Skip to main content
SpringerLink
Log in

On the theory of Besov–Herz spaces and Euler equations

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we consider Euler equations (E) for an incompressible ideal fluid filling the whole space ℝn for n ≥ 2. We prove local-in-time wellposedness and give a blow-up criterion for (E) in a new framework, namely Besov type spaces based on Herz spaces. Solutions are global-in-time when n = 2. Our results cover critical and supercritical cases of the regularity. For that, we develop properties and estimates in those spaces such as product and commutator-type estimates, interpolation, duality, among others. For the blow-up result, another ingredient is a logarithmic type inequality in our spaces.

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. J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Communications in Mathematical Physics 94 (1984), 61–66.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, Vol. 223, Springer-Verlag, Berlin–New York, 1976.

  3. F. Bernicot and S. Keraani, On the global well-posedness of the 2D Euler equations for a large class of Yudovich type data, Annales Scientifiques de l’École Normale Supérieure 47 (2014), 559–576.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales Scientifiques de lÉcole Normale Superieure 14 (1981), 209–246.

    Article  MATH  Google Scholar 

  5. J. Bourgain and D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Inventiones Mathematicae 201 (2015), 97–157.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis 4 (1980), 677–681.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations 5 (1980), 773–789.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptotic Analysis 38 (2004), 339–358.

    MathSciNet  MATH  Google Scholar 

  10. D. Chae, On the Euler equations in the critical Triebel–Lizorkin spaces, Archive for Rational Mechanics and Analysis 170 (2003), 185–210.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Chae, On the well-posedness of the Euler equations in the Triebel–Lizorkin spaces, Communications on Pure and Applied Mathematics 55 (2002), 654–678.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications, Vol. 14, Clarendon Press, Oxford University Press, New York, 1998.

  13. J.-Y. Chemin, Régularité de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace, Journal de Mathématiques Pures et Appliquées 71 (1992), 407–417.

    MathSciNet  MATH  Google Scholar 

  14. J.-Y. Chemin, Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel, Inventiones Mathematicae 103 (1991), 599–629.

    Article  MathSciNet  MATH  Google Scholar 

  15. Y. Z. Chen and K. S. Lau, On some new classes of Hardy spaces, Journal of Functional Analysis 84 (1989), 255–278.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones mathematicae 98 (1989), 511–547.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics 92 (1970), 102–163.

    Article  MathSciNet  MATH  Google Scholar 

  18. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 2010.

  19. J. García-Cuerva and M.-J. L. Herrero, A theory of Hardy spaces associated to the Herz spaces, Proceedings of the London Mathematical Society 69 (1994), 605–628.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Grafakos, X. Li and D. Yang, Bilinear Operators on Herz-type Hardy spaces, Transactions of the American Mathematical Society 350 (1998), 1249–1275.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, Vol. 38, SIAM, Philadelphia, PA, 2002.

  22. E. Hernandez and D. Yang, Interpolation of Herz spaces and applications, Mathematische Nachrichten 205 (1999), 69–87.

    Article  MathSciNet  MATH  Google Scholar 

  23. C. S. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, Transylvanian Journal of Mathematics and Mechanics 18 (1968/69), 83–323.

    MathSciNet  MATH  Google Scholar 

  24. R. Johnson, Lipschitz spaces, Littlewood–Paley spaces, and convoluteurs, Proceedings of the London Mathematical Society 29 (1974), 127–141.

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Kato, Nonstationary flows of viscous and Ideal fluids in R3, Journal of Functional Analysis 9 (1972), 296–305.

    Article  MATH  Google Scholar 

  26. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Communications on Pure and Applied Mathematics 41 (1988), 891–907.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Kato and G. Ponce, Well-Posedness of the Euler and Navier–Stokes equations in the Lebesgue espaces L p s(R 2), Revista Matemática Iberoamericana 2 (1986), 73–88.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Mathematische Zeitschrift 242 (2002), 251–278.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations, Communications in Mathematical Physics 214 (2000), 191–200.

    Article  MathSciNet  MATH  Google Scholar 

  30. P. G. Lemarié-Rieusset, Recent Development in the Navier–Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, Vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.

  31. X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois Journal of Mathematics 40 (1996), 484–501.

    MathSciNet  MATH  Google Scholar 

  32. S. Lu and D. Yang, Herz-type Sobolev and Bessel potential spaces and their applications, Science in China. Series A. Mathematics 40 (1997), 113–129.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Frontiers of the mathematical sciences: 1985 (New York, 1985), Communications on Pure and Applied Mathematics 39 (1986), S, suppl., S187–S220.

    MathSciNet  Google Scholar 

  34. F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Archive for Rational and Mechanical Analysis 27 (1967), 329–348.

    MathSciNet  MATH  Google Scholar 

  35. H. Pak and Y. Park, Existence of solution for the Euler equations in a critical Besov space B ∞,1 1(R n), Communications in Partial Differential Equations 29 (2004), 1149–1166.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. Pak and Y. Park, Persistence of the incompressible Euler equations in a Besov space B 1,1 d+1 (Rd), Advances in Difference Equations (2013), 2013:153, 18 pp.

    Article  Google Scholar 

  37. O. Sawada and R. Takada, On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity, Journal of Functional Analysis 260 (2011), 2148–2162.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Takada, Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, Journal of Evolution Equations 8 (2008), 693–725.

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Temam, On the Euler equations of incompressible perfect fluids, Journal of Functional Analysis 20 (1975), 32–43.

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Vol. 78, Birkhäuser Verlag, Basel, 1983.

  41. Y. Tsutsui, The Navier–Stokes equations and weak Herz spaces, Advances in Differential Equations 16 (2011), 1049–1085.

    MathSciNet  MATH  Google Scholar 

  42. M. Vishik, Hydrodynamics in Besov spaces, Archive for Rational andMechanical Analysis 145 (1998), 197–214.

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Annales Scientifiques de l’École Normale Supérieure 32 (1999), 769–812.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Xu, Equivalent norms of Herz-type Besov and Triebel–Lizorkin spaces, Journal of Function Spaces and Applications 3 (2005), 17–31.

    Article  MathSciNet  MATH  Google Scholar 

  45. V. Yudovich, Nonstationary flow of an ideal incompressible liquid, (Russian) Žurnal Vyčislitel’noĭ Matematiki i Matematičeskoĭ Fiziki 3 (1963), 1032–1066.

    Google Scholar 

  46. Y. Zhou, Local well-posedness for the incompressible Euler equations in the critical Besov spaces, Université de Grenoble. Annales de l’Institut Fourier 54 (2004), 773–786.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidade Estadual de Campinas, IMECC - Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil

    Lucas C. F. Ferreira & J. E. Pérez-López

Authors
  1. Lucas C. F. Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. E. Pérez-López
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lucas C. F. Ferreira.

Additional information

L. Ferreira (corresponding author) was supported by FAPESP and CNPQ, Brazil.

J. Pérez-López was supported by CAPES, Brazil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferreira, L.C.F., Pérez-López, J.E. On the theory of Besov–Herz spaces and Euler equations. Isr. J. Math. 220, 283–332 (2017). https://doi.org/10.1007/s11856-017-1519-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-017-1519-6

Search

Navigation