Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909569595211776 |
|---|---|
| author | Ifrim, Mihaela Tataru, Daniel |
| author_facet | Ifrim, Mihaela Tataru, Daniel |
| contents | The first target of this article is the local well-posedness question for 1D quasilinear Schrödinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a \emph{sharp local well-posedness} result.
The second goal of this article is to consider the long time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: ``\emph{Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions}''; the conjecture was initially proved for a well chosen semilinear model of Schrödinger type.
Our work here establishes the above conjecture for 1D quasilinear Schrödinger flows. Precisely, we show that if the problem has \emph{phase rotation symmetry} and is \emph{conservative and defocusing}, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows. Furthermore, we prove it at the minimal Sobolev regularity in our local well-posedness result.
The defocusing condition is essential in our global result. Without it, the authors have conjectured that \emph{small, $ε$ size data yields long time solutions on the $ε^{-8}$ time-scale}. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schrödinger flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00570 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow Ifrim, Mihaela Tataru, Daniel Analysis of PDEs 35Q55 The first target of this article is the local well-posedness question for 1D quasilinear Schrödinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a \emph{sharp local well-posedness} result. The second goal of this article is to consider the long time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: ``\emph{Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions}''; the conjecture was initially proved for a well chosen semilinear model of Schrödinger type. Our work here establishes the above conjecture for 1D quasilinear Schrödinger flows. Precisely, we show that if the problem has \emph{phase rotation symmetry} and is \emph{conservative and defocusing}, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows. Furthermore, we prove it at the minimal Sobolev regularity in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that \emph{small, $ε$ size data yields long time solutions on the $ε^{-8}$ time-scale}. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schrödinger flows. |
| title | Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2306.00570 |