Journal of Mathematical Fluid Mechanics

, Volume 16, Issue 2, pp 375–406 | Cite as

Topography Influence on the Lake Equations in Bounded Domains

  • Christophe Lacave
  • Toan T. Nguyen
  • Benoit Pausader


We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and L p perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and Métivier treating the lake equations with a fixed topography and by Gérard-Varet and Lacave treating the Euler equations in singular domains.

Mathematics Subject Classification (2010)

Primary 35Q35 Secondary 76D03 


Existence and uniqueness global weak solutions Hausdorff approximations singular domains rough bottoms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bresch D., Métivier G.: Global existence and uniqueness for the Lake equations with vanishing topography : elliptic estimates for degenerate equations. Nonlinearity 19(3), 591–610 (2006)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Camassa R., Holm D.D., Levermore C.D.: Long-time effects of bottom topography in shallow water (Los Alamos, NM, 1995). Nonlinear phenomena in ocean dynamics Physica D 98(2–4), 258–286 (1996)ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Galdi G.P.: An introduction to the mathematical theory of the Navier-Stokes equations Steady-state problems. Springer Monographs in Mathematics, 2nd edn. Springer, New York (2011)Google Scholar
  4. 4.
    Gérard-Varet D., Lacave C.: The two dimensional Euler equation on singular domains. Arch. Ration. Mech. Anal. 209(1), 131–170 (2013)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Greenspan H.P.: The theory of rotating fluids. Cambridge University Press, London (1968)MATHGoogle Scholar
  6. 6.
    Henrot A., Pierre M.: Variation et optimisation de formes. Une analyse géométrique (French) [Shape variation and optimization. A geometric analysis], Mathématiques Applications 48. Springer, Berlin (2005)Google Scholar
  7. 7.
    Iftimie D., Lopes Filho M.C., Nussenzveig Lopes H.J., Sueur F., Weak vorticity formulation for the incompressible 2D Euler equations in domains with boundary, arXiv:1305.0905 (preprint 2013)Google Scholar
  8. 8.
    Lacave C., Uniqueness for Two Dimensional Incompressible Ideal Flow on Singular Domains, arXiv:1109.1153 (preprint 2011)Google Scholar
  9. 9.
    Levermore C.D., Oliver M., Titi E.S.: Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J. 45(2), 479–510 (1996)MATHMathSciNetGoogle Scholar
  10. 10.
    Levermore C.D., Oliver M., Titi E.S.: Global well-posedness for the lake equation. Physica D 98(2–4), 492–509 (1996)Google Scholar
  11. 11.
    Sverák V.: On optimal shape design. J. Math. Pures Appl.(9) 72(6), 537–551 (1993)MATHMathSciNetGoogle Scholar
  12. 12.
    Wolibner W.: Un théorème sur l’existence du mouvement plan d’un fluide parfait homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698–726 (1933)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Yudovich V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyc̆isl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)MATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Christophe Lacave
    • 1
  • Toan T. Nguyen
    • 2
  • Benoit Pausader
    • 3
  1. 1.UMR 7586-CNRS, Institut de Mathématiques de Jussieu-Paris Rive GaucheUniversité Paris-Diderot (Paris 7)Paris Cedex 13France
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity Park, State CollegeUSA
  3. 3.Laboratoire Analyse, Géométrie et ApplicationsUMR 7539, Institut GaliléeUniversité Paris 13France

Personalised recommendations