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 on the initial data we prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equations in 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 - A, 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. doi: 10.1080/03605302.2012.669443.

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).

[3]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields,, Invent. Math., 158 (2004), 227. doi: 10.1007/s00222-004-0367-2.

[4]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields,, in, 1927 (2008), 1. doi: 10.1007/978-3-540-75914-0_1.

[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. doi: 10.1137/S0036139995294111.

[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. doi: 10.2307/1971509.

[7]

L. Caffarelli, Boundary regularity of maps with convex potentials. II.,, Ann. of Math. (2), 144 (1996), 453. doi: 10.2307/2118564.

[8]

L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation,, Ann. of Math. (2), 131 (1990), 135. doi: 10.2307/1971510.

[9]

L. Caffarelli, Some regularity properties of solutions to Monge-Ampère equations,, Comm. Pure Appl. Math., 44 (1991), 965. doi: 10.1002/cpa.3160440809.

[10]

L. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. doi: 10.1090/S0894-0347-1992-1124980-8.

[11]

M. Cullen, "A Mathematical Theory of Large-scale Atmosphere/Ocean Flow,", Imperial College Press, (2006).

[12]

M. Cullen and M. Feldman, Lagrangian solutions of semigeostrophic equations in physical space,, SIAM J. Math. Anal., 37 (2006), 1371. doi: 10.1137/040615444.

[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. doi: 10.1007/s002050000124.

[14]

M. Cullen and R. J. Purser, An extended Lagrangian theory of semi-geostrophic frontogenesis,, J. Atmos. Sci., 41 (1984), 1477. doi: 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2.

[15]

G. De Philippis and A. Figalli, $W^{2,1}$ regularity for solutions of the Monge-Ampère equation,, Invent. Math., 192 (2013), 55. doi: 10.1007/s00222-012-0405-4.

[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.

[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. doi: 10.1007/s00208-012-0895-9.

[18]

R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835.

[19]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).

[20]

G. Loeper, On the regularity of the polar factorization for time dependent maps,, Calc. Var. Partial Differential Equations, 22 (2005), 343. doi: 10.1007/s00526-004-0280-y.

[21]

G. Loeper, A fully nonlinear version of the incompressible Euler equations: The semi-geostrophic system,, SIAM J. Math. Anal., 38 (2006), 795. doi: 10.1137/050629070.

[22]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps,, Duke Math. J., 80 (1995), 309. doi: 10.1215/S0012-7094-95-08013-2.

[23]

T. Schmidt, $W^{2,1+\e}$ estimates for the Monge-Ampère equation,, Adv. Math., 240 (2013), 672. doi: 10.1016/j.aim.2012.07.034.

[24]

G. J. Shutts and M. Cullen, Parcel stability and its relation to semi-geostrophic theory,, J. Atmos. Sci., 44 (1987), 1318.

[25]

J. Urbas, On the second boundary value problem for equations of Monge-Ampère type,, J. Reine Angew. Math., 487 (1997), 115. doi: 10.1515/crll.1997.487.115.

[26]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9.

show all references

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. doi: 10.1080/03605302.2012.669443.

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).

[3]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields,, Invent. Math., 158 (2004), 227. doi: 10.1007/s00222-004-0367-2.

[4]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields,, in, 1927 (2008), 1. doi: 10.1007/978-3-540-75914-0_1.

[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. doi: 10.1137/S0036139995294111.

[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. doi: 10.2307/1971509.

[7]

L. Caffarelli, Boundary regularity of maps with convex potentials. II.,, Ann. of Math. (2), 144 (1996), 453. doi: 10.2307/2118564.

[8]

L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation,, Ann. of Math. (2), 131 (1990), 135. doi: 10.2307/1971510.

[9]

L. Caffarelli, Some regularity properties of solutions to Monge-Ampère equations,, Comm. Pure Appl. Math., 44 (1991), 965. doi: 10.1002/cpa.3160440809.

[10]

L. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. doi: 10.1090/S0894-0347-1992-1124980-8.

[11]

M. Cullen, "A Mathematical Theory of Large-scale Atmosphere/Ocean Flow,", Imperial College Press, (2006).

[12]

M. Cullen and M. Feldman, Lagrangian solutions of semigeostrophic equations in physical space,, SIAM J. Math. Anal., 37 (2006), 1371. doi: 10.1137/040615444.

[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. doi: 10.1007/s002050000124.

[14]

M. Cullen and R. J. Purser, An extended Lagrangian theory of semi-geostrophic frontogenesis,, J. Atmos. Sci., 41 (1984), 1477. doi: 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2.

[15]

G. De Philippis and A. Figalli, $W^{2,1}$ regularity for solutions of the Monge-Ampère equation,, Invent. Math., 192 (2013), 55. doi: 10.1007/s00222-012-0405-4.

[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.

[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. doi: 10.1007/s00208-012-0895-9.

[18]

R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835.

[19]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).

[20]

G. Loeper, On the regularity of the polar factorization for time dependent maps,, Calc. Var. Partial Differential Equations, 22 (2005), 343. doi: 10.1007/s00526-004-0280-y.

[21]

G. Loeper, A fully nonlinear version of the incompressible Euler equations: The semi-geostrophic system,, SIAM J. Math. Anal., 38 (2006), 795. doi: 10.1137/050629070.

[22]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps,, Duke Math. J., 80 (1995), 309. doi: 10.1215/S0012-7094-95-08013-2.

[23]

T. Schmidt, $W^{2,1+\e}$ estimates for the Monge-Ampère equation,, Adv. Math., 240 (2013), 672. doi: 10.1016/j.aim.2012.07.034.

[24]

G. J. Shutts and M. Cullen, Parcel stability and its relation to semi-geostrophic theory,, J. Atmos. Sci., 44 (1987), 1318.

[25]

J. Urbas, On the second boundary value problem for equations of Monge-Ampère type,, J. Reine Angew. Math., 487 (1997), 115. doi: 10.1515/crll.1997.487.115.

[26]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9.

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