Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4604217 | Annales de l'Institut Henri Poincare (C) Non Linear Analysis | 2015 | 14 Pages |
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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Klaus Widmayer,