Interfaces and Free Boundaries in Heterogeneous Media

本项目旨在开发数学工具来研究具有小尺度异质性的物理系统的大尺度特征。总的目标是从复杂的微观结构中推导出更简单的“均质化”宏观规律。重点是涉及相界面的系统：多孔介质中流体-流体界面的运动，湍流中的火焰传播，特别是粗糙表面上的液滴接触线。尽管这些系统差别很大，但它们的数学模型具有重要的共同特征。这个项目试图建立一套统一的思想来研究这些问题。宏观均匀化定律的理论推导允许有效的数值模拟，这对相关的物理系统具有预测能力。通过与工程应用（例如防水表面设计和基于液滴的印刷）的联系，对上述系统的更深入了解有可能带来更广泛的社会效益。该项目为博士生提供了潜在的研究机会，除了研究活动外，首席研究员还将教授和指导本科生和研究生。在更精确的术语，该项目是有关几个偏微分方程（PDE）模型的传播和静止锋在异质介质。主要目标是严格的分析性质，但这些目标受到物理应用的深刻推动。从广义上讲，该项目研究(i)周期性介质中可能大规模出现的奇异和各向异性特征，以及（ii）随机介质中的问题，其中需要开发新的定量估计，以控制PDE解对底层介质变化的依赖。第一部分着重讨论了一组模型自由边界问题（FBP），这些问题是在研究图案表面上的毛细液滴和研究某些离散粒子系统时出现的。微观结构的均匀化导致了一类新的FBP。这部分项目的主要目标是严格建立这个缩放/均匀化极限，并研究极限FBP的适定性。第二部分重点讨论了几个缺乏均匀化理论经典矫顽力假设的几何Hamilton-Jacobi （HJ）方程的物理动机例子。人们期望通过随机介质的平均效应在大尺度上恢复矫顽力，但这只在有限的情况下得到了严格的证实。这部分项目的目标是开发研究弱强制HJ方程的新技术；在物理术语中，这通常对应于在去钉住转变附近研究界面传播。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to develop mathematical tools to study the large-scale features of physical systems with small scale heterogeneities. The general goal is the derivation of simpler "homogenized" macroscopic laws from complicated microscopic structures. The focus is on systems involving phase interfaces: motion of fluid-fluid interfaces in porous media, flame propagation in turbulent flow, and, especially, liquid droplet contact lines on rough surfaces. Although these systems are widely different, their mathematical models have important common features. This project seeks to build a unified set of ideas to study these problems. The theoretical derivation of macroscopic homogenized laws allows for effective numerical simulations, which have predictive capacity for the associated physical systems. Deeper understanding of the above-mentioned systems has potential for broader societal benefits through connections with engineering applications, for example, design of water repellant surfaces and liquid droplet-based printing. The project contains potential research opportunities for doctoral students, and, in addition to research activities, the principal investigator will teach and mentor undergraduate and graduate students.In more precise terms the project is concerned with several partial differential equation (PDE) models of propagating and stationary fronts in heterogeneous media. The primary goals are of a rigorous analytical nature, but these aims are deeply motivated by physical applications. In broad terms, the project studies (i) singular and anisotropic features which can arise at large scales in periodic media and (ii) problems in random media where novel quantitative estimates need to be developed to control the dependence of PDE solutions on the variations in the underlying medium. The first part focuses on a set of model free-boundary problems (FBP) which arise in the study of capillary droplets on patterned surfaces and in the study of certain discrete particle systems. Homogenization of the microscopic structure leads to a new class of FBP. The main objective of this part of the project is to establish rigorously this scaling/homogenization limit and study the well-posedness properties of the limiting FBP. The second part focuses on several physically motivated examples of geometric Hamilton-Jacobi (HJ) equations which lack the classical coercivity assumption of homogenization theory. The expectation is that coercivity is recovered at large scales through the averaging effect of the random medium, but this has only been rigorously established in a limited number of cases. The objective of this part of the project is to develop new techniques to study weakly coercive HJ equations; in physical terms this often corresponds to studying interface propagation near a de-pinning transition.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.