1461. doi: 10.1137/S0036139995294111.  Google Scholar [6] L. Caffarelli, A localization property of viscosity solutions to the Monge-Amp\ere equation and their strict convexity, Ann. of Math. (2), 131 (1990), 129-134. doi: 10.2307/1971509.  Google Scholar [7] L. Caffarelli, Boundary regularity of maps with convex potentials. II., Ann. of Math. (2), 144 (1996), 453-496. doi: 10.2307/2118564.  Google Scholar [8] L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2), 131 (1990), 135-150. doi: 10.2307/1971510.  Google Scholar [9] L. Caffarelli, Some regularity properties of solutions to Monge-Ampère equations, Comm. Pure Appl. Math., 44 (1991), 965-969. doi: 10.1002/cpa.3160440809.  Google Scholar [10] L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. doi: 10.1090/S0894-0347-1992-1124980-8.  Google Scholar [11] M. Cullen, "A Mathematical Theory of Large-scale Atmosphere/Ocean Flow," Imperial College Press, 2006. Google Scholar [12] M. Cullen and M. Feldman, Lagrangian solutions of semigeostrophic equations in physical space, SIAM J. Math. Anal., 37 (2006), 1371-1395. doi: 10.1137/040615444.  Google Scholar [13] M. Cullen and W. Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Ration. Mech. Anal., 156 (2001), 241-273. doi: 10.1007/s002050000124.  Google Scholar [14] M. Cullen and R. J. Purser, An extended Lagrangian theory of semi-geostrophic frontogenesis, J. Atmos. Sci., 41 (1984), 1477-1497. doi: 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2.  Google Scholar [15] G. De Philippis and A. Figalli, $W^{2,1}$ regularity for solutions of the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69. doi: 10.1007/s00222-012-0405-4.  Google Scholar [16] G. De Philippis and A. Figalli, Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optimal transport maps, Anal. PDE, 6 (2013), 993-1000. Google Scholar [17] G. De Philippis, A. Figalli and O. Savin, A note on interior $W^{2,1+\e}$ estimates for the Monge-Ampère equation, Math. Ann., 357 (2013), 11-22. doi: 10.1007/s00208-012-0895-9.  Google Scholar [18] R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar [19] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar [20] G. Loeper, On the regularity of the polar factorization for time dependent maps, Calc. Var. Partial Differential Equations, 22 (2005), 343-374. doi: 10.1007/s00526-004-0280-y.  Google Scholar [21] G. Loeper, A fully nonlinear version of the incompressible Euler equations: The semi-geostrophic system, SIAM J. Math. Anal., 38 (2006), 795-823. doi: 10.1137/050629070.  Google Scholar [22] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.  Google Scholar [23] T. Schmidt, $W^{2,1+\e}$ estimates for the Monge-Ampère equation, Adv. Math., 240 (2013), 672-689. doi: 10.1016/j.aim.2012.07.034.  Google Scholar [24] G. J. Shutts and M. Cullen, Parcel stability and its relation to semi-geostrophic theory, J. Atmos. Sci., 44 (1987), 1318-1330. Google Scholar [25] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997), 115-124. doi: 10.1515/crll.1997.487.115.  Google Scholar [26] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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##### References:
 [1] L. Ambrosio, M. Colombo, G. De Philippis and A. Figalli, Existence of Eulerian solutions to the semigeostrophic equations in physical space: The 2-dimensional periodic case, Comm. Partial Differential Equations, 37 (2012), 2209-2227. doi: 10.1080/03605302.2012.669443.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar [3] L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.  Google Scholar [4] L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in "Calculus of Variations and Non-Linear Partial Differential Equations'' (eds. B. Dacorogna and P. Marcellini), Lecture Notes in Mathematics, 1927, Springer, Berlin, (2008), 1-41. doi: 10.1007/978-3-540-75914-0_1.  Google Scholar [5] J.-D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equation formulated as a coupled Monge-Ampère/transport problem, SIAM J. Appl. Math., 58 (1998), 1450-1461. doi: 10.1137/S0036139995294111.  Google Scholar [6] L. Caffarelli, A localization property of viscosity solutions to the Monge-Amp\ere equation and their strict convexity, Ann. of Math. (2), 131 (1990), 129-134. doi: 10.2307/1971509.  Google Scholar [7] L. Caffarelli, Boundary regularity of maps with convex potentials. II., Ann. of Math. (2), 144 (1996), 453-496. doi: 10.2307/2118564.  Google Scholar [8] L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2), 131 (1990), 135-150. doi: 10.2307/1971510.  Google Scholar [9] L. Caffarelli, Some regularity properties of solutions to Monge-Ampère equations, Comm. Pure Appl. Math., 44 (1991), 965-969. doi: 10.1002/cpa.3160440809.  Google Scholar [10] L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. doi: 10.1090/S0894-0347-1992-1124980-8.  Google Scholar [11] M. Cullen, "A Mathematical Theory of Large-scale Atmosphere/Ocean Flow," Imperial College Press, 2006. Google Scholar [12] M. Cullen and M. Feldman, Lagrangian solutions of semigeostrophic equations in physical space, SIAM J. Math. Anal., 37 (2006), 1371-1395. doi: 10.1137/040615444.  Google Scholar [13] M. Cullen and W. Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Ration. Mech. Anal., 156 (2001), 241-273. doi: 10.1007/s002050000124.  Google Scholar [14] M. Cullen and R. J. Purser, An extended Lagrangian theory of semi-geostrophic frontogenesis, J. Atmos. Sci., 41 (1984), 1477-1497. doi: 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2.  Google Scholar [15] G. De Philippis and A. Figalli, $W^{2,1}$ regularity for solutions of the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69. doi: 10.1007/s00222-012-0405-4.  Google Scholar [16] G. De Philippis and A. Figalli, Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optimal transport maps, Anal. PDE, 6 (2013), 993-1000. Google Scholar [17] G. De Philippis, A. Figalli and O. Savin, A note on interior $W^{2,1+\e}$ estimates for the Monge-Ampère equation, Math. Ann., 357 (2013), 11-22. doi: 10.1007/s00208-012-0895-9.  Google Scholar [18] R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar [19] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar [20] G. Loeper, On the regularity of the polar factorization for time dependent maps, Calc. Var. Partial Differential Equations, 22 (2005), 343-374. doi: 10.1007/s00526-004-0280-y.  Google Scholar [21] G. Loeper, A fully nonlinear version of the incompressible Euler equations: The semi-geostrophic system, SIAM J. Math. Anal., 38 (2006), 795-823. doi: 10.1137/050629070.  Google Scholar [22] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.  Google Scholar [23] T. Schmidt, $W^{2,1+\e}$ estimates for the Monge-Ampère equation, Adv. Math., 240 (2013), 672-689. doi: 10.1016/j.aim.2012.07.034.  Google Scholar [24] G. J. Shutts and M. Cullen, Parcel stability and its relation to semi-geostrophic theory, J. Atmos. Sci., 44 (1987), 1318-1330. Google Scholar [25] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997), 115-124. doi: 10.1515/crll.1997.487.115.  Google Scholar [26] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar
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