Invariant manifolds for flows in Banach spaces.

*(English)*Zbl 0691.58034We present a theory of smooth invariant manifolds based on the classical method of Lyapunov-Perron for continuous semiflows in Banach spaces. Basic hypotheses for these semiflows will be satisfied by semilinear parabolic equations on bounded or unbounded domains or hyperbolic equations. Examples of these continuous semigroups from evolution equations may be found in P. Bates and C. Jones [The center manifold theorem with applications (preprint)]. The two basic theorems are stated for nonlinear integral equations. One is on the existence of smooth invariant manifolds (Theorem 4.4) and the other is on exponential attractivity of invariant manifolds (Theorem 5.1).

##### MSC:

37C80 | Symmetries, equivariant dynamical systems (MSC2010) |

35K55 | Nonlinear parabolic equations |

35L70 | Second-order nonlinear hyperbolic equations |

##### Keywords:

inertial manifolds; smooth invariant manifolds; Lyapunov-Perron; continuous semiflows; semilinear parabolic equations; hyperbolic equations
PDF
BibTeX
XML
Cite

\textit{S.-N. Chow} and \textit{K. Lu}, J. Differ. Equations 74, No. 2, 285--317 (1988; Zbl 0691.58034)

Full Text:
DOI

##### References:

[1] | Adams, R.A, Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 |

[2] | Babin, A.V; Vishik, M.I, Unstable invariant sets of semigroups of nonlinear operators and their perturbations, Russian math. surveys, 41, 1-41, (1986) · Zbl 0624.47065 |

[3] | \scP. Bates and C. K. R. T. Jones, The center manifold theorem with applications, preprint. |

[4] | Carr, J, Application of center manifold theory, () |

[5] | Chow, S.-N; Hale, J.K, Methods of bifurcation theory, (1982), Springer-Verlag New York |

[6] | \scS.-N. Chow and K. Lu, Ck center unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, in press. |

[7] | \scS.-N. Chow, K. Lu, and G. R. Sell, to appear. |

[8] | \scP. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, to appear. · Zbl 0683.58002 |

[9] | Constantin, P; Foias, C; Temam, R, Attractors representing turbulent flows, Mem. amer. math. soc., 314, (1985) · Zbl 0567.35070 |

[10] | Conway, E; Hoff, D; Smoller, J, Large time behavior of solutions of non-linear reaction-diffusion equations, SIAM J. appl. math., 35, 1-16, (1978) · Zbl 0383.35035 |

[11] | \scC. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low dimensional behavior in the complex Giuzburg-Landau equation, preprint. · Zbl 0741.35079 |

[12] | Foias, C; Nicolaenko, B; Sell, G.R; Temam, R, Variétés inertielles pour l’équation de Kuramoto-Sivashinsky, C. R. acad. sci. Paris, Sér. I math., 301, 285-288, (1985) · Zbl 0591.35063 |

[13] | \scC. Foias, B. Nicolaenko, G. R. Sell, and R. Temam, Inertial manifold for the Kuramoto Sivashinsky equation. IMA preprint, No. 285. · Zbl 0591.35063 |

[14] | \scC. Foias, G. R. Sell, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, in press. · Zbl 0643.58004 |

[15] | van Gils, S.A; Vanderbauwhede, A, Center manifolds and contractions on a scale of Banach spaces, J. funct. anal., 72, 209-224, (1987) · Zbl 0621.47050 |

[16] | Hale, J.K, Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048 |

[17] | \scJ. K. Hale, Asymptotic behavior of dissipative systems, Amer. Math. Soc., in press. · Zbl 0642.58013 |

[18] | Hale, J.K; Lin, X.B, Symbolic dynamics and nonlinear flows, Ann. mat. pura appl. (4), 144, 229-260, (1986) |

[19] | Hale, J.K; Magalhaes, L.T; Oliva, W.M, An introduction to infinite dimensional dynamical systems-geometric theory, () · Zbl 0533.58001 |

[20] | \scJ. K. Hale and G. Raugel, Upper semi continuity of the attractor for a singularly perturbed hyperbolic equation, preprint. · Zbl 0666.35012 |

[21] | Henry, D, Geometric theory of parabolic equation, () |

[22] | Hirsch, M; Pugh, C, Stable manifolds and hyperbolic sets, (), 133-1163 |

[23] | Mallet-Paret, J, Negatively invariant sets of compact maps and an extension of a theorem of cartwright, J. differential equations, 22, 331-348, (1976) · Zbl 0354.34072 |

[24] | \scJ. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, IMA preprint, No. 331. · Zbl 0674.35049 |

[25] | Mane, R, On the dimension of the compact invariant sets of certain nonlinear maps, (), 230-242 |

[26] | \scJ. Marsden and J. Scheurle, The construction and smoothness of invariant manifolds by the deformation method, preprint. · Zbl 0551.58024 |

[27] | Nicolaenko, B; Scheurer, B; Temam, R, Some global dynamical properties of the Kuramoto Sivashinsky equations: nonlinear stability and attractors, Phys. D, 16, 155-183, (1985) · Zbl 0592.35013 |

[28] | \scX. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations. In “Contribution to Nonlinear Partial Differential Equations” (I. Diaz and P. L. Lions, Eds.), Longmans, Green, New York, to appear. · Zbl 0642.35061 |

[29] | Mora, X; Sola-Morales, J, Existence and nonexistence of finite dimensional globally attracting invariant manifolds in semilinear damped wave equations, () · Zbl 0642.35062 |

[30] | \scX. Mora and J. Sola-Morales, Diffusion equations as singular limits of damped wave equations, to appear. · Zbl 0699.35178 |

[31] | Pazy, A, Semigroups of linear operators and applications to partial differential equations, () · Zbl 0516.47023 |

[32] | Wells, J.C, Invariant manifolds of nonlinear operators, Pacific J. math., 62, 285-293, (1976) · Zbl 0343.58010 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.