A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909106828214272 |
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| author | Ohlmann, Gaspard |
| author_facet | Ohlmann, Gaspard |
| contents | In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension $1$. The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise as the continuum limit of Calogero-Moser classical spin systems. In higher dimension $d\geq 5$, it has been shown that (HWM) is well-posed by Krieger and Sire. This result has been improved by Krieger and Kiesenhofer to $d = 4$ but the Strichartz estimate on which the argument is built no longer holds for smaller dimensions. A Lax-pair structure has been revealed for this equation by Lenzmann and Gérard, indicating complete integrability and the fact that rational solutions stay rational for all time. The well-posedness of the (HWM) equation in lower dimensions remains an open problem. Here, we show the well-posedness of (HWM) in the rational case for finite times with separated poles, and for large and infinite times with distinct speeds of propagation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13442 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations Ohlmann, Gaspard Analysis of PDEs In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension $1$. The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise as the continuum limit of Calogero-Moser classical spin systems. In higher dimension $d\geq 5$, it has been shown that (HWM) is well-posed by Krieger and Sire. This result has been improved by Krieger and Kiesenhofer to $d = 4$ but the Strichartz estimate on which the argument is built no longer holds for smaller dimensions. A Lax-pair structure has been revealed for this equation by Lenzmann and Gérard, indicating complete integrability and the fact that rational solutions stay rational for all time. The well-posedness of the (HWM) equation in lower dimensions remains an open problem. Here, we show the well-posedness of (HWM) in the rational case for finite times with separated poles, and for large and infinite times with distinct speeds of propagation. |
| title | A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.13442 |