Advertisement

Theoretical and Mathematical Physics

, Volume 188, Issue 3, pp 1296–1304 | Cite as

Dispersive deformations of the Hamiltonian structure of Euler’s equations

  • M. CasatiEmail author
Article

Abstract

Euler’s equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie–Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie–Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler’s equation: we show that they are trivial up to the second order.

Keywords

Euler’s equations Poisson bracket Poisson vertex algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Arnold, Ann. Inst. Fourier (Grenoble), 16, 319–361 (1996).CrossRefGoogle Scholar
  2. 2.
    P. J. Olver, J. Math. Anal. Appl., 89, 233–250 (1982).MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. P. Novikov, Russian Math. Surv., 37, 1–56 (1982).ADSCrossRefGoogle Scholar
  4. 4.
    D. D. Holm and J. E. Marsden, “Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,” in: The Breadth of Symplectic and Poisson Geometry (Progr. Math., Vol. 232), Birkhäuser, Boston, Mass. (2005), pp. 203–235.Google Scholar
  5. 5.
    M. Casati, Commun. Math. Phys., 335, 851–894 (2015).ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Barakat, A. De Sole, and V. G. Kac, Japan J. Math., 4, 141–252 (2009).CrossRefGoogle Scholar
  7. 7.
    H. Lamb, Hydrodynamics, Cambrige Univ. Press Cambridge (1932).zbMATHGoogle Scholar
  8. 8.
    B. A. Dubrovin and Y. Zhang, “Normal forms of hierarchies of integrable PDEs, Frobenius manifolds, and Gromov–Witten invariants,” arXiv:math/0108160v1 (2001).Google Scholar
  9. 9.
    S.-Q. Liu and Y. Zhang, Adv. Math., 227, 73–130 (2011).MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Getzler, Duke Math. J., 111, 535–560 (2002).MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Degiovanni, F. Magri, and V. Sciacca, Commun. Math. Phys., 253, 1–24 (2005).ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    M. Casati, “Multidimensional Poisson vertex algebras and the Poisson cohomology of Hamiltonian structures of hydrodynamic type,” Doctoral dissertation, Scuola Internazionale Superiore di Studi Avanzati di Trieste, Trieste (2015).zbMATHGoogle Scholar
  13. 13.
    G. Carlet, M. Casati, and S. Shadrin, “Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets,” arXiv:1512.05744v1 [math.DG] (2015).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly

Personalised recommendations