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The Hopf bifurcation for nonlinear semigroups

Marsden, J. (1973) The Hopf bifurcation for nonlinear semigroups. Bulletin of the American Mathematical Society, 79 (3). pp. 537-541. ISSN 0273-0979. doi:10.1090/S0002-9904-1973-13191-X.

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Several authors, have shown by perturbation techniques that the Hopf theorem on the development of periodic stable solutions is valid for the Navier-Stokes equations; in particular, solutions near the stable periodic ones remain defined and smooth for all t ≥ 0 . The principal difficulty is that the Hopf theorem deals with flows of smooth vector fields on finite-dimensional spaces, whereas the Navier-Stokes equations define a flow (or evolution operator) for a nonlinear partial differential operator (actually it is a nonlocal operator). The aim of this note is to outline a method for overcoming this difficulty which is entirely different in appearance from the perturbation approach. The method depends on invariant manifold theory plus certain smoothness properties of the flow which actually hold for the Navier-Stokes flow.

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Additional Information:© American Mathematical Society 1973. Communicated by M. H. Protter, November 6, 1972. The author thanks D. Ruelle for his hospitality at I.H.E.S. in Bures-sur-Yvette where this work was begun, N. Kopell, M. Hirsch and C. Pugh for valuable conversations, and J. Glimm for encouragement.
Subject Keywords:Nonlinear semigroups, Navier-Stokes equations, loss of stability, bifurcation, center manifold theory
Issue or Number:3
Classification Code:AMS (MOS): (1970). Primary 76E30, 35L60, 58BIO
Record Number:CaltechAUTHORS:20100824-120043727
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19637
Deposited By: Ruth Sustaita
Deposited On:01 Sep 2010 21:44
Last Modified:08 Nov 2021 23:53

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