An Introduction to Semiflows

An Introduction to Semiflows

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This book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs. The authors concentrate on three types of absorbing sets: attractors, exponential attractors, and inertial manifolds. They present the fundamental properties of these sets, and then proceed to show the existence of some of these sets for a number of dynamical systems generated by well-known physical models. In particular, they consider in full detail two particular PDEEs: a semilinear version of the heat equation and a corresponding version of the dissipative wave equation. These examples illustrate the most important features of the theory of semiflows and provide a sort of template that can be applied to the analysis of other models. The material builds in a careful, gradual progression, developing the background needed by newcomers to the field, and culminating in a more detailed presentation of the main topics than found in most sources. The authors' approach to and treatment of the subject builds the foundation for more advanced references and research on global attractors, exponential attractors, and inertial more

Product details

  • Hardback | 386 pages
  • 160 x 234 x 26mm | 698.54g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • New.
  • 26 black & white illustrations, 4 black & white tables, 7 black & white halftones
  • 1584884584
  • 9781584884583

Review quote

"I am convinced that the authors have made this mathematically-demanding area as accessible as possible to the newcomer in the field. The pictures and 'simple ODE applications' help the reader gain intuition. This book does a fine job of presenting difficult material on applications to readers without in-depth knowledge of partial differential equations in a painless and enjoyable way." - Hal Smith, Arizona State Universityshow more

Table of contents

DYNAMICAL PROCESSES Introduction Ordinary Differential Equations Attracting Sets Iterated Sequences Lorenz' Equations Duffing's Equation Summary ATTRACTORS OF SEMIFLOWS Distance and Semidistance Discrete and Continuous Semiflows Invariant Sets Attractors Dissipativity Absorbing Sets and Attractors Attractors via a-Contractions Fractal Dimension A Priori Estimates ATTRACTORS FOR SEMILINEAR EVOLUTION EQUATIONS PDEEs as Dynamical Systems Functional Framework The Parabolic Problem The Hyperbolic Problem Regularity Upper Semicontinuity of the Global Attractors EXPONENTIAL ATTRACTORS Introduction The Discrete Squeezing Property The Parabolic Problem The Hyperbolic Problem Proof of Theorem 4.4 Concluding Remarks INERTIAL MANIFOLDS Introduction Definitions and Comparisons Geometric Assumptions on the Semiflow Strong Squeezing Property and Inertial Manifolds A Modification Inertial Manifolds for Evolution Equations Applications Semilinear Evolution Equations in One Space Dimension EXAMPLES Cahn-Hilliard Equations Beam and von Karman Equation Navier-Stokes Equations Maxwell's Equations A NON-EXISTENCE RESULT FOR INERTIAL MANIFOLDS The Initial-Boundary Value Problem Overview of the Argument The Linearized Problem Inertial Manifolds for the Linearized Problem C1 Linearization Equivalence Perturbations of the Nonlinear Flow Asymptotic Properties of the Perturbed Flow The Non-Existence Result Proof of Proposition 7.17 The C1 Linearization Equivalence Theorems. APPENDIX: SELECTED RESULTS FROM ANALYSIS A.1 Ordinary Differential Equations A.2 Linear Spaces and their Duals A.3 Semigroups of Linear Operators A.4 Lebesgue Spaces A.5 Sobolev Spaces of Scalar Valued Functions A.6 Sobolev Spaces of Vector Valued Functions A.7 The Spaces H(div,W) and H(curl,W) A.8 Almost Periodic Functions BIBLIOGRAPHY INDEX NOMENCLATUREshow more