Geschäftskunden Kundenprogramme
Vor Ort
Merkzettel Warenkorb
Produktbild: Infinite-Dimensional Dynamical
Band 68

Infinite-Dimensional Dynamical

Aus der Reihe Applied Mathematical Sciences

Fr. 198.00

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

01.07.2012

Verlag

Springer

Seitenzahl

500

Maße (L/B/H)

23.4/15.6/2.7 cm

Gewicht

721 g

Auflage

Softcover reprint of the original 1st ed. 1988

Sprache

Englisch

ISBN

978-1-4684-0315-2

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

01.07.2012

Verlag

Springer

Seitenzahl

500

Maße (L/B/H)

23.4/15.6/2.7 cm

Gewicht

721 g

Auflage

Softcover reprint of the original 1st ed. 1988

Sprache

Englisch

ISBN

978-1-4684-0315-2

Noch keine Bewertungen vorhanden

Verfassen Sie die erste Bewertung zu diesem Artikel

Helfen Sie anderen Kundinnen und Kunden durch Ihre Meinung.

Kundinnen und Kunden meinen

Bewertungen (0)
  • Produktbild: Infinite-Dimensional Dynamical
  • 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.