Skip to main content
SpringerLink
Log in

Well-Posedness of the Prandtl Equations Without Any Structural Assumption

  • Manuscript
  • Published:
Annals of PDE Aims and scope Submit manuscript

Abstract

We show the local in time well-posedness of the Prandtl equations for data with Gevrey 2 regularity in x and Sobolev regularity in y. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of Gérard-Varet and Dormy (J Am Math Soc 23(2):591–609, 2010).

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. Alexandre, R., Wang, Y.-G., Xu, C.-J., Yang, T.: Well-posedness of the Prandtl equation in Sobolev spaces. J. Am. Math. Soc. 28(3), 745–784 (2015)

    Article  MathSciNet  Google Scholar 

  2. Camliyurt, G., Kukavica, I., Vicol, V.: Gevrey regularity for the Navier–Stokes in a half-space. J. Differ. Equ. 265(9), 4052–4075 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  3. Dalibard, A.-L., Dietert, H., Gérard-Varet, D., Marbach, F.: High frequency analysis of the unsteady interactive boundary layer model. SIAM J. Math. Anal. 50(4), 4203–4245 (2018)

    Article  MathSciNet  Google Scholar 

  4. Dalibard, A.-L., Masmoudi, N.: Phénomène de séparation pour l’équation de Prandtl stationnaire. In Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2014–2015, pages Exp. No. IX, 18. Ed. École Polytechnique, Palaiseau (2016)

  5. Drazin, P., Reid, W.: Hydrodynamic Stability. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2004). (With a foreword by John Miles)

    Book  Google Scholar 

  6. Weinan, E.: Boundary layer theory and the zero-viscosity limit of the Navier–Stokes equation. Acta Math. Sin. (Engl. Ser.) 16(2), 207–218 (2000)

    Article  MathSciNet  Google Scholar 

  7. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989)

    Article  MathSciNet  Google Scholar 

  8. Gargano, F., Lombardo, M.C., Sammartino, M., Sciacca, V.: Singularity formation and separation phenomena in boundary layer theory. In: Fefferman, C.L., Robinson, J.C., Rodrigo, J.L. (eds.) Partial Differential Equations and Fluid Mechanics, London Mathematical Society Lecture Note series, vol. 364, pp. 81–120. Cambridge University Press, Cambridge (2009)

    Chapter  Google Scholar 

  9. Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)

    Article  MathSciNet  Google Scholar 

  10. Gerard-Varet, D., Maekawa, Y.: Sobolev stability of Prandtl expansions for the steady Navier–Stokes equations. Preprint arXiv (2018)

  11. Gérard-Varet, D., Maekawa, Y., Masmoudi, N.: Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows. Duke Math. J. 167(13), 2531–2631 (2018)

    Article  MathSciNet  Google Scholar 

  12. Gerard-Varet, D., Masmoudi, N.: Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. Éc. Norm. Supér. (4) 48(6), 1273–1325 (2015)

    Article  MathSciNet  Google Scholar 

  13. Gérard-Varet, D., Nguyen, T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77(1–2), 71–88 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Grenier, E.: On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53(9), 1067–1091 (2000)

    Article  MathSciNet  Google Scholar 

  15. Grenier, E., Guo, Y., Nguyen, T.T.: Spectral instability of characteristic boundary layer flows. Duke Math. J. 165, 3085–3146 (2016)

    Article  MathSciNet  Google Scholar 

  16. Grenier, E., Nguyen, T.T.: \({L}^\infty \) instability of Prandtl layers. Preprint arXiv (2018)

  17. Guo, Y., Iyer, S.: Validity of steady Prandtl expansions. Preprint arXiv (2018)

  18. Guo, Y., Nguyen, T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64(10), 1416–1438 (2011)

    Article  MathSciNet  Google Scholar 

  19. Guo, Y., Nguyen, T.T.: Prandtl boundary layer expansions of steady Navier–Stokes flows over a moving plate. Ann. PDE 3(1), 58 (2017). Art. 10

    Article  MathSciNet  Google Scholar 

  20. Hong, L., Hunter, J.K.: Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations. Commun. Math. Sci. 1(2), 293–316 (2003)

    Article  MathSciNet  Google Scholar 

  21. Kukavica, I., Masmoudi, N., Vicol, V., Wong, T.K.: On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J. Math. Anal. 46(6), 3865–3890 (2014)

    Article  MathSciNet  Google Scholar 

  22. Kukavica, I., Vicol, V.: On the local existence of analytic solutions to the Prandtl boundary layer equations. Commun. Math. Sci. 11(1), 269–292 (2013)

    Article  MathSciNet  Google Scholar 

  23. Li, W.-X., Yang, T.: Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points. J. Eur. Math. Soc. (2017). arXiv:1609.08430

  24. Liu, C.-J., Yang, T.: Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay. J. Math. Pures Appl. (9) 108(2), 150–162 (2017)

    Article  MathSciNet  Google Scholar 

  25. Lombardo, M.C.: Analytic solutions of the Navier–Stokes equations. Rend. Circ. Mat. Palermo (2) 50(2), 299–311 (2001)

    Article  MathSciNet  Google Scholar 

  26. Maekawa, Y., Mazzucato, A.: The inviscid limit and boundary layers for Navier–Stokes flows. In: Giga, Y., Novotný, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 1–48. Springer, Berlin (2017)

    Google Scholar 

  27. Masmoudi, N., Wong, T.K.: Local-in-time existence and uniqueness of solutions to the prandtl equations by energy methods. Commun. Pure Appl. Math. 68(10), 1683–1741 (2015)

    Article  MathSciNet  Google Scholar 

  28. Métivier, G.: Small Viscosity and Boundary Layer Methods: Theory, Stability Analysis, and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston (2004)

    Book  Google Scholar 

  29. Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation, vol. 15. Chapman & Hall/CRC, Boca Raton (1999)

    MATH  Google Scholar 

  30. Sammartino, M., Caflisch, R.E.: Zero viscosity limit for analytic solutions, of the Navier–Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192(2), 433–461 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  31. Sammartino, M., Caflisch, R.E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II. Construction of the Navier–Stokes solution. Commun. Math. Phys. 192(2), 463–491 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  32. Wang, C., Wang, Y., Zhang, Z.: Zero-viscosity limit of the Navier–Stokes equations in the analytic setting. Arch. Ration. Mech. Anal. 224(2), 555–595 (2017)

    Article  MathSciNet  Google Scholar 

  33. Xu, C.-J., Zhang, X.: Long time well-posedness of Prandtl equations in Sobolev space. J. Differ. Equ. 263(12), 8749–8803 (2017)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank Weiren Zhao for useful remarks on the first version of this manuscript. They acknowledge the support of the Université Sorbonne Paris Cité, through the funding “Investissements d’Avenir”, convention ANR-11-IDEX-0005. This work was also supported by the SingFlows Project, Grant ANR-18-CE40-0027 of the French National Research Agency (ANR). H.D. is grateful to the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France. D.G.-V. acknowledges the support of the Institut Universitaire de France.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche (UMR 7586), CNRS, Sorbonne Université, Université Paris Diderot, 75013, Paris, France

    Helge Dietert

  2. Institut Universitaire de France, 75231, Paris Cedex 05, France

    David Gérard-Varet

  3. Institut de Mathématiques de Jussieu-Paris Rive Gauche (UMR 7586), Université Paris Diderot, Sorbonne Paris Cité, 75205, Paris, France

    David Gérard-Varet

Authors
  1. Helge Dietert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Gérard-Varet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Gérard-Varet.

Additional information

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

Dietert, H., Gérard-Varet, D. Well-Posedness of the Prandtl Equations Without Any Structural Assumption. Ann. PDE 5, 8 (2019). https://doi.org/10.1007/s40818-019-0063-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40818-019-0063-6

Keywords

Search

Navigation