Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866913350583058432 |
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| author | Bulut, Aynur |
| author_facet | Bulut, Aynur |
| contents | We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrödinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates.
As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_05477 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds Bulut, Aynur Analysis of PDEs Mathematical Physics We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrödinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates. As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate. |
| title | Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2001.05477 |