Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 431–443 | Cite as

Well-Posedness for the Navier-Stokes Equations with Datum in the Sobolev Spaces

Article

Abstract

In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces \(\dot {H}^{s}_{p}(\mathbb {R}^{d})\) for \(d \geq 2, p > \frac {d}{2}\), and \(\frac {d}{p} - 1 \leq s < \frac {d}{2p}\). The obtained result improves the known ones for p > d and s = 0 (see [4, 6]). In the case of critical indexes \(s=\frac {d}{p}-1\), we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the one in [5] in which p = d and s = 0.

Keywords

Navier-Stokes equations Existence and uniqueness of local and global mild solutions Critical Sobolev and Besov spaces 

Mathematics Subject Classification (2010)

Primary 35Q30 Secondary 76D05 76N10 

References

  1. 1.
    Bourgain, J., Pavloviéc, N.: Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J. Funct. Anal. 255(9), 2233–2247 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bourdaud, G.: Ce qu’il faut savoir sur les espaces de Besov Prépublication de l’Universitéde Paris 7 (janvier 1993)Google Scholar
  3. 3.
    Bourdaud, G.: Réalisation des espaces de Besov homogènes. Ark. Mat. 26(1), 41–54 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cannone, M.: Ondelettes, Paraproduits et Navier-Stokes, p 191. Diderot Editeur, Paris (1995)MATHGoogle Scholar
  5. 5.
    Cannone, M.: A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat. Iberoam. 13(3), 515–541 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cannone, M., Meyer, Y.: Littlewood-Paley decomposition and the Navier-Stokes equations. Methods Appl. Anal. 2, 307–319 (1995)MathSciNetMATHGoogle Scholar
  7. 7.
    Chemin, J.M.: Remarques sur l’existence globale pour le système de Navier-Stokes incompressible. SIAM J. Math. Anal. 23, 20–28 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fabes, E., Jones, B., Riviere, N.N.: The initial value problem for the Navier-Stokes equations with data in L p. Arch. Rat. Mech. Anal. 45, 222–240 (1972)CrossRefMATHGoogle Scholar
  9. 9.
    Fujita, H., Kato, T.: On the Navier-Stokes initial value problem I. Arch. Rat. Mech. Anal. 16, 269–315 (1964)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Giga, Y.: Solutions of semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62, 186–212 (1986)CrossRefMATHGoogle Scholar
  11. 11.
    Giga, Y., Miyakawa, T.: Solutions in L r of the Navier-Stokes initial value problem. Arch. Rat. Mech. Anal. 89, 267–281 (1985)CrossRefMATHGoogle Scholar
  12. 12.
    Khai, D.Q., Tri, N.M.: Solutions in mixed-norm Sobolev-Lorentz spaces to the initial value problem for the Navier-Stokes equations. J. Math. Anal. Appl. 417, 819–833 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Khai, D.Q., Tri, N.M.: Well-posedness for the Navier-Stokes equations with datum in Sobolev-Fourier-Lorentz spaces. J. Math. Anal. Appl. 437, 754–781 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khai, D.Q., Tri, N.M.: On the Hausdorff dimension of the singular set in time for weak solutions to the nonstationary Navier-Stokes equation on torus. Vietnam J. Math. 43, 283–295 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Khai, D.Q., Tri, N.M.: On the initial value problem for the Navier-Stokes equations with the initial datum in critical Sobolev and Besov spaces. J. Math. Sci. Univ. Tokyo 23, 499–528 (2016)MathSciNetMATHGoogle Scholar
  16. 16.
    Khai, D.Q., Tri, N.M.: Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces. preprint, arXiv:1601.01742
  17. 17.
    Khai, D.Q., Tri, N.M.: The existence and decay rates of strong solutions for Navier-Stokes Equations in Bessel-potential spaces. preprint, arXiv:1603.01896
  18. 18.
    Khai, D.Q., Tri, N.M.: The existence and space-time decay rates of strong solutions to Navier-Stokes equations in weighed L (|x|γdx)L (|x|βdx) spaces. preprint, arXiv:1601.01723
  19. 19.
    Kato, T., Fujita, H.: On the non-stationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32, 243–260 (1962)MathSciNetMATHGoogle Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    Kato, T.: Strong solutions of the Navier-Stokes equations in Morrey spaces. Bol. Soc. Brasil. Math. 22, 127–155 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lemarie-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC Research Notes in Mathematics, vol. 431, p 395. Chapman and Hall/CRC, Boca Raton, FL (2002)Google Scholar
  24. 24.
    Taylor, M.E.: Analysis on Morrey spaces and applications to Navier-Stokes equations and other evolution equations. Comm. P. D. E. 17, 1407–1456 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Weissler, F.B.: The Navier-Stokes initial value problem in L p. Arch. Rat. Mech. Anal. 74, 219–230 (1981)CrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Institute of Mathematics, Vietnam Academy of Science and TechnologyCau GiayVietnam

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