Trends in Nonlinear Analysis

By: M. Kirkilionis, S. Kromker, R. Rannacher

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Hardcover | 27 November 2002

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442 Pages

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23.5 x 15.88 x 2.54

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Applied mathematics is a central connecting link between scientific observations and their theoretical interpretation. Nonlinear analysis has surely contributed major developments which nowadays shape the face of applied mathematics. At the beginning of the millennium, all sciences are expanding at increased speed. Technological, ecological, economical and medical problem solving is a central issue of every modern society. Mathematical models help to expose fundamental structures hidden in these problems and serve as unifying tools to deepen our understanding. What are the new challenges applied mathematics has to face with the increased diversity of scientific problems? In which direction should the classical tools of nonlinear analysis be developed further? How do new available technologies influence the development of the field? How can problems be solved which have been beyond reach in former times? It is the aim of this book to explore new developments in the field by way of discussion of selected topics from nonlinear analysis.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Science Physics Electricity Differential Calculus & Equations Biology, Life Sciences Applied Mathematics Nonlinear Science Computing & I.T.

Computer Science Engineering & Technology Environmental Science Environmental Monitoring

Interview with Willi Jager	p. 1
Spatio-Temporal Dynamics of Reaction-Diffusion Patterns	p. 23
Introduction and Overview	p. 23
One Space Dimension: Global Attractors	p. 27
Lyapunov Functions, Comparison Principles, and Sturm Property	p. 27
Lyapunov functions	p. 27
Comparison principles	p. 29
Sturm property, revisited	p. 30
Sturm Attractors on the Interval	p. 33
Global attractors	p. 33
Sturm attractors and Sturm permutations	p. 34
Sturm permutations and heteroclinics	p. 38
Combinatorics of Sturm attractors	p. 41
Sturm Attractors on the Circle	p. 45
Poincare-Bendixson theory	p. 45
Heteroclinic connections of rotating waves	p. 47
One Unbounded Space-Dimension: Travelling Waves	p. 51
Unbounded Domains and Essential Spectra	p. 51
From bounded to unbounded domains	p. 51
Spectra of travelling waves: group velocities and Fredholm indices	p. 53
Instabilities of Travelling Waves	p. 60
Instability of a front caused by point spectrum	p. 61
The Turing instability	p. 61
Essential Hopf instability of a front	p. 64
Instability of a pulse caused by the essential spectrum	p. 67
Fredholm indices and essential instabilities	p. 67
Spatial dynamics and essential instabilities	p. 70
From Unbounded to Large Domains: Absolute Versus Essential Spectra	p. 74
Two Space Dimensions: Existence of Spiral Waves	p. 80
Kinematics and its Defects	p. 80
Curvature flow of Archimedean spirals	p. 82
The front-back matching problem	p. 83
Archimedean Spiral Waves in Radial Dynamics	p. 86
Rigid rotation and asymptotic wavetrains	p. 86
Linear and nonlinear group velocities	p. 88
Characterizing Archimedean spirals	p. 89
Bifurcation to Spiral Waves	p. 91
Two Space Dimensions: Bifurcations from Spiral Waves	p. 93
Phenomenology of Spiral Instabilities	p. 93
Meandering Spirals and Euclidean Symmetry	p. 95
Euclidean equivariance	p. 96
Relative center manifolds	p. 97
Palais coordinates	p. 98
Spiral tip motion, Hopf meandering, and drift resonance	p. 99
Relative normal forms	p. 102
Relative Hopf resonance	p. 104
Relative Takens-Bogdanov bifurcation	p. 105
Spectra of Spiral Waves	p. 107
The eigenvalue problem for spiral waves: core versus farfield	p. 107
Spatial Floquet theory and the dispersion relation of wavetrains	p. 109
Relative Morse indices and essential spectra of spiral waves	p. 111
Absolute spectra of spiral waves	p. 114
Point spectrum and the shape of eigenfunctions	p. 116
Comparison with Experiments	p. 118
Meander instabilities	p. 118
Farfield and core breakup	p. 120
Three Space Dimensions: Scroll Waves	p. 123
Filaments, Scrolls, and Twists	p. 123
Spirals, tips, and Brouwer degree	p. 124
Scroll waves, filaments, and twists	p. 124
Generic Changes of Scroll Filament Topology	p. 127
Generic level sets	p. 128
Sturm property, revisited	p. 129
Comparison principle and nodal domains	p. 130
Annihilation of spiral tips	p. 131
Collisions of scroll wave filaments	p. 131
Numerical Simulations	p. 134
References	p. 140
Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions	p. 153
Introduction	p. 153
The Simplest Nonlocal Parabolic-like Evolution and its Relatives	p. 154
Comparison Between the Local and Nonlocal Equations	p. 156
Models from Statistical Mechanics	p. 157
Related Nonlocal Evolutions	p. 158
Digression on the Role of Gradient Flows in Modeling	p. 158
The Issue of Discontinuous Profiles in the Nonlocal Problem	p. 160
The Simplest Pattern-Forming Parabolic Equation	p. 160
Overview	p. 160
Spinodal Decomposition in Higher Dimensions	p. 162
Layer Phenomena Related to the Cahn-Hilliard Equation	p. 164
The Slowness of Some Motions	p. 164
Phenomena in 1D	p. 164
Bubbles and such	p. 165
Reduction to the Mullins-Sekerka Problem	p. 165
Further Reductions: Ripening	p. 166
Patterning Due to Competition in General Gradient Systems	p. 168
An Abstract Setting	p. 168
Examples	p. 169
Threshold results	p. 170
Properties of the minimizers	p. 170
Restriction to real-valued functions	p. 170
Conserved Evolutions	p. 171
A Paradigm	p. 171
Ginzburg-Landau Energies with Nonlocal Additions	p. 171
A Prototypical Inverse Elliptic Reduction	p. 172
Free Boundary Reductions	p. 173
Another Kind of Competition	p. 175
Models for Copolymers	p. 176
Conclusion	p. 177
References	p. 178
Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics	p. 193
Introduction	p. 193
Creating a Prescribed Intensity Distribution in the Far-Field	p. 195
Statement of the Problem	p. 195
Weak Formulation of the Problem	p. 196
Strong Solutions of the Reflector Problem	p. 200
Existence, Uniqueness and Regularity	p. 200
Computational Methods	p. 203
The method of supporting paraboloids (SP method)	p. 203
Open Problems	p. 204
Creating a Prescribed Intensity Distribution in the Near-Field	p. 205
Statement of the Near-Field (NF) Reflector Problem	p. 206
Weak Formulation and Solution of the NF Reflector Problem	p. 207
Some Open Problems	p. 210
Two-Reflector System for Transforming a Beam of Parallel Rays	p. 210
Statement of the Problem	p. 211
Properties of Reflectors R[subscript 1] and R[subscript 2]	p. 214
Weak Formulation and Weak Solutions	p. 215
Regularity and Numerics	p. 218
Two-Reflector System with a Point Source	p. 219
References	p. 222
Recent Developments in Multiscale Problems Coming from Fluid Mechanics	p. 225
Homogenization of Flow Problems in the Presence of Rough Boundaries and Interfaces	p. 226
Wall Laws at Rough Boundaries	p. 226
Introduction	p. 226
Navier's boundary layer	p. 228
Justification of the Navier's slip condition for the laminar 3D Couette flow	p. 230
Drag Reduction and Homogenization	p. 236
Law of Beavers and Joseph	p. 238
Introduction	p. 238
Modeling of the experiment by Beavers and Joseph	p. 241
Navier's boundary layer	p. 243
Justification of the law by Beavers and Joseph	p. 245
Interactions Flow-Structures	p. 249
Introduction	p. 249
Biot's Model Without Dissipation	p. 252
Biot's Model with Dissipation	p. 259
References	p. 264
From Molecular Dynamics to Conformation Dynamics in Drug Design	p. 269
Introduction	p. 269
Classical Molecular Dynamics	p. 270
Hamiltonian Differential Equations	p. 270
Condition of Molecular Initial Value Problems	p. 271
Example: Trinucleotide ACC	p. 272
Metastable Conformations as Almost Invariant Sets	p. 272
Perron-Frobenius Operator	p. 274
Stochastic Transition Operator	p. 274
Perron Cluster Analysis (PCCA)	p. 276
Approximation of the Transition Operator	p. 280
Example: HIV protease inhibitor VX-478	p. 284
Perspectives	p. 286
References	p. 286
A Posteriori Error Estimates and Adaptive Methods for Hyperbolic and Convection Dominated Parabolic Conservation Laws	p. 289
Introduction	p. 289
A Posteriori Error Estimates for Scalar Hyperbolic Conservation Laws	p. 291
Cell Centered Finite Volume Approximations	p. 292
Staggered Lax-Friedrichs Approximations	p. 295
A Posteriori Error Estimates for Weakly Coupled Systems	p. 298
The finite volume scheme	p. 299
Numerical Experiments	p. 302
Transport of Contaminants with Degradation	p. 302
Conclusion	p. 304
References	p. 305
On Anisotropic Geometric Diffusion in 3D Image Processing and Image Sequence Analysis	p. 307
Introduction	p. 307
Review of Related Work	p. 308
Anisotropic Geometric Diffusion on Still Images	p. 311
Processing Image Sequences via Coupled Anisotropic Geometric Diffusion	p. 315
Local Curvature and Motion Evaluation	p. 316
Finite Element Discretization	p. 317
References	p. 320
Population Dynamics: A Mathematical Bird's Eye View	p. 323
The Chemostat	p. 323
Consumer-Resource Interaction	p. 324
Competition for Substrate in the Chemostat	p. 327
A Chemostat Containing a Food-Chain	p. 328
Infectious Agents and the Art of Averaging	p. 331
Heterogeneity	p. 333
Heterogeneity Deriving from Physiological Differences	p. 333
Heterogeneity Deriving from Spatial Position	p. 334
The gradostat and the creation of niches	p. 335
The Pecularities of Semelparity	p. 335
Concluding Sermon	p. 336
References	p. 337
Did Something Change? Thresholds in Population Models	p. 341
Introduction	p. 341
Mathematical Background on Bifurcations	p. 343
Disease Thresholds	p. 347
Kermack-McKendrick	p. 347
Schistosomiasis	p. 350
Predator-Prey Systems	p. 352
The Basic Model	p. 353
Subcritical Bifurcation	p. 356
Bifurcation from a Limit Cycle	p. 359
Chaos	p. 361
Iterating Reproduction Curves	p. 361
Random Perturbations of Ecological Systems	p. 364
Lotka-Volterra Model with Random Perturbations	p. 365
The Basic Model with Random Perturbations	p. 371
Summary	p. 372
References	p. 373
Multiscale Modeling of Materials--the Role of Analysis	p. 375
Introduction	p. 375
Soft Magnetic Films	p. 377
Micromagnetics	p. 378
Thin Film Limit	p. 380
Numerical Results and Comparison with Experiment	p. 383
Discussion	p. 385
Nematic Elastomers	p. 386
Microscopic Model	p. 387
Quasiconvexification	p. 391
Finite-Element Computations	p. 396
Attainment Results	p. 399
Attainment and non-attainment for Dirichlet boundary conditions	p. 399
Attainment for a Dirichlet-Neumann problem	p. 401
Discussion and Perspectives	p. 404
References	p. 406
Color Plates	p. 409
Table of Contents provided by Rittenhouse. All Rights Reserved.

Trends in Nonlinear Analysis

At a Glance

Hardcover

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