Archive for Rational Mechanics and Analysis

, Volume 204, Issue 1, pp 231–271 | Cite as

On the H s Theory of Hydrostatic Euler Equations

  • Nader Masmoudi
  • Tak Kwong Wong


In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H s solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.


Vorticity Euler Equation Local Existence Strong Uniqueness Unique Classical Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brenier Y.: Homogeneous hydrostatic flows with convex velocity profiles. Nonlinearity 12(3), 495–512 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    E W.: Boundary layer theory and the zero-viscosity limit of the Navier–Stokes equation. Acta Math. Sin. (Engl. Ser.) 16(2), 207–218 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Lions P.L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1996)Google Scholar
  4. 4.
    Grenier E.: On the derivation of homogeneous hydrostatic equations. Math. M2AN Model. Numer. Anal. 33(5), 965–970 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brenier Y.: Remarks on the derivation of the hydrostatic Euler equations. Bull. Sci. Math. 127(7), 585–595 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Arnold V.I., Khesin B.A.: Topological methods in hydrodynamics Applied Mathematical Sciences, vol. 125. Springer, New York (1998)Google Scholar
  7. 7.
    Arnold, V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications á l’hydrodynamique des fluides parfaits. (French). Ann. Inst. Fourier (Grenoble) 16(fasc. 1), 319–361 (1966)Google Scholar
  8. 8.
    Arnold V.: Les méthodes mathématiques de la mécanique classique. (French). Traduit du russe par Djilali Embarek. Éditions Mir, Moscow (1976)Google Scholar
  9. 9.
    Drazin P.G., Reid W.H.: Hydrodynamic Stability Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge-New York (1981)Google Scholar
  10. 10.
    Friedlander S., Strauss W., Vishik M.: Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 187–209 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Rayleigh L.: On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 57–70 (1880)zbMATHCrossRefGoogle Scholar
  12. 12.
    Grenier E.: On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53(9), 1067–1091 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Renardy M.: Ill-posedness of the Hydrostatic Euler and Navier–Stokes Equations. Arch. Rational Mech. Anal. 194(3), 877–886 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Gérard-Varet D., Dormy E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)zbMATHCrossRefGoogle Scholar
  15. 15.
    Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Evolutionary equations. Handbook of Differential Equations, vol. III, pp. 195–276. Elsevier/North- Holland, Amsterdam, 2007Google Scholar
  16. 16.
    Kukavica I., Temam R., Vicol V.C., Ziane M.: Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data. C. R. Math. Acad. Sci. Paris 348(11–12), 639–645 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kukavica I., Temam R., Vicol V.C., Ziane M.: Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain. J. Diff. Equ. 250(3), 1719–1746 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Brenier Y.: Generalized solutions and hydrostatic approximation of the Euler equations. Phys. D 237(14–17), 1982–1988 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hong L., Hunter J.K.: Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations. Commun. Math. Sci. 1(2), 293–316 (2003)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Renardy M.: On hydrostatic free surface problems. J. Math. Fluid Mech. 13, 89–93 (2011)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Majda A.J., Bertozzi A.L.: Vorticity and incompressible flow Cambridge. Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations