Attractors for infinite-dimensional non-autonomous dynamical systems.(English)Zbl 1263.37002

Applied Mathematical Sciences 182. Berlin: Springer (ISBN 978-1-4614-4580-7/hbk; 978-1-4614-4581-4/ebook). xxxvi, 409 p. (2013).
This monograph not only summarizes the research of the authors over the last decade, but also provides an accessible and well-written approach to the recent theory of non-autonomous dynamical systems in infinite dimensions with a focus on corresponding attractors and invariant manifolds. The non-autonomous systems under consideration are formulated in terms of processes (2-parameter semigroups) rather than as skew-product flows, and the work focuses on pullback attractors and their structure. While the results are applied to various differential equations generating infinite-dimensional systems, the authors also provide various immediately accessible illustrations using ordinary differential equations.
The contents of the book is subdivided into three parts titled “Abstract theory” (5 chapters), “Invariant manifolds of hyperbolic solutions” (4 chapters) and finally ”Applications” (7 chapters):
The first part begins with an instructive discussion of processes (solution operators to non-autonomous evolutionary equations), their pullback attractors and how they extend the classical autonomous theory – one furthermore finds motivating examples and remarks on the connection to the related notion of a random attractor. Based on non-autonomous omega-limit sets, as well as corresponding attraction and compactification notions, existence results for pullback attractors are deduced. One also finds a discussion of non-autonomous point dissipativity and remarks on more general basins of attraction. Upper- and lower-semicontinuity questions from a further central topic, and also the equivalence between continuity and equi-attraction of attractors is addressed. The chapter on finite-dimensional attractors includes a discussion of the box-counting dimension (of invariant sets), the necessary multi-linear algebra in Hilbert spaces, embedding results into Euclidean spaces and a corresponding theory for non-autonomous sets. The first part closes with a chapter on gradient semigroups and their dynamical properties, where a large part is devoted to the autonomous theory. It contains results on attractor-repeller pairs, Morse decompositions of attractors and associated Lyapunov-functions, as well as perturbation results for gradient systems. Finally, non-autonomous problems of the above type are tackled via perturbation arguments.
In part II the book continues with the necessary existence and uniqueness theory for solutions to evolutionary (partial) differential equations of semilinear form. This includes basics on linear operators, their fractional powers and strongly continuous linear semigroups. As explicit applications, the authors present self-adjoint operators, the Laplacian on $$L^2$$ and $$L^p$$, as well as the Stokes and the wave operators. Essential tools for nonlinear equations are presented next in form of Gronwall inequalities, well-posedness issues, energy estimates yielding global existence, and differentiability or monotonicity properties of the generated processes. Furthermore, the finite-dimensionality of pullback attractors for semilinear equations is addressed. Exponential dichotomies are discussed as appropriate non-autonomous hyperbolicity concept. Here, as in [D. Henry, Geometric theory of semilinear parabolic equations. Berlin etc.: Springer (1981; Zbl 0456.35001)] the discrete time theory is used to obtain robustness and admissibility properties for time-continuous problems being the main focus of the book. Hyperbolic solutions are characterized via a dichotomic variational equation and shown to be robust under bounded perturbations. The associated stable and unstable manifolds are constructed via a Lyapunov-Perron technique, which also yields perturbation results and smooth dependence on parameters. The applications to semilinear evolutionary differential equations include a characterisation of attractors under non-autonomous perturbation (plus continuity results) and corresponding results for asymptotically autonomous equations.
The final part is devoted to various evolutionary differential equations. A warm-up form finite-dimensional problems like the logistic or the Lotka-Volterra equation. Non-autonomous attractors of delay equations are addressed next. Then the Navier-Stokes equations under non-autonomous forcing are studied – the presentation consists of the essential functional-analytical setting, the existence of a pullback attractor, its finite-dimensionality and the situation, where the attractor is a single trajectory. Abstract parabolic equations are tackled next including the necessary comparison results, global well-posedness yielding pullback attraction and eventually the autonomous case with its gradient structure. The Chafee-Infante equation historically serves as a prototype example where a precise characterization is possible in the autonomous case. The authors discuss a non-autonomous generalization, where the $$3$$rd order term in the nonlinearity is equipped with a bounded time-dependent coefficient. After reviewing the autonomous theory (lap numbers etc.), upper- and lower-bounds for the dimension of the pullback attractor are given. Its structure is discussed in terms of non-autonomous equilibria (i.e. entire bounded solutions) and associated unstable manifolds forming the attractor. Here, particular focus is on the case where the non-autonomous term is close to a constant or slowly varying – both cases allow a complete description of the pullback attractor. The next chapter is devoted to perturbations of the diffusive linear part in autonomous reaction-diffusion equations. Here, convergence results on eigenvalues and eigenfunctions are obtained. One also finds results on the rate of convergence for equilibria and invariant manifolds. As final applications the authors address wave equations with a time-depending damping term. After presenting the theoretical basics, a pullback attractor is shown to exist and its regularity properties, as well as a gradient-like structure are deduced and investigated.
The appendix essentially deals with skew-product flows and uniform attractors, as well as their relationship to the central objects of the book, namely processes and pullback attractors.
This book is a well-written and carefully prepared text appropriate for advanced classes on dynamical systems and seminars. It nicely illustrates the progress in infinite-dimensional dynamical systems since the monographs by Henry [loc. cit., Zbl 0456.35001] or J. K. Hale [Asymptotic behavior of dissipative systems. Providence, RI: American Mathematical Society (AMS) (1988, Zbl 0642.58013)]. Consequently, also the experienced researcher in this field will benefit from, and find a useful reference in this work, whose contents are pleasantly disjoint from, e.g., [G. R. Sell and Y. You, Dynamics of evolutionary equations. New York, NY: Springer (2002; Zbl 1254.37002)] or [P. E. Kloeden and M. Rasmussen, Nonautonomous dynamical systems. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1244.37001)].

MSC:

 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory 37B55 Topological dynamics of nonautonomous systems 37C60 Nonautonomous smooth dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37D05 Dynamical systems with hyperbolic orbits and sets 37D10 Invariant manifold theory for dynamical systems 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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