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

本研究致力于了解小尺度非均质性对偏微分方程组(PDE)解的影响。具有若干空间和时间尺度、微观、中观或宏观尺度结构的系统普遍存在于工业(复合材料、微流体)、生物(组织、细胞膜、脑)、地球物理(海底)、气象(云)、流体力学(湍流)和物理学(颗粒材料、物质结构)。数学研究的总体精神是弄清楚如何将这些小尺度整合成渐近简化的模型。此外，本工作着重从动力学的角度来理解不同尺度之间的相互作用：记忆效应、能量传递、不稳定性和非平衡动力学。这项基础性研究具有深远的影响。这项工作是设计新的数值方法的基础，目的是证明数值格式的精度，并使其能够提高效率。这一建议侧重于研究解的边界行为，以及分析在系数中或在边界上具有低正则性的方程和系统。许多已有的偏微分方程组理论，甚至对于椭圆型问题，都是针对光滑区域上的常系数或光滑系数的对称方程而发展起来的。同样，渐近模型的推导往往依赖于强结构假设，如周期性。新的应用使得放松这些假设的必要性变得更加重要。这些问题导致了许多具有挑战性的公开问题。主要目标是(I)发展非对称椭圆型方程和变系数方程组的工具，(Ii)研究高振荡边界条件，(Iii)放宽与振荡边界有关的问题的结构假设，(Iv)在无限能量空间中的定常线性或非线性系统的分析方面取得进展，(V)为数值齐次化提供易于处理的结果，以及(Vi)严格证明海洋学和粘弹性流体理论中的某些渐近模型。这一领域的研究目前非常活跃，提出的问题很重要。PI和他的合作者在最近的著作中阐述了处理这些问题的新方法。这些工具的进一步发展不仅将有助于解决问题(I)-(VI)，而且将为偏微分方程组的许多领域带来新的思想：齐次化、调和分析、椭圆型方程和系统以及流体力学。

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.