Mathematical Sciences: A Combinatorial Method for Large Vortex Systems and Systematic Time-Reversible Dynamics in Particle Sedimentation

在具体问题的背景下，将发展两种不同的研究流体流动动力学性质的方法。在第一部分中，将考虑具有大量粒子和不受限制涡量的涡系统的准周期和混沌运动。这些涡旋系统包括点涡旋模型和去具体化的涡旋团模型。这是一种组合方法，它将首次使与涡旋方法和粒子方法相关的动力学问题的研究得以应用。这种方法将克服Kolmogorov-Arnold-Moser理论中对微扰大小的严格限制，这一限制迄今为止将其应用于自由度较小的问题。第二部分将研究沉降理论中斯托克斯球的周期运动和混沌运动。将强调相同沉积颗粒的近平衡动力学。Melnikov理论的等变版本将被尝试。

Two separate methods for the study of dynamical properties of fluid-flows will develop in the context of specific problems. In part I, the quasi-periodic and chaotic motions of vortex systems with a large number of particles and unrestricted vorticity will be considered. These vortex systems include the point-vortex model, and desingularized versions such as the vortex blob model. A combinatorial approach that for the first time will enable the study of dynamical problems that are relevant vortex methods and particle methods in general to be used. This approach will overcome the severe restriction on the size of perturbations in Kolmogorov-Arnold-Moser theory, which has hitherto limited its applications to problems with few degrees of freedom. In part II, periodic and chaotic motions of stokes spheres in sedimentation theory will be studied. Near-equilibrium dynamics of identical sedimenting particles will emphasized. Equivariant version of the Melnikov theory will be attempted.