Geba D.-A., Grillakis M.G. An Introduction to the Theory of Wave Maps and Related Geometric Problems

Simgapore: World Scientific, 2017. — 491 p.The wave map problem is one of the most beautiful and challenging nonlinear hyperbolic problems, which has kept the attention of mathematicians for more than thirty years now. The study of the problem involves diverse issues, e.g., well-posedness, regularity, formation of singularities, and stability of solitons, and combines intricate tools from analysis, geometry, and topology. Moreover, the wave map system has a natural formulation as the Euler-Lagrange system for a map between manifolds, a special case being the nonlinear sigma model, which is one of the fundamental objects in classical field theory. One of the goals of this book is to offer an up-to-date and self-contained overview of the main regularity theory for wave maps. Another goal is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave map system (e.g., the Skyrme model), which are extremely interesting and pose challenging new questions in their own right.
The topic of wave maps has experienced an incredible advancement in the past ten years. This is precisely the time passed from the moment when the last monograph (i.e., Tao [171]) which tried to give a state-of-the-art for this topic appeared. Our book tries to fill this gap by presenting the most recent developments in the field, e.g., the resolution of the large data global regularity theory for wave maps. These results are very technical, being accessible only to experts in their current format. Our goal is to try to explain them to a wider group, which includes advanced graduate students, in the hope of stimulating new research ideas. Moreover, this book is the first one which discusses, from a mathematical point of view, the time evolution for the models which are extensions of the nonlinear sigma model: the Skyrme, Faddeev, and Adkins-Nappi theories.