PDE dynamics : an introduction
Author(s)
Bibliographic Information
PDE dynamics : an introduction
（Mathematical modeling and computation, 23）
Society for Industrial and Applied Mathematics, c2019
- : pbk
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Note
Includes bibliographical references (p. 211-240) and index
Description and Table of Contents
Description
This book provides an overview of the myriad methods for applying dynamical systems techniques to PDEs and highlights the impact of PDE methods on dynamical systems. Also included are many nonlinear evolution equations, which have been benchmark models across the sciences, and examples and techniques to strengthen preparation for research.
PDE Dynamics: An Introduction is intended for senior undergraduate students, beginning graduate students, and researchers in applied mathematics, theoretical physics, and adjacent disciplines. Structured as a textbook or seminar reference, it can be used in courses titled Dynamics of PDEs, PDEs 2, Dynamical Systems 2, Evolution Equations, or Infinite-Dimensional Dynamics.
Table of Contents
- 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.
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