Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation

Abstract

The inverse power potential $U(r)=r^{-1/s},0<s<1$, generates the Boltzmann kernel $B^s=|v-v_{\ast }{|}^{1-4s}{b}_s(\theta )$ with an angular singularity as $\theta \to 0$. Jang-Kepka-Nota-Velázquez (2023) proved the limit $B^s\to \frac{1}{4}|v-v_{\ast }|$ as $s\to 0$, as well as weak convergence of solutions based on this kernel convergence. In this work we establish the following sharp quantitative estimate: $$|b_s(\theta )-\frac{1}{4}|\le Cs{\theta }^{-2-2s}.$$ In particular, this sharp estimate yields the optimal $O(s)$ convergence rate for solutions of the homogeneous Boltzmann equation with large initial data in suitable Sobolev spaces; i.e., for any $t\in [0,T]$, we have $$f^s(t)=f^0(t)+O(s),$$ quantified by the $L_k^1$ norm for $k\ge 2.$

27 pages