Abstract
The Cauchy problem for the Hunter–Saxton equation is known to be locally well posed in Besov spaces \(B^s_{2,r} \) on the circle. We prove that the data-to-solution map is not uniformly continuous from any bounded subset of \(B^s_{2,r} \) to \(C([0, T]; B^s_{2,r} )\). We also show that the solution map is Hölder continuous with respect to a weaker topology.
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Holmes, J., Tiglay, F. Non-uniform dependence of the data-to-solution map for the Hunter–Saxton equation in Besov spaces. J. Evol. Equ. 18, 1173–1187 (2018). https://doi.org/10.1007/s00028-018-0436-4
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DOI: https://doi.org/10.1007/s00028-018-0436-4
Keywords
- Well-posedness
- Initial value problem
- Cauchy problem
- Besov spaces
- Sobolev spaces
- Multi-linear estimates
- Hunter–Saxton equation