Developments in the Theory of Homogenization (15w5164)

Arriving in Banff, Alberta Sunday, July 26 and departing Friday July 31, 2015

Organizers

(University of California, Los Angeles)

Claude Le Bris (Ecole Nationale des Ponts et Chaussees)

(New York University)

Panagiotis Souganidis (University of Chicago)

Yifeng Yu (University of California Irvine)

Objectives

The aim of this workshop is threefold: (i) to discuss the latest significant discoveries in the field among experts with different perspectives, (ii) to identify important open problems and new directions in both the theory and computations of homogenization, and (iii) to develop plans to integrate approaches and ideas on significant and challenging topics such as stochastic homogenization of interfacial motion and non-convex Hamilton-Jacobi equations, percolation phenomena, and quantitative properties of the effective equations. The opportunity offered by the proposed BIRS five-day workshop will significantly enhance our understanding of the field. Special attention will be given to young researchers and under-represented groups.The specific topics we plan to cover in this workshop include:(1)Properties of the effective equations: In applications it is often necessary to go beyond the existence of homogenized equations and better understand properties of homogenized quantities. One example is understanding the dependence of the effective Hamiltonian on the potential function in classical mechanics. Another is deriving precise bounds on the effective velocity which arises in several significant engineering applications, such as the turbulent flame speed in combustion theory and the long time average of flux for the traffic flow. In the context of contact angle dynamics, one would like to understand how the small-scale rough structure of the surface propagates to the shape of the liquid drop moving on the surface. These types of questions are still largely open and the literature in the theoretical setting is limited. In the numerical setting more efficient methods remain to be developed for computing the homogenized nonlinear equation. (2) Homogenization of Interfacial motions: Interfacial motions, like crystal growth or the spreading of water droplets, arise in abundance in nature. Finding the effective speed of the front propagation when the environment is inhomogeneous is an important problem for both theory and application. The first step in the investigation is to rigorously establish the existence of effective speeds which are observed in experiments. In the last decade, much progress has been made in understanding mathematically difficult situations, such as when the self-propagation speed of the interface depends on front curvature in periodic media or a strong advection field in random environment. Despite this progress, there are still many other open problems where compactness is lost due to various other physical reasons. For example, little is known about the averaging properties of curvature-dependent interfacial motions in random media. Interfacial motion is a topic of interest in diverse group of experts, and we hope that the workshop will generate new observations and motivating discussions. (3)Homogenization in random geometry: There are many multi-scale problems where the rough structure of the domain or the boundary data may influence the behavior of solutions in a given PDE system. Most known results so far rely strongly on the periodicity assumption, and little is known for the non-periodic settings. An example is the narrow escape problem in biology in which one has a diffusion or control process in a bounded domain with mostly reflecting boundaries except for a small portion of sporadic exit points on the boundary. Here the interest is in measuring the mean escape time from the domain which depends on the configuration and frequency of the exits. Another problem is to understand the long time behavior of diffusion processes in random geometry. Given recent developments on random homogenization, progress is expected through discussions on different perspectives among experts from PDEs and probability. (4) Issues in scientific computing: An important goal in this workshop is to discuss the current status and challenges related to problems of homogenization theory in scientific computing. There has been considerable ongoing effort in approximating the solutions of PDEs with rough or oscillatory coefficients by multi-scale finite element and collocation methods, Wiener chaos expansion, heterogeneous multi-scale method, multi-grid and multi-scale domain decomposition methods, dynamic bi-orthogonal method, sparse dynamics method. Still many open questions remain, such as the approximation of the solutions of PDEs with no scale-separation or with dynamic singular structure. Though several numerical schemes have been proposed and studied recently, mathematical analysis will be integral to understanding and improving these methods. We will strive to fill the gap between theory and computations through lectures and discussions from participants in both theory and practice.