On the ill-posedness of the Prandtl equation

Gérard-Varet, David; Dormy, Emmanuel

doi:10.1090/S0894-0347-09-00652-3

On the ill-posedness of the Prandtl equation
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by David Gérard-Varet and Emmanuel Dormy;

J. Amer. Math. Soc. 23 (2010), 591-609

DOI: https://doi.org/10.1090/S0894-0347-09-00652-3

Published electronically: November 24, 2009

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Abstract:

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.

References

Yann Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity 12 (1999), no. 3, 495–512. MR 1690189, DOI 10.1088/0951-7715/12/3/004
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 69338
Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207–218. MR 1778702, DOI 10.1007/s101140000034
Weinan E and Bjorn Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287–1293. MR 1476316, DOI 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4
Emmanuel Grenier, On the derivation of homogeneous hydrostatic equations, M2AN Math. Model. Numer. Anal. 33 (1999), no. 5, 965–970. MR 1726718, DOI 10.1051/m2an:1999128
Emmanuel Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091. MR 1761409, DOI 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.3.CO;2-H
E. Guyon, J.P. Hulin, and L. Petit, Hydrodynamique physique, EDP Sciences, vol. 142, CNRS Editions, Paris, 2001.
Lan Hong and John K. Hunter, Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci. 1 (2003), no. 2, 293–316. MR 1980477
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
Maria Carmela Lombardo, Marco Cannone, and Marco Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal. 35 (2003), no. 4, 987–1004. MR 2049030, DOI 10.1137/S0036141002412057
O. A. Oleinik and V. N. Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1697762
L. Prandtl, Uber flüssigkeits-bewegung bei sehr kleiner reibung., Actes du 3ème Congrés international dse Mathématiciens, Heidelberg, Teubner, Leipzig, 1904, pp. 484–491.
Marco Sammartino and Russel 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 (1998), no. 2, 433–461. MR 1617542, DOI 10.1007/s002200050304
Marco Sammartino and Russel E. Caflisch, 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. 192 (1998), no. 2, 463–491. MR 1617538, DOI 10.1007/s002200050305
Zhouping Xin and Liqun Zhang, On the global existence of solutions to the Prandtl’s system, Adv. Math. 181 (2004), no. 1, 88–133. MR 2020656, DOI 10.1016/S0001-8708(03)00046-X