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Two-bubble dynamics for threshold solutions to the wave maps equation

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Abstract

We consider the energy-critical wave maps equation \(\mathbb {R}^{1+2} \rightarrow \mathbb {S}^2\) in the equivariant case, with equivariance degree \(k \ge 2\). It is known that initial data of energy \(< 8\pi k\) and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy \(8 \pi k \). We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines the classical concentration-compactness techniques of Kenig–Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.

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Acknowledgements

J. Jendrej was supported by the ERC Grant 291214 BLOWDISOL and by the NSF Grant DMS-1463746. This work was completed during his postdoc at the University of Chicago. A. Lawrie was supported by NSF Grant DMS-1700127. We would like to thank Raphaël Côte for many helpful discussions. And lastly, we would like to thank the anonymous referees for their careful reading of an earlier version of the manuscript and for suggesting substantial improvements.

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  1. LAGA, UMR 7539, CNRS and Université Paris 13, 99 av J.-B. Clément, 93430, Villetaneuse, France

    Jacek Jendrej

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, 2-267, Cambridge, MA, 02139, USA

    Andrew Lawrie

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  1. Jacek Jendrej
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Jendrej, J., Lawrie, A. Two-bubble dynamics for threshold solutions to the wave maps equation. Invent. math. 213, 1249–1325 (2018). https://doi.org/10.1007/s00222-018-0804-2

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