未正则化Vlasov--Riesz--Fokker--Planck系统在总变差距离下的定量混沌传播
Quantitative propagation of chaos in total variation for unregularized Vlasov--Riesz--Fokker--Planck systems
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中文总结 AI 辅助
本文针对未正则化三维排斥Vlasov-Poisson-Fokker-Planck粒子系统，在总变差距离下建立了定量混沌传播，通过条件化、辅助Fokker-Planck方程和加权L^q估计，证明了误差界，并推广到Riesz势情形。
中文摘要 AI 辅助
我们建立了未正则化三维排斥Vlasov--Poisson--Fokker--Planck（VPFP）粒子系统在总变差距离下的定量混沌传播。对于满足加权Sobolev正则性和高斯矩的张量积初始数据，我们证明对每个$0<b<1/3$，存在与$N$和$k$无关的$\tau_b>0$和$C_b\ge1$，使得\\[ \sup_{0\le t\le\tau_b}\\|F_{N,k}(t)-f_t^{\otimes k}\\|_{\mathrm{TV}} \le C_b^kN^{-b}, \qquad N\ge1,\quad 1\le k\le N. \\]更一般地，对于在$\mathbb{T}^d$上具有奇性$|x|^{-s}$的排斥Riesz势，以及$\mathbb{R}^d$，$0<s<d/2$，我们获得$C^k\bigl(N^{-1/q}+N^{-(d/s-2)}\bigr)$，其中$q\ge2$和$q>d/(d-s-1)$。证明结合了乘积初始律的条件化、辅助$k$粒子Fokker--Planck方程以及BBGKY层次的加权$L^q$估计。在局部条件$K\in L^{q'}_{\mathrm{loc}}$下，通过加权Hölder估计控制第$(k+1)$层相互作用项；对于$d=3$和$s=1$，该条件为$q>3$。在$\mathbb{R}^d$上，多项式空间权重控制远场。对于每个固定的粒子数，奇异粒子动力学全局适定且无碰撞。这允许移除力的正则化。
英文摘要
We establish quantitative propagation of chaos in total variation for the unregularized three-dimensional repulsive Vlasov--Poisson--Fokker--Planck (VPFP) particle system. For tensorized initial data satisfying weighted Sobolev regularity and a Gaussian moment, we prove that for every $0<b<1/3$ there exist $τ_b>0$ and $C_b\ge1$, independent of $N$ and $k$, such that \[ \sup_{0\le t\leτ_b}\|F_{N,k}(t)-f_t^{\otimes k}\|_{\mathrm{TV}} \le C_b^kN^{-b}, \qquad N\ge1,\quad 1\le k\le N. \] More generally, for repulsive Riesz potentials with singularity $|x|^{-s}$ on $\mathbb{T}^d$ and $\mathbb{R}^d$, $0<s<d/2$, we obtain $C^k\bigl(N^{-1/q}+N^{-(d/s-2)}\bigr)$ for $q\ge2$ and $q>d/(d-s-1)$. The proof combines conditioning of the product initial law, auxiliary $k$-particle Fokker--Planck equations, and weighted $L^q$ estimates for the BBGKY hierarchy. The level-$(k+1)$ interaction term is controlled by a weighted Hölder estimate under the local condition $K\in L^{q'}_{\mathrm{loc}}$; for $d=3$ and $s=1$, this condition is $q>3$. On $\mathbb{R}^d$, polynomial spatial weights control the far field. For each fixed particle number, the singular particle dynamics are globally well posed and have no collisions. This permits the force regularization to be removed.
发表机构
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
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