Uniform collision estimates for concentrated packets in non-cutoff collisional models

Establish the required uniform collision estimate for the concentrated packet family in the Landau equation and the Boltzmann equation without angular cutoff, particularly at a fixed positive observation time as the packet radius tends to zero, in order to determine whether the packet-based discontinuity mechanism persists for these equations.

Background

The paper’s discontinuity construction relies on transporting spatially sparse packets so that their velocity envelope becomes large, while showing that the self-consistent dynamics perturb the relevant characteristics by less than the packet widths. For cutoff Boltzmann equations, a gain–loss representation can potentially preserve this mechanism if the packet family has uniform existence, velocity-tail, and loss-frequency bounds.

For the Landau equation and the Boltzmann equation without angular cutoff, gain and loss cannot be separated in the same way. The authors note that velocity derivatives of the free transported packet profile become large as the packet radius decreases, leading heuristically to an accumulated second-order collision error of order h+h^3/r^2. Proving the uniform collision estimate needed to control this error, especially for a fixed positive observation time while the packet radius tends to zero, is left unresolved.