Analysis of Complex Fluids and Moving Phase Boundaries

这个项目将研究一大组描述复杂流体和沉浸在低维几何物体中的动力学的非线性偏微分方程组。这些偏微分方程包括输运方程、相场方程、Fokker-Planck方程和几何演化方程的非线性耦合，以及Navier-Stokes方程。它们既可以是抛物线的，也可以是双曲线的，并且具有尖锐的界面、奇点或多个尺度。这些都是令人着迷和具有挑战性的问题，直接受到生物、物理、流体力学和材料科学问题的推动。对它们的透彻理解需要新的思想和方法。该项目包含一组具体的问题，建立在一些部分成果的基础上，并根据一项详细的计划进行，该计划描述了要做出的努力和采取的方法。人们期待着这一理论方面令人兴奋的新发展，并将其有趣地应用于其他领域。许多生物流体(例如血液)属于“复杂流体”的标题。对复杂流体的研究也出现在材料科学、医学和物理学中。了解这些流体的动力学，特别是在存在浸没的几何物体的情况下，是一个具有挑战性的问题，对于包括医疗和高科技设备的设计在内的各种应用来说非常重要。一般来说，用复杂的流体进行实验以收集关于它们的一组良好的数据是昂贵的。对它们的动态进行建模也不容易，也不容易对它们的行为进行准确的计算机模拟。对复杂流体的数学模型进行一些理论分析(如本项目中将进行的数学模型)不仅将导致对它们更好的定性理解，这对我们基础知识的进步非常重要，而且还将提供对涉及此类流体的问题的本质的洞察，有助于建立可靠和有效的数值方案来研究它们。后者将提供可能被证明对应用有用的数据(相当于许多物理实验产生的数据)。

This project will study a large set of nonlinear partial differential equations that describe the dynamics of complex fluids and immersed lower-dimensional geometric objects. These partial differential equations involve nonlinear couplings of transport, phase-field, Fokker-Planck, and geometric evolutionary equations, along with the Navier-Stokes equations. They may be of both parabolic and hyperbolic nature and possess sharp interfaces, singularities, or multiple scales. These are fascinating and challenging problems that are directly motivated by problems in biology, physics, fluid mechanics, and materials science. A thorough understanding of them will require new ideas and methods. The project encompasses a concrete set of problems, builds on some partial results, and proceeds under a detailed plan that describes the efforts to be made and approaches to be taken. One expects exciting new developments in the theory and interesting applications to other fields. Many biological fluids (blood, for example) fall under the heading of "complex fluids." The study of complex fluids also arises in materials science, medicine, and physics. Understanding the dynamics of these fluids, particularly in the presence of immersed geometric objects, is a challenging problem and is very important for various applications, including the design of medical and high-tech devices. It is, in general, expensive to perform experiments with complex fluids in order to collect a good set of data about them. It is also not easy to model their dynamics nor to do accurate computer simulations of their behavior. Some theoretical analysis of mathematical models of complex fluids (such as that to be undertaken in this project) will not only lead to improved qualitative understanding of them, which is important for advances in our basic knowledge, but also provide insights into the nature of problems involving such fluids in a way that could help to set up reliable and effective numerical schemes for investigating them. The latter would provide data (equivalent to that produced by numerous physical experiments) that could prove useful for applications.