论文标题
通过中等相互作用的粒子的汉堡和凯勒 - 塞格方程的定量近似
Quantitative approximation of the Burgers and Keller-Segel equations by moderately interacting particles
论文作者
论文摘要
在这项工作中，我们获得了两个中等相互作用的随机粒子系统的收敛速率，该系统与粘性汉堡和凯勒 - 塞格方程相关的单数内核。 这项工作的主要新颖性是考虑一个非局部整合的内核。也就是说，对于$ \ mathbb {r} $中的粘性汉堡方程，我们几乎可以在任何时间间隔内获得较小的经验度量与pde解决方案的互惠率，并在某些贝塞尔空间中融合了pde解决方案。以相同的速率，在瓦斯坦斯坦距离和粒子轨迹的水平以及标准耦合与麦基恩 - 维拉索夫粒子的轨迹的水平上，收敛也适用于真正的经验度量。 如果在$ d $维圆环上的Keller-Segel方程式中，我们几乎可以确保将易流的经验度量与PDE在某些$ l^q $空间中的解决方案，并以$ n^{ - \ frac {1} {1} {2（d+1）}} $。对于任何化学吸引剂$χ$的任何值，结果都达到了PDE的最大存在时间。
In this work we obtain rates of convergence for two moderately interacting stochastic particle systems with singular kernels associated to the viscous Burgers and Keller-Segel equations. The main novelty of this work is to consider a non-locally integrable kernel. Namely for the viscous Burgers equation in $\mathbb{R}$, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some Bessel space with a rate of convergence of order $N^{-1/6}$, on any time interval. With the same rate, convergence also holds for the genuine empirical measure in Wasserstein distance, and at the level of the trajectories of the particles with the standard coupling to McKean-Vlasov particles. In the case of the Keller-Segel equation on a $d$-dimensional torus, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some $L^q$ space with a rate of order $N^{-\frac{1}{2(d+1)}}$. The result holds up to the maximal existence time of the PDE, for any value of the chemo-attractant sensitivity $χ$.
