Non-uniqueness of weak solutions to the wave map problem
Klaus Widmayer
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York 10012-1185 NY, USA
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Abstract
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.
Cite this article
Klaus Widmayer, Non-uniqueness of weak solutions to the wave map problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 3, pp. 519–532
DOI 10.1016/J.ANIHPC.2014.02.001