Interfaces and Free Boundaries in Heterogeneous Media

该项目旨在开发数学工具来研究具有小尺度非均匀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“均匀化”宏观定律。 重点是涉及相界面的系统：多孔介质中的流体-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。 虽然这些系统差异很大，但它们的数学模型具有重要的共同特征。本项目旨在建立一套统一的思路来研究这些问题。 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。 对上述系统的更深入了解有可能通过与工程应用的联系，例如防水表面的设计和基于液滴的打印，产生更广泛的社会效益。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。更准确地说，该项目涉及非均匀介质中传播和静止前沿的几个偏微分方程（PDE）模型。 主要目标是严格的分析性质，但这些目标是由物理应用深深激励。从广义上讲，该项目研究（i）奇异和各向异性的功能，可能会出现在大规模的周期性介质和（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解决方案对底层介质变化的依赖性。 第一部分集中在一组模型自由边界问题（FBP），出现在图案化表面上的毛细液滴的研究和某些离散颗粒系统的研究。 微观结构的均匀化导致一类新的FBP。该项目的这一部分的主要目标是严格建立这个缩放/均匀化限制，并研究限制FBP的适定性。第二部分主要讨论了几个缺乏均匀化理论中经典的非线性假设的几何Hamilton-Jacobi（HJ）方程的物理例子。 期望的是，通过随机介质的平均效应，在大尺度上可以恢复非线性，但这只在有限的情况下得到了严格的证实。 该项目的这一部分的目标是开发新的技术来研究弱强制HJ方程;在物理方面，这往往对应于研究界面传播附近的脱钉transition.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.