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

TL;DR

The paper proves a sharp quantitative link between inverse-power collision potentials and the hard-sphere Boltzmann model in the spatially homogeneous setting. It first establishes an explicit rate $|b_s(\theta)-\tfrac{1}{4}| \le C\,s\,\theta^{-2-2s}$ for the angular kernel as $s\to0$, by analyzing fourfold compositions of implicit functions and their endpoint degeneracies. Using this kernel control, the authors derive an optimal $O(s)$ convergence for solutions: for $t\in[0,T]$, $f^s(t)=f^0(t)+O(s)$ in weighted $L^1_k$ spaces ($k\ge 2$), with a precise equation for the scaled difference $F^s=(f^s-f^0)/s$ and a priori moment bounds. The analysis combines kernel-decomposition, Povzner-type moment inequalities, and large-data techniques to propagate moments and control nonlinear interactions uniformly in $s$, thereby establishing sharp convergence both at the kernel and equation levels. This work quantitatively clarifies the hard-sphere limit for inverse-power potentials and provides a rigorous foundation for using hard-sphere dynamics as an accurate approximation in kinetic theory, potentially informing extensions to the spatially inhomogeneous Boltzmann equation.

Abstract

The inverse power potential $U(r)=r^{-1/s}, 0<s<1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(θ)$ with an angular singularity as $θ\to 0$. Jang-Kepka-Nota-Velázquez (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: $$ |b_s(θ)-\tfrac14| \le C\, s\,θ^{-2-2s}. $$ 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 $k\ge 2.$

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.1: Optimality
• Theorem 1.2
• Remark 1.2: Optimality of the solution convergence
• Lemma 2.1
• proof : Proof of Lemma \ref{['diff-varphi-and-limit']}
• Lemma 2.2
• Remark 2.1
• proof : Proof of Lemma \ref{['diff-y-and-limit']}
• Lemma 2.3