Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916388004691968 |
|---|---|
| author | Hirayama, Hiroyuki Kinoshita, Shinya |
| author_facet | Hirayama, Hiroyuki Kinoshita, Shinya |
| contents | In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^{s}$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_06563 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations Hirayama, Hiroyuki Kinoshita, Shinya Analysis of PDEs 35Q55 In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^{s}$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$. |
| title | Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/1802.06563 |