Non-uniqueness of weak solutions to the wave map problem
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 3, pp. 519-532.

In this note we show that weak solutions to the wave map problem in the energy-supercritical dimension 3 are not unique. On the one hand, we find weak solutions using the penalization method introduced by Shatah [12] and show that they satisfy a local energy inequality. On the other hand we build on a special harmonic map to construct a weak solution to the wave map problem, which violates this energy inequality.Finally we establish a local weak-strong uniqueness argument in the spirit of Struwe [15] which we employ to show that one may even have a failure of uniqueness for a Cauchy problem with a stationary solution. We thus obtain a result analogous to the one of Coron [2] for the case of the heat flow of harmonic maps.

DOI: 10.1016/j.anihpc.2014.02.001
Classification: 35L05,  35L71
Keywords: Wave maps, Weak solutions, Weak-strong uniqueness
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Widmayer, Klaus. Non-uniqueness of weak solutions to the wave map problem. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 3, pp. 519-532. doi : 10.1016/j.anihpc.2014.02.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.02.001/

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