Geodesic
On the ill-posedness of the Prandtl equation
Gérard-Varet, David 1   ; Dormy, Emmanuel 2
1 DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm,75005 Paris, France
2 ENS/IPGP/CNRS, Ecole Normale Supérieure, 29 rue Lhomond, 75005 Paris, France
Journal of the American Mathematical Society, Tome 23 (2010) no. 2, pp. 591-609 Cet article a éte moissonné depuis la source American Mathematical Society

Voir la notice de l'article

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with nondegenerate critical points. Interestingly, the strong instability is due to viscosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.
DOI : 10.1090/S0894-0347-09-00652-3

Gérard-Varet, David  1   ; Dormy, Emmanuel  2

1 DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm,75005 Paris, France
2 ENS/IPGP/CNRS, Ecole Normale Supérieure, 29 rue Lhomond, 75005 Paris, France 
@article{10_1090_S0894_0347_09_00652_3,
     author = {G\'erard-Varet, David and Dormy, Emmanuel},
     title = {On the ill-posedness of the {Prandtl} equation},
     journal = {Journal of the American Mathematical Society},
     pages = {591--609},
     year = {2010},
     volume = {23},
     number = {2},
     doi = {10.1090/S0894-0347-09-00652-3},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-09-00652-3/}
}
TY  - JOUR
AU  - Gérard-Varet, David
AU  - Dormy, Emmanuel
TI  - On the ill-posedness of the Prandtl equation
JO  - Journal of the American Mathematical Society
PY  - 2010
SP  - 591
EP  - 609
VL  - 23
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-09-00652-3/
DO  - 10.1090/S0894-0347-09-00652-3
ID  - 10_1090_S0894_0347_09_00652_3
ER  - 
%0 Journal Article
%A Gérard-Varet, David
%A Dormy, Emmanuel
%T On the ill-posedness of the Prandtl equation
%J Journal of the American Mathematical Society
%D 2010
%P 591-609
%V 23
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-09-00652-3/
%R 10.1090/S0894-0347-09-00652-3
%F 10_1090_S0894_0347_09_00652_3
Gérard-Varet, David; Dormy, Emmanuel. On the ill-posedness of the Prandtl equation. Journal of the American Mathematical Society, Tome 23 (2010) no. 2, pp. 591-609. doi: 10.1090/S0894-0347-09-00652-3

[1] Brenier, Yann Homogeneous hydrostatic flows with convex velocity profiles Nonlinearity 1999 495 512

[2] Coddington, Earl A., Levinson, Norman Theory of ordinary differential equations 1955

[3] E, Weinan Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation Acta Math. Sin. (Engl. Ser.) 2000 207 218

[4] E, Weinan, Engquist, Bjorn Blowup of solutions of the unsteady Prandtl’s equation Comm. Pure Appl. Math. 1997 1287 1293

[5] Grenier, Emmanuel On the derivation of homogeneous hydrostatic equations M2AN Math. Model. Numer. Anal. 1999 965 970

[6] Grenier, Emmanuel On the nonlinear instability of Euler and Prandtl equations Comm. Pure Appl. Math. 2000 1067 1091

[7] Hong, Lan, Hunter, John K. Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations Commun. Math. Sci. 2003 293 316

[8] Kato, Tosio Perturbation theory for linear operators 1995

[9] Lombardo, Maria Carmela, Cannone, Marco, Sammartino, Marco Well-posedness of the boundary layer equations SIAM J. Math. Anal. 2003 987 1004

[10] Oleinik, O. A., Samokhin, V. N. Mathematical models in boundary layer theory 1999

[11] Sammartino, Marco, Caflisch, Russel E. 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. 1998 433 461

[12] Sammartino, Marco, Caflisch, Russel E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution Comm. Math. Phys. 1998 463 491

[13] Xin, Zhouping, Zhang, Liqun On the global existence of solutions to the Prandtl’s system Adv. Math. 2004 88 133

Cité par Sources :