Côte, R.; Kenig, C. E.; Lawrie, A.; Schlag, W. Characterization of large energy solutions of the equivariant wave map problem. I. (English) Zbl 1315.35130 Am. J. Math. 137, No. 1, 139-207 (2015). The first important result refers to the global existence and scattering for wave maps in the zero topological class, below a sharp energy threshold. In fact one proves that for smooth data in this class, there exist a unique global evolution and it scatters to zero. The result is sharp in the sense that twice the energy of the harmonic map given by stereographic projection is a true threshold. The proof is based on the concentration compactness/rigidity method developed by C. E. Kenig and F. Merle, in two papers: [Invent. Math. 166, No. 3, 645–675 (2006; Zbl 1115.35125)] and [Acta Math. 201, No. 2, 147–212 (2008; Zbl 1183.35202)]. Further, wave maps of topological degree one are studied. For this case, a result on classification of blow-up solutions with energies below three time the energy of the harmonic map is shown. One provides a list of all known results used in the proofs of theorems. Preliminaries are presented in Section 2, a brief outline of the concentration compactness /rigidity method is given in Section 3. Next sections are devoted to the proofs. For Part II see [the authors, ibid. 137, No. 1, 209–250 (2015; Zbl 1315.35131)]. Reviewer: Claudia Simionescu-Badea (Wien) Cited in 2 ReviewsCited in 39 Documents MSC: 35L71 Second-order semilinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35P25 Scattering theory for PDEs 47A40 Scattering theory of linear operators 35B44 Blow-up in context of PDEs Keywords:energy solutions; wave maps; energy threshold; well-posedness; scattering; blow-up; stereographic projection; concentration compactness/rigidity method Citations:Zbl 1115.35125; Zbl 1183.35202; Zbl 1315.35131 PDFBibTeX XMLCite \textit{R. Côte} et al., Am. J. Math. 137, No. 1, 139--207 (2015; Zbl 1315.35130) Full Text: DOI arXiv Link