Stability and Dynamics of Traveling Waves, and Boundary Layer Theory

行波的稳定性和动力学以及边界层理论

• 批准号: 1338643
• 负责人: Toan Nguyen
• 金额: $ 4.93万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-01-02 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1338643&HistoricalAwards=false
• 关键词: Stability; Dynamics; Traveling; Waves; Boundary

The principal investigator will study mathematical problems regarding the existence, stability and instability, asymptotic analysis, and long-time dynamics of structural solutions of systems of partial differential equations. The study is divided into two independent projects. The first project concerns the stability properties of one-dimensional sources of reaction-diffusion systems that are temporally- and spatially-periodic traveling waves with defects. Pointwise Green function and Evans function techniques are exploited to investigate nonlinear stability and long-time dynamics of sources. The second project regards the vanishing viscosity limit problem of the incompressible Navier-Stokes equations in the presence of impermeable boundaries. Topics that will be addressed include the well-posedness of the Prandtl boundary-layer equation, the validity of asymptotic boundary-layer expansions, and fluid-structure interactions. The primary goal of these projects is to provide a mathematically rigorous investigation into these structural solutions and boundary-layer phenomena in fluid dynamics. The mathematical research projects that will be undertaken are motivated by many scientific disciplines including oceanography, aerodynamics, fluid dynamics, and the dynamics of biological and chemical reactions. The objects of this study, namely traveling waves, boundary layers, and coherent structures, their stability properties and their dynamics are of fundamental importance in biology, engineering, and physics. The primary goal of this research is to provide mathematical understanding of the stability properties of these structural solutions and to develop analytical methods that can be of practical use in biology, engineering, physics, and manufacturing. Among many others, one particular practical use of the study of boundary layers is to provide fundamental principles that help engineers to calculate the friction drag of a ship, an airfoil, or the body of an airplane, and to help determining an efficient shape of the body in order to minimize the friction drag and to reduce turbulence. Another objective of this research is to study the effect of viscosity in fluid motion, and to mathematically justify phenomena related to boundary layers that have been observed in experiments. Results of this research will be disseminated through presentations at national and international conferences, seminars and publications in scientific journals.

首席研究者将研究有关偏微分方程系统结构解决方案的存在，稳定性和不稳定性，渐近分析以及长期动力学的数学问题。该研究分为两个独立的项目。第一个项目涉及在时间和空间周期性的行进波中具有缺陷的一维反应扩散系统的稳定性。利用点式绿色功能和Evans功能技术来研究非线性稳定性和源的长期动力学。第二个项目认为，在存在不可压缩的边界的情况下，不可压缩的Navier-Stokes方程的消失粘度极限问题。将要解决的主题包括prandtl边界层方程的适当性，渐近边界层扩展的有效性以及流体结构相互作用。这些项目的主要目标是对流体动力学中的这些结构解决方案和边界层现象进行数学上的严格研究。将要进行的数学研究项目是由许多科学学科的动机，包括海洋学，空气动力学，流体动力学以及生物学和化学反应的动力学。这项研究的对象，即行进波，边界层和连贯的结构，它们的稳定性及其动力学在生物学，工程和物理学中至关重要。这项研究的主要目标是对这些结构解决方案的稳定性特性提供数学理解，并开发在生物学，工程，物理和制造中可以实际使用的分析方法。除其他许多外，边界层研究的一种特殊实际用途是提供基本原理，以帮助工程师计算船，机翼或飞机的身体的摩擦拖动，并帮助确定身体有效形状，以最大程度地减少摩擦力的阻力并减少湍流。这项研究的另一个目的是研究粘度在流体运动中的影响，并在实验中观察到的与边界层相关的现象是合理的。这项研究的结果将通过在科学期刊的国家和国际会议，研讨会和出版物上的演讲来传播。