Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915380961738752 |
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| author | Hirayama, Hiroyuki Kinoshita, Shinya Okamoto, Mamoru |
| author_facet | Hirayama, Hiroyuki Kinoshita, Shinya Okamoto, Mamoru |
| contents | We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic setting, the authors proved some well-posedness results, which contain the scaling critical case for $d\geq 2$. In the present paper, we prove the well-posedness of this system for the periodic setting. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some conditions for the coefficients of the Laplacian. We also prove some ill-posedness results. As long as we use an iteration argument, our well-posedness results are optimal except for some critical cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03565 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations Hirayama, Hiroyuki Kinoshita, Shinya Okamoto, Mamoru Analysis of PDEs 35Q55 We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic setting, the authors proved some well-posedness results, which contain the scaling critical case for $d\geq 2$. In the present paper, we prove the well-posedness of this system for the periodic setting. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some conditions for the coefficients of the Laplacian. We also prove some ill-posedness results. As long as we use an iteration argument, our well-posedness results are optimal except for some critical cases. |
| title | Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2407.03565 |