Abstract
In this article we consider large data Wave-Maps from \({\mathbb R^{2+1}}\) into a compact Riemannian manifold \({(\mathcal{M},g)}\), and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. This is a companion to our concurrent article [21], which together with the present work establishes a full regularity theory for large data Wave-Maps.
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Acknowlegements
The authors would like to thank Manos Grillakis, Sergiu Klainerman, Joachim Krieger, Matei Machedon, Igor Rodnianski, and Wilhelm Schlag for many stimulating discussions over the years regarding the wave-map problem. We would also especially like to thank Terry Tao for several key discussions on the nature of induction-on-energy type proofs.
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Communicated by P. Constantin
The first author was supported in part by the NSF grant DMS-0701087.
The second author was supported in part by the NSF grant DMS-0801261.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Sterbenz, J., Tataru, D. Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions. Commun. Math. Phys. 298, 139–230 (2010). https://doi.org/10.1007/s00220-010-1061-4
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DOI: https://doi.org/10.1007/s00220-010-1061-4