PDE Dynamics

Christian Kuehn is Lichtenberg Professor for Multiscale and Stochastic Dynamics at Technical University of Munich and has worked at the Max Planck Institute for Physics of Complex Systems and Vienna University of Technology as a postdoctoral fellow. He has been an MFO Leibniz Fellow and an APART Fellow of the Austrian Academy of Sciences, and he is a recipient of the Richard von Mises Prize for his contributions to nonlinear dynamics. His research interests lie at the interface of differential equations, dynamical systems, and mathematical modelling, and his key goal is analyzing multiscale problems and the effect of noise/uncertainty in various classes of ordinary, partial, and stochastic differential equations as well as in adaptive networks.

• Preface;Course Design;1 A Whirlwind Introduction;2 Some ODE Theory and Geometric Dynamics;3 Some PDE Theory and Functional Analysis;4 Implicit Functions and Lyapunov-Schmidt;5 Crandall-Rabinowitz and Local Bifurcations;6 Stability and Spectral Theory;7 Existence of Travelling Waves;8 Pushed and Pulled Fronts;9 Sturm-Liouville and Stability of Travelling Waves;10 Exponential Dichotomies and Evans Function;11 Characteristics and Shocks;12 Onset of Patterns and Multiple Scales;13 Validity of Amplitude Equations;14 Semigroups and Sectorial Operators;15 Dissipation and Absorbing Sets;16 Nonlinear Saddles and Invariant Manifolds;17 Spectral Gap and Inertial Manifolds;18 Attractors and the Variational Equation;19 Lyapunov Exponents and Fractal Dimension;20 Metastability and Manifolds;21 Exponentially Small Terms;22 Coarsening Bounds and Scaling;23 Gradient Flows and Lyapunov Functions;24 Entropies and Global Decay;25 Convexity and Minimizers;26 Mountain Passes and Periodic Waves;27 Hamiltonian Dynamics and Normal Forms;28 Empirical Measures and the Mean Field;29 Two Effects: Hypocoercivity and Turing;30 Blow-up in Cross-Diffusion Systems;31 Self-Similarity and Free Boundaries;32 Spirals and Symmetry;33 Averaging and Ergodicity;34 Two-Scale Convergence;35 Asymptotics and Layers;36 Fast-Slow Systems: Periodicity and Chaos;A Finite Differences and Simulation;B Finite Elements and Continuation; Bibliography;Index.