Global well-posedness of the energy-critical nonlinear Schrödinger equations on $\mathbb{T}^{d}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915036655517696 |
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| author | Kwak, Beomjong |
| author_facet | Kwak, Beomjong |
| contents | In this paper, we prove the global well-posedness of the energy-critical nonlinear Schrödinger equations on the torus $\mathbb{T}^{d}$ for general dimensions. This result is new for dimensions $d\ge5$, extending previous results for $d=3,4$ [10,22]. Compared to the cases $d=3,4$, the regularity theory for higher $d$, developed in the underlying local well-posedness result [17], is less understood. In particular, stability theory and inverse inequalities, which are ingredients in [10,22] and more generally in the widely used concentration compactness framework since [13], are too weak to be applied to higher dimensions.
Our proof introduces a new strategy for addressing global well-posedness problems. Without relying on perturbation theory, we develop tools to analyze the concentration dynamics of the nonlinear flow. On the way, we show the formation of a nontrivial concentration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of the energy-critical nonlinear Schrödinger equations on $\mathbb{T}^{d}$ Kwak, Beomjong Analysis of PDEs 35Q55 In this paper, we prove the global well-posedness of the energy-critical nonlinear Schrödinger equations on the torus $\mathbb{T}^{d}$ for general dimensions. This result is new for dimensions $d\ge5$, extending previous results for $d=3,4$ [10,22]. Compared to the cases $d=3,4$, the regularity theory for higher $d$, developed in the underlying local well-posedness result [17], is less understood. In particular, stability theory and inverse inequalities, which are ingredients in [10,22] and more generally in the widely used concentration compactness framework since [13], are too weak to be applied to higher dimensions. Our proof introduces a new strategy for addressing global well-posedness problems. Without relying on perturbation theory, we develop tools to analyze the concentration dynamics of the nonlinear flow. On the way, we show the formation of a nontrivial concentration. |
| title | Global well-posedness of the energy-critical nonlinear Schrödinger equations on $\mathbb{T}^{d}$ |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2411.18163 |