Skip to main content
SpringerLink
Log in

Regularity and Expansion for Steady Prandtl Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Due to degeneracy near the boundary, the question of high regularity for solutions to the steady Prandtl equations has been a longstanding open question since the celebrated work of Oleinik. We settle this open question in affirmative in the absence of an external pressure. Our method is based on energy estimates for the quotient, \(q = \frac{v}{\bar{u}}\), \(\bar{u}\) being the classical Prandtl solution, via the linear derivative Prandtl (LDP) equation. As a consequence, our regularity result leads to the construction of Prandtl layer expansion up to any order.

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. Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Elsevier Science Ltd, Kidlington, Oxford (2003)

    MATH  Google Scholar 

  2. 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 

  3. Asano, A.: Zero viscosity limit of incompressible Navier–Stokes equations. In: Conference at the Fourth Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, Kyoto (1991)

  4. Bardos, C., Titi, E.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42–76 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  5. Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Meth. Appl. Sci. 2, 556–581 (1980)

    Article  MathSciNet  Google Scholar 

  6. Dalibard, A.L., Masmoudi, N.: Phenomene de separation pour l’equation de Prandtl stationnaire. Seminaire Laurent Schwartz - EDP et applications: 1–18 (2014–2015)

  7. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to fractional Sobolev spaces. Bull. des Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Weinan, E., Engquist, B.: Blowup of solutions of the unsteady Prandtl’s equation. Comm. Pure Appl. Math. 50, 1287–1293 (1997)

    Article  MathSciNet  Google Scholar 

  10. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, RI (2010)

    Google Scholar 

  11. Lopes Filho, M.C., Mazzucato, A.L., Nussenzveig Lopes, H.J.: Vanishing viscosity limit for incompressible flow inside a rotating circle. Physica D 237, 1324–1333 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  12. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Second Edition. Springer Monographs in Mathematics (2011)

  13. Gerard-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 

  14. Gerard-Varet, D., Masmoudi, N.: Well-posedness for the Prandtl system without analyticity or monotonicity. arXiv:1305.0221v1 (2013)

  15. Gerard-Varet, D., Maekawa, Y., Masmoudi, N.: Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows. arXiv:1607.06434v1 (2016)

  16. Gerard-Varet, D., Maekawa, Y.: Sobolev stability of Prandtl expansions for the steady Navier–Stokes equations. arXiv:1805.02928 (2018)

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

    Article  MathSciNet  Google Scholar 

  18. Gie, G.-M., Jung, C.-Y., Temam, R.: Recent progresses in the boundary layer theory. Disc. Cont. Dyn. Syst. A 36(5), 2521–2583 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Grenier, E., Guo, Y., Nguyen, T.: Spectral instability of symmetric shear flows in a two-dimensional channel. Adv. Math. 292(9), 52–110 (2016)

    Article  MathSciNet  Google Scholar 

  20. Grenier, E., Guo, Y., Nguyen, T.: Spectral instability of characteristic boundary layer flows. Duke Math. J 165(16), 3085–3146 (2016)

    Article  MathSciNet  Google Scholar 

  21. Grenier, E., Guo, Y., Nguyen, T.: Spectral instability of Prandtl boundary layers: an overview. Analysis (Berlin) 35(4), 343–355 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Grenier, E., Nguyen, T.: On nonlinear instability of Prandtl’s boundary layers: the case of Rayleigh’s stable shear flows. arXiv:1706.01282 (2017)

  23. Grenier, E., Nguyen, T.: Sublayer of Prandtl boundary layers. arXiv:1705.04672 (2017)

  24. Grenier, E., Nguyen, T.: \(L^\infty \) Instability of Prandtl layers. arXiv:1803.11024 (2018)

  25. Guo, Y., Iyer, S.: Validity of steady Prandtl layer expansions. arXiv:1805.05891 (2018)

  26. Guo, Y., Iyer, S.: Steady Prandtl layer expansions with external forcing. arXiv:1810.06662 (2018)

  27. Guo, Y., Nguyen, T.: Prandtl boundary layer expansions of steady Navier–Stokes flows over a moving plate. Ann. PDE 3, 10 (2017). https://doi.org/10.1007/s40818-016-0020-6

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Ignatova, M., Vicol, V.: Almost global existence for the Prandtl boundary layer equations. Arch. Rational Mech. Anal. 220, 809–848 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  31. Iyer, S.: Steady Prandtl boundary layer expansion of Navier–Stokes flows over a rotating disk. Arch. Ration. Mech. Anal. 224(2), 421–469 (2017)

    Article  MathSciNet  Google Scholar 

  32. Iyer, S.: Global steady Prandtl expansion over a moving boundary. arXiv:1609.05397 (2016)

  33. Iyer, S.: Steady Prandtl layers over a moving boundary: non-shear euler flows. arXiv:1705.05936v2 (2017)

  34. Iyer, S.: On global-in-x stability of Blasius profiles. Arch. Rational Mech. Anal. 237, 951–998 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  35. Iyer, S., Masmoudi, N.: Global-in-x stability of steady Prandtl expansions for 2D Navier–Stokes flows. arXiv:2008.12347 (2020)

  36. Kelliher, J.: On the vanishing viscosity limit in a disk. Math. Ann. 343, 701–726 (2009)

    Article  MathSciNet  Google Scholar 

  37. 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 

  38. 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 

  39. Kukavica, I., Vicol, V., Wang, F.: The van Dommelen and Shen singularity in the Prandtl equations. arXiv:1512.07358v1 (2015)

  40. Liu, C.J., Xie, F., Yang, T.: Justification of Prandtl Ansatz for MHD Boundary Layer. arXiv:1704.00523v4 (2017)

  41. Lombardo, M.C., Cannone, M., Sammartino, M.: Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 35(4), 987–1004 (2003)

    Article  MathSciNet  Google Scholar 

  42. Maekawa, Y.: On the inviscid problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure. Appl. Math. 67, 1045–1128 (2014)

    Article  MathSciNet  Google Scholar 

  43. Maekawa, Y., Mazzucato, A.: The inviscid limit and boundary layers for Navier-Stokes flows. Handb. Math. Anal. Mech. Viscous Fluids 1–48 (2017)

  44. Mazzucato, A., Taylor, M.: Vanishing viscocity plane parallel channel flow and related singular perturbation problems. Anal. PDE 1(1), 35–93 (2008)

    Article  MathSciNet  Google Scholar 

  45. 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, 1683–1741 (2015)

    Article  MathSciNet  Google Scholar 

  46. Nirenberg, L.: On elliptic partial differential equations. Annali Della Scoula Normale Superiore Di Pisa., (1959)

  47. Oleinik, O.A., Samokhin, V.N.: Mathematical models in boundary layer theory Applied Mathematics and Mathematical Computation, vol. 15. Champan and Hall/CRC, Boca Raton (1999)

    MATH  Google Scholar 

  48. Oleinik, O.A.: On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid. J. Appl. Math. Mech. 30, 951–974 (1967)

    Article  MathSciNet  Google Scholar 

  49. Orlt, M.: Regularity for Navier-Stokes in domains with corners, Ph.D. Thesis, (1998). (in German)

  50. Orlt, M., Sandig, A.M.: Regularity of viscous Navier–Stokes flows in non smooth domains. Boundary value problems and integral equations in non smooth domains (Luminy, 1993). Lecture Notes in Pure and Appl. Math., 167, Dekker, New York, 1995

  51. Prandtl, L.: Uber flussigkeits-bewegung bei sehr kleiner reibung. In Verhandlungen des III Internationalen Mathematiker-Kongresses, Heidelberg. Teubner, Leipzig,, pp. 484–491 (1904). English Translation: “Motion of fluids with very little viscosity,” Technical Memorandum No. 452 by National Advisory Committee for Aeuronautics

  52. Sammartino, M., Caflisch, R.: 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 

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

    Article  ADS  Google Scholar 

  54. Schlichting, H., Gersten, K.: Boundary Layer Theory, 8th edn. Springer, Berlin (2000)

    Book  Google Scholar 

  55. Serrin, J.: Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory. Proc. R. Soc. Lond. A 299, 491–507 (1967)

    Article  ADS  Google Scholar 

  56. Temam, R., Wang, T.: Boundary layers associated with Incompressible Navier–Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179, 647–686 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  57. Xin, Z., Zhang, L.: On the global existence of solutions to the Prandtl’s system. Adv. Math. 181(1), 88–133 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research of Yan Guo is supported in part by NSF Grants DMS-1611695, DMS-1810868. Sameer Iyer was also supported in part by NSF Grant DMS 1802940.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02906, USA

    Yan Guo

  2. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544, USA

    Sameer Iyer

Authors
  1. Yan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sameer Iyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan Guo.

Additional information

Communicated by C. Mouhot

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

Guo, Y., Iyer, S. Regularity and Expansion for Steady Prandtl Equations. Commun. Math. Phys. 382, 1403–1447 (2021). https://doi.org/10.1007/s00220-021-03964-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-03964-9

Search

Navigation