论文标题
测量具有单数和基质值通信的动力学cucker-smale模型的解决方案
Measure solutions to a kinetic Cucker-Smale model with singular and matrix-valued communication
论文作者
论文摘要
我们引入了具有单数和基质值的通信重量的动力学cucker-smale模型的多维变体，在一个维情况下，它减少了单数动力学cucker-smale方程。我们提出了适当的弱度量值解决方案的概念，并提出了适当的一阶减少，这持续超出了经典规范的爆炸时间。本文的核心是表明这两个配方在这个奇异的制度中均等效，从而将先前的结果从常规到弱奇异的通信权重扩展。结果，我们获得：（i）弱度量值解决方案的全球及时良好性，（ii）定量收敛速率与平衡，以及（iii）由于上述等效性以及我们最近在光纤梯度流上的工作，均匀的均值平均场限制。
We introduce a multi-dimensional variant of the kinetic Cucker-Smale model with singular and matrix-valued communication weight, which reduces to the singular kinetic Cucker-Smale equation in the one-dimensional case. We propose an appropriate notion of weak measure-valued solution to this second-order system and a suitable first-order reduction, which persist beyond the blow-up time in classical norms. The core of the paper is to show that both formulations are equivalent in this singular regime, thus extending the previous results from regular to weakly singular communication weights. As a consequence, we obtain: (i) global-in-time well-posedness of weak measure-valued solutions, (ii) quantitative convergence rates to equilibrium, and (iii) uniform-in-time mean-field limits thanks to the above-mentioned equivalence and our recent work on fibered gradient flows.
