Skip to main content
SpringerLink
Log in

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

It is shown both locally and globally that \({L_t^{\infty}(L_x^{3,q})}\) solutions to the three-dimensional Navier–Stokes equations are regular provided \({q\neq\infty}\). Here \({L_x^{3,q}}\), \({0 < q \leq\infty}\), is an increasing scale of Lorentz spaces containing \({L^3_x}\). Thus the result provides an improvement of a result by Escauriaza et al. (Uspekhi Mat Nauk 58:3–44, 2003; translation in Russ Math Surv 58, 211–250, 2003), which treated the case q = 3. A new local energy bound and a new \({\epsilon}\)-regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray–Hopf weak solutions in \({L_t^{\infty}(L_x^{3,q})}\), \({q\neq\infty}\), is also obtained as a consequence.

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Chemin J.-Y., Planchon F.: Self-improving bounds for the Navier–Stokes equations. Bull. Soc. Math. France 140, 583–597 (2012)

    MATH  MathSciNet  Google Scholar 

  3. Chen C.-C., Strain R.M., Tsai T.-P., Yau H.-T.: Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations II. Commun. Partial Differ. Equ. 34, 203–232 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, C.-C., Strain, R.M., Yau, H.-T., Tsai, T.-P.: Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations. Int. Math. Res. Not. IMRN (2008) (no. 9, art. ID rnn016)

  5. Escauriaza, L., Seregin, G., Šverák, V.: \({L_{3, \infty}}\)-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk. (Russian) 58, 3–44 (2003) (translation in Russian Math Surv, 58, 211–250, 2003)

  6. Escauriaza L., Seregin G., Šverák V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169, 147–157 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gallagher I., Koch G., Planchon F.: A profile decomposition approach to the \({L^{\infty}_t(L^{3}_{x})}\) Navier–Stokes regularity criterion. Math. Ann. 355, 1527–1559 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gallagher, I., Koch, G., Planchon, F.: Blow-up of critical Besov norms at a potential Navier–Stokes singularity (2014). arXiv:1407.4156 (preprint)

  9. Giga Y., Sohr H.: Abstract L p-estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing, River Edge (2003)

    Book  MATH  Google Scholar 

  11. Grafakos, L.: Classical Fourier analysis, 2nd edn. In: Graduate Texts in Mathematics, vol. 249. Springer, New York (2008)

  12. Hopf E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kato T.: Strong L p-solutions of the Navier–Stokes equations in R m with applications to weak solutions. Math. Z. 187, 471–480 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kenig C., Koch G.: An alternative approach to the Navier–Stokes equations in critical spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 159–187 (2011)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Kim H., Kozono H.: Interior regularity criteria in weak spaces for the Navier–Stokes equations. Manuscr. Math. 115, 85–100 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kozono H., Sohr H.: Remark on uniqueness of weak solutions to the Navier–Stokes equations. Analysis 16, 255–271 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ladyzhenskaya, O.A.: Mathematical problems of the dynamics of viscous incompressible fluids. Gos. Izdat. Fiz. Mat. Lit. Moscow (1961) (2nd rev. Aug. ed. of English transl., The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1969)

  18. Ladyzhenskaya, O.: On uniqueness and smoothness of generalized solutions to the Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185 (1967) (English transl., Sem. Math. V.A. Steklov Math. Inst. Leningrad, 5, 60–66, 1969)

  19. Ladyzhenskaya O., Seregin G.: On partial regularity of suitable weak solutions to the three-dimensional Navier Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lin F.-H.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)

    Article  MATH  Google Scholar 

  22. Luo, Y., Tsai, T.-P.: Regularity criteria in weak L 3 for 3D incompressible Navier–Stokes equations (2014). arXiv:1310.8307v5 (preprint)

  23. Maremonti, P., Solonnikov, V.A.: On estimates for the solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces with a mixed norm. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222, 124–150 (1995) (English transl., J. Math. Sci. (New York), 87, 3859–3877, 1997)

  24. Nečas J., Růžička M., Šverák V.: On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176, 283–294 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Prodi G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. (4) 48, 173–182 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  26. Scheffer V.: Partial regularity of solutions to the Navier–Stokes equations. Pac. J. Math. 66, 535–552 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Scheffer V.: The Navier–Stokes equations in a bounded domain. Commun. Math. Phys. 73, 1–42 (1980)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Scheffer V.: Boundary regularity for the Navier–Stokes equations in a half-space. Commun. Math. Phys. 85, 275–299 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Seregin G., Šverák V.: Navier–Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163, 65–86 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Seregin G., Šverák V.: On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations. Commun. Partial Differ. Equ. 34, 171–201 (2009)

    Article  MATH  Google Scholar 

  32. Seregin G.: A certain necessary condition of potential blow up for Navier–Stokes equations. Commun. Math. Phys. 312, 833–845 (2012)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. Seregin, G.: Selected topics of local regularity theory for the Navier–Stokes equations. Topics in mathematical fluid mechanics. In: Lecture Notes in Mathematics, vol. 2073, pp. 239–313. Springer, Heidelberg (2013)

  34. Serrin J.: On the interior regularity of weak solutions of the Navier Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  35. Serrin, J.: The initial value problem for the Navier Stokes equations. In: Langer, R. (ed.) Nonlinear Problems, pp. 69–98. University of Wisconsin Press, Madison (1963)

  36. Sohr H.: A regularity class for the Navier–Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441–467 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Solonnikov, V.A.: Estimates of the solutions of a nonstationary linearized system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 70, 213–317 (1964) (English transl., Am. Math. Soc. Transl. (2), 75, 1–116, 1968)

  38. Struwe M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41, 437–458 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  39. Takahashi S.: On interior regularity criteria for weak solutions of the Navier–Stokes equations. Manuscr. Math. 69, 237–254 (1990)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA, 70803, USA

    Nguyen Cong Phuc

Authors
  1. Nguyen Cong Phuc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Cong Phuc.

Additional information

Communicated by R. Shvydkoy

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Phuc, N.C. The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces. J. Math. Fluid Mech. 17, 741–760 (2015). https://doi.org/10.1007/s00021-015-0229-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-015-0229-2

Mathematics Subject Classification

Keywords

Search

Navigation