CAREER: Analysis of Partial Differential Equations in Moving Interface Problems

多相流体流动中的运动界面非常重要，因为它们发生在从微观到宇宙的各种自然现象中。弹性动脉中的血液流动，咖啡表面，海浪，以及与太空真空相遇的太阳等离子体只是几个例子。流体界面在工业和技术应用中也发挥着关键作用，从工业乳液制造中的气泡形成到聚变反应堆中的瑞利-泰勒不稳定性。该项目旨在通过对这些模型背后的非线性偏微分方程组(PDE)的数学分析，帮助理解这些不同和重要的现象。该项目的主要目标有两个。首先，研究人员将开发新的数学工具和技术来研究与几个特定模型相关的偏微分方程。其次，研究人员将通过开发本科生协作阅读和研究计划，以及通过课程开发和研究生研究指导来培养新一代研究人员。本项目侧重于粘性流体流动的几个具体模型：接触线动力学、表面活性剂驱动的流动，以及天体物理学中的气态恒星和相关模型。研究这些模型的数学目的是证明适定性(解的存在、唯一性和估计)，确定特殊平衡构型的稳定性或不稳定性，并确定解在稳定区域中的长期行为。对每个模型的分析提出了新的困难，需要开发新的技术、先验估计方案和处理不同类型的耦合PDE系统的基本思想。

Moving interfaces in multi-phase fluid flow are of significant importance, as they occur in a huge range of natural phenomena at scales from microscopic to cosmic. Blood flow in elastic arteries, the surface of a cup of coffee, ocean waves, and solar plasma meeting the vacuum of space are just a few examples. Fluid interfaces also play a key role in industrial and technological applications, from bubble formation in industrial emulsion manufacturing to Rayleigh-Taylor instabilities in fusion reactors. This project aims to contribute to the understanding of these diverse and important phenomena through the mathematical analysis of the nonlinear systems of partial differential equations (PDEs) underlying these models. The main goals of the project are two-fold. First, the investigator will develop new mathematical tools and techniques for studying the PDEs associated with several specific models. Second, the investigator will foster the development of a new generation of researchers through the development of an undergraduate collaborative reading and research program, as well as through course development and graduate research mentoring.This project focuses on several specific models of viscous fluid flow: contact line dynamics, surfactant-driven flows, and gaseous stars and related models in astrophysics. The mathematical aim in studying these models is to prove well-posedness (existence, uniqueness, and estimates of solutions), determine the stability or instability of special equilibrium configurations, and to determine the long-time behavior of solutions in the stable regime. Analysis of each model presents novel difficulties that require the development of new techniques, schemes of a priori estimates, and basic ideas for dealing with coupled PDE systems of different type.