Global regularity of wave maps III. Large energy from \R¹⁺² to hyperbolic spaces

2008/05/30 by Terence Tao, Tao, Terence · 2 citations
Mathematics · Physics and Astronomy · #35L70 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.0805.4666

openalex publication_date 2008/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that wave maps ϕ from two-dimensional Minkowski space \R¹⁺² to hyperbolic spaces \H^m are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the self-similar and travelling (or stationary solutions); we will address these claims in the sequels \citetao:heatwave2, \citetao:heatwave3, \citetao:heatwave4 to this paper. Following recent work in critical dispersive equations, the strategy is to reduce matters to the study of an almost periodic maximal Cauchy development in the energy class. We then repeatedly analyse the stress-energy tensor of this development (as in \citetao:forges) to extract either a self-similar, travelling, or degenerate non-trivial energy class solution to the wave maps equation. We will then rule out such solutions in the sequels to this paper, establishing the desired global regularity result for wave maps.

Citations

• On the regularity properties of a model problem related to wave maps
• Weak solutions and development of singularities of the SU(2) σ‐model
• Harmonic Mappings of Riemannian Manifolds

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• Schrödinger maps to a Kähler manifold in two dimensions
• Universality of blow up profile for small blow up solutions to the\n energy critical wave map equation

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