Exact value of the L2-equicontinuity mass threshold M* for the focusing continuum Calogero–Moser model

Determine the exact value of the sharp mass threshold M*, defined for the focusing continuum Calogero–Moser model iu_t + u_{xx} - 2 i u Π^+ ∂_x(|u|^2) = 0, with the property that whenever a bounded family U ⊂ H^∞_+ of initial data is L2-equicontinuous (i.e., uniformly translation-continuous in L2 or, equivalently, with uniformly vanishing high-frequency tails) and satisfies sup_{u0∈U} M(u0) < M*, then the associated set of partial orbits U_T^* = {u(t) : u solves the model with u(0) ∈ U, t ∈ [0, T)} is L2-equicontinuous for all T > 0.

Background

Laurens, Killip, and Vișan introduced the notion of L2-equicontinuity for families of initial data in H^∞_+ and defined a sharp mass threshold M* ensuring that if sup mass is below M*, then the entire set of partial orbits remains L2-equicontinuous for all times. They proved M* ≥ 2π and global well-posedness below M* for all H^s_+, s ≥ 0.

The present paper proves that for any mass just above 2π there exist initial data whose maximal orbit fails to be L2-equicontinuous, showing that global-in-time L2-equicontinuity fails in general for mass > 2π. Despite this, the precise value of M* is not determined in the paper and remains explicitly unidentified.