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| HAL: hal-00595434, version 2 |
| arXiv: 1105.4954 |
| Detailed view |  Export this paper |
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| Available versions: | v1 (2011-05-25) | v2 (2011-05-27) | v3 (2011-05-31) |
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| On Schrodinger equations with modified dispersion |
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| Rémi Carles 1 |
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| (2011-05-24) |
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| We consider the nonlinear Schrodinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness results which are sometimes sharp. When the Fourier multiplier is bounded, we infer that no Strichartz-type estimate improving on Sobolev embedding is available. Finally, we show that when the symbol is bounded, the Cauchy problem may be ill-posed in the case of critical regularity, with arbitrarily small initial data. The same is true when the symbol is homogeneous of degree one, where scaling arguments may not even give the right critical value. |
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| 1: | Institut de Mathématiques et de Modélisation de Montpellier (I3M) |
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CNRS : UMR5149 – Université Montpellier II - Sciences et techniques |
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| Subject | : | Mathematics/Analysis of PDEs |
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| hal-00595434, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00595434 | |
| oai:hal.archives-ouvertes.fr:hal-00595434 | |
| From: Rémi Carles | |
| Submitted on: Wednesday, 25 May 2011 11:15:04 | |
| Updated on: Friday, 27 May 2011 15:41:51 | |
