A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations

2023/10/20 by Ohlmann, Gaspard
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.13442

Abstract

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≥ 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.

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