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

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 $$\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{L^2}^2<2\pi \mathrm{.}$$ Our proof is based on a parabolic regularization of the equation, uniform $\mathit{\text{a–priori}}$ estimates for the regularized solutions, and a Bona–Smith type approximation argument.