Perturbations of foliated bundles and evolutionary equations
Tóm tắt
In two earlier papers, we presented a perturbation theory for laminated, or foliated, invariant sets
$\mathcal{K}^o$
for a given finite-dimensional system of ordinary differential equations, see [20,21]. The main objective in that perturbation theory is to show that: if the given vector field has a suitable exponential trichotomy on
$\mathcal{K}^o$
, then any perturbed system that is C1-close to the given vector field near
$\mathcal{K}^o$
has an invariant set
$\mathcal{K}^n$
, where
$\mathcal{K}^n$
is homeomorphic to
$\mathcal{K}^o$
and where the perturbed vector field has an exponential trichotomy on
$\mathcal{K}^n$
. In this paper we present a dual-faceted extension of this perturbation theory to include: (1) a class of infinite-dimensional evolutionary equations that arise in the study of reaction diffusion equations and the Navier–Stokes equations and (2) nonautonomous evolutionary equations in both finite and infinite dimensions. For the nonautonomous problem, we require that the time-dependent terms in the problem lie in a compact, invariant set M. For example, M may be the hull of an almost periodic, or a quasiperiodic, function of time.
Tài liệu tham khảo
Agmon, S., Nirenberg, L.: Properties of solutions of ordinary differential equations in Banach space. Commun. Pure Appl. Math. 16, 121–239 (1963)
Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov 90, 209 (1967)
Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer 1983
Bardos, C., Tartar, L.: Sur l’unicité rétrograde des équations paraboliques et quelques équations voisine. Arch. Ration. Mech. Anal. 50, 10–25 (1973)
Bates, P.W., Lu, K., Zeng, C.: Existence and persistence of invariant manifolds for semiflows in Banach spaces. Mem. Am. Math. Soc. 135 no. 645 (1998)
Besicovitch, A.S.: Almost Periodic Functions. New York: Dover 1932/1955
Cartwright, M.L.: Almost periodic flows and solutions of differential equations. Proc. Lond. Math. Soc. 17, 355–380 (1967)
Cartwright, M.L.: Almost periodic equations and almost periodic flows. J. Differ. Equations 5, 167–181 (1969)
Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)
Fink, A.M.: Almost Periodic Differential Equations. Lect. Notes Math., vol. 377. Berlin, Heidelberg, New York: Springer 1974
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Math Surveys and Monographs, vol. 25. Providence, RI: Am. Math. Soc. 1988
Henry, D.B.: Geometric Theory of Semilinear Parabolic Equations. Lect. Notes Math., vol. 840. Berlin, Heidelberg, New York: Springer 1981
Henry, D.B.: Some infinite dimensional Morse–Smale systems defined by parabolic partial differential equations. J. Differ. Equations 53, 401–458 (1985)
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lect. Notes Math., vol. 583. Berlin, Heidelberg, New York: Springer 1977
Kamaev, D.A.: Hyperbolic limit sets of evolutionary equations and the Galerkin method. Russ. Math. Surv. 35, 239–243 (1980)
Kelley, J.L., Namioka, I.: Linear Topological Spaces. Grad. Texts Math., no. 36. Berlin, Heidelberg, New York: Springer 1976
Lions, J.L., Malgrange, B.: Sur l’unicité rétrograde dans les problèmes mixtes paraboliques. Math. Scand. 8, 277–286 (1960)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci., vol. 44. Berlin, Heidelberg, New York: Springer 1983
Pilyugin, S.Y.: Introduction to Structurally Stable Systems of Differential Equations. Russian version: Vvedenie v grubye sistemy differentsial’nykh uravneniy (1988). Boston: Birkhäuser 1992
Pliss, V.A., Sell, G.R.: Perturbations of attractors of differential equations. J. Differ. Equations 92, 100–124 (1991)
Pliss, V.A., Sell, G.R.: Approximation dynamics and the stability of invariant sets. J. Differ. Equations 149, 1–51 (1998)
Pliss, V.A., Sell, G.R.: Robustness of exponential dichotomies in infinite dimensional dynamical systems. J. Dyn. Differ. Equations 11, 471–513 (1999)
Pliss, V.A., Sell, G.R.: On the stability of normally hyperbolic sets of smooth flows. Dokl. Akad. Nauk 378, 153–154 (2000)
Pliss, V.A., Sell, G.R.: Perturbations of normally hyperbolic manifolds with applications to the Navier–Stokes equations. J. Differ. Equations 169, 396–492 (2001)
Pugh, C., Shub, M., Starkov, A.: Stable ergodicity. Bull. Am. Math. Soc. 41, 1–41 (2004)
Raugel, G., Sell, G.R.: Navier–Stokes equations on thin 3D domains III: Global and local attractors. In: Sell, G.R., Foias, C., Temam, R. (eds.): Turbulence in Fluid Flows: A Dynamical Systems Approach. IMA Volumes in Mathematics and its Applications, vol. 55 pp. 137–163. Berlin, Heidelberg, New York: Springer 1993
Robinson, C., Young, L.-S.: Nonabsolutely continuous foliations for an Anosov diffeomorphism. Invent. Math. 61, 159–176 (1980)
Sacker, R.J.: A perturbation theorem for invariant manifolds and Hölder continuity. J. Math. Mech. 18, 705–762 (1969)
Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems, I. J. Differ. Equations 15, 429–458 (1974)
Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems, II. J. Differ. Equations 22, 478–496 (1976)
Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems, III. J. Differ. Equations 22, 497–522 (1976)
Sell, G.R.: Nonautonomous differential equations and topological dynamics: I. The basic theory; and II. Limiting equations. Trans. Am. Math. Soc. 127, 241–262, 263–283 (1967)
Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Appl. Math. Sci., vol. 143. Berlin, Heidelberg, New York: Springer 2002
Shen, W., Yi, Y.: Almost automorphic and almost periodic dynamics in skew product semiflows. Mem. Am. Math. Soc. 136, no. 647 (1998)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)
Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Berlin, Heidelberg, New York: Springer 1988