| Abstract: |
| The inverse power potential $U(r)=r^{-1/s}$, $0\lt s\lt 1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(\theta)$ with an angular singularity as $\theta\to 0$.
Jang et~al.~\cite{Jang2023-df} 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 talk I further introduce recent establishment of the following sharp quantitative estimate:
\[
|b_s(\theta)-\tfrac14| \le C\, s\,\theta^{-2-2s}.
\]
In particular, this sharp estimate yields the \emph{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 $k\ge 2$. This is a joint work with Zheng-Nan Hu, Zheng-An Yao, and Yu-Long Zhou. |
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