×

On the nonlinear instability of Euler and Prandtl equations. (English) Zbl 1048.35081

The author proves that if the stationary solution of Euler equations is linearly unstable (in the sense that there exists an eigenvalue of the linearized Euler operator with nonnegative real part), then the time-dependent solution is nonlinearly unstable. This result is valid for the whole space, the half-space, for periodic and non-periodic strips, and for the periodic torus. An application to nonlinear stability of regularized vortex sheets is given. Similar results are also established for the nonlinear instability of Prandtl boundary layers, with application to the construction of asymptotic solutions for Navier-Stokes equations.

MSC:

35Q35 PDEs in connection with fluid mechanics
76E30 Nonlinear effects in hydrodynamic stability
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arnold, Ann Inst Fourier (Grenoble) 16 pp 319– (1966) · Zbl 0148.45301
[2] Les méthodes mathématiques de la mécanique classique. Éditions Mir, Moscow, 1976.
[3] Brenier, Nonlinearity 12 pp 495– (1999)
[4] Desjardins, Adv Differential Equations 3 pp 715– (1998)
[5] ; Hydrodynamic stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University, Cambridge-New York, 1981.
[6] E, Comm Pure Appl Math 50 pp 1287– (1997)
[7] Friedlander, Ann Inst H Poincaré Anal Non Linéaire 14 pp 187– (1997)
[8] The theory of rotating fluids. Reprint of the 1968 original. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University, Cambridge-New York, 1980
[9] Grenier, M2AN Math Model Numer Anal 33 pp 965– (1999)
[10] Grenier, J Math Pures Appl (9) 7610 pp 965– (1997)
[11] Grenier, J Differential Equations
[12] Grenier, J Differential Equations 143 pp 110– (1998)
[13] Guo, Ann Inst H Poincaré Anal Non Linéaire 12 pp 339– (1995)
[14] Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), 85-98. Math Sci Res Inst Publ, 2. Springer, New York-Berlin, 1984.
[15] ; Fluid mechanics. Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6. Pergamon, London-Paris-Frankfurt; Addison-Wesley, Reading, Mass., 1959.
[16] Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and Its Applications, 3. Oxford Science Publications. Clarendon, Oxford University, New York, 1996.
[17] ; Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences, 96. Springer, New York, 1994. · Zbl 0789.76002
[18] Ole??nik, Dokl Akad Nauk SSSR 168 pp 751– (1966)
[19] Rayleigh, Proc London Math Soc 11 pp 57– (1880)
[20] Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992.
[21] Sammartino, Comm Math Phys 192 pp 433– (1998)
[22] Boundary layer theory. Translated by J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill, New York-Toronto-London; G. Braun, Karlsruhe, 1960.
[23] Sulem, Comm Math Phys 80 pp 485– (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.