Derivation of the Half-Wave Maps Equation from Calogero--Moser Spin Systems

2020/07/30 by Lenzmann, Enno, Sok, Jérémy · 1 citation
#35Q99 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.15323

Abstract

We prove that the energy-critical half-wave maps equation ∂_t S =S × |∇| S, (t,x) ∈ ℝ × \mathbbT arises as an effective equation in the continuum limit of completely integrable Calogero-Moser classical spin systems with inverse square 1/r² interactions on the circle. We study both the convergence to global-in-time weak solutions in the energy class as well as short-time strong solutions of higher regularity. The proofs are based on Fourier methods and suitable discrete analogues of fractional Leibniz rules and Kato-Ponce-Vega commutator estimates. In a companion paper, we further extend our arguments to study the real line case and more general spin interactions.

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