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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.
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},
pages = {519--532},
publisher = {Elsevier},
volume = {32},
number = {3},
year = {2015},
doi = {10.1016/j.anihpc.2014.02.001},
mrnumber = {3353699},
zbl = {1320.35006},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.02.001/}
}
TY - JOUR AU - Widmayer, Klaus TI - Non-uniqueness of weak solutions to the wave map problem JO - Annales de l'I.H.P. Analyse non linéaire PY - 2015 SP - 519 EP - 532 VL - 32 IS - 3 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.02.001/ DO - 10.1016/j.anihpc.2014.02.001 LA - en ID - AIHPC_2015__32_3_519_0 ER -
%0 Journal Article %A Widmayer, Klaus %T Non-uniqueness of weak solutions to the wave map problem %J Annales de l'I.H.P. Analyse non linéaire %D 2015 %P 519-532 %V 32 %N 3 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.02.001/ %R 10.1016/j.anihpc.2014.02.001 %G en %F 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, Tome 32 (2015) no. 3, pp. 519-532. doi: 10.1016/j.anihpc.2014.02.001
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