Well--posedness for the continuum Calogero--Moser equations by parabolic regularization

Abstract

We prove that the focusing and defocusing continuum Calogero--Moser equations are globally well-posed in $H_{+}^s(\mathbb{R})$ for every integer $s\ge 3$. In the focusing case, this requires the initial data to satisfy the mass condition $$\Vert q_0{\Vert }_{L^2}^2<2\pi .$$ Our proof is based on a parabolic regularization of the equation, uniform \textit{a--priori} estimates for the regularized solutions, and a Bona--Smith type approximation argument.