Global well-posedness for strong and classical kinetic SL solutions

Establish global well-posedness for strong and classical solutions of the kinetic Schrödinger–Lohe equation on the infinite-dimensional phase space consisting of wave functions and one-body potentials.

Background

The paper derives a Vlasov–McKean-type kinetic Schrödinger–Lohe equation on the phase space $\mathbf S\mathfrak H\times C(\mathbb T^d)$, proves existence and uniqueness through a characteristic-flow formulation, and establishes quantitative mean-field and synchronization estimates. However, the authors distinguish these results from a full strong or classical well-posedness theory for the kinetic transport equation itself. Developing such a theory would clarify regularity, uniqueness, and global continuation properties beyond the Lagrangian weak-solution framework used in the paper.