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 \(B^{s}=|v-v_*|^{1-4s} b_s(\theta )\) with an angular singularity as \(\theta \rightarrow 0\). Jang et al. [7] proved the limit \(B^{s}\rightarrow \frac{1}{4}|v-v_*|\) as \(s\rightarrow 0\), as well as weak convergence of solutions, but without a rate. In this work we establish the sharp estimate

$$ |b_s(\theta )-\tfrac{1}{4}| \le C\, s\,\theta ^{-2-2s}. $$

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

$$\begin{aligned} f^s(t)=f^0(t)+O(s), \end{aligned}$$

quantified by the \(L^1_k\) norm for \(k\ge 2.\)