Skip to main content
SpringerLink
Log in

Global Weak Besov Solutions of the Navier–Stokes Equations and Applications

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

Abstract

We introduce a notion of global weak solution to the Navier–Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces \({\dot{B}^{-1+\frac{3}{p}}_{p,\infty}}\), p >  3. These solutions satisfy a certain stability property with respect to the weak-\({\ast}\) convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.

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. Albritton D.: Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces. Anal. PDE 11(6), 1415–1456 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubin J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)

    MathSciNet  MATH  Google Scholar 

  3. Auscher P., Dubois S., Tchamitchian P.: On the stability of global solutions to Navier–Stokes equations in the space. J. Math. Pures Appl. (9) 83(6), 673–697 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343. Springer, Heidelberg 2011

  5. Bahouri H., Cohen A., Koch G.: A general wavelet-based profile decomposition in the critical embedding of function spaces. Confluentes Math. 3(3), 387–411 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barker, T.: Existence and Weak* Stability for the Navier–Stokes System with Initial Values in Critical Besov Spaces. ArXiv e-prints 2017

  7. Barker, T.: Uniqueness results for weak leray–hopf solutions of the navier–stokes system with initial values in critical spaces. J. Math. Fluid Mech. 20(1), 133–160 2018. https://doi.org/10.1007/s00021-017-0315-8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Barker T., Seregin G.: A necessary condition of potential blowup for the navier–stokes system in half-space. Mathematische Annalen 369(3), 1327–1352 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Barker, T., Seregin, G., S̆verák, V.: On stability of weak Navier–Stokes solutions with large \({L^{3,\infty}}\) initial data. Commun. Partial Differ. Equ. 2018. https://doi.org/10.1080/03605302.2018.1449219

  10. Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin, New York 1976

  11. Bradshaw, Z., Tsai, T.-P.: Discretely self-similar solutions to the Navier–Stokes equations with Besov space data. ArXiv e-prints 2017

  12. Bradshaw Z., Tsai T.-P.: Forward discretely self-similar solutions of the Navier–Stokes equations II. Ann. Henri Poincaré 18(3), 1095–1119 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Bradshaw Z., Tsai T.-P.: Rotationally corrected scaling invariant solutions to the Navier–Stokes equations. Commun. Partial Differ. Equ. 42(7), 1065–1087 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commum. Pure Appl. Math. 35(6), 771–831 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Calderón C.P.: Existence of weak solutions for the Navier–Stokes equations with initial data in L p. Trans. Am. Math. Soc. 318(1), 179–200 (1990)

    MATH  Google Scholar 

  16. Cannone M.: A generalization of a theorem by Kato on Navier–Stokes equations. Rev. Mat. Iberoamericana 13(3), 515–541 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chae, D., Wolf, J.: Existence of discretely self-similar solutions to the Navier–Stokes equations for initial value in \({ L^2\_{ loc}({\mathbb{R}}^{3})}\). ArXiv e-prints 2016

  18. Choe, H.J., Wolf, J., Yang, M.: On regularity and singularity for \({L^\infty(0,T;L^{3,w}(\mathbb{R}^3))}\) solutions to the Navier–Stokes equations. ArXiv e-prints 2016

  19. Dong B.-Q., Zhang Z.: On the weak–strong uniqueness of Koch–Tataru’s solution for the Navier–Stokes equations. J. Differ. Equ. 256(7), 2406–2422 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Escauriaza L., Seregin G., S̆verák V.: \({L_{3,\infty}}\)-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk, 58(2(350)), 3–44 (2003)

    Article  MathSciNet  Google Scholar 

  21. Fujita H., Kato T.: On the Navier–Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269–315 (1964)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Gallagher I., Iftimie D., Planchon F.: Asymptotics and stability for global solutions to the Navier–Stokes equations. Ann. Inst. Fourier (Grenoble) 53(5), 1387–1424 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gallagher I., Koch G.S., Planchon F.: A profile decomposition approach to the \({L^\infty_t(L^3_x)}\) Navier–Stokes regularity criterion. Math. Ann. 355(4), 1527–1559 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gallagher I., Koch G.S., Planchon F.: Blow-up of critical Besov norms at a potential Navier–Stokes singularity. Commun. Math. Phys. 343(1), 39–82 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Guillod, J., S̆verák, V.: Numerical investigations of non-uniqueness for the Navier–Stokes initial value problem in borderline spaces. ArXiv e-prints 2017

  26. Heywood J.G.: The Navier–Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29(5), 639–681 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jia H., S̆verák V.: Minimal L 3-initial data for potential Navier–Stokes singularities. SIAM J. Math. Anal. 45(3), 1448–1459 (2013)

    Article  MathSciNet  Google Scholar 

  28. Jia H., S̆verák V.: Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions. Invent. Math. 196(1), 233–265 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  29. Jia H., S̆verák V.: Are the incompressible 3d Navier–Stokes equations locally ill-posed in the natural energy space?. J. Funct. Anal. 268(12), 3734–3766 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kato T.: Strong L p-solutions of the Navier–Stokes equation in R m, with applications to weak solutions. Math. Z. 187(4), 471–480 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kenig C.E., Koch G.S.: An alternative approach to regularity for the Navier–Stokes equations in critical spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2), 159–187 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Kenig C.E., Merle F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Koch, H., Tataru, D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157(1), 22–35 2001

    Article  MathSciNet  MATH  Google Scholar 

  34. Korobkov M., Tsai T.-P.: Forward self-similar solutions of the Navier–Stokes equations in the half space. Anal. PDE 9(8), 1811–1827 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Krepkogorskiĭ V.L.: Interpolation in Lizorkin–Triebel and Besov spaces. Mat. Sb. 185(7), 63–76 (1994)

    MathSciNet  Google Scholar 

  36. Ladyzhenskaya O.A., Seregin G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1(4), 356–387 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lemarié-Rieusset, P.G.: Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton 2002

  38. Lemarié-Rieusset P.G.: The Navier–Stokes problem in the 21st century. CRC Press, Boca Raton (2016)

    Book  MATH  Google Scholar 

  39. Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lin F.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51(3), 241–257 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nec̆as, J., R\({\mathring{\text{u}}}\)z̆ic̆ka, M., S̆verák, V.: On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176(2), 283–294 1996

  42. Phuc N.C.: The Navier–Stokes equations in nonendpoint borderline Lorentz spaces. J. Math. Fluid Mech. 17(4): 741–760 (2015)

  43. Planchon F.: Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier–Stokes equations in R 3. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(3), 319–336 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Rusin W.: Navier–Stokes equations, stability and minimal perturbations of global solutions. J. Math. Anal. Appl. 386(1), 115–124 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Rusin W., S̆verák V.: Minimal initial data for potential Navier–Stokes singularities. J. Funct. Anal. 260(3), 879–891 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Seregin G.: A certain necessary condition of potential blow up for Navier–Stokes equations. Commum. Math. Phys. 312(3), 833–845 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Seregin G., S̆verák V.: On global weak solutions to the Cauchy problem for the Navier-Stokes equations with large L 3-initial data. Nonlinear Anal. 154, 269–296 (2017)

    Article  MathSciNet  Google Scholar 

  48. Seregin, G.A.: Necessary conditions of potential blow up for Navier–Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 385(Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 41), 187–199, 236, 2010

  49. Seregin, G.: Leray-Hopf solutions to Navier–Stokes equations with weakly converging initial data. In: Robinson J.C., Rodrigo, J.L., Sadowski, W. (eds.) Mathematical Aspects of Fluid Mechanics. London Math. Soc. Lecture Note Ser., vol. 402, pp. 251–258. Cambridge Univ. Press, Cambridge, 2012

  50. Seregin, G.: Lecture Notes on Regularity Theory for the Navier–Stokes Equations. World Scientific Publishing Co. Pte. Ltd., Hackensack 2015

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Sohr, H.: The Navier–Stokes equations. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2001. An elementary functional analytic approach, [2013 reprint of the 2001 original] [MR1928881]

  53. Tsai T.-P.: On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal., 143(1), 29–51 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Tsai T.-P.: Forward discretely self-similar solutions of the Navier–Stokes equations. Commun. Math. Phys. 328(1), 29–44 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Wang W.D., Zhang Z.F.: Blow-up of critical norms for the 3-D Navier–Stokes equations. Sci. China Math. 60(4), 637–650 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author was partially supported by the NDSEG Graduate Fellowship. He also thanks his advisor, Vladimír Šverák, as well as Simon Bortz and Raghavendra Venkatraman for helpful suggestions. The second author was supported by an EPSRC Doctoral Prize award. We thank P. G. Lemarié-Rieusset for pointing out the reference [35], as well as the anonymous referee for his or her work reviewing the paper.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Dallas Albritton

  2. OxPDE, Mathematical Institute, University of Oxford, Oxford, UK

    Tobias Barker

Authors
  1. Dallas Albritton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Barker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dallas Albritton.

Additional information

Communicated by F. Lin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Albritton, D., Barker, T. Global Weak Besov Solutions of the Navier–Stokes Equations and Applications. Arch Rational Mech Anal 232, 197–263 (2019). https://doi.org/10.1007/s00205-018-1319-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-018-1319-0

Search

Navigation