Skip to main content
SpringerLink
Log in

On the Euler Equations in the Critical Triebel-Lizorkin Spaces

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper we study the initial value problem of the incompressible Euler equations in ℝn for initial data belonging to the critical Triebel-Lizorkin spaces, i.e., v 0 F n+1 1,q , q[1, ∞]. We prove the blow-up criterion of solutions in F n+1 1,q for n=2,3. For n=2, in particular, we prove global well-posedness of the Euler equations in F 3 1,q , q[1, ∞]. For the proof of these results we establish a sharp Moser-type inequality as well as a commutator-type estimate in these spaces. The key methods are the Littlewood-Paley decomposition and the paradifferential calculus by J. M. Bony.

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. Bahouri, H., Chemin, J.Y.: Equations de transport relatives à des champs de vecteurs non-lipshitziens et mé canique des fluides. Arch. Rational Mech. Anal. 127, 159–199 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Bahouri, H., Dehman, B.: Remarques sur l’apparition de singularités dans les écoulements Eulériens incompressibles à donnée initiale Höldérienne. J. Math. Pures Appl. 73, 335–346 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94, 61–66 (1984)

    MathSciNet  MATH  Google Scholar 

  4. Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. de l’Ecole Norm. Sup. 14, 209–246 (1981)

    MathSciNet  MATH  Google Scholar 

  5. Cannone, M.: Ondelettes, Paraproduits et Navier-Stokes. Diderot Editeur (1995)

  6. Cannone, M.: Nombres de Reynolds, stabilité et Navier-Stokes. Banach Center Publication, (1999)

  7. Chae, D.: On the Well-Posedness of the Euler Equations in the Besov and the Triebel-Lizorkin Spaces. Proceedings of the Conference, ‘‘Tosio Kato’s Method and Principle for Evolution Equations in Mathematical Physics’’, June 27–29 (2001)

  8. Chae, D.: On the Well-Posedness of the Euler Equations in the Triebel-Lizorkin Spaces. Comm. Pure Appl. Math. 55, 654–678 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Chemin, J.Y.: Perfect incompressible fluids. Clarendon Press, Oxford (1998)

  10. Chemin, J.Y.: Régularité des trajectoires des particules d’un fluide incompressible remplissant l’espace. J. Math. Pures Appl. 71, 407–417 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley theory and the study of function spaces. AMS-CBMS Regional Conference Series in Mathematics, 79 (1991)

  12. Frazier, M., Torres, R., Weiss, G.: The boundedness of Caleron-Zygmund operators on the spaces \(\dot{F}^{\alpha, q}_{p}\). Revista Mathemática Ibero-Americana 4, 41–72 (1988)

    MATH  Google Scholar 

  13. Jawerth, B.: Some observations on Besov and Lizorkin-Triebel Spaces. Math. Sacand. 40, 94–104 (1977)

    MATH  Google Scholar 

  14. Kato, T.: Nonstationary flows of viscous and ideal fluids in ℝ3. J. Functional Anal. 9, 296–305 (1972)

    MATH  Google Scholar 

  15. Kato, T.: untitled manuscript. Proceedings of the Conference, ‘‘Tosio Kato’s Method and Principle for Evolution Equations in Mathematical Physics’’, June 27–29 (2001)

  16. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)

    MathSciNet  MATH  Google Scholar 

  17. Kato, T., Ponce, G.: On nonstationary flows of viscous and ideal fluids in L p s (ℝ2 ). Duke Math. J. 55, 487–499 (1987)

    MathSciNet  MATH  Google Scholar 

  18. Kato, T., Ponce, G.: Well posedness of the Euler and Navier-Stokes equations in Lebesgue spaces L s p (ℝ2 ). Revista Mathemática Ibero-Americana 2, 73–88 (1986)

    MATH  Google Scholar 

  19. Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations. Comm. Math. Phys. 214, 191–200 (2000)

    MathSciNet  MATH  Google Scholar 

  20. Kozono, H., Ogawa, T., Taniuchi, Y.: The Critical Sobolev Inequalities in Besov Spaces and Regularity Criterion to Some Semilinear Evolution Equations. Preprint (2001)

  21. Lichtenstein, L.: Über einige Existenzprobleme der Hydrodynamik homogener unzusammendrückbarer, reibunglosser Flüssikeiten und die Helmholtzschen Wirbelsalitze. Math. Zeit. 23, 89–154 (1925); 26, 193–323, 387–415 (1927); 32, 608–725 (1930)

    MATH  Google Scholar 

  22. Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39, 187–220 (1986)

    Google Scholar 

  23. Majda, A.: Mathematical theory of fluid mechanics. Princeton University Graduate Course Lecture Note (1986)

  24. Sickel, W., Triebel, H.: Hölder inequalities and sharp embeddings in function spaces of B s p,q and F s p,q type. Z. Anal. Anwendungen. 14, 105–140 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Stein, E.M.: Harmonic analysis; Real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43 1993

  26. Triebel, H.: Theory of Function Spaces. Birkhäuser, (1983)

  27. Triebel, H.: Theory of Function Spaces II. Birkhäuser, (1992)

  28. Yudovich, V.: Nonstationary flow of an ideal incompressible liquid. Zh. Vych. Mat. 3, 1032–1066 (1963)

    MATH  Google Scholar 

  29. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145, 197–214 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. École Norm. Sup. (4) 32, 769–812 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Seoul National University, Seoul, 151-742, Korea

    Dongho Chae

Authors
  1. Dongho Chae
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongho Chae.

Additional information

Communicated by C. M. Dafermos

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chae, D. On the Euler Equations in the Critical Triebel-Lizorkin Spaces. Arch. Rational Mech. Anal. 170, 185–210 (2003). https://doi.org/10.1007/s00205-003-0271-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-003-0271-8

Keywords

Search

Navigation