Topics in Mathematical Fluid Dynamics

DMS Award AbstractAward #： 0202767 PI： Friedlander，SusanInstitution： 伊利诺伊大学芝加哥分校课程： 应用数学项目经理：Catherine Mavriplis职务：数学流体动力学主题本项目使用非线性偏微分方程中的各种技术来研究与描述流体运动的方程（即欧拉方程和纳维-斯托克斯方程）相关的几个不同的开放性问题。流体流动的稳定性是流体动力学中最基本的问题之一：稳定的流动在环境中不可避免的扰动下是鲁棒的，而不稳定的流动可能会破裂，有时很快。研究者将继续探索不同类型的不稳定之间的关系，目的是定义不稳定的规模或程度。寻求充分条件来证明非线性不稳定类无粘流体。该项目研究的另一个主题是当应力和应变张量之间存在非线性关系时粘性流体（所谓的非牛顿流体）的方程。对任何流体方程提出的一个基本数学问题是奇点在有限时间内发展的可能性。受非牛顿方程并矢模型结果的启发，研究者和合作者使用小波和Littlewood -Paley理论的技术，在具有非线性粘性的Navier Stokes方程的情况下，给出了奇异集维数的上界。 我们世界的大部分是由流体组成的：例如，大气层海洋甚至我们自己的身体然而，流体的行为非常复杂，目前只在很小的程度上了解。这个项目使用严格的数学来研究几个问题，这些问题对流体运动的本质是至关重要的。这些包括流体配置的稳定性或不稳定性。在许多科学家看来，波浪和不稳定性是长期天气预报的核心，对全球变化和世界经济有着实际的影响。另一个正在研究的主题涉及具有非线性粘性力关系的流体。例如，这种情况发生在湍流涡流模型中，也发生在具有特殊分子结构的流体中，如血液或某些聚合物。研究人员试图限制这种流体运动中奇点（例如无限能量尖峰）的发展。日期：二○ ○二年四月二十六日

DMS Award AbstractAward #: 0202767PI: Friedlander, SusanInstitution: University of Illinois, ChicagoProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Topics in Mathematical Fluid DynamicsThis project uses a variety of techniques in nonlinear partial differential equations to study several different open questions connected with the equations that describe the motion of a fluid, namely the Euler and the Navier-Stokes equations. Stability of a fluid flow is one of the most basic problems in fluid dynamics: stable flows are robust under the inevitable disturbances in the environment, while unstable flows may break up, sometimes rapidly. The investigator will continue to explore the relations between different types of instability with the goal of defining scales or degrees of instability. Sufficient conditions are sought to demonstrate nonlinear instability for classes of inviscid fluids. Another topic studied in the project is the equation for a viscous fluid when there is a nonlinear relation between the stress and strain tensors (a so-called non-Newtonian fluid ). A fundamental mathematical question asked for any of the fluid equations is the possibility of the development of singularities in finite time. Motivated by results from a dyadic model for the non-Newtonian equations, the investigator and collaborators use techniques of wavelets and Littlewood -Paley theory to give an upper bound on the dimension of the singular set in the case of the Navier Stokes equations with nonlinear viscosity. The greater portion of our world is composed of fluids: e.g., the atmosphere, the oceans, even our own bodies. However fluids behave in very complex ways that are presently understood only to a very minor degree. This project uses rigorous mathematics to examine several questions that are fundamantal to the nature of fluid motion. These include the stability or instability of a fluid configuration. In the view of many scientists, waves and instabilities lie at the heart of long term weather prediction with all its practical implications for global change and the world economy. Another topic under investigation concerns fluids with a nonlinear viscous force relation. This happens, for example, in models for turbulent eddies and also in fluids with a special molecular structure such a blood or some polymers. The investigator seeks to bound the development of singularities (e.g. infinite energy spikes) in the motion of such fluids. Date: April 26, 2002