Skip to main content
SpringerLink
Log in

Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data

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

Abstract

Local in time weak solutions to the 3D Navier–Stokes are constructed for a class of initial data in \(L^2_\mathrm {loc}\). In contrast to other constructions (e.g. Lemarié-Rieusset in Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol 431. Chapman & Hall/CRC, Boca Raton, 2002; Kikuchi and Seregin in Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality. Nonlinear equations and spectral theory, American Mathematical Society translations: series 2, vol 220. American Mathematical Society, Providence, pp 141–164, 2007; Kwon and Tsai in Global Navier–Stokes flows for non-decaying initial data with slowly decaying oscillation. arXiv:1811.03249 ), the initial data is not required to be uniformly locally square integrable and, in particular, can exhibit growth in a local \(L^2\) sense. This class of initial data includes vector fields in the critical Morrey space and discretely self-similar vector fields in \(L^2_\mathrm {loc}\).

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

Fig. 1

Similar content being viewed by others

References

  1. Basson, A.: Solutions spatialement homogénes adaptées au sens de Caffarelli, Kohn et Nirenberg des équations de Navier–Stokes, Thèse, Université d’Évry (2006)

  2. Bogovskiĭ, M.E.: Solutions of some problems of vector analysis, associated with the operators \({\rm div}\) and \({\rm grad}\), Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, no. 1, vol. 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5–40, 149

  3. 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  Google Scholar 

  4. Bradshaw, Z., Tsai, T.P.: Discretely self-similar solutions to the Navier–Stokes equations with data in \(L_{{\rm loc}}^2\) satisfying the local energy inequality, Analysis & PDE (to appear)

  5. Bradshaw, Z., Tsai, T.P.: Global existence, regularity, and uniqueness of infinite energy solutions to the Navier–Stokes equations. arXiv:1907.00256

  6. Constantin, P., Foias, C.: Navier–Stokes equations. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1988)

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

    Article  ADS  MathSciNet  Google Scholar 

  8. Chae, D., Wolf, J.: Existence of discretely self-similar solutions to the Navier–Stokes equations for initial value in \(L_{{\rm loc}}^2({\mathbb{R}}^3)\). Ann. Inst. H. Poincaré Anal. Non Linéaire 35(4), 1019–1039 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  9. Dascaliuc, R., Grujić, Z.: Energy cascades and flux locality in physical scales of the 3D NSE. Commun. Math. Phys. 305, 199–220 (2011)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Fernández-Dalgo, P.G., Lemarié-Rieusset, P.G.: Weak solutions for Navier–Stokes equations with initial data in weighted \(L^2\) spaces. arXiv:1906.11038

  12. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. In: Steady-state problems, 2nd edn. Springer Monographs in Mathematics. Springer, New York (2011). ISBN: 978-0-387-09619-3

  13. Guillod, J., Šverák, V.: Numerical investigations of non-uniqueness for the Navier–Stokes initial value problem in borderline spaces. arXiv:1704.00560

  14. Hao, J., Šverák, 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  Google Scholar 

  15. Jia, H., Šverák, 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 

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

    Article  MathSciNet  Google Scholar 

  17. Kang, K., Miura, H., Tsai, T.-P.: Short time regularity of Navier–Stokes flows with locally \(L^3\) initial data and applications, preprint. arXiv:1812.10509

  18. Kikuchi, N., Seregin, G.: Weak Solutions to the Cauchy Problem for the Navier–Stokes Equations Satisfying the Local Energy Inequality. Nonlinear Equations and Spectral Theory. American Mathematical Society Translations: Series 2, vol. 220, pp. 141–164. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Kukavica, I.: On partial regularity for the Navier–Stokes equations. Discrete Contin. Dyn. Syst. 21(3), 717–728 (2008)

    Article  MathSciNet  Google Scholar 

  21. Kukavica, I., Vicol, V.: On local uniqueness of weak solutions to the Navier–Stokes system with \({\rm BMO}^{-1}\) initial datum. J. Dyn. Differ. Equ. 20(3), 719–732 (2008)

    Article  MathSciNet  Google Scholar 

  22. Kwon, H., Tsai, T.-P.: Global Navier–Stokes flows for non-decaying initial data with slowly decaying oscillation. arXiv:1811.03249

  23. 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)

    Book  Google Scholar 

  24. Lemarié-Rieusset, P.G.: The Navier–Stokes Problem in the 21st Century. CRC Press, Boca Raton, FL (2016)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Maekawa, Y., Miura, H., Prange, C.: Local energy weak solutions for the Navier–Stokes equations in the half-space. Commun. Math. Phys. 367(2), 517–580 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  27. Robinson, J.C., Rodrigo, J.L., Sadowski, W.: The Three-Dimensional Navier–Stokes Equations. Cambridge Studies in Advanced Mathematics, Classical Theory, vol. 157. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Stein, E.M.: Note on singular integrals. Proc. Am. Math. Soc. 8, 250–254 (1957)

    Article  MathSciNet  Google Scholar 

  30. Temam, R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence, RI, (2001). Theory and Numerical Analysis, Reprint of the 1984 edition

  31. Tsai, T.-P.: Lectures on Navier–Stokes equations, Graduate Studies in Mathematics, vol. 192. American Mathematical Society, Providence (2018)

    Book  Google Scholar 

Download references

Acknowledgements

ZB was supported in part by the Simons Foundation, while IK was supported in part by the NSF Grants DMS-1615239 and DMS-1907992.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Arkansas, Fayetteville, AR,  72701, USA

    Zachary Bradshaw

  2. Department of Mathematics, University of Southern California, Los Angeles, CA,  90089, USA

    Igor Kukavica

Authors
  1. Zachary Bradshaw
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Kukavica
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zachary Bradshaw.

Ethics declarations

Conflict of interest

The authors have no conflict of interest to report.

Additional information

Communicated by R. Shvydkoy

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bradshaw, Z., Kukavica, I. Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data. J. Math. Fluid Mech. 22, 3 (2020). https://doi.org/10.1007/s00021-019-0462-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-019-0462-1

Mathematics Subject Classification

Search

Navigation