Calogero-Moser自旋系统半波映象方程的推导
Derivation of the Half-Wave Maps Equation from Calogero--Moser Spin Systems
摘要：
我们证明了能量临界半波映射方程\[ \partial_t \mathbf{S} =\mathbf{S} \times |\nabla| \mathbf{S}, \quad (t,x) \in \mathbb{R} \times \mathbb{T} \]在圆上具有平方反比$1/r^2$相互作用的完全可积Calogero-Moser经典自旋系统的连续极限中是一个有效方程.我们研究了在能量类中的全局时间弱解的收敛性以及具有更高正则性的短时强解的收敛性。证明是基于傅立叶方法和适当的离散模拟分数莱布尼茨规则和加藤庞塞Vega交换子估计。在配套文件中，我们进一步扩展我们的论点，研究真正的线的情况下，更一般的自旋相互作用。
英文摘要：
We prove that the energy-critical half-wave maps equation \[ \partial_t \mathbf{S} =\mathbf{S} \times |\nabla| \mathbf{S}, \quad (t,x) \in \mathbb{R} \times \mathbb{T} \] arises as an effective equation in the continuum limit of completely integrable Calogero-Moser classical spin systems with inverse square $1/r^2$ 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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