Optimal one-dimensional weak-flocking decay rate

Determine the optimal one-dimensional decay rate for the weak-flocking functional of the phase-spatially extended kinetic Cucker–Smale model with a confining force, particularly whether the dimension-independent polynomial upper bound established for dimensions at least two is nonoptimal when the spatial dimension is one.

Background

The paper proves that, for polynomial mechanical-energy tails and communication weights with far-field decay exponent β, the weak-flocking functional decays at an algebraic rate proportional to (1+t)^-2(q-1/β), and establishes sharpness in dimensions d≥2 using persistent rotating modes. The construction relies on angular momentum and rotating shells, which are unavailable in one dimension.

The authors explain that one-dimensional confined trajectories repeatedly cross the central region, where the communication strength is larger than its far-field value. This suggests that the general upper bound may not be sharp in d=1, but the paper does not determine the correct one-dimensional rate.