This is the first attempt at a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics. Other areas of science and technology are included where appropriate. The relation between infinite and finite dimensional systems is presented from a synthetic viewpoint and equations considered include reaction-diffusion, Navier-Stokes and other fluid mechanics equations, magnetohydrodynamics, thermohydraulics, pattern formation, Ginzburg-Landau, damped wave and an introduction to inertial manifolds.

General Introduction. The User's Guide.- 1. Mechanism and Description of Chaos. The Finite-Dimensional Case.- 2. Mechanism and Description of Chaos. The Infinite-Dimensional Case.- 3. The Global Attractor. Reduction to Finite Dimension.- 4. Remarks on the Computational Aspect.- 5. The User's Guide.- I General Results and Concepts on Invariant Sets and Attractors.- 1. Semigroups, Invariant Sets, and Attractors.- 1.1. Semigroups of Operators.- 1.2. Functional Invariant Sets.- 1.3. Absorbing Sets and Attractors.- 1.4. A Remark on the Stability of the Attractors.- 2. Examples in Ordinary Differential Equations.- 2.1. The Pendulum.- 2.2. The Minea System.- 2.3. The Lorenz Model.- 3. Fractal Interpolation and Attractors.- 3.1. The General Framework.- 3.2. The Interpolation Process.- 3.3. Proof of Theorem 3.1.- II Elements of Functional Analysis.- 1. Function Spaces.- 1.1. Definition of the Spaces. Notations.- 1.2. Properties of Sobolev Spaces.- 1.3. Other Sobolev Spaces.- 1.4. Further Properties of Sobolev Spaces.- 2. Linear Operators.- 2.1. Bilinear Forms and Linear Operators.- 2.2. "Concrete" Examples of Linear Operators.- 3. Linear Evolution Equations of the First Order in Time.- 3.1. Hypotheses.- 3.2. A Result of Existence and Uniqueness.- 3.3. Regularity Results.- 3.4. Time-Dependent Operators.- 4. Linear Evolution Equations of the Second Order in Time.- 4.1. The Evolution Problem.- 4.2. Another Result.- 4.3. Time-Dependent Operators.- III Attractors of the Dissipative Evolution Equation of the First Order in Time: Reaction-Diffusion Equations. Fluid Mechanics and Pattern Formation Equations.- 1. Reaction-Diffusion quations.- 1.1. Equations with a Polynomial Nonlinearity.- 1.2. Equations with an Invariant Region.- 2. Navier-Stokes Equations (n = 2).- 2.1. The Equations and Their Mathematical Setting.- 2.2. Absorbing Sets and Attractors.- 2.3. Proof of Theorem 2.1.- 3. Other Equations in Fluid Mechanics.- 3.1. Abstract Equation. General Results.- 3.2. Fluid Driven by Its Boundary.- 3.3. Magnetohydrodynamics (MHD).- 3.4. Geophysical Flows (Flows on a Manifold).- 3.5. Thermohydraulics.- 4. Some Pattern Formation Equations.- 4.1. The Kuramoto-Sivashinsky Equation.- 4.2. The Cahn-Hilliard Equation.- 5. Semilinear Equations.- 5.1. The Equations. The Semigroup.- 5.2. Absorbing Sets and Attractors.- 5.3. Proof of Theorem 5.2.- 6. Backward Uniqueness.- 6.1. An Abstract Result.- 6.2. Applications.- IV Attractors of Dissipative Wave Equations.- 1. Linear Equations: Summary and Additional Results.- 1.1. The General Framework.- 1.2. Exponential Decay.- 1.3. Bounded Solutions on the Real Line.- 2. The Sine-Gordon Equation.- 2.1. The Equation and Its Mathematical Setting.- 2.2. Absorbing Sets and Attractors.- 2.3. Other Boundary Conditions.- 3. A Nonlinear Wave Equation of Relativistic Quantum Mechanics.- 3.1. The Equation and Its Mathematical Setting.- 3.2. Absorbing Sets and Attractors.- 4. An Abstract Wave Equation.- 4.1. The Abstract Equation. The Group of Operators.- 4.2. Absorbing Sets and Attractors.- 4.3. Examples.- 4.4. Proof of Theorem 4.1 (Sketch).- 5. A Nonlinear SchrÖdinger Equation.- 5.1. The Equation and Its Mathematical Setting.- 5.2. Absorbing Sets and Attractors.- 6. Regularity of Attractors.- 6.1. A Preliminary Result.- 6.2. Example of Partial Regularity.- 6.3. Example of ?? Regularity.- 7. Stability of Attractors.- V Lyapunov Exponents and Dimension of Attractors.- 1. Linear and Multilinear Algebra.- 1.1. Exterior Product of Hilbert Spaces.- 1.2. Multilinear Operators and Exterior Products.- 1.3. Image of a Ball by a Linear Operator.- 2. Lyapunov Exponents and Lyapunov Numbers.- 2.1. Distortion of Volumes Produced by the Semigroup.- 2.2. Definition of the Lyapunov Exponents and Lyapunov Numbers.- 2.3. Evolution of the Volume Element and Its Exponential Decay: The Abstract Framework.- 3. Hausdorff and Fractal Dimensions of Attractors.- 3.1. Hausdorff and Fractal Dimensions.- 3.2. Covering Lemmas.- 3.3. The Main Results.- 3.4. Application to Evolution Equations.- VI Explicit Bounds on the Number of Degrees of Freedom and the Dimension of Attractors of Some Physical Systems.- 1. The Lorenz Attractor.- 2. Reaction-Diffusion quations.- 2.1. Equations with a Polynomial Nonlinearity.- 2.2. Equations with an Invariant Region.- 3. Navier-Stokes Equations (n =2).- 3.1. General Boundary Conditions.- 3.2. Improvements for the Space-Periodic Case.- 4. Other Equations in Fluid Mechanics.- 4.1. The Linearized Equations (The Abstract Framework).- 4.2. Fluid Driven by Its Boundary.- 4.3. Magnetohydrodynamics.- 4.4. Flows on a Manifold.- 4.5. Thermohydraulics.- 5. Pattern Formation quations.- 5.1. The Kuramoto-Sivashinsky Equation.- 5.2. The Cahn-Hilliard Equations.- 6. Dissipative Wave quations.- 6.1. The Linearized Equation.- 6.2. Dimension of the Attractor.- 6.3. Sine-Gordon Equations.- 6.4. Some Lemmas.- 7. A Nonlinear chrÖdinger Equation.- 7.1. The Linearized Equation.- 7.2. Dimension of the Attractor.- 8. Differentiability of the emigroup.- VII Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions.- A: NON-WELL-POSED PROBLEMS.- 1. Dissipativity and Well Posedness.- 1.1. General Definitions.- 1.2. The Class of Problems Studied.- 1.3. The Main Result.- 2. Estimate of Dimension for Non-Well-Posed Problems: Examples in Fluid Dynamics.- 2.1. The Equations and Their Linearization.- 2.2. Estimate of the Dimension of X.- 2.3. The Three-Dimensional Navier-Stokes Equations.- B: UNSTABLE MANIFOLDS, LYAPUNOV FUNCTIONS, AND LOWER BOUNDS ON DIMENSIONS.- 3. Stable and Unstable Manifolds.- 3.1. Structure of a Mapping in the Neighborhood of a Fixed Point.- 3.2. Application to Attractors.- 3.3. Unstable Manifold of a Compact Invariant Set.- 4. The Attractor of a Semigroup with a Lyapunov Function.- 4.1. A General Result.- 4.2. Additional Results.- 4.3. Examples.- 5. Lower Bounds on imensions of Attractors: An Example.- VIII The Cone and Squeezing Properties. Inertial Manifolds.- 1. The Cone Property.- 1.1. The Cone Property.- 1.2. Generalizations.- 1.3. The Squeezing Property.- 2. Construction of an Inertial Manifold: Description of the Method.- 2.1. Inertial Manifolds: The Method of Construction.- 2.2. The Initial and Prepared Equations.- 2.3. The Mapping.- 3. Existence of an Inertial Manifold.- 3.1. The Result of Existence.- 3.2. First Properties of ? ?.- 3.3. Utilization of the Cone Property.- 3.4. Proof of Theorem 3.1 (End).- 3.5. Another Form of Theorem 3.1.- 4. Examples.- 4.1. Example 1: The Kuramoto-Sivashinsky Equation.- 4.2. Example 2: Approximate Inertial Manifolds for the Navier-Stokes Equations.- 4.3. Example 3: Reaction-Diffusion Equations.- 4.4. Example 4: The Ginzburg-Landau Equation.- 5. Approximation and Stability of the Inertial Manifold with Respect to Perturbations.- APPENDIX Collective Sobolev Inequalities.- 1. Notations and Hypotheses.- 1.1. The Operator 31.- 1.2. The SchrÖdinger-Type Operators.- 2. Spectral Estimates for SchrÖdinger-Type Operators.- 2.1. The Birman-Schwinger Inequality.- 2.2. The Spectral Estimate.- 3. Generalization of the Sobolev-Lieb-Thirring Inequality (I).- 4. Generalization of the Sobolev-Lieb-Thirring Inequality (II).- 4.1. The Space-Periodic Case.- 4.2. The General Case.- 4.3. Proof of Theorem 4.1.- 5. Examples.