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Joseph, D. D.; Sattinger, D. H.

Bifurcating time periodic solutions and their stability. (English) Zbl 0239.76057

Arch. Ration. Mech. Anal. 45, 79-109 (1972).
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Cited in 2 Reviews
Cited in 63 Documents

MSC:

76E05 Parallel shear flows in hydrodynamic stability
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[1] Joseph, D. D., Stability of convection in containers of arbitrary shape. J. Fluid Mech. 47, 257–282 (1971). · Zbl 0216.52803 · doi:10.1017/S0022112071001046
[2] Sattinger, D., Stability of bifurcating solutions by Leray-Schauder degree. Arch. Rational Mech. Anal. 43, 154–166 (1971). · Zbl 0232.34027 · doi:10.1007/BF00252776
[3] Meksyn, D., & J. T. Stuart, Stability of viscous motion between parallel planes for finite disturbances. Proc. Roy. Soc. A 208, 517–526 (1951). · Zbl 0043.40003 · doi:10.1098/rspa.1951.0177
[4] Stuart, J. T., On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows; Part I: The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353 (1960). · Zbl 0096.21102 · doi:10.1017/S002211206000116X
[5] Landau, L., On the problem of turbulence. C. R. Acad. Sci., U.R.S.S. 44, 311 (1944). See also Landau, L., & E. M. Lifschitz, Fluid Mechanics. Oxford: Pergamon Press 1959.
[6] Hopf, E., A mathematical example displaying features of turbulence. Comm. Pure. Appl. Math. 1, 303 (1948). See also Hopf, E., Repeated branching through loss of stability, an example. Proc. of the conf. on Diff. Eqs. Univ. of Maryland, 49, 1956. · Zbl 0031.32901 · doi:10.1002/cpa.3160010401
[7] Sattinger, D., Bifurcation of periodic solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal. 41, 66–80 (1971). · Zbl 0222.76022 · doi:10.1007/BF00250178
[8] Fife, P., & D. Joseph, Existence of convective solutions of the generalized Benard problem which are analytic in their norm. Arch. Rational Mech. Anal. 33, 116–138 (1969). · Zbl 0193.56601 · doi:10.1007/BF00247756
[9] Hopf, E., Abzweigung einer periodischen Lösung eines Differentialsystems. Aus den Berichten der Mathematisch-Physikalischen Klasse der Sächsischen Akademie der Wissenschaften zu Leipzig XCIV, 1–22 (1942).
[10] Watson, J., On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows: Part II: The development of a solution for plane Poiseuille flow and plane Couette flow. J. Fluid Mech. 14, 336 (1960). · Zbl 0096.21103
[11] Reynolds, W. C., & M. C. Potter, Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465–492. · Zbl 0166.46102
[12] Ladyshenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, 2nd edition. New York: Gordon and Breach 1968.
[13] Serrin, J., On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1–13 (1959). · Zbl 0086.20001 · doi:10.1007/BF00284160
[14] Sattinger, D., The mathematical problem of hydrodynamic stability. J. Math and Mech. 19, 797–819 (1970). · Zbl 0198.30401
[15] Coddington, E., & N. Levinson, Theory of Ordinary Differential Equations. New York: McGraw-Hill 1955 (see Chapter 13). · Zbl 0064.33002
[16] Vainberg, M. M., & V. A. Trenogin, The methods of Lyapunov and Schmidt in the theory of nonlinear equations and their further development. Russ. Math. Survey 17, 1–60 (1962). · Zbl 0117.31904 · doi:10.1070/RM1962v017n02ABEH001127
[17] Schlichting, H., Boundary Layer Theory. New York: McGraw-Hill 1968 (see Chapter XVI). · Zbl 0096.20105
[18] Krishnamurti, R., Finite amplitude convection with changing mean temperature, Part II. J. Fluid Mech. 33, 457–463 (1968). · Zbl 0182.28803 · doi:10.1017/S0022112068001448
[19] Joseph, D. D., On the Place of Energy Methods in a Global Theory of Hydrodynamic stability, in ”Instability of Continuous Systems”, p. 132–143, H. Leipholz, Ed. Berlin-Heidelberg-New York: Springer 1971.
[20] Fife, P., The Bénard problem for general fluid dynamical equations and remarks on the Boussinesq equation. Indiana Univ. Math. J. 20, 303–326 (1970). · Zbl 0211.29302 · doi:10.1512/iumj.1971.20.20026
[21] Sattinger, D. H., Stability and Bifurcation Theory for Nonlinear Partial Differential Equations. Lecture Notes, University of Minnesota, 1971–72.
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