U.S.-Polish Collaborative Research: Multidimensional Nonlinear Fractal Dynamics in Continuous Media

这是一个美国与波兰的合作研究项目，将研究连续介质中多维非线性分形动力学的分析和计算工具。非线性和非局部平流色散方程涉及连续时间随机游走的发生器也将被研究。将利用概率论和非线性偏微分方程理论的工具，包括在NSF先前支持下获得的Levy守恒定律临界非线性的理论结果。主要研究人员是凯斯西储大学的Wojbor Woyczynski博士和弗罗茨瓦夫大学的Grzegorz Karch教授。所提出的活动产生的更广泛的影响是由待研究的动力学反映了各种复杂系统的行为这一事实所保证的，这些系统由大量非线性相互作用的组件组成，这些组件在不确定的环境中运行，并且关于它们的初始状态的信息不完全。此类系统的例子包括互联网上的服务器网络、分配电力的电网、存在湍流大气和复杂海洋流的环境系统，以及用于制造半导体薄膜的化学气相沉积系统。该数学研究项目实现了将美国和中欧/东欧的顶尖专家聚集在一起的计划目标，在平等、互惠和互利的基础上，在共同感兴趣和能力强的领域结合互补的努力和能力。

This is a U.S.-Polish cooperative research project that will investigate analytical and computational tools for multidimensional nonlinear fractal dynamics in continuous media. Nonlinear and nonlocal advection-dispersion equations involving generators of continuous-time random walks will also be studied. The tools from both probability theory and the theory of nonlinear partial differential equations will be utilized, including theoretical results on critical nonlinearities for Levy conservations laws obtained under prior NSF support. The principal investigators are Dr. Wojbor Woyczynski from Case Western Reserve University and Professor Grzegorz Karch from the University of Wroclaw. The broader impact resulting from the proposed activity is assured by the fact that the dynamics to be studied reflect behavior of a variety of complex systems of a large number of nonlinearly interacting components which operate in uncertain environments and with imperfect information about their initial states. Examples of such systems are networks of servers on the internet, power grids distributing electric power, environmental systems in the presence of turbulent atmosphere and complex oceanic flows, and chemical vapor deposition systems for manufacturing thin semiconductor films.This project in mathematics research fulfills the program objectives of bringing together leading experts in the U.S. and Central/Eastern Europe to combine complementary efforts and capabilities in areas of strong mutual interest and competence on the basis of equality, reciprocity, and mutuality of benefit.