Stability and Dynamics of Traveling Waves, and Boundary Layer Theory

主要研究人员将研究关于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的数学问题。这项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，该系统是具有缺陷的时间和空间周期行波。利用点态格林函数和Evans函数技术研究了源的非线性稳定性和长时间动力学。第二个方案研究了具有不可渗透边界的不可压缩Navier-Stokes方程的零粘性极限问题。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和边界层现象进行严格的数学研究。将开展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。本文的研究对象是行波、边界层和相干结构，它们的稳定性和动力学性质在生物学、工程学和物理学中都具有重要意义。这项研究的主要目标是提供对这些结构解的稳定性属性的数学理解，并开发可在生物学、工程学、物理学和制造业中实际使用的分析方法。其中，边界层研究的一个特别实际用途是提供基本原理，帮助工程师计算船舶、翼型或飞机机身的摩擦阻力，并帮助确定有效的机身形状，以便将摩擦阻力降至最低并减少湍流。这项研究的另一个目的是研究流体运动中粘度的影响，并从数学上证明实验中观察到的与边界层有关的现象。这项研究的结果将通过在国家和国际会议、研讨会和科学期刊上发表的报告来传播。

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.