Interfaces and Free Boundaries in Heterogeneous Media

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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.