AI 中文总结
本文针对有界密度下的排斥库仑流，建立定量平均场收敛与混沌传播，证明Riesz弱-强稳定性，通过离群点构造分离两类收敛性，给出不同维度下的误差界。
AI 中文摘要
我们针对排斥库仑梯度流，在极限方程的有界密度正则性下建立了定量平均场收敛性与混沌传播特性。该论证将耗散调制能量恒等式与全N粒子律的归一化二次输运代价相结合，通过磨光后的完全平方公式处理剩余的负均方力误差项：非Lipschitz余项被该负项吸收，同时对磨光后的Lipschitz场应用了精确的一阶对易子估计。对于库仑方程，精确的L^∞衰减给出密度包络m(t)=‖ρ₀‖_{L^∞}/(1+t‖ρ₀‖_{L^∞})。适配密度的输运权重与磨光尺度m(t)^{-1/d}得到Osgood比较。因此，对所有d≥2且ρ₀∈𝒫₂(ℝ^d)∩L^∞(ℝ^d)，我们在每个给定的有限区间上获得与全局有界密度库仑解的定量比较。对于张量积初始数据，全N粒子律的归一化平方Wasserstein距离、期望调制能量及时间积分均方力误差，在d≥3时被N^{-2γ_{T,d}/d}界定，在d=2时被((1+log N)/N)^{γ_{T,2}}界定，其中γ_{T,d}=(1+T‖ρ₀‖_{L^∞})^{-c_d}。对于d-2<s<d，我们还证明了在给定参考解属于L^∞(0,T;B^{s-d+2}_{∞,q})时的Riesz弱-强稳定性，根据q分别得到Gronwall、Bihari与Osgood比较，并在所述Besov类中证明唯一性。最后，通过离群点构造分离了调制能量收敛性与Kac混沌及全N粒子律的归一化Wasserstein收敛性。
英文摘要
We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^\infty$ decay gives the density envelope $m(t)=\|ρ_0\|_{L^\infty}/(1+t\|ρ_0\|_{L^\infty})$. A density-adapted transport weight and mollification scale $m(t)^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge2$ and $ρ_0\in\mathcal P_2(\mathbb R^d)\cap L^\infty(\mathbb R^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2γ_{T,d}/d}$ for $d\ge3$ and $((1+\log N)/N)^{γ_{T,2}}$ for $d=2$, where $γ_{T,d}=(1+T\|ρ_0\|_{L^\infty})^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak--strong stability for prescribed reference solutions in $L^\infty(0,T;B^{s-d+2}_{\infty,q})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.
Comments60 pages
