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

这是一个美国-波兰合作研究项目，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。非线性和非局部对流-弥散方程，包括连续时间随机游动的发电机也将进行研究。 从概率论和非线性偏微分方程理论的工具将被利用，包括理论结果的关键非线性利维保守法律下获得的前NSF支持。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。拟议的活动产生的更广泛的影响是有保证的事实，即要研究的动态反映了大量的非线性相互作用的组成部分，在不确定的环境中运行的各种复杂系统的行为，并与不完善的信息，其初始状态。 这种系统的例子是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动的环境系统，和化学气相沉积系统制造薄半导体薄膜。这个数学研究项目实现了汇集美国和中央/地区领先专家的计划目标。- 东欧将在平等、互惠和互利的基础上，在具有强烈共同利益和能力的领域，将联合收割机的互补努力和能力结合起来。

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.