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The Mathematical Theory of Dilute Gases
Perturbations of Equilibria and Space Homogeneous Solutions
other
Author(s):
Carlo Cercignani
,
Reinhard Illner
,
Mario Pulvirenti
Publication date
(Print):
1994
Publisher:
Springer New York
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Author and book information
Book Chapter
Publication date (Print):
1994
Pages
: 191-225
DOI:
10.1007/978-1-4419-8524-8_8
SO-VID:
b1883566-922d-40cb-9f5d-0dd7169c712c
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Book chapters
pp. 1
Introduction
pp. 1
Group and Hamiltonian Structures of Fluid Dynamics
pp. 1
Introduction
pp. 1
Introduction and Basic Concepts
pp. 1
Introduction
pp. 1
Introduction
pp. 1
Normed Spaces
pp. 1
Pseudodifferential Operators
pp. 1
Overview of Book
pp. 1
Introduction
pp. 1
The Direct Method of the Calculus of Variations
pp. 1
The Single First-Order Equation
pp. 1
Inverse Problems
pp. 1
Introduction
pp. 1
Basic Material
pp. 1
Introduction
pp. 1
Introduction
pp. 1
Introduction and Basic Functional Analysis
pp. 3
Manifold, Divergence and Dually Flat Structure
pp. 3
Introduction to Transient Chaos
pp. 3
Preview
pp. 3
Finite Element Interpolation
pp. 4
Historical Introduction
pp. 4
Presentation of the Approach and of the Main Results
pp. 5
Background Material and Notation
pp. 7
Reader’s Guide
pp. 9
Asymptotics of Slow-time Processes, First Steps
pp. 9
Historical Notes
pp. 10
Some Mathematical Tools
pp. 11
A General Initial Value Problem
pp. 12
Genesis of Hysteresis
pp. 13
Existence
pp. 13
Informal Derivation of the Boltzmann Equation
pp. 13
Bounded and Compact Operators
pp. 15
The Transport of Finite-Dimensional Contact Elements
pp. 16
Continuation of Solutions
pp. 17
Bounded and Compact Operators
pp. 20
Ill-Posed Problems and Regularization
pp. 21
Continuous Dependence and Uniqueness
pp. 21
Spectral Blocking Property
pp. 22
Hysteresis Operators
pp. 23
Regularization Theory for Equations of the First Kind
pp. 24
Backward Continuation
pp. 24
Jacobi’s Elliptic Functions
pp. 25
Strong Squeezing Property
pp. 25
The Riesz Theory
pp. 28
The Fenchel Transform and Duality
pp. 29
Cone Invariance Properties
pp. 30
Caratheodory Conditions
pp. 31
Exponential Families and Mixture Families of Probability Distributions
pp. 32
Rheological and Circuital Models
pp. 32
Remarks on the Map Defined by Solutions
pp. 33
Elementary Properties of the Solutions
pp. 33
Riesz Theory
pp. 33
Consequences Regarding the Global Attractor
pp. 33
The Theory of Averaging
pp. 33
Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables
pp. 35
Critical Points and Local Behavior
pp. 36
Local Exponential Decay Toward Blocked Integral Surfaces
pp. 36
Limit Process Expansions for Ordinary Differential Equations
pp. 36
Dual Systems and Fredholm Theory
pp. 37
Nonlinear hyperbolic systems in one space dimension
pp. 37
Transient Chaos in Low-Dimensional Systems
pp. 38
Exponential Decay of Volume Elements and the Dimension of the Global Attractor
pp. 39
Cubical Homology
pp. 39
Uniqueness and Stability in the Cauchy Problem
pp. 39
Essential and Absolute Spectra
pp. 42
Choice of the Initial Manifold
pp. 42
Minimization of the Dual Action
pp. 43
Autonomous Systems
pp. 45
Dual Systems and Fredholm Alternative
pp. 47
Construction of the Inertial Manifold
pp. 47
Definitions of Stability
pp. 50
Regularization in Dual Systems
pp. 50
Elliptic Integrals
pp. 51
Sufficient Conditions for Stability of General Systems
pp. 51
Invariant Geometry of Manifold of Probability Distributions
pp. 52
Lower Bound for the Exponential Rate of Convergence to the Attractor
pp. 54
Characteristic Manifolds and the Cauchy Problem
pp. 55
Asymptotic Completeness: Preparation
pp. 58
Potential Theory
pp. 59
Plays, Stops and Prandtl-Ishlinskiĭ Models
pp. 61
Asymptotic Completeness: Proof of Theorem 12.1
pp. 63
Regularization in Dual Systems
pp. 63
Rigorous Validity of the Boltzmann Equation
pp. 65
Sufficient Conditions for Instability
pp. 65
Regularization by Discretization
pp. 67
Attraction
pp. 68
Stability with Respect to Perturbations
pp. 69
Imperfection Sensitivity Laws
pp. 69
Stability in Autonomous Systems
pp. 71
$$\alpha $$ α -Geometry, Tsallis q-Entropy and Positive-Definite Matrices
pp. 72
Application: The Kuramoto—Sivashinsky Equation
pp. 72
An Example of Levin and Nohel
pp. 73
Elliptic Equations. Single Boundary Measurements
pp. 73
Minimax Theorems for Indefinite Functionals
pp. 74
Spectral Theory
pp. 75
Potential Theory
pp. 75
Asymptotic Stability of Waves in Dissipative Systems
pp. 78
An Equation of Volterra
pp. 79
Crises
pp. 80
Nonhomogeneous Linear Systems
pp. 81
Approximation in Banach Spaces by Galerkin Methods
pp. 82
Singular Integral Equations
pp. 82
Application: A Nonlocal Burgers Equation
pp. 83
Averaging over Spatial Variables: Systems with Slowly Varying Frequency and Passage through Resonance
pp. 87
Worst Imperfection (I)
pp. 88
The “Adjoint” Equation and Representation of Solutions
pp. 91
Stability of Perturbed Systems
pp. 91
Application: The Cahn—Hilliard Equation
pp. 93
Computing Homology Groups
pp. 94
Linear Autonomous Equations. The Semigroup and Infinitesimal Generator
pp. 94
The Laplace Equation
pp. 97
The Preisach Model
pp. 98
The Eigenvalues of a Linear Autonomous Equation. Decomposition of C.
pp. 99
Gas dynamics and reacting flows
pp. 103
Singular Boundary Integral Equations
pp. 104
Decomposing C with the Adjoint Equation
pp. 105
Elliptic Equations: Many Boundary Measurements
pp. 105
Application: A Parabolic Equation in Two Space Variables
pp. 107
Random Imperfection (I)
pp. 107
Noise and Transient Chaos
pp. 108
Sobolev Spaces
pp. 109
Elements of Differential Geometry
pp. 111
Coercive Problems
pp. 111
Application: The Chaffee—Infante Reaction—Diffusion Equation
pp. 111
A Borsuk-Ulam Theorem and Index Theories
pp. 112
Estimates on the Complementary Subspace
pp. 116
An Example
pp. 117
Orbital Stability of Waves in Hamiltonian Systems
pp. 118
Limit Process Expansions for Partial Differential Equations
pp. 120
The Decomposition in the Variation of Constants Formula
pp. 122
Hysteresis and Differential Equations
pp. 124
Normal Forms
pp. 125
Experimentally Observed Bifurcation Diagrams
pp. 125
Forced Linear Systems
pp. 125
Inverse Eigenvalue Problems
pp. 126
Hyperbolic Equations in Higher Dimensions
pp. 126
Lusternik-Schnirelman Theory and Multiple Periodic Solutions with Fixed Energy
pp. 130
The Duhem Model
pp. 130
Rarity and Exponentiality
pp. 131
The Saddle Point Property
pp. 131
Dual Affine Connections and Dually Flat Manifold
pp. 132
The Heat Equation
pp. 133
Existence and Uniqueness Results
pp. 139
Applications of Estimation Theory to Numerical Weather Prediction
pp. 141
Sobolev Spaces
pp. 142
A Fixed Point Theorem for Cones
pp. 142
Operator Approximations
pp. 143
Hamiltonian Systems
pp. 143
Chain Maps and Reduction Algorithms
pp. 144
Scattering problems
pp. 145
Scattering by Obstacles
pp. 147
Fractal Basin Boundaries
pp. 150
Phase Transitions and Hysteresis
pp. 151
Group-Theoretic Bifurcation Theory
pp. 151
Discontinuous Hysteresis
pp. 152
A Periodicity Theorem for Functional Equations
pp. 153
Morse-Ekeland Index and Multiple Periodic Solutions with Fixed Period
pp. 154
Degenerate Kernel Approximation
pp. 155
Singularities and Similarities in Interface Flows
pp. 156
The Equation $${\rm{\dot x}}\left( {\rm{t}} \right) = - \alpha {\rm{x}}\left( {{\rm{t}} - 1} \right)\left[ {{\rm{1}} + {\rm{x}}\left( {\rm{t}} \right)} \right]$$
pp. 159
Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems
pp. 162
The Equation $${\rm{\dot x}}\left( {\rm{t}} \right) = - \alpha {\rm{x}}\left( {{\rm{t}} - 1} \right)\left[ {{\rm{l}} - {\rm{x}}^2 \left( {\rm{t}} \right)} \right]$$
pp. 163
Integral Geometry and Tomography
pp. 164
The Equation $${\rm{\ddot x}}\left( {\rm{t}} \right) + {\rm{f}}\left( {{\rm{x}}\left( {\rm{t}} \right){\rm{\dot x}}\left( {\rm{t}} \right)} \right) + {\rm{g}}\left( {{\rm{x}}\left( {{\rm{t}} - {\rm{r}}} \right)} \right) = 0$$
pp. 165
Asymptotic Theory of Statistical Inference
pp. 167
Finite difference schemes for one-dimensional systems
pp. 167
The Initial Value Problem for the Homogeneous Boltzmann Equation
pp. 167
Morse Theory
pp. 168
Quadrature Methods
pp. 171
The Heat Equation
pp. 173
An Inverse Scattering Problem
pp. 173
Preview of Maps
pp. 175
Mixed Problems
pp. 175
Hysteresis Effects in Shape Memory Alloys
pp. 177
The “Adjoint” Equation for General Linear Systems
pp. 177
Point Spectrum: Linear Hamiltonian Systems
pp. 179
Estimation in the Presence of Hidden Variables
pp. 181
Appendices
pp. 182
The True Adjoint of a Linear System
pp. 183
Operator Approximations
pp. 184
Projection Methods
pp. 184
Hyperbolic Problems
pp. 185
Higher-Order Elliptic Equations with Constant Coefficients
pp. 186
P.D.E. Models of Elasto-Plasticity
pp. 187
Chaotic Scattering
pp. 187
Boundary Value Problems
pp. 191
Neyman-Scott Problem: Estimating Function and Semiparametric Statistical Model
pp. 191
Perturbations of Equilibria and Space Homogeneous Solutions
pp. 195
Differential Geometry of Diffeomorphism Groups
pp. 196
Linear Periodic Systems. General Theory
pp. 199
Bifurcation Behavior of D n -Equivariant Systems
pp. 199
Degenerate Kernel Approximation
pp. 199
Homology of Maps
pp. 203
Decomposition of Linear Periodic Systems
pp. 205
Applications of Morse Theory to Second Order Systems
pp. 206
Parabolic Equations
pp. 206
Iterative Solution and Stability
pp. 211
Hysteresis and Semigroups
pp. 213
Nondegenerate Periodic Orbits
pp. 215
The Evans Function for Boundary-Value Problems
pp. 215
Linear Systems and Time Series
pp. 217
Nondegenerate Critical Manifolds
pp. 218
Phase Field Models with Non-Conserving Kinetics
pp. 218
Inverse parabolic problems
pp. 219
First-Order PDEs
pp. 219
Quadrature Methods
pp. 221
Notes and Remarks
pp. 221
Equations of the First Kind
pp. 226
Boundary Conditions
pp. 231
Machine Learning
pp. 235
Computing Homology of Maps
pp. 235
H. Lewey’s Example of a Linear Equation without Solutions
pp. 239
Quantum Chaotic Scattering and Conductance Fluctuations in Nanostructures
pp. 241
Projection Methods
pp. 241
Dirac Operators and Index Theory
pp. 243
Tikhonov Regularization
pp. 244
Existence Results for Initial-Boundary and Boundary Value Problems
pp. 247
Some Numerical Methods
pp. 249
The Evans Function for Sturm–Liouville Operators on the Real Line
pp. 253
Worst Imperfection (II)
pp. 257
Quasilinear P.D.E.s with Memory
pp. 257
Prospects in Digital Image Processing
pp. 259
Regularization by Discretization
pp. 265
Transient Chaos in Higher Dimensions
pp. 267
The Method of Multiple Scales for Ordinary Differential Equations
pp. 270
Inverse Scattering Theory
pp. 271
Phase Field Models With Conserved Order Parameters
pp. 271
Random Imperfection (II)
pp. 279
Natural Gradient Learning and Its Dynamics in Singular Regions
pp. 279
Iterative Solution and Stability
pp. 279
Time-Dependent Problems
pp. 279
Homological Algebra
pp. 286
Particle Simulation of the Boltzmann Equation
pp. 287
Description and Computation of Bifurcation Behaviors
pp. 295
Semilinear P.D.E.s with Memory
pp. 297
Equations of the First Kind
pp. 301
Brownian Motion and Potential Theory
pp. 303
The case of multidimensional systems
pp. 304
Phase Transitions in Eutectoid Carbon Steels
pp. 305
The Evans Function for nth-Order Operators on the Real Line
pp. 307
Nonlinear Dynamics
pp. 311
Transient Chaos in Spatially Extended Systems
pp. 312
Hydrodynamical Limits
pp. 315
Signal Processing and Optimization
pp. 323
Tikhonov Regularization
pp. 323
Efficient Transformation for Block-Diagonalization
pp. 325
P.D.E.s with Discontinuous Hysteresis
pp. 336
Open Problems and New Directions
pp. 337
Data Structuring and Mesh Generation
pp. 343
Chaotic Advection in Fluid Flows
pp. 351
Regularization by Discretization
pp. 357
Quadratures, Assembling, and Storage
pp. 362
Some Tools
pp. 365
Inverse Boundary Value Problems
pp. 367
Bifurcation of Cylindrical Sand Specimens
pp. 377
Homology of Topological Polyhedra
pp. 383
Linear Algebra
pp. 385
Controlling Transient Chaos and Applications
pp. 389
The ̄∂-Neumann Problem
pp. 389
Conclusion
pp. 395
Echelon-Mode Formation
pp. 397
Topology
pp. 410
Near-Identity Averaging Transformations: Transient and Sustained Resonance
pp. 413
Transient Chaotic Time-Series Analysis
pp. 417
An introduction to boundary conditions
pp. 419
Algebra
pp. 421
A Posteriori Error Estimates and Adaptive Meshes
pp. 451
Bifurcation of Steel Specimens
pp. 451
Syntax of Algorithms
pp. 460
Connections and Curvature
pp. 471
Flower Patterns on Honeycomb Structures
pp. 522
Multiple-Scale Expansions for Partial Differential Equations
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