Abstract
In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in \(L^\infty \) in the inviscid limit. In this paper we prove that, for a class of smooth solutions of Navier Stokes equations, namely for shear layer profiles which are unstable for Rayleigh equations, this Ansatz is false if we consider solutions with Sobolev regularity, in strong contrast with the analytic case, pioneered by Sammartino and Caflisch (Commun Math Phys 192(2)433–461, 1998; Commun Math Phys 192(2)463–491, 1998). Meanwhile we address the classical problem of the nonlinear stability of shear layers near a boundary and prove that if a shear flow is spectrally unstable for Euler equations, then it is non linearly unstable for the Navier Stokes equations provided the viscosity is small enough.
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References
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)
Drazin, P.G., Reid, W. H.: Hydrodynamic stability, 2nd edition. In: With a foreword by John Miles (ed.) Cambridge mathematical library. Cambridge University Press, Cambridge (2004)
Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)
Gerard-Varet, D., Maekawa, Y., Masmoudi, N.: Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows. Duke Math. J. 167(13), 2531–2631 (2018)
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)
Gérard-Varet, D., Nguyen, T.T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77(1–2), 71–88 (2012)
Grenier, E.: On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53(9), 1067–1091 (2000)
Grenier, E., Guo, Y., Nguyen, T.T.: Spectral stability of Prandtl boundary layers: an overview. Analysis (Berlin) 35(4), 343–355 (2015)
Grenier, E., Guo, Y., Nguyen, T.T.: Spectral instability of characteristic boundary layer flows. Duke Math. J. 165(16), 3085–3146 (2016)
Grenier, E., Nguyen, T.T.: Green function of Orr-Sommerfeld equations away from critical layers SIAM. J. Math. Anal. 51(2), 1279–1296 (2019)
Grenier, E., Nguyen, T.T.: Sublayer of prandtl boundary layers. Arch. Ration. Mech. Anal. 229(3), 1139–1151 (2018)
Grenier, E., Nguyen, T. T.: On nonlinear instability of Prandtl’s boundary layers: the case of Rayleigh’s stable shear flows. arXiv:1706.01282 (2017)
Grenier, E., Nguyen, T.T.: Green function of Orr-Sommerfeld equations away from critical layers. SIAM J. Math. Anal. 51(2), 1279–1296 (2019)
Guo, Y., Nguyen, T.T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64(10), 1416–1438 (2011)
Iftimie, D., Sueur, F.: Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145–175 (2011)
Maekawa, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure Appl. Math. 67(7), 1045–1128 (2014)
Masmoudi, N., Rousset, F.: Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal. 203(2), 529–575 (2012)
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)
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)
Paddick, M.: Stability and instability of Navier boundary layers. Differ. Integr. Eq. 27(9–10), 893–930 (2014)
Prandtl, L. : Uber flüssigkeits-bewegung bei sehr kleiner reibung. pp. 484–491, (1904)
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)
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)
Schlichting, H., Gersten, K.: Boundary-layer theory, 8th revised and enlarged edition. Springer, Heidelberg (2000)
Acknowledgements
TN’s research was partly supported by the NSF under grant DMS-1764119 and by an AMS Centennial Fellowship.
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Grenier, E., Nguyen, T.T. \(L^\infty \) Instability of Prandtl Layers. Ann. PDE 5, 18 (2019). https://doi.org/10.1007/s40818-019-0074-3
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DOI: https://doi.org/10.1007/s40818-019-0074-3