Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces
By Terence Tao. 2008
85 citations recorded by the source corpus.
We show that wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ 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 \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} to this paper. Following recent work in critical dispersive equations, the strategy is to reduce matters to the study of an \emph{almost periodic} maximal Cauchy development in the energy class. We then repeatedly analyse the stress-energy tensor of this development (as in \cite{tao: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.
Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/b5317d826e85780be6c55f756a909569d9ac3168
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Citations in the corpus
- Concentration Compactness for Critical Wave Maps (2009, background): “[50] Tao, T.”
- Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications (2008, methodology): “For instance, it was applied to the “mass-critical” NLS in the radial case, in dimension d ≥ 2, in both focusing and defocusing cases, in works of Killip, Tao, Vişan and Zhang, [38], [21], [23], to the focusing cubic NLS for d = 3, by Holmer–Roudenko [13] and Duyckaerts–Holmer–Roudenko [6], to coro”
- Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, background): “Proposition 4.2 is due to Tao [44]; we give a complete proof of Proposition 4.2 in Section 8.”