An introduction to the theory of wave maps and related geometric problems (Q2818806)

The monograph is an introduction to the theory of wave maps and related geometric problems, in particular to their local and global well-posedness and scattering theory.NEWLINENEWLINEThe wave map problem is one of the most beautiful and challenging nonlinear hyperbolic problems. 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. The wave map system has a natural formulation as the Euler-Lagrange system for a map \(\phi\), from \(n+1\)-dimensional Minkowski spacetime, \((\mathbb{R}^{n+1}, m_{\mu\nu}=\operatorname{diag}(-1,1,\cdots,1))\) to a \(k\)-dimensional Riemannian manifold \((N, h_{ab})\), with Lagrangian \(\frac{1}{2}m^{\mu\nu}h_{ab}\partial_\mu \phi^a \partial_\nu \phi^b\). Written down in local coordinates, it takes the form of semilinear wave system for \(\phi=(\phi^1,\cdots, \phi^k)\), \(\square\phi^a=\Gamma^{a}_{bc}(\phi)m^{\mu\nu}\partial_\mu \phi^b \partial_\nu \phi^c\) with \(\square=\partial_t^2-\Delta_x\).NEWLINENEWLINEThe topic of wave maps has experienced vast advancement in the past three decades. 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).NEWLINENEWLINEChapter 1 gives an introduction of the physical motivation and the mathematical formulation of the wave map problem and its generalizations.NEWLINENEWLINEIn Chapter 2, the authors developed the analytic background needed in the investigation of these problems, which include Strichartz estimates (homogeneous and inhomogeneous), hyperbolic Sobolev spaces (introduced by Klainerman-Machedon, which is also known as the Bourgain space in the field of dispersive equations) and Tataru's \(F\)-spaces.NEWLINENEWLINEChapter 3 is devoted to the study of the local and small data global well-posedness theories for the wave map equation, where one can see the motivation for the emergence of more and more powerful analytic techniques needed in handling the challenging nature of these topics. For local well-posedness, various arguments were presented: energy argument (local well posed in \(H^s(\mathbb{R}^n)\) with \(s>(n+2)/2\)), Strichartz estimates (\(s>\max((n+1)/2, (n+5)/4)\) [\textit{G. Ponce} and \textit{T. C. Sideris}, Commun. Partial Differ. Equations 18, No. 1--2, 169--177 (1993; Zbl 0803.35096)]), hyperbolic Sobolev spaces (\(s>n/2\)) [\textit{S. Klainerman} and \textit{S. Selberg}, Commun. Partial Differ. Equations 22, No. 5--6, 901--918 (1997; Zbl 0884.35102)]. For small data global well-posedness when \(n\geq 4\), it also includes detailed expositions for two results, due to \textit{T. Tao} (\(n\geq 5\)) [Commun. Math. Phys. 224, No. 2, 443--544 (2001; Zbl 1020.35046)] and \textit{J. Shatah} and \textit{M. Struwe} (\(n= 4\)) [Int. Math. Res. Not. 2002, No. 11, 555--571 (2002; Zbl 1024.58014)], respectively, which make the case for the important role played by the intrinsic geometric aspect of the wave map problem.NEWLINENEWLINEConcerning the resolution of the large data regularity theory for wave maps in the energy-critical case (\(n=2\)), there are three major results obtained by Tao, Krieger-Schlag and Sterbenz-Tataru. In Chapter 4, the authors focus on the Sterbenz-Tataru's program [\textit{J. Sterbenz} and \textit{D. Tataru}, Commun. Math. Phys. 298, No. 1, 139--230 (2010; Zbl 1218.35129); ibid. 298, No. 1, 231--264 (2010; Zbl 1218.35057)], which provides a complete description of the regimes when a large data, finite energy wave map blows up and when it is global-in-time. The presentation of this topic reworks Sterbenz-Tataru's argument, including a significant number of refinements and extra details.NEWLINENEWLINEChapter 5 addresses well-posedness questions for the classical Skyrme model and its extensions, including \textit{W. W. Y. Wong}'s result [Classical Quantum Gravity 28, No. 21, Article ID 215008, 23 p. (2011; Zbl 1230.83020)] on the Skyrme model using Christodoulou's regular hyperbolicity framework, Lei-Lin-Zhou [\textit{Z. Lei} et al., Acta Math. Sin., Engl. Ser. 27, No. 2, 309--328 (2011; Zbl 1209.35073)] global regularity theorem for the \(2+1\)-dimensional Faddeev problem, which relies on an adaptation of Klainerman's vector field method.NEWLINENEWLINEIn Chapter 6, the authors discuss equivariant results for all the equations considered in this book, which include non-concentration of energy, small data global well-posedness in critical Besov spaces, and global regularity for sufficiently smooth large data. The final chapter is devoted to the phenomenon of collapse for wave maps and examine \textit{P. Raphaël} and \textit{I. Rodnianski}'s result [Publ. Math., Inst. Hautes Étud. Sci. 115, 1--122 (2012; Zbl 1284.35358)] on the topic.NEWLINENEWLINEThe book also includes an appendix detailing the basic differential geometry concepts needed for following the presentation of the material.