Trends in Nonlinear Analysis

by Kirkilionis, Markus; Kromer, Susanne; Rannacher, Rolg; Tomi, Friedrich; Kramker, Susanne

• ISBN13: 9783540441984
• ISBN10: 3540441980
• Format: Hardcover
• Copyright: 2003-02-01
• Publisher: Springer Verlag

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 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.

Table of Contents

Interview with Willi Jager	1	(22)
Willi Jager
Markus Kirkilionis
Susanne Kromker
Rolf Rannacher
Friedrich Tomi
Spatio-Temporal Dynamics of Reaction-Diffusion Patterns	23	(130)
Bernold Fiedler
Arnd Scheel
Introduction and Overview	23	(4)
One Space Dimension: Global Attractors	27	(24)
Lyapunov Functions, Comparison Principles, and Sturm Property	27	(1)
Lyapunov functions	27	(2)
Comparison principles	29	(1)
Sturm property, revisited	30	(3)
Sturm Attractors on the Interval	33	(1)
Global attractors	33	(1)
Sturm attractors and Sturm permutations	34	(4)
Sturm permutations and heteroclinics	38	(3)
Combinatorics of Sturm attractors	41	(4)
Sturm Attractors on the Circle	45	(1)
Poincare-Bendixson theory	45	(2)
Heteroclinic connections of rotating waves	47	(4)
One Unbounded Space-Dimension: Travelling Waves	51	(29)
Unbounded Domains and Essential Spectra	51	(1)
From bounded to unbounded domains	51	(2)
Spectra of travelling waves: group velocities and Fredholm indices	53	(7)
Instabilities of Travelling Waves	60	(1)
Instability of a front caused by point spectrum	61	(1)
The Turing instability	61	(3)
Essential Hopf instability of a front	64	(3)
Instability of a pulse caused by the essential spectrum	67	(1)
Fredholm indices and essential instabilities	67	(3)
Spatial dynamics and essential instabilities	70	(4)
From Unbounded to Large Domains: Absolute Versus Essential Spectra	74	(6)
Two Space Dimensions: Existence of Spiral Waves	80	(13)
Kinematics and its Defects	80	(2)
Curvature flow of Archimedean spirals	82	(1)
The front-back matching problem	83	(3)
Archimedean Spiral Waves in Radial Dynamics	86	(1)
Rigid rotation and asymptotic wavetrains	86	(2)
Linear and nonlinear group velocities	88	(1)
Characterizing Archimedean spirals	89	(2)
Bifurcation to Spiral Waves	91	(2)
Two Space Dimensions: Bifurcations from Spiral Waves	93	(30)
Phenomenology of Spiral Instabilities	93	(2)
Meandering Spirals and Euclidean Symmetry	95	(1)
Euclidean equivariance	96	(1)
Relative center manifolds	97	(1)
Palais coordinates	98	(1)
Spiral tip motion, Hopf meandering, and drift resonance	99	(3)
Relative normal forms	102	(2)
Relative Hopf resonance	104	(1)
Relative Takens-Bogdanov bifurcation	105	(2)
Spectra of Spiral Waves	107	(1)
The eigenvalue problem for spiral waves: core versus farfield	107	(2)
Spatial Floquet theory and the dispersion relation of wavetrains	109	(2)
Relative Morse indices and essential spectra of spiral waves	111	(3)
Absolute spectra of spiral waves	114	(2)
Point spectrum and the shape of eigenfunctions	116	(2)
Comparison with Experiments	118	(1)
Meander instabilities	118	(2)
Farfield and core breakup	120	(3)
Three Space Dimensions: Scroll Waves	123	(30)
Filaments, Scrolls, and Twists	123	(1)
Spirals, tips, and Brouwer degree	124	(1)
Scroll waves, filaments, and twists	124	(3)
Generic Changes of Scroll Filament Topology	127	(1)
Generic level sets	128	(1)
Sturm property, revisited	129	(1)
Comparison principle and nodal domains	130	(1)
Annihilation of spiral tips	131	(1)
Collisions of scroll wave filaments	131	(3)
Numerical Simulations	134	(6)
References	140	(13)
Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions	153	(40)
Paul Fife
Introduction	153	(1)
The Simplest Nonlocal Parabolic-like Evolution and its Relatives	154	(6)
Comparison Between the Local and Nonlocal Equations	156	(1)
Models from Statistical Mechanics	157	(1)
Related Nonlocal Evolutions	158	(1)
Digression on the Role of Gradient Flows in Modeling	158	(2)
The Issue of Discontinuous Profiles in the Nonlocal Problem	160	(1)
The Simplest Pattern-Forming Parabolic Equation	160	(4)
Overview	160	(2)
Spinodal Decomposition in Higher Dimensions	162	(2)
Layer Phenomena Related to the Cahn-Hilliard Equation	164	(4)
The Slowness of Some Motions	164	(1)
Phenomena in 1D	164	(1)
Bubbles and such	165	(1)
Reduction to the Mullins-Sekerka Problem	165	(1)
Further Reductions: Ripening	166	(2)
Patterning Due to Competition in General Gradient Systems	168	(3)
An Abstract Setting	168	(1)
Examples	169	(1)
Threshold results	170	(1)
Properties of the minimizers	170	(1)
Restriction to real-valued functions	170	(1)
Conserved Evolutions	171	(1)
A Paradigm	171	(1)
Ginzburg-Landau Energies with Nonlocal Additions	171	(2)
A Prototypical Inverse Elliptic Reduction	172	(1)
Free Boundary Reductions	173	(2)
Another Kind of Competition	175	(2)
Models for Copolymers	176	(1)
Conclusion	177	(16)
References	178	(15)
Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics	193	(32)
Vladimir Oliker
Introduction	193	(2)
Creating a Prescribed Intensity Distribution in the Far-Field	195	(10)
Statement of the Problem	195	(1)
Weak Formulation of the Problem	196	(4)
Strong Solutions of the Reflector Problem	200	(1)
Existence, Uniqueness and Regularity	200	(3)
Computational Methods	203	(1)
The method of supporting paraboloids (SP method)	203	(1)
Open Problems	204	(1)
Creating a Prescribed Intensity Distribution in the Near-Field	205	(5)
Statement of the Near-Field (NF) Reflector Problem	206	(1)
Weak Formulation and Solution of the NF Reflector Problem	207	(3)
Some Open Problems	210	(1)
Two-Reflector System for Transforming a Beam of Parallel Rays	210	(9)
Statement of the Problem	211	(3)
Properties of Reflectors R1 and R2	214	(1)
Weak Formulation and Weak Solutions	215	(3)
Regularity and Numerics	218	(1)
Two-Reflector System with a Point Source	219	(6)
References	222	(3)
Recent Developments in Multiscale Problems Coming from Fluid Mechanics	225	(44)
Andro Mikelic
Homogenization of Flow Problems in the Presence of Rough Boundaries and Interfaces	226	(23)
Wall Laws at Rough Boundaries	226	(1)
Introduction	226	(2)
Navier's boundary layer	228	(2)
Justification of the Navier's slip condition for the laminar 3D Couette flow	230	(6)
Drag Reduction and Homogenization	236	(2)
Law of Beavers and Joseph	238	(1)
Introduction	238	(3)
Modeling of the experiment by Beavers and Joseph	241	(2)
Navier's boundary layer	243	(2)
Justification of the law by Beavers and Joseph	245	(4)
Interactions Flow-Structures	249	(20)
Introduction	249	(3)
Biot's Model Without Dissipation	252	(7)
Biot's Model with Dissipation	259	(5)
References	264	(5)
From Molecular Dynamics to Conformation Dynamics in Drug Design	269	(20)
Peter Deuflhard
Introduction	269	(1)
Classical Molecular Dynamics	270	(2)
Hamiltonian Differential Equations	270	(1)
Condition of Molecular Initial Value Problems	271	(1)
Example: Trinucleotide ACC	272	(1)
Metastable Conformations as Almost Invariant Sets	272	(8)
Perron-Frobenius Operator	274	(1)
Stochastic Transition Operator	274	(2)
Perron Cluster Analysis (PCCA)	276	(4)
Approximation of the Transition Operator	280	(6)
Example: HIV protease inhibitor VX-478	284	(2)
Perspectives	286	(3)
References	286	(3)
A Posteriori Error Estimates and Adaptive Methods for Hyperbolic and Convection Dominated Parabolic Conservation Laws	289	(18)
Dietmar Kroner
Marc Kuther
Mario Ohlberger
Christian Rohde
Introduction	289	(2)
A Posteriori Error Estimates for Scalar Hyperbolic Conservation Laws	291	(7)
Cell Centered Finite Volume Approximations	292	(3)
Staggered Lax-Friedrichs Approximations	295	(3)
A Posteriori Error Estimates for Weakly Coupled Systems	298	(4)
The finite volume scheme	299	(3)
Numerical Experiments	302	(2)
Transport of Contaminants with Degradation	302	(2)
Conclusion	304	(3)
References	305	(2)
On Anisotropic Geometric Diffusion in 3D Image Processing and Image Sequence Analysis	307	(16)
Karol Mikula
Tobias Preußer
Martin Rumpf
Fiorella Sgallari
Introduction	307	(1)
Review of Related Work	308	(3)
Anisotropic Geometric Diffusion on Still Images	311	(4)
Processing Image Sequences via Coupled Anisotropic Geometric Diffusion	315	(1)
Local Curvature and Motion Evaluation	316	(1)
Finite Element Discretization	317	(6)
References	320	(3)
Population Dynamics: A Mathematical Bird's Eye View	323	(18)
Odo Diekmann
Markus Kirkilionis
The Chemostat	323	(1)
Consumer-Resource Interaction	324	(3)
Competition for Substrate in the Chemostat	327	(1)
A Chemostat Containing a Food-Chain	328	(3)
Infectious Agents and the Art of Averaging	331	(2)
Heterogeneity	333	(2)
Heterogeneity Deriving from Physiological Differences	333	(1)
Heterogeneity Deriving from Spatial Position	334	(1)
The gradostat and the creation of niches	335	(1)
The Pecularities of Semelparity	335	(1)
Concluding Sermon	336	(5)
References	337	(4)
Did Something Change? Thresholds in Population Models	341	(34)
Frank Hoppensteadt
Paul Waltman
Introduction	341	(2)
Mathematical Background on Bifurcations	343	(4)
Disease Thresholds	347	(5)
Kermack-McKendrick	347	(3)
Schistosomiasis	350	(2)
Predator-Prey Systems	352	(9)
The Basic Model	353	(3)
Subcritical Bifurcation	356	(3)
Bifurcation from a Limit Cycle	359	(2)
Chaos	361	(3)
Iterating Reproduction Curves	361	(3)
Random Perturbations of Ecological Systems	364	(8)
Lotka-Volterra Model with Random Perturbations	365	(6)
The Basic Model with Random Perturbations	371	(1)
Summary	372	(3)
References	373	(2)
Multiscale Modeling of Materials -- the Role of Analysis	375	(34)
Sergio Conti
Antonio DeSimone
Georg Dolzmann
Stefan Muller
Felix Otto
Introduction	375	(2)
Soft Magnetic Films	377	(9)
Micromagnetics	378	(2)
Thin Film Limit	380	(3)
Numerical Results and Comparison with Experiment	383	(2)
Discussion	385	(1)
Nematic Elastomers	386	(23)
Microscopic Model	387	(4)
Quasiconvexification	391	(5)
Finite-Element Computations	396	(3)
Attainment Results	399	(1)
Attainment and non-attainment for Dirichlet boundary conditions	399	(2)
Attainment for a Dirichlet-Neumann problem	401	(3)
Discussion and Perspectives	404	(2)
References	406	(3)
Appendix Color Plates	409