Stochastic Analysis of Vortex Filaments

研究者研究随机涡旋细丝及其与湍流的关系。她发展了细丝结构的严格随机描述，研究了细丝动力学，并研究了相互作用的涡旋细丝的平均场极限及其与欧拉方程的关系。布朗半鞅是其他随机涡丝研究中受到较多关注的随机过程，本文考虑将其扩展到具有多重分形结构或间歇性特征的随机过程。潜在的湍流问题是根本性的。学生们参与了这个项目。随机涡旋细丝——粗略地说，在随机过程的影响下运动的流体的垂直管——是流体运动的模型，可以揭示流体流动中湍流的发展和演变。研究者研究随机涡旋细丝，考虑它们的结构，它们的动力学，涡旋细丝的大集合的行为，以及它们与欧拉方程的关系，一个众所周知的流体流动模型。学生们参与了这个项目。流体流动中湍流的更好的数学模型可以使我们更好地理解湍流的机制。反过来，这将提高我们对大气和海洋环流的理解，并可能对天气预报和工程应用中的流体流动产生影响，例如设计更快、更经济地运行的船舶、飞机和车辆。

BessaihDMS-0608494 The investigator studies random vortex filaments and theirrelationship to turbulent flows. She develops a rigorousstochastic description of filament structures, studies filamentdynamics, and examines mean field limits for interacting vortexfilaments and their relation to the Euler equations. WhileBrownian semimartingales are the stochastic processes that havereceived more attention in other studies of random vortexfilaments, here extension to stochastic processes withmultifractal structure or intermittency features is considered. The underlying problem of turbulence is fundamental. Studentsare involved in the project. Random vortex filaments -- roughly speaking, vortical tubesof fluid moving under the influence of a random process -- are amodel for the motion of fluids that could shed some light on thedevelopment and evolution of turbulence in fluid flows. Theinvestigator studies random vortex filaments, considering theirstructure, their dynamics, the behavior of large collections ofvortex filaments, and their relationship to the Euler equations,a well-known model of fluid flows. Students are involved in theproject. Better mathematical models of turbulence in fluid flowscould lead to a better understanding of the mechanisms ofturbulence. This in turn would improve our understanding ofatmosphere and ocean circulation, with possible consequences forweather prediction, and of fluid flows in engineeringapplications, such as the design of faster and more economicallyoperated ships, planes, and vehicles.