Regularity estimates for the non-cutoff soft potential Boltzmann equation with typical rough and slowly decaying data

In the present work, we investigate estimates of regularity for weak solutions to the non-cutoff Boltzmann equation with soft potentials. We restrict our focus to the so-called "typically rough and slowly decaying data", which is constructed to satisfy typical properties: low regularity and having exact polynomial decay in high velocity regimes. By exploring the degenerate and non-local properties of the collision operator, we demonstrate that (i) such data induce only finite smoothing effects for weak solutions in Sobolev spaces; (ii) this finite smoothing property implies that the Leibniz rule does not hold for high derivatives of the collision operator (even in the weak sense). Moreover, we can also prove that the average of the solution or the average of the collision operator on a special domain in $\mathbb{R}^3_v$ will induce discontinuity in the $x$ variable. These facts present major obstacles to proving the conjecture that solutions to the equation will instantly become infinitely smooth for both spatial and velocity variables at any positive time if the initial data has only polynomial decay in high velocity regimes.