Uniqueness under polynomial mechanical-energy tails

Establish uniqueness of global Lagrangian weak solutions to the phase-spatially extended kinetic Cucker–Smale equation with a confining force when the initial data have only polynomial mechanical-energy tails.

Background

The paper proves global existence for centered initial data with finite second moments and establishes uniqueness through an Osgood-type Wasserstein stability estimate under exponential mechanical-energy tails. It explicitly notes that polynomial tails do not yield the modulus needed for the presented uniqueness argument.

The polynomial weak-flocking decay results therefore apply to every global Lagrangian weak solution furnished by the existence theorem, without asserting that such a solution is uniquely determined by its initial datum. Extending uniqueness from exponential to polynomial tails is identified as a remaining problem.