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PDE dynamics : an introduction
Kuehn C., SIAM-Society for Industrial and Applied Mathematics, Philadelphia, PA, 2019. 245 pp. Type: Book (978-1-611975-65-9)
Date Reviewed: Feb 12 2020

Interactions between partial differential equations (PDEs) and dynamical systems are explored in this book. PDEs involving time intervals are used to explain many problems. The study of dynamics can help readers understand PDEs via time-based equations. The book provides introductory notes on this aspect of PDEs. How to analytically visualize data related to PDEs is described using various visual effects found in data. Bifurcations, manifolds, waves, spirals, and so on are referred to as visual effects produced while plotting PDEs.

Chapter 1 explains the book’s goal through examples. It gives a list of more than ten different types of PDEs found in dynamics. Chapter 2 introduces geometrical dynamics. It includes various theorems related to the stability of manifolds and bifurcations in ordinary differential equations (ODEs). Chapter 3 gives an overview of PDE theory, including elliptical PDEs and parabolic PDEs.

Chapter 4 proves the Lyapunov-Schmidt reduction to reduce an infinite dimensional problem into a more tractable finite dimensional problem. Chapter 5 explains first-order and second-order bifurcation curve approximations. Chapter 6, on spectral theory, provides eigenvalue equations–based conditions for determining linear stability.

Chapters 7 and 8 cover PDEs with respect to waves. They explain the basic properties of waves, including determining the existence of waves. Chapter 9 is on determining the spectral stability of waves, and Chapter 10 defines exponential dichotomies and uses them for the stability of waves.

Chapter 11 introduces a method of characteristics for studying PDEs. It explains various related concepts, conditions, and theorems. Chapters 12 and 13 are on amplitude equations, which are also known as modulation equations. Apart from an explanation of various such equations, they give a validation theorem for amplitude equations. Chapter 14 describes sectorial operators and proves their connection with analytic semigroups. Chapter 15 applies absorbing sets for checking whether a PDE is finite dimensional.

Chapter 16 is on the PDE stable/unstable manifold theorem using spectral gap conditions. Chapter 17 then provides theorems on the existence of inertial manifolds using spectral gap conditions. Chapters 18 and 19 introduce the theory of attractors for PDEs. Chapter 20 is on conditions for metastable layers in motion, and chapter 21 presents the related exponentially small terms theorem.

Chapter 22 provides theorems to determine coarsening bounds using energies. Chapter 23 introduces gradient flow analysis for ODEs/PDEs. Chapter 24 gives steps for analysis based on entropy functionals. Chapter 25 presents analysis based on energy minimizers. Chapter 26 is on the mountain pass theorem for gradient flow analysis. Chapter 27 introduces Hamiltonian PDEs.

Chapter 28 is on empirical measurements. Chapter 29 introduces the hypocoercivity effect. Chapter 30 describes a non-smooth blowup observation. Chapter 31 describes self-similarity solutions for PDEs. It also covers the related free boundary growth theorem. Chapter 32 describes the formation of spirals through equations. Chapters 33 and 34 view PDEs with ergodic theory. Chapter 35 is on spikes analysis, providing some equations to prove spikes. Chapter 36 is on what is known as fast-slow PDEs, which have both fast and slow pulses.

Part of SIAM’s “Mathematical Modeling and Computation” series, it is an applied mathematics book with an analytical focus. There are two appendices: one on the finite difference method and another related to the finite element method on the computation side. It is clear from the preface that this textbook is designed for a course on PDE dynamics. The chapters are full of theorems, proofs, definitions, examples, and illustrations. Each chapter ends with a section on further reading. The book does not have any exercises because it is intended for a preparatory research course. Readers are assumed to have good knowledge of ODEs, PDEs, algebra, and geometry. However, all of the many topics are covered at an introductory level.

Reviewer:  Maulik A. Dave Review #: CR146887 (2007-0151)
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