Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T03:59:52.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Applications of Leray–Schauder degree theory to problems of hydrodynamic stability

Published online by Cambridge University Press:  24 October 2008

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Fluid Mechanics Research Institute, University of Essex
Get access Rights & Permissions [Opens in a new window]

Abstract

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 2 , March 1976 , pp. 373 - 392
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Benjamin, T. B.A unified theory of conjugate flows. Phil. Trans. R. Soc. Lond. A 269 (1971), 587–647.Google Scholar
(2)Berger, M. and Berger, M.Perspectives in nonlinearity (Benjamin, New York, 1968).Google Scholar
(3)Caandrasekhar, S.Hydrodynamic and hydromagnetic stability (Oxford University Press, 1961).Google Scholar
(4)Crandall, M. G. and Rabinowitz, P. H.Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rat. Mech. Anal. 52 (1974), 161–180.CrossRefGoogle Scholar
(5)Finn, R.Stationary solutions of the Navier-Stokes equations. Proc. Symposia Appl. Math. 17 Amer. Math. Soc. (1965), 121–153.CrossRefGoogle Scholar
(6)Gavalas, G. R.Nonlinear differential equations of chemically reacting systems (Springer-Verlag, Berlin, 1968).CrossRefGoogle Scholar
(7)Joseph, D. D.Stability of convection in containers of arbitrary shape. J. Fluid Mech. 47 (1971), 257–282.CrossRefGoogle Scholar
(8)Joseph, D. D. and Sattinger, D. H.Bifurcating time periodic solutions and their stability. Arch. Rat. Mech. Anal. 45 (1972), 75–109.CrossRefGoogle Scholar
(9)Kirchgässner, K. and Sorger, P.Branching analysis for the Taylor problem. Quart. J. Mech. Appl. Math. 22 (1969), 183–209.CrossRefGoogle Scholar
(10)Krasnosel'skii, M. A.Topological methods in the theory of nonlinear integral equations (Pergamon, London, 1964).Google Scholar
(11)Krasnosel'skii, M. A.Positive solutions of operator equations (Noordhoff, Groningen, 1964).Google Scholar
(12)Ladyzhenskaya, O. A.The mathematical theory of viscous incompressible flow, 2nd edn. (Gordon and Breach, New York, 1969).Google Scholar
(13)Leray, J. and Schauder, J.Topologie et équations fonctionnelles. Ann. Sci. Ecole Norm. Sup., série 3, 51 (1934), 45–78.CrossRefGoogle Scholar
(14)Prodi, G.Theorem di tipo locale per il sistema di Navier-Stokes e stabilità delle soluzioni stazionarie. Rend. Sem. Mat. Univ. Padova 32 (1962), 374–397.Google Scholar
(15)Sattinger, D. H.The mathematical problem of hydrodynamic stability. J. Math. Mech. 19 (1970), 797–817.Google Scholar
(16)Sattinger, D. H.Stability of bifurcating solutions by Leray-Schauder degree. Arch. Rat. Mech. Anal. 43 (1971), 154–166.CrossRefGoogle Scholar
(17)Sattinger, D. H.Topics in stability and bifurcation theory (Springer-Verlag, Berlin, 1973).CrossRefGoogle Scholar
(18)Schwartz, J. T.Non-linear functional analysis. (Gordon and Breach, New York, 1969).Google Scholar
(19)Serrin, J.On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3 (1959), 1–13.CrossRefGoogle Scholar
(20)Velte, W.Stabilitätsverhalten und Verzweigung stationärer Losungen der Navier-Stokes-schen Gleichungen. Arch. Rat. Mech. Anal. 16 (1964), 97–125.CrossRefGoogle Scholar
(21)Yudovich, V. I.Stability of stress flows of viscous incompressible fluids. Dokl. Akad. Nauh. SSSR 161 (1965), 1037–1040.Google Scholar
(22)Yudovich, V. I.On the origin of convection. Prikl. Mat. Mek. (J. Appl. Math. Mech.) 30 no. 6 (1966), 1193–1199.Google Scholar
(23)Zabreiko, P. P. and Krasnosel'skii, M. A.Calculation of the index of a fixed point of a vector field. Sibirsk. Mat. Z. 5 (1964), 509–531.Google Scholar