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On the Semi-geostrophic System in Physical Space with General Initial Data

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Abstract

In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic 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 said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space; as consequences we get conservation of Hamiltonian energy and some weak time-regularity of solutions.

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References

  1. Ambrosio L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math., 158, 227–260 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. 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. Part. Differ. Equ. 37(12), 2209–2227 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ambrosio L., Colombo M., De Philippis G., Figalli A.: A global existence result for the semigeostrophic equations in three dimensional convex domains. Discret. Cont. Dyn. Syst. 34(4), 1251–1268 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosio L., Gangbo W.: Hamiltonian ODE in the Wasserstein spaces of probability measures. Commun. Pure Appl. Math. 61, 18–53 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and the Wasserstein spaces of probability measures. Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cullen, M.: Private communication.

  9. Cullen M., Feldman M.: Lagrangian solutions of Semi-Geostrophic equations in physical space. SIAM J. Math. Anal. 37(5), 1371–1395 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cullen M., Gangbo W.: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Rat. Mech. Anal. 156, 241–273 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cullen M., Gangbo W., Pisante G.: The Semigeostrophic Equations discretized in reference and dual variables. Arch. Ration. Mech. Anal. 185, 341–363 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cullen M., Maroofi H.: The fully compressible semi-geostrophic system from meteorology. Arch. Ration. Mech. Anal. 167(4), 309–336 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. De Philippis, G., Figalli, A.: W 2,1 regularity for solutions of the Monge-Ampère equation. Invent. Math. (2012) doi:10.1007/s00222-012-0405-4

  14. O Faria J.C., Lopes Filho M.C., Nussenzveig Lopes H.J.: Weak stability of Lagrangian solutions to the Semi-Geostrophic equations. Nonlinearity 22, 2521–2539 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Feldman M., Tudorascu A.: On Lagrangian solutions for the Semi-Geostrophic system with singular initial data. SIAM J. Math. Anal., 45(3), 1616–1640 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gangbo W., Nguyen T., Tudorascu A.: Euler-Poisson systems as action-minimizing paths in the Wasserstein space. Arch. Ration. Mech. Anal. 192(3), 419–452 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hoskins B.: The Geostrophic Momentum Approximation and the Semi-Geostrophic Equations. J. Atmos. Sci. 32(2), 233–242 (1975)

    Article  ADS  Google Scholar 

  18. Loeper G.: A fully nonlinear version of the incompressible Euler equations: the Semigeostrophic system. SIAM J. Math. Anal. 38(3), 795–823 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lopes Filho M.C., Nussenzveig Lopes H.J.: Existence of a weak solution for the semigeostrophic equation with integrable initial data. Proc. Roy. Soc. Edinb. Sect. A 132(2), 329–339 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tudorascu A.: On the velocities of flows consisting of cyclically monotone maps. Indiana Univ. Math. J., 59(3), 929–955 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics 58, American Mathematical Society, 2003

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Correspondence to Adrian Tudorascu.

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Communicated by D. Kinderlehrer

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Feldman, M., Tudorascu, A. On the Semi-geostrophic System in Physical Space with General Initial Data. Arch Rational Mech Anal 218, 527–551 (2015). https://doi.org/10.1007/s00205-015-0865-y

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  • DOI: https://doi.org/10.1007/s00205-015-0865-y

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