Items related to Infinite-Dimensional Dynamical Systems in Mechanics...
Synopsis
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.-
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