Suitable Solutions for the Navier–Stokes Problem with an Homogeneous Initial Value

Abstract

This paper is devoted to the study of strong or weak solutions of the Navier–Stokes equations in the case of an homogeneous initial data. The case of small initial data is discussed. For large initial data, an approximation is developed, in the spirit of a paper of Vishik and Fursikov. Qualitative convergence is obtained by use of the theory of Muckenhoupt weights.

This is a preview of subscription content, access via your institution.

References

  1. BAS 06

    Basson A.: Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations. Commun. Math. Phys. 266, 17–35 (2006)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  2. CAF 82

    Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  3. CAN 95

    Cannone M.: Ondelettes, paraproduits et Navier–Stokes. Diderot Editeur, Paris (1995)

    MATH  Google Scholar 

  4. GRU 06

    Grujić Z.: Regularity of forward-in-time self-similar solutions to the 3D Navier–Stokes equations. Discrete Cont. Dyn. Systems 14, 837–843 (2006)

    MATH  Article  Google Scholar 

  5. KAT 84

    Kato T.: Strong L p solutions of the Navier–Stokes equations in \({\mathbb R^m}\) with applications to weak solutions. Math. Zeit. 187, 471–480 (1984)

    MATH  Article  Google Scholar 

  6. LEL 10

    Lelièvre, F.: A scaling and energy equality preserving approximation for the 3D Navier–Stokes equations in the finite energy case. Nonlinear Anal. TMA (2011, to appear)

  7. LEM 99

    Lemarié-Rieusset, P.G.: Solutions faibles d’énergie infinie pour les équations de Navier–Stokes dans \({\mathbb R^3}\) . C. R. Acad. Sc. Paris 328, série I, 1133–1138 (1999)

  8. LEM 02

    Lemarié-Rieusset P.G.: Recent developments in the Navier–Stokes problem. Boca Raton, FL, Chapman & Hall/CRC (2002)

    MATH  Book  Google Scholar 

  9. LEM 07

    Lemarié-Rieusset P.G.: The Navier–Stokes equations in the critical Morrey–Campanato space. Rev. Mat. Iberoamer. 23, 897–930 (2007)

    MATH  Article  Google Scholar 

  10. LER 34

    Leray J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    MathSciNet  MATH  Article  Google Scholar 

  11. PLE 03

    Plecháč P., Šverák V.: Singular and regular solutions of a nonlinear parabolic system. Nonlinearity 16, 2093–2097 (2003)

    ADS  Google Scholar 

  12. PRA 07

    Pradolini G., Salinas O.: Commutators of singular integrals on spaces of homogeneous type. Czech. Math. J. 57, 75–93 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  13. SCH 77

    Scheffer V.: Hausdorff measures and the Navier–Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  14. STE 93

    Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993)

    MATH  Google Scholar 

  15. VIS 77

    Vishik M.I., Fursikov A.V.: Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier–Stokes. Ann. Scuola Norm. Sup. Pisa, série IV IV, 531–576 (1977)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pierre Gilles Lemarié–Rieusset.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lemarié–Rieusset, P.G., Lelièvre, F. Suitable Solutions for the Navier–Stokes Problem with an Homogeneous Initial Value. Commun. Math. Phys. 307, 133 (2011). https://doi.org/10.1007/s00220-011-1291-0

Download citation