Trends in nonlinear analysis / Markus Kirkilionis ... [and others] (eds.).

Format:

Book

Contributor:

Kirkilionis, Markus, 1962-

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Differential equations, Partial.

Differentiable dynamical systems.

Ergodic theory.

Electromagnetic theory.

Physical Description:

xv, 419 pages : illustrations (some color) ; 24 cm

Place of Publication:

Berlin ; New York : Springer, [2003]

Summary:

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, most dominantly the modern computer, influence the development of the field? How can problems be solved which have been beyond reach in former times? What are the other major trends that currently transform the appearance of nonlinear analysis?

Contents:

1 Interview with Willi Jager / Markus Kirkilionis, Susanne Kromker, Rolf Rannacher, Friedrich Tomi 1

2 Spatio-Temporal Dynamics of Reaction-Diffusion Patterns / Bernold Fiedler, Arnd Scheel 23

2.2 One Space Dimension: Global Attractors 27

2.2.1 Lyapunov Functions, Comparison Principles, and Sturm Property 27

Lyapunov functions 27

Comparison principles 29

Sturm property, revisited 30

2.2.2 Sturm Attractors on the Interval 33

Global attractors 33

Sturm attractors and Sturm permutations 34

Sturm permutations and heteroclinics 38

Combinatorics of Sturm attractors 41

2.2.3 Sturm Attractors on the Circle 45

Poincare-Bendixson theory 45

Heteroclinic connections of rotating waves 47

2.3 One Unbounded Space-Dimension: Travelling Waves 51

2.3.1 Unbounded Domains and Essential Spectra 51

From bounded to unbounded domains 51

Spectra of travelling waves: group velocities and Fredholm indices 53

2.3.2 Instabilities of Travelling Waves 60

Instability of a front caused by point spectrum 61

The Turing instability 61

Essential Hopf instability of a front 64

Instability of a pulse caused by the essential spectrum 67

Fredholm indices and essential instabilities 67

Spatial dynamics and essential instabilities 70

2.3.3 From Unbounded to Large Domains: Absolute Versus Essential Spectra 74

2.4 Two Space Dimensions: Existence of Spiral Waves 80

2.4.1 Kinematics and its Defects 80

Curvature flow of Archimedean spirals 82

The front-back matching problem 83

2.4.2 Archimedean Spiral Waves in Radial Dynamics 86

Rigid rotation and asymptotic wavetrains 86

Linear and nonlinear group velocities 88

Characterizing Archimedean spirals 89

2.4.3 Bifurcation to Spiral Waves 91

2.5 Two Space Dimensions: Bifurcations from Spiral Waves 93

2.5.1 Phenomenology of Spiral Instabilities 93

2.5.2 Meandering Spirals and Euclidean Symmetry 95

Euclidean equivariance 96

Relative center manifolds 97

Palais coordinates 98

Spiral tip motion, Hopf meandering, and drift resonance 99

Relative normal forms 102

Relative Hopf resonance 104

Relative Takens-Bogdanov bifurcation 105

2.5.3 Spectra of Spiral Waves 107

The eigenvalue problem for spiral waves: core versus farfield 107

Spatial Floquet theory and the dispersion relation of wavetrains 109

Relative Morse indices and essential spectra of spiral waves 111

Absolute spectra of spiral waves 114

Point spectrum and the shape of eigenfunctions 116

2.5.4 Comparison with Experiments 118

Meander instabilities 118

Farfield and core breakup 120

2.6 Three Space Dimensions: Scroll Waves 123

2.6.1 Filaments, Scrolls, and Twists 123

Spirals, tips, and Brouwer degree 124

Scroll waves, filaments, and twists 124

2.6.2 Generic Changes of Scroll Filament Topology 127

Generic level sets 128

Sturm property, revisited 129

Comparison principle and nodal domains 130

Annihilation of spiral tips 131

Collisions of scroll wave filaments 131

2.6.3 Numerical Simulations 134

3 Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions / Paul Fife 153

3.2 The Simplest Nonlocal Parabolic-like Evolution and its Relatives 154

3.2.1 Comparison Between the Local and Nonlocal Equations 156

3.2.2 Models from Statistical Mechanics 157

3.2.3 Related Nonlocal Evolutions 158

3.2.4 Digression on the Role of Gradient Flows in Modeling 158

3.2.5 The Issue of Discontinuous Profiles in the Nonlocal Problem 160

3.3 The Simplest Pattern-Forming Parabolic Equation 160

3.3.2 Spinodal Decomposition in Higher Dimensions 162

3.4 Layer Phenomena Related to the Cahn-Hilliard Equation 164

3.4.1 The Slowness of Some Motions 164

Phenomena in 1D 164

Bubbles and such 165

3.4.2 Reduction to the Mullins-Sekerka Problem 165

3.4.3 Further Reductions: Ripening 166

3.5 Patterning Due to Competition in General Gradient Systems 168

3.5.1 An Abstract Setting 168

Threshold results 170

Properties of the minimizers 170

Restriction to real-valued functions 170

3.5.2 Conserved Evolutions 171

3.5.3 A Paradigm 171

3.6 Ginzburg-Landau Energies with Nonlocal Additions 171

3.6.1 A Prototypical Inverse Elliptic Reduction 172

3.7 Free Boundary Reductions 173

3.8 Another Kind of Competition 175

3.8.1 Models for Copolymers 176

4 Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics / Vladimir Oliker 193

4.2 Creating a Prescribed Intensity Distribution in the Far-Field 195

4.2.1 Statement of the Problem 195

4.2.2 Weak Formulation of the Problem 196

4.2.3 Strong Solutions of the Reflector Problem 200

4.2.4 Existence, Uniqueness and Regularity 200

4.2.5 Computational Methods 203

The method of supporting paraboloids (SP method) 203

4.2.6 Open Problems 204

4.3 Creating a Prescribed Intensity Distribution in the Near-Field 205

4.3.1 Statement of the Near-Field (NF) Reflector Problem 206

4.3.2 Weak Formulation and Solution of the NF Reflector Problem 207

4.3.3 Some Open Problems 210

4.4 Two-Reflector System for Transforming a Beam of Parallel Rays 210

4.4.1 Statement of the Problem 211

4.4.2 Properties of Reflectors R[subscript 1] and R[subscript 2] 214

4.4.3 Weak Formulation and Weak Solutions 215

4.4.4 Regularity and Numerics 218

4.5 Two-Reflector System with a Point Source 219

5 Recent Developments in Multiscale Problems Coming from Fluid Mechanics / Andro Mikelic 225

5.1 Homogenization of Flow Problems in the Presence of Rough Boundaries and Interfaces 226

5.1.1 Wall Laws at Rough Boundaries 226

Navier's boundary layer 228

Justification of the Navier's slip condition for the laminar 3D Couette flow 230

5.1.2 Drag Reduction and Homogenization 236

5.1.3 Law of Beavers and Joseph 238

Modeling of the experiment by Beavers and Joseph 241

Navier's boundary layer 243

Justification of the law by Beavers and Joseph 245

5.2 Interactions Flow-Structures 249

5.2.2 Biot's Model Without Dissipation 252

5.2.3 Biot's Model with Dissipation 259

6 From Molecular Dynamics to Conformation Dynamics in Drug Design / Peter Deuflhard 269

6.2 Classical Molecular Dynamics 270

6.2.1 Hamiltonian Differential Equations 270

6.2.2 Condition of Molecular Initial Value Problems 271

Example: Trinucleotide ACC 272

6.3 Metastable Conformations as Almost Invariant Sets 272

6.3.1 Perron-Frobenius Operator 274

6.3.2 Stochastic Transition Operator 274

6.3.3 Perron Cluster Analysis (PCCA) 276

6.4 Approximation of the Transition Operator 280

Example: HIV protease inhibitor VX-478 284

6.5 Perspectives 286

7 A Posteriori Error Estimates and Adaptive Methods for Hyperbolic and Convection Dominated Parabolic Conservation Laws / Dietmar Kroner, Marc Kuther, Mario Ohlberger, Christian Rohde 289

7.2 A Posteriori Error Estimates for Scalar Hyperbolic Conservation Laws 291

7.2.1 Cell Centered Finite Volume Approximations 292

7.2.2 Staggered Lax-Friedrichs Approximations 295

7.3 A Posteriori Error Estimates for Weakly Coupled Systems 298

The finite volume scheme 299

7.4 Numerical Experiments 302

7.4.1 Transport of Contaminants with Degradation 302

8 On Anisotropic Geometric Diffusion in 3D Image Processing and Image Sequence Analysis / Karol Mikula, Tobias Preusser, Martin Rumpf, Fiorella Sgallari 307

8.2 Review of Related Work 308

8.3 Anisotropic Geometric Diffusion on Still Images 311

8.4 Processing Image Sequences via Coupled Anisotropic Geometric Diffusion 315

8.5 Local Curvature and Motion Evaluation 316

8.6 Finite Element Discretization 317

9 Population Dynamics: A Mathematical Bird's Eye View / Odo Diekmann, Markus Kirkilionis 323

9.1 The Chemostat 323

9.2 Consumer-Resource Interaction 324

9.3 Competition for Substrate in the Chemostat 327

9.4 A Chemostat Containing a Food-Chain 328

9.5 Infectious Agents and the Art of Averaging 331

9.6 Heterogeneity 333

9.6.1 Heterogeneity Deriving from Physiological Differences 333

9.6.2 Heterogeneity Deriving from Spatial Position 334

The gradostat and the creation of niches 335

9.7 The Pecularities of Semelparity 335

9.8 Concluding Sermon 336

10 Did Something Change? Thresholds in Population Models / Frank Hoppensteadt, Paul Waltman 341

10.2 Mathematical Background on Bifurcations 343

10.3 Disease Thresholds 347

10.3.1 Kermack-McKendrick 347

10.3.2 Schistosomiasis 350

10.4 Predator-Prey Systems 352

10.4.1 The Basic Model 353

10.4.2 Subcritical Bifurcation 356

10.4.3 Bifurcation from a Limit Cycle 359

10.5 Chaos 361

10.5.1 Iterating Reproduction Curves 361

10.6 Random Perturbations of Ecological Systems 364

10.6.1 Lotka-Volterra Model with Random Perturbations 365

10.6.2 The Basic Model with Random Perturbations 371

11 Multiscale Modeling of Materials

the Role of Analysis / Sergio Conti, Antonio DeSimone, Georg Dolzmann, Stefan Muller, Felix Otto 375

11.2 Soft Magnetic Films 377

11.2.1 Micromagnetics 378

11.2.2 Thin Film Limit 380

11.2.3 Numerical Results and Comparison with Experiment 383

11.3 Nematic Elastomers 386

11.3.1 Microscopic Model 387

11.3.2 Quasiconvexification 391

11.3.3 Finite-Element Computations 396

11.3.4 Attainment Results 399

Attainment and non-attainment for Dirichlet boundary conditions 399

Attainment for a Dirichlet-Neumann problem 401.

Notes:

Includes bibliographical references.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

3540441980

OCLC:

51022652