Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude
We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space Hs with s > 3/2. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in Hs,...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:150692 |
| Acceso en línea: | https://ddd.uab.cat/record/150692 https://dx.doi.org/urn:doi:10.1016/j.nonrwa.2013.12.007 |
| Access Level: | acceso abierto |
| Palabra clave: | Camassa-Holm equation Flow map Non-uniform continuity Water waves |
| Sumario: | We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space Hs with s > 3/2. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in Hs, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in Hs on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in Hs at any positive time in this interval. The result is obtained by approximating the solutions corresponding to these initial data by explicit formulae and by estimating the approximation error in suitable Sobolev norms. |
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