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

TL;DR

This paper provides a precise quantitative estimate on how the inverse power potential kernel converges to the hard sphere limit in the Boltzmann equation, establishing an optimal convergence rate of order s for solutions with large initial data.

Contribution

It derives a sharp estimate for the angular kernel difference, leading to the first optimal convergence rate result for solutions of the Boltzmann equation as the potential parameter s approaches zero.

Findings

Established a sharp estimate: |b_s(θ) - 1/4| ≤ C s θ^{-2-2s}.

Proved the optimal O(s) convergence rate for solutions in Sobolev spaces.

Quantified the convergence of solutions as s→0 with explicit bounds.

Abstract

The inverse power potential $U(r)=r^{-1/s}, 0<s<1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(\theta)$ with an angular singularity as $\theta\to 0$. Jang-Kepka-Nota-Vel\'azquez (2023) proved the limit $B^{s}\to \frac14|v-v_*|$ 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: $$ |b_s(\theta)-\tfrac14| \le C\, s\,\theta^{-2-2s}. $$ 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 $L^1_k$ norm for $k\ge 2.$