Stability and Dynamics of Traveling Waves, and Boundary Layer Theory

主要研究者将研究关于偏微分方程系统的结构解的存在性，稳定性和不稳定性，渐近分析和长时间动态的数学问题。该研究分为两个独立的项目。第一个项目涉及的一维源的反应扩散系统的时间和空间周期行波缺陷的稳定性。利用逐点绿色函数和Evans函数技术研究了源的非线性稳定性和长时间动力学。第二个项目是考虑不可压缩Navier-Stokes方程在不可渗透边界存在下的粘性消失极限问题。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提供一个数学上严格的调查，这些结构的解决方案和边界层现象的流体动力学。将进行的数学研究项目是由许多科学学科，包括海洋学，空气动力学，流体动力学，以及生物和化学反应的动力学的动机。本研究的对象，即行波，边界层和相干结构，它们的稳定性和动力学特性在生物学，工程学和物理学中具有根本的重要性。本研究的主要目标是提供这些结构解决方案的稳定性的数学理解，并开发分析方法，可以在生物学，工程，物理学和制造业的实际用途。其中，边界层研究的一个特殊的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和减少湍流。本研究的另一个目的是研究粘性在流体运动中的作用，并从数学上证明实验中观察到的与边界层相关的现象。这项研究的结果将通过在国家和国际会议、研讨会上的介绍以及在科学杂志上发表文章的方式传播。

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.