Advertisement

Continuous dependence of the pressure field with respect to endpoints for ideal incompressible fluids

  • Aymeric BaradatEmail author
Article
  • 21 Downloads

Abstract

In the Brenier variational model for perfect fluids, the datum is the joint law of the initial and final positions of the particles. In this paper, we show that both the optimal action and the pressure field are Hölder continuous with respect to this datum for the Monge–Kantorovich distance.

Mathematics Subject Classification

49N15 49N60 49S05 

Notes

Acknowledgements

This work is part of my Ph.D. thesis supervised by Yann Brenier and Daniel Han-Kwan. I would like to thank both of them for the careful reading and advice.

References

  1. 1.
    Ambrosio, L.: Transport equation and Cauchy problem for non-smooth vector fields. In: Dacorogna, B., Marcellini, P. (eds.) Calculus of Variations and Non-Linear Partial Differential Equations, pp. 1–41. Springer, Berlin (2008)CrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Figalli, A.: On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations. Calc. Var. Partial Differ. Equ. 31(4), 497–509 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Figalli, A.: Geodesics in the space of measure-preserving maps and plans. Arch. Ration. Mech. Anal. 194(2), 421–462 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. In Annales de l’institut Fourier, vol. 16, pp. 319–361. Institut Fourier (1966)Google Scholar
  5. 5.
    Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics, vol. 125. Springer, Berlin (1999)zbMATHGoogle Scholar
  6. 6.
    Baradat, A.: Work in progressGoogle Scholar
  7. 7.
    Bernot, M., Figalli, A., Santambrogio, F.: Generalized solutions for the Euler equations in one and two dimensions. Journal de mathématiques pures et appliquées 91(2), 137–155 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brenier, Y.: The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2(2), 225–255 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brenier, Y.: The dual least action problem for an ideal, incompressible fluid. Arch. Ration. Mech. Anal. 122(4), 323–351 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brenier, Y.: On the motion of an ideal incompressible fluid. In: Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math., vol. XXXV, pp. 123–148. Cambridge University Press, Cambridge (1994)Google Scholar
  11. 11.
    Brenier, Y.: Extended Monge–Kantorovich Theory. Lecture Notes in Mathematics, pp. 91–122. Springer, Berlin (2003)Google Scholar
  12. 12.
    Brenier, Y.: Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics. Calcul. Var. Partial Differ. Equ. 47(1), 55–64 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. In: Annales de l’Institut Henri Poincaré. Analyse non linéaire, vol. 7, pp. 1–26. Elsevier, Amsterdam (1990)Google Scholar
  14. 14.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2015)zbMATHGoogle Scholar
  15. 15.
    Shnirelman, A.I.: On the geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. Math. USSR-Sbornik 56(1), 79 (1987)CrossRefGoogle Scholar
  16. 16.
    Shnirelman, A.I.: Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4(5), 586–620 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CMLSPolytechniquePalaiseauFrance

Personalised recommendations