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，将发展两种不同的方法来研究流体流动的动力学性质。在第一部分中，将考虑具有大量粒子和无限制涡量的涡旋系统的准周期和混沌运动。这些涡系包括点-涡模型和去奇异版本，如涡团模型。一种组合方法，它将第一次使动力学问题的研究成为可能，通常使用相关的涡旋方法和质点方法。这种方法克服了Kolmogorov-Arnold-Moser理论对摄动大小的严格限制，到目前为止，它的应用仅限于几个自由度的问题。在第二部分，将研究沉积理论中Stokes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。将尝试梅尔尼科夫理论的等变版本。

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.