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

2008/05/30 by Terence Tao, Tao, Terence · 8 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 #math.AP #math.DG #msc:35L70

paper · pdf · doi:10.48550/arxiv.0805.4666

35 pages, no figures. References updated. Will not be published in current form, pending future reorganisation of the heatwave project

openalex publication_date 2008/05/30 · arxiv created 2009/08/08 · arxiv updated 2009/12/01 · 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

Cited by

• Schrödinger maps to a Kähler manifold in two dimensions
• Universality of blow up profile for small blow up solutions to the energy critical wave map equation
• Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in H¹
• Long time solutions for wave maps with large data
• Bilinear Strichartz estimates for the Schrödinger map problem
• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• An alternative approach to regularity for the Navier–Stokes equations in critical spaces
• Regularity of Wave-Maps in Dimension 2 + 1

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