Global well-posedness for the defocusing 3D quadratic NLS in the sharp critical space
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| Format: | Preprint |
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2024
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| _version_ | 1866912060645834752 |
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| author | Shen, Jia Wu, Yifei |
| author_facet | Shen, Jia Wu, Yifei |
| contents | In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schrödinger equation \begin{align*} i\partial_t u + \frac12Δu = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot H_x^{1/2}$ for radial data.
Killip, Masaki, Murphy, and Visan [2017, NoDEA] have proved its global well-posedness and scattering, if the $\mathcal F \dot H_x^{1/2}$-norm of the solution is bounded in the maximal lifespan. Now, we remove this a priori assumption for the global well-posedness statement in the radial case.
Our method is based on the almost conservation of pseudo conformal energy. This energy scales like $\dot H_x^{-1}$, which is supercritical. We are still able to derive the global well-posedness using this monotone quantity. The main observation is that we can establish the local solution in supercritical weighted space when the initial time is away from the origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness for the defocusing 3D quadratic NLS in the sharp critical space Shen, Jia Wu, Yifei Analysis of PDEs 35Q55, 35B40 In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schrödinger equation \begin{align*} i\partial_t u + \frac12Δu = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot H_x^{1/2}$ for radial data. Killip, Masaki, Murphy, and Visan [2017, NoDEA] have proved its global well-posedness and scattering, if the $\mathcal F \dot H_x^{1/2}$-norm of the solution is bounded in the maximal lifespan. Now, we remove this a priori assumption for the global well-posedness statement in the radial case. Our method is based on the almost conservation of pseudo conformal energy. This energy scales like $\dot H_x^{-1}$, which is supercritical. We are still able to derive the global well-posedness using this monotone quantity. The main observation is that we can establish the local solution in supercritical weighted space when the initial time is away from the origin. |
| title | Global well-posedness for the defocusing 3D quadratic NLS in the sharp critical space |
| topic | Analysis of PDEs 35Q55, 35B40 |
| url | https://arxiv.org/abs/2410.04337 |