Skip to main content
SpringerLink
Log in

Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a ``physical condition'', related to the fact that the pressure of a fluid has to be positive.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alinhac, S., Gerard, P.: Operateurs pseudo-differentiels et theorem de Nash-Msoer. Paris: Inter Editions and CNRS, 1991

  2. Baouendi, M.S., Gouaouic, C.: Remarks on the abstract form of nonlinear Cauchy-Kovalevsky theorems. Comm. Part. Diff. Eq. 2, 1151–1162 (1977)

    Google Scholar 

  3. Christodoulou, D.: Self-Gravitating Relativistic Fluids: A Two-Phase Model. Arch. Rat. Mech. Anal. 130, 343–400 (1995)

    Article  Google Scholar 

  4. Christodoulou, D.: Oral Communication, August, 1995

  5. Christodoulou, D., Klainerman, S.: The Nonlinear Stability of the Minkowski space-time. Princeton, NJ: Princeton Univ. Press, 1993

  6. Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Comm. Pure Appl. Math. 53 1536–1602 (2000)

    Google Scholar 

  7. Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. Berlin-Heidelberg-NewYork: Springer-Verlag, 1977

  8. Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-deVries scaling limits. Comm. in P. D. E. 10, 787–1003 (1985)

    Google Scholar 

  9. Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincare Anal. Non. Lineaire 7, 1–26 (1990)

    Google Scholar 

  10. Dain, S., Nagy, G.: Initial data for fluid bodies in general relativity. Phys. Rev. D 65(8), 084020 (2002)

    Article  Google Scholar 

  11. Ebin, D.: The equations of motion of a perfect fluid with free boundary are not well posed. Comm. Part. Diff. Eq. 10, 1175–1201 (1987)

    Google Scholar 

  12. Ebin, D.: Oral communication, November, 1997

  13. Friedrich, H.: Evolution equations for gravitating ideal fluid bodies in general relativity. Phys. Rev. D 57, 2317 (1998)

    Article  Google Scholar 

  14. Friedrich, H., Nagy, G.: The initial boundary value problem for Einstein's vacuum field equation. Commmun. Math. Phys. 201, 619–655 (1999)

    Article  Google Scholar 

  15. Hamilton, R.: Nash-Moser Inverse Function Theorem. Bull. Amer. Math. Soc. (N.S.) 7, 65–222 (1982)

    Google Scholar 

  16. Hörmander, L.: The boundary problem of Physical geodesy. Arch. Rat. Mech. Anal. 62, 1–52 (1976)

    Google Scholar 

  17. Hörmander, L. : Implicit function theorems. Lecture Notes, Stanford Univ, 1977

  18. Hörmander, L.: The analysis of Linear Partial Differential Operators III. Berlin-Heidelberg-NewYork: Springer Verlag, 1985

  19. Klainerman, S.: On the Nash-Moser-Hörmander scheme. Unpublished lecture notes

  20. Klainerman, S.: Global solutions of nonlinar wave equations. Comm. Pure Appl. Math. 33, 43–101 (1980)

    Google Scholar 

  21. Lindblad, H.: Well posedness for the linearized motion of an incompressible liquid with free surface boundary. Comm. Pure Appl. Math. 56, 153–197 (2003)

    Article  Google Scholar 

  22. Lindblad, H.: Well posedness for the linearized motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 236, 281–310 (2003)

    Article  Google Scholar 

  23. Lindblad, H.: Well posedness for the motion of an incompressible liquid with free surface boundary. To appear in Annals of Math., July 2005

  24. Nalimov, V.I.: The Cauchy-Poisson Problem (in Russian). Dynamika Splosh. Sredy 18, 104–210 (1974)

    Google Scholar 

  25. Nishida, T.: A note on a theorem of Nirenberg. J. Diff. Geom. 12, 629–633 (1977)

    Google Scholar 

  26. Rendall, A.D.: The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys. 33, 1047–1053 (1992)

    Article  Google Scholar 

  27. Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. Cambridge, MA: International Press, 1994

  28. Stein, E.: Singular Integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press, 1970

  29. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997)

    Article  Google Scholar 

  30. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12, 445–495 (1999)

    Article  Google Scholar 

  31. Yosihara, H.: Gravity Waves on the Free Surface of an Incompressible Perfect Fluid. Publ. RIMS Kyoto Univ. 18, 49–96 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA

    Hans Lindblad

Authors
  1. Hans Lindblad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans Lindblad.

Additional information

Communicated by P. Constantin

The author was supported in part by the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lindblad, H. Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary. Commun. Math. Phys. 260, 319–392 (2005). https://doi.org/10.1007/s00220-005-1406-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1406-6

Keywords

Search

Navigation