Skip to main content
SpringerLink
Log in

Global Well-Posedness for the 2-D Inhomogeneous Incompressible Navier-Stokes System with Large Initial Data in Critical Spaces

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

Abstract

Without any smallness assumption, we prove the global unique solvability of the 2-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov space, which is almost the energy space in the sense that they have the same scaling in terms of this 2-D system.

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. Abidi, H.: Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique. Rev. Mat. Iberoam. 23(2), 537–586, 2007

  2. Abidi, H., Gui, G., Zhang, P.: On the wellposedness of \(3-\)D inhomogeneous Navier-Stokes equations in the critical spaces. Arch. Rational Mech. Anal. 204, 189–230, 2012

  3. Abidi, H., Paicu, M.: Existence globale pour un fluide inhomogéne. Ann. Inst. Fourier (Grenoble) 57, 883–917, 2007

  4. Abidi, H., Zhang, P.: On the global well-posedness of \(2\)-D inhomogeneous incompressible Navier-Stokes system with variable viscous coefficient. J. Differ. Equ. 259, 3755–3802, 2015

  5. Kazhikov, A.V.: Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, (Russian) Dokl. Akad. Nauk SSSR 216, 1008–1010, 1974

    ADS  MathSciNet  Google Scholar 

  6. Bahouri, H., Chemin, J.Y.; Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Grundlehren der Mathematischen Wissenschaften, 2010

  7. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Cham, 1976

  8. Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246, 1981

  9. Chemin, J.-Y.: Théorémes d’unicité pour le systéme de Navier-Stokes tridimensionnel. J. Anal. Math. 77, 27–50, 1999

  10. Chemin, J.-Y., Lerner, N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differ. Equ. 121, 314–328, 1995

  11. Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differ. Equ. 26, 1183–1233, 2001

    Article  MathSciNet  Google Scholar 

  12. Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. R. Soc. Edin. Sect. A 133, 1311–1334, 2003

    Article  MathSciNet  Google Scholar 

  13. Danchin, R.: Local and global well-posedness resultats for flows of inhomogenenous viscous fluids. Adv. Differ. Equ. 9, 353–386, 2004

    MATH  Google Scholar 

  14. Danchin, R.: The inviscid limit for density-dependent incompressible fluids. Anneles de la Faculté de Sciences des Toulouse Sér. 15, 637–688, 2006

  15. Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Comm. Pure. Appl. Math. 72(7), 1351–1385, 2019

  16. Danchin, R., Mucha, P.B.: A Lagrangian approach for the incompressible Navier-Stokes equations with variable density. Comm. Pure. Appl. Math. 65, 1458–1480, 2012

  17. DiPerna, R. J., Lions, P. L.: Equations différentielles ordinaires et équations de transport avec des coefficients irréguliers. In Séminaire EDP 1988-1989, Ecole Polytechnique, Palaiseau, 1989

  18. Fleet, T.M.: Differential analysis. Cambridge University Press, Cambridge, 1980

  19. Haspot, B.: Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity. Annales de l’Institut Fourier 62(5), 1717–1763, 2012

  20. Ladyženskaja, O.A., Solonnikov, V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. (Russian) Boundary value problems of mathematical physics, and related questions of the theory of functions. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52(52–109), 218–219, 1975

  21. Lions, P. L.: Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996

  22. Paicu, M., Zhang, P., Zhang, Z.: Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density. Commun. Partial Differ. Equ. 38(7), 1208–1234, 2013

  23. Peetre, J.: New thoughts on Besov spaces, Duke University Mathematical Series 1, Durham N. C., 1976

  24. Planchon, F.: An extension of the Beale-Kato-Majda criterion for the Euler equations. Comm. Math. Phys. 232, 319–326, 2003

    Article  ADS  MathSciNet  Google Scholar 

  25. Triebel, H.: Theory of Function Spaces, Monograph in mathematics, vol. 78. Birkhauser Verlag, Basel, 1983

Download references

Funding

We thank the anonymous referee for the profitable suggestions. G. Gui is supported in part by the National Natural Science Foundation of China under the Grants 11571279 and 11931013.

Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia

    Hammadi Abidi

  2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Guilong Gui

  3. School of Mathematics, Northwest University, Xi’an, 710069, China

    Guilong Gui

Authors
  1. Hammadi Abidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guilong Gui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guilong Gui.

Additional information

Communicated by P.-L. Lions

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

Abidi, H., Gui, G. Global Well-Posedness for the 2-D Inhomogeneous Incompressible Navier-Stokes System with Large Initial Data in Critical Spaces. Arch Rational Mech Anal 242, 1533–1570 (2021). https://doi.org/10.1007/s00205-021-01710-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-021-01710-y

Search

Navigation