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Conservation laws and Lie symmetries a (2+1)-dimensional thin film equation

  • Elena Recio
  • Tamara M. Garrido
  • Rafael de la Rosa
  • María S. Bruzón
Original Paper
  • 21 Downloads

Abstract

This paper considers a generalized thin film equation in two spatial dimensions depending on three arbitrary functions. This equation describes the time evolution of a Newtonian liquid that is considerably thinner in one direction than in the other directions. We include a classification of point symmetries and the corresponding transformation groups. We derive all low-order local conservation laws of the equation in terms of the arbitrary functions. In addition, we discuss the physical meaning of the conserved quantities and provide a useful conservation identity.

Keywords

Thin film equation Lie symmetries Conservation laws Conservation identity 

Notes

Acknowledgements

The authors gratefully acknowledge Dr. Stephen Anco from Brock University for his expert guidance and help during his visit to Universidad de Cádiz. The authors express their sincerest gratitude to the Plan Propio de Investigación de la Universidad de Cádiz.

References

  1. 1.
    L.M. Alonso, On the Noether map. Lett. Math. Phys. 3, 419–424 (1979)CrossRefGoogle Scholar
  2. 2.
    S.C. Anco, G.W. Bluman, Direct constrution of conservation laws from field equations. Phys. Rev. Lett. 78, 2869–2942 (1997)CrossRefGoogle Scholar
  3. 3.
    S.C. Anco, G.W. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566 (2002)Google Scholar
  4. 4.
    S.C. Anco, G.W. Bluman, Direct construction method for conservation laws of partial differential equations. Part II: general treatment. Eur. J. Appl. Math. 13, 567–585 (2002)Google Scholar
  5. 5.
    S.C. Anco, Symmetry properties of conservation laws. Int. J. Mod. Phys. B 30, 1640003 (2016)CrossRefGoogle Scholar
  6. 6.
    S.C. Anco, Generalization of Noether’s Theorem in modern form to non-variational partial differential equations, in Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Field Institute Communications, vol. 79 (Springer, New York, 2017), pp. 119–182CrossRefGoogle Scholar
  7. 7.
    A.L. Bertozzi, The mathematics of moving contact lines in thin liquid films. Not. Am. Math. Soc. 45, 689–697 (1998)Google Scholar
  8. 8.
    A.L. Bertozzi, M. Pugh, Long-wave instabilities and saturation in thin film equations. Commun. Pure Appl. Math. 51, 625–661 (1998)CrossRefGoogle Scholar
  9. 9.
    G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematics Series, 154 (Springer, New York, 2002)Google Scholar
  10. 10.
    G.W. Bluman, A. Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematics Series, 168 (Springer, New York, 2010)CrossRefGoogle Scholar
  11. 11.
    J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  12. 12.
    R. Cherniha, L. Myroniuk, Lie symmetries and exact solutions of a class of thin film equations. J. Phys. Math. 2, 1–19 (2010)CrossRefGoogle Scholar
  13. 13.
    V.A. Galaktionov, S.R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman & Hall/CRC, Boca Raton, 2017)Google Scholar
  14. 14.
    M.L Gandarias, M.S. Bruzón, Symmetry analysis for a thin film equation, Proceedings in Applied Mathematics and Mechanics, PAMM. Wiley (2007), pp. 267–274Google Scholar
  15. 15.
    P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, 2nd edn. (Springer, Berlin, 1993)CrossRefGoogle Scholar
  16. 16.
    E. Recio, S. Anco, M.S. Bruzón, Conservation laws and potential system for a generalized thin film equation. AIP Conf. Proc. 1863, 280002-1–280002-4 (2017)Google Scholar
  17. 17.
    R. Tracinà, M.S. Bruzón, M.L. Gandarias, On the nonlinear self-adjointness of a class of fourth-order. Appl. Math. Comput. 275, 299–304 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de CádizCádizSpain

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