Spatial Locality of Boltzmann Initial Layers: A Cutoff-Potential Phase Diagram

We compare the fast-time mild evolutions generated by L + εv·∇ₓ and by L, the latter being the spatially frozen collision flow. For cutoff collision models with frequency ν(v) ≍ ⟨v⟩^γ, we prove a phase diagram in polynomially weighted, spatially Lipschitz L^∞ spaces. At the hard-sphere endpoint γ = 1, the frozen approximation is uniformly accurate to order ε down to the initial time, and a fixed datum gives a matching order-ε lower bound along times tending to zero. For 0 < γ < 1, the critical fast and physical times are respectively εγ/(1−γ) and ε1/(1−γ): an order-one operator-norm effect occurs at this scale, while localization holds above it on bounded fast-time intervals. For Maxwell molecules, nonlocality persists at a fixed positive fast time. For soft potentials, spatial freezing fails in operator norm at every positive fast time. The collision-time lower bounds require relative high-output smallness, whereas fixed-time Maxwell and soft winding require only absolute high-output decay. On the proper tail-vanishing subspace, every fixed spatially Lipschitz datum nevertheless admits uniform-on-compact-fast-time freezing. The lower bounds survive the full signed integral part of the linearized operator and can be realized by one fixed spatially prepared datum. The result yields a conditional obstruction to modeling a normalized weighted Hilbert initial layer by a spatially frozen collision problem.