Asymptotic Analysis of Partial Differential Equations and Systems with Emphasis on Boundary Layers

University Of Chicago, Chicago IL

Investigators

Abstract

This research proposal is devoted to the understanding of the effect of small scale heterogeneities on solutions of Partial Differential Equations (PDEs). Systems having structures at several spatial and temporal scales, micro-, meso- or macroscopic scales, are ubiquitous in industry (composite materials, microfluidics), in biology (tissues, cell membranes, brain), in geophysics (seabed), in meteorology (clouds), in fluid mechanics (turbulence) and in physics (granular materials, structure of matter). The general spirit of the mathematical study is to figure out how one can integrate these small scales into asymptotic simplified models. Moreover, this work focuses on the understanding of the interactions between the different scales from a dynamical point of view: memory effects, energy transfers, instabilities and out of equilibrium dynamics. This fundamental research has far reaching consequences. This work underlies the design of new numerical methods, aims at proving the accuracy of numerical schemes and enables to improve their efficiency. This proposal focuses on the study of the boundary behavior of solutions and on the analysis of equations and systems with low regularity, either in the coefficients, or in the boundary. A lot of the existing theory of PDEs, even for elliptic problems, has been developed for equations, in smooth domains, with constant or smooth coefficients, with symmetry. Similarly, the derivation of asymptotic models often relies on strong structure assumptions such as periodicity. New applications have made the need for relaxing these assumptions even more important. These questions lead to many challenging open problems. The primary goals are to (i) develop the tools for non symmetric elliptic equations and systems with non constant coefficients, (ii) investigate highly oscillating boundary conditions, (iii) relax structure assumptions in problems concerned with oscillating boundaries, (iv) make progress in the analysis of stationary linear or nonlinear systems in infinite energy spaces motivated by the study of boundary layers, (v) provide tractable results for numerical homogenization and (vi) justify rigorously some asymptotic models in oceanography and in the theory of viscoelastic fluids. This area of research is currently very active, and the proposed problems are important. The PI and his collaborators have elaborated new methods in recent works to deal with such questions. Developing these tools further will not only help solve the problems (i)-(vi) but also bring new ideas to many fields of PDEs: homogenization, harmonic analysis, elliptic equations and systems, and fluid mechanics.