Non-uniqueness of weak solutions to the wave map problem
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 3, p. 519-532
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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 : https://doi.org/10.1016/j.anihpc.2014.02.001
Classification:  35L05,  35L71
Keywords: Wave maps, Weak solutions, Weak-strong uniqueness
@article{AIHPC_2015__32_3_519_0,
     author = {Widmayer, Klaus},
     title = {Non-uniqueness of weak solutions to the wave map problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {3},
     year = {2015},
     pages = {519-532},
     doi = {10.1016/j.anihpc.2014.02.001},
     zbl = {1320.35006},
     mrnumber = {3353699},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_3_519_0}
}
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://www.numdam.org/item/AIHPC_2015__32_3_519_0/

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