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A centre manifold theorem for hyperbolic PDEs

Published online by Cambridge University Press:  14 November 2011

Michael Renardy
Michael Renardy
Affiliation:
Department of Mathematics and ICAM, Virginia Tech, Blacksburg, VA 24061–0123, U.S.A.
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Synopsis

A version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations. As an application, the Benard problem for a viscoelastic fluid is discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 363 - 377
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Ball, J. M.. Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam. In Nonlinear Elasticity, ed. W, R.. Dickey, 93–160 (New York: Academic Press, 1973).Google Scholar
2Bates, P. and Jones, C. K.. Invariant manifolds for semilinear partial differential equations. Dynamics Reported 2 (1989), 1–38.CrossRefGoogle Scholar
3Carr, J.. Applications of Centre Manifold Theory (Berlin: Springer, 1981).CrossRefGoogle Scholar
4Chow, S. N. and Lu, K.. Ck center unstable manifolds. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 303–320.CrossRefGoogle Scholar
5Hughes, T. J. R., Kato, T. and Marsden, J. E.. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63 (1976), 273–284.CrossRefGoogle Scholar
6Kelley, A.. The stable, center-stable, center, center-unstable, unstable manifolds. J. Differential Equations 3 (1967), 546–570.CrossRefGoogle Scholar
7Kato, T.. Linear evolution equations of “hyperbolic” type II. J. Math. Soc. Japan 25 (1973) 648–666.CrossRefGoogle Scholar
8Kato, T.. Quasi-linear equations of evolution with application to partial differential equations. In Spectral Theory of Differential Equations, ed. Everitt, W. N.. Lecture Notes in Mathematics 448, 25–70 (Berlin: Springer, 1975).CrossRefGoogle Scholar
9Mielke, A.. Locally invariant manifolds for quasilinear parabolic equations. Rocky Mountain J. Math. 21 (1991), 707–714.CrossRefGoogle Scholar
10Da, G. Prato and Lunardi, A.. Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations. Arch. Rational Mech. Anal. 101 (1988), 115–141.Google Scholar
11and, M.Renardy, Y.. Pattern selection in the Benard problem for a viscoelastic fluid. Z. Angew. Math. Phys. 43 (1992), 154–180.Google Scholar
12Scarpellini, B.. Center manifolds of infinite dimensions. Z. Angew. Math. Phys. 42 (1991), 1–32; 280–314.CrossRefGoogle Scholar