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: 1108821
• 负责人: Toan Nguyen
• 金额: $ 11.4万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2013-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108821&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.

首席研究员将研究偏微分方程组结构解的存在性、稳定性和不稳定性、渐近分析和长期动力学等数学问题。该研究分为两个独立的项目。第一个项目涉及反应扩散系统的一维源的稳定性特性，该系统是具有缺陷的时间和空间周期性行波。利用逐点格林函数和埃文斯函数技术来研究源的非线性稳定性和长期动态。第二个项目涉及不可压缩纳维-斯托克斯方程在存在不可渗透边界的情况下的消失粘度极限问题。将讨论的主题包括普朗特边界层方程的适定性、渐近边界层展开的有效性以及流体-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严格的数学研究。将进行的数学研究项目受到许多科学学科的推动，包括海洋学、空气动力学、流体动力学以及生物和化学反应动力学。这项研究的对象，即行波、边界层和相干结构，它们的稳定性特性和动力学在生物学、工程和物理学中具有根本重要性。这项研究的主要目标是提供对这些结构解的稳定性特性的数学理解，并开发可在生物学、工程、物理学和制造中实际使用的分析方法。其中，边界层研究的一个具体实际用途是提供基本原理，帮助工程师计算船舶、机翼或飞机机身的摩擦阻力，并帮助确定有效的机身形状，以最大限度地减少摩擦阻力并减少湍流。这项研究的另一个目标是研究粘度对流体运动的影响，并从数学上证明与实验中观察到的边界层相关的现象。这项研究的结果将通过在国内和国际会议、研讨会和科学期刊上发表的演讲来传播。