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  • Séminaire Laurent Schwartz — EDP et applications
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Well-posedness issues for the Prandtl boundary layer equations
David Gérard-Varet1; Nader Masmoudi2
1 Univ Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ Paris 06, F-75013, Paris, France
2 Courant Institute, NYU, 251 Mercer Street, New-York 10012 USA
Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 15, 10 p.
  • Abstract

These notes are an introduction to the recent paper [7], about the well-posedness of the Prandtl equation. The difficulties and main ideas of the paper are described on a simpler linearized model.

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DOI: 10.5802/slsedp.59
Author's affiliations:
David Gérard-Varet 1; Nader Masmoudi 2

1 Univ Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ Paris 06, F-75013, Paris, France
2 Courant Institute, NYU, 251 Mercer Street, New-York 10012 USA
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@article{SLSEDP_2013-2014____A15_0,
     author = {David G\'erard-Varet and Nader Masmoudi},
     title = {Well-posedness issues for the {Prandtl} boundary layer equations},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:15},
     pages = {1--10},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2013-2014},
     doi = {10.5802/slsedp.59},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.59/}
}
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JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:15
PY  - 2013-2014
SP  - 1
EP  - 10
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.59/
DO  - 10.5802/slsedp.59
LA  - en
ID  - SLSEDP_2013-2014____A15_0
ER  - 
%0 Journal Article
%A David Gérard-Varet
%A Nader Masmoudi
%T Well-posedness issues for the Prandtl boundary layer equations
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:15
%D 2013-2014
%P 1-10
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.59/
%R 10.5802/slsedp.59
%G en
%F SLSEDP_2013-2014____A15_0
David Gérard-Varet; Nader Masmoudi. Well-posedness issues for the Prandtl boundary layer equations. Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 15, 10 p. doi : 10.5802/slsedp.59. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.59/
  • References
  • Cited by

[1] R. Alexandre, Y.-G. Wang, C.-J. Xu, and T. Yang. Well-posedness of the Prandtl equation in sobolev spaces. J. Amer. Math. Soc., 2014.

[2] J. Bedrossian and N. Masmoudi. Asymptotic stability for the Couette flow in the 2D Euler equations. Appl. Math. Res. Express. AMRX, 1:157–175, 2014. | MR | Zbl

[3] S. J. Cowley, L. M. Hocking, and O. R. Tutty. The stability of solutions of the classical unsteady boundary-layer equation. Phys. Fluids, 28(2):441–443, 1985. | MR | Zbl

[4] A. B. Ferrari and E. S. Titi. Gevrey regularity for nonlinear analytic parabolic equations. Comm. Partial Differential Equations, 23(1-2):1–16, 1998. | MR | Zbl

[5] C. Foias and R. Temam. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal., 87(2):359–369, 1989. | MR | Zbl

[6] D. Gérard-Varet and E. Dormy. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 23(2):591–609, 2010. | MR | Zbl

[7] D. Gerard-Varet and N. Masmoudi. Well-posedness for the prandtl system without analyticity or monotonicity. 2014. | arXiv

[8] D. Gérard-Varet and T. Nguyen. Remarks on the ill-posedness of the Prandtl equation. Asymtotic Analysis, 77:71–88, 2012. | MR | Zbl

[9] E. Grenier. On the stability of boundary layers of incompressible Euler equations. J. Differential Equations, 164(1):180–222, 2000. | MR | Zbl

[10] E. Guyon, J. Hulin, and L. Petit. Hydrodynamique physique, volume 142 of EDP Sciences. CNRS Editions, Paris, 2001.

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

[12] J. P. Kelliher. Bonus to the paper : Vanishing viscosity and the accumulation of vorticity on the boundary. Available at http://math.ucr.edu/~kelliher/JKPapers.html. | Zbl

[13] I. Kukavica, N. Masmoudi, V. Vicol, and T. K. Wong. On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions. 2014. | arXiv

[14] I. Kukavica and V. Vicol. On the local existence of analytic solutions to the Prandtl boundary layer equations. Communications in Mathematical Sciences, 11(1):269–292, 2013. | MR | Zbl

[15] C. D. Levermore and M. Oliver. Analyticity of solutions for a generalized Euler equation. J. Differential Equations, 133(2):321–339, 1997. | MR | Zbl

[16] M. C. Lombardo, M. Cannone, and M. Sammartino. Well-posedness of the boundary layer equations. SIAM J. Math. Anal., 35(4):987–1004 (electronic), 2003. | MR | Zbl

[17] N. Masmoudi and T. K. Wong. Local in time existence and uniqueness of solutions to the Prandtl equations by energy methods. to appear in CPAM, 2012.

[18] N. Masmoudi and T. K. Wong. On the H s theory of hydrostatic Euler equations. Arch. Ration. Mech. Anal., 204(1):231–271, 2012. | MR

[19] O. A. Oleinik and V. N. Samokhin. Mathematical models in boundary layer theory, volume 15 of Applied Mathematics and Mathematical Computation. Chapman & Hall/CRC, Boca Raton, FL, 1999. | MR | Zbl

[20] M. Oliver and E. S. Titi. Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schrödinger equation. Indiana Univ. Math. J., 47(1):49–73, 1998. | MR | Zbl

[21] L. Prandtl. Boundary layer. Verhandlung Internationalen Mathematiker-Kongresses, Heidelberg,, pages 484–491, 1904.

[22] O. Reynolds. n experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. Philos. Trans. R. Soc., 174:935–82, 1883.

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

[24] Z. Xin and L. Zhang. On the global existence of solutions to the Prandtl’s system. Adv. Math., 181(1):88–133, 2004. | MR | Zbl

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