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On the stability of boundary layers of incompressible Euler equations. (English) Zbl 0958.35106
The author studies the stability of the boundary layer of 2D-incompressible Euler equations in a periodic half strip. Using energy methods, he shows under certain assumptions (which roughly speaking say that there is no inflection point in the boundary layer profile) the linear stability of the Euler equations and, under some additional assumptions, he also proves the nonlinear stability. As an application, the asymptotic expansion of the boundary layer is proved.
The other part deals with instability results; using the result of Lord Rayleigh [On the stability, or instability of certain fluid motions, Proc. London Math. Soc. 11, 57-70 (1880)], where the linear instability is shown, the author constructs an example where the solution to the nonlinear system is unstable. Indeed, in this case there is an inflection point in the boundary layer profile.

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q05 Euler-Poisson-Darboux equations
76E17 Interfacial stability and instability in hydrodynamic stability
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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[1] V. I. Arnold, Méthodes mathématiques de la mécanique classique, Mir.
[2] Arnold, V.I., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. institut Fourier, 16, 319-361, (1996) · Zbl 0148.45301
[3] Y. Brenier, Reformulations of some inviscid hydrostatic models, manuscript, 1997.
[4] R. E. Caflisch, and, M. Sanmartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, preprint.
[5] R. E. Caflisch, and, M. Sanmartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution, preprint.
[6] B. Desjardins, and, E. Grenier, On the homogeneous model of wind driven ocean circulation, preprint, 1997. · Zbl 0958.76092
[7] Drazin, P.G.; Reid, W.H., Hydrodynamic stability, Cambridge monographs on mechanics and applied mathematics, (1981) · Zbl 0449.76027
[8] W. E. B. Engquist, Blow up of solutions to the unsteady Prandtl equations, preprint, 1996.
[9] Friedlander, S.; Strauss, W.; Vishik, M., Nonlinear instability in an ideal fluid, Ann. inst. H. Poincaré anal. non linéaire, 14, 187-209, (1997) · Zbl 0874.76026
[10] Guo, Y.; Strauss, W., Nonlinear instability of double-humped equilibria, Ann. inst. H. Poincaré anal. non linéaire, 12, 339-352, (1995) · Zbl 0836.35130
[11] Greenspan, H.P., The theory of rotating fluids, Cambridge monographs on mechanics and applied mathematics, (1969) · Zbl 0181.54303
[12] E. Grenier, Boundary layers of characteristic nonlinear parabolic equations, J. Math. Pures Appl, to appear. · Zbl 0914.35032
[13] E. Grenier, On the instability of boundary layers of incompressible Navier Stokes equations, in preparation.
[14] E. Grenier, and, O. Guès, On the inviscid limit of noncharacteristic nonlinear parabolic systems, J. Differential Equations, to appear.
[15] Grenier, E.; Masmoudi, N., Ekman layers of rotating fluids, the case of well prepared initial data, Comm. partial differential equations, 22, 953-975, (1997) · Zbl 0880.35093
[16] Hoskins, B.J., The geostrophic momentum approximation and the semi-geostrophic equations, J. atmospheric sci., 32, 233-242, (1975)
[17] Kato, T., Remarks on zero viscosity limit for nonstationary Navier Stokes flows with boundary, Seminar on nonlinear partial differential equations (Berkeley, CA, 1983), Math. sci. res. inst. publ., 2, (1984), Springer-Verlag New York/Berlin, p. 85-98
[18] Landau, L.; Lifchitz, E., Mécanique des fluides, (1959), Mir
[19] Marsden, J.; Raţiu, T., Nonlinear stability in fluids and plasmas, Seminar on new results in nonlinear partial differential equations (Bonn, 1984), Aspects of math., E10, (1987), Vieweg Brauschweig, p. 101-134
[20] Marchioro, C.; Pulvirenti, M., Mathematical theory of incompressible nonviscous fluids, Applied mathematical sciences, 96, (1994), Springer-Verlag · Zbl 0789.76002
[21] Rayleigh, Lord, On the stability, or instability, of certain fluid motions, Proc. London math. soc., 11, 57-70, (1880) · JFM 12.0711.02
[22] Schlichting, H., Boundary layer theory, (1968), McGraw-Hill New York
[23] Wang, Temam, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana univ. math. J., 45, (1996) · Zbl 0881.35097
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