Skip to main content
SpringerLink
Log in

Spatial asymptotic expansions in the incompressible Euler equation

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

In this paper we prove that the Euler equation describing the motion of an ideal fluid in \({\mathbb{R}^d}\) is well-posed in a class of functions allowing spatial asymptotic expansions as \({|x|\to\infty}\) of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with rapidly decaying initial data develop non-trivial spatial asymptotic expansions of the type considered here.

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. Arnold V.: Sur la geometrié differentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluids parfaits. Annales de l’institut Fourier 16(1), 319–361 (1966)

    Article  Google Scholar 

  2. Bardos C., Titi E.: Euler equations for an ideal incompressible fluid. Russian Mathematical Surveys 62(3), 409–451 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bondareva I., Shubin M.: Growing asymptotic solutions of the Korteweg-de Vries equation and of its higher analogues. Doklady Akademii Nauk SSSR 267(5), 1035–1038 (1982)

    MathSciNet  MATH  Google Scholar 

  4. I. Bondareva and M. Shubin, Uniqueness of the solution of the Cauchy problem for the Korteweg-de Vries equation in classes of increasing functions. Vestnik Moskov. Univ. Ser. I Mat. Mekh, (3)1985, 35–38

  5. Bondareva I., Shubin M.: Equations of Korteweg-de Vries type in classes of increasing functions. Journal of Soviet Mathematics 51(3), 2323–2332 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourguignon J.P., Brezis H.: Remarks on the Euler equation. Journal of Functional Analysis 15, 341–363 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brandolese L.: Space-time decay of Navier–Stokes flows invariant under rotations. Mathematische Annalen 329(4), 685–706 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brandolese L., Meyer Y.: On the instantaneous spreading for the Navier–Stokes system in the whole space A tribute to J. L. Lions. ESAIM: Control, Optimisation and Calculus of Variations 8, 273–285 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cantor M.: Perfect fluid flows over \({\mathbb{R}^n}\) with asymptotic conditions. Journal of Functional Analysis 18, 73–84 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Physical Review Letters 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Constantin A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Annales de l’institut Fourier, Grenoble 50(2), 321–362 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. C. De Lellis, T. Kappeler and P. Topalov. Low regularity solutions of the Camassa–Holm equation. Comm. Partial Differential Equations, (1–3)32 (2007), 87–126

  13. J. Dieudonné. Foundations of Modern Analysis. Academic Press, Cambridge (1969)

  14. Ebin D., Marsden J.: Groups of Diffeomorphisms and the Motion of an Incompressible Fluid. Annals of Mathematics 92, 102–163 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dobrokhotov S., Shafarevich A.: Some integral identities and remarks on the decay at infinity of the solutions of the Navier–Stokes Equations in the entire space. Russian Journal of Mathematical Physics 2(1), 133–135 (1994)

    MathSciNet  MATH  Google Scholar 

  16. A. Erdélyi. Asymptotic Expansions. Dover Publications, Inc., Mineola (1956)

  17. Inci H., Kappeler T., Topalov P.: On the regularity of the composition of diffeomorphisms. Memoirs of the American Mathematical Society 226, 1062 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kappeler T., Loubet E., Topalov P.: Analyticity of Riemannian exponential maps on Diff\({(\mathbb{T})}\). Journal of Lie Theory 17(3), 481–503 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Kappeler T., Loubet E., Topalov P.: Riemannian exponential maps of the diffeomorphism groups of \({\mathbb{T}^2}\). Asian Journal of Mathematics 12(3), 391–420 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kappeler T., Perry P., Shubin M., Topalov P.: Solutions of mKdV in classes of functions unbounded at infinity. Journal Of Geometric Analysis 18(2), 443–477 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato T.: On classical solutions of the two-dimensional non-stationary Euler equation. Archive for Rational Mechanics and Analysis 25, 188–200 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  22. T. Kato. Quasi-linear equations of evolution, with applications to partial differential equations. Lecture Notes in Math., 448, Springer, Berlin, (1975)

  23. Kukavica I., Reis E.: Asymptotic expansion for solutions of the Navier–Stokes equations with potential forces. Journal of Differential Equations 250(1), 607–622 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Lang. Differential Manifolds. Addison-Wesley Series in Mathematics, Boston (1972)

  25. McOwen R.: The behavior of the Laplacian on weighted Sobolev spaces. Communications on Pure and Applied Mathematics 32(6), 783–795 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. McOwen R., Topalov P.: Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems 35(7), 3103–3131 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. R. McOwen and P. Topalov, Groups of asymptotic diffeomorphisms. Discrete and Continuous Dynamical Systems, (11)36 (2016), 6331–6377. arXiv:1412.4732

  28. Menikoff A.: The existence of unbounded solutions of the Korteweg-de Vries equation. Communications on Pure and Applied Mathematics 25, 407–432 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Misiolek G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. Journal of Geometric Analysis 24, 203–208 (1998)

    MathSciNet  MATH  Google Scholar 

  30. P. Serfati. Équations d’Euler et holomorphies à faible regularité spatiale. Comptes Rendus de l’AcadTmie des Sciences, (2)320 (1994), Série I, 175–180

  31. A. Shnirelman. On the analyticity of particle trajectories in the ideal incompressible fluid. arXiv:1205.5837v

  32. M. Shubin. Pseudodifferential Operators and Spectral Theory. Second edition, Springer, Berlin (2001)

  33. Wolibner W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Mathematische Zeitschrift 37(1), 698–726 (1933)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeastern University, 360 Huntington Ave., Boston, MA, 02115, USA

    Robert McOwen & Petar Topalov

Authors
  1. Robert McOwen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petar Topalov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petar Topalov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

McOwen, R., Topalov, P. Spatial asymptotic expansions in the incompressible Euler equation. Geom. Funct. Anal. 27, 637–675 (2017). https://doi.org/10.1007/s00039-017-0410-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-017-0410-2

Search

Navigation