AI 中文总结
研究具有适度软势的玻尔兹曼方程相关Kac粒子系统，在初始数据有特定条件下，证明其分布唯一性，建立从经验测度到弱解在Wasserstein - 2距离下的显式收敛速率，是软势设置下该系统首个定量收敛速率。
AI 中文摘要
我们研究了与空间均匀玻尔兹曼方程相关的Kac粒子系统，其具有在适度软势区域\(-1 < \gamma < 0\)中的非截断碰撞核。我们证明了粒子系统在分布上的唯一性，特别是在初始数据\(f_0\)具有有限Fisher信息和多项式矩的假设下，建立了从Kac粒子系统的经验测度到玻尔兹曼方程在Wasserstein - 2距离下弱解的显式收敛速率。据我们所知，这在软势设置中为Kac粒子系统提供了第一个定量收敛速率。
英文摘要
We study the Kac particle system associated with the spatially homogeneous Boltzmann equation with non-cutoff collision kernels in the moderately soft potential regime $-1<γ<0$. We prove uniqueness in law for the particle system and, in particular, establish an explicit convergence rate from the empirical measure of the Kac particle system to the weak solution of the Boltzmann equation in Wasserstein-2 distance, assuming that the initial datum $f_0$ has finite Fisher information and polynomial moments. To the best of our knowledge, this provides the first quantitative convergence rate for the Kac particle system in the soft potential setting.
Comments48 pages
