论文标题
适用于在几个空间维度中细长颗粒的正规惯性院长卡瓦萨基模型的适应性
Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions
论文作者
论文摘要
在任何有限的尺寸$ d \ in \ mathbb {n} $中，提出了一个随机PDE，描述了有限体积弱相互作用的惯性粒子系统中的介观波动。它是Dean-Kawasaki模型的正规化和惯性版本。开发了该模型的高概率能力理论。该理论在同一作者的早期作品中施加的空间缩放限制大大提高，该限制仅适用于在一个维度上明显更大的粒子。现在，当粒子宽度$ε$与$ n^{ - 1/θ} $成正比时，适用的理论现在适用于$ d $ dimensions，对于$θ> 2d $，而$ n $是粒子的数量。在某个Sobolev规范中，此缩放是最佳的。分析的关键工具是分数Sobolev空间，贝塞尔功能上的尖锐界限，$ d $空间维度中正则化的可分离性以及使用faàdibruno公式的使用。
A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the Dean-Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in $d$-dimensions when the particle-width $ε$ is proportional to $N^{-1/θ}$ for $θ>2d$ and $N$ is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the $d$-spatial dimensions, and use of the Faà di Bruno's formula.
