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

本研究计划致力于了解小尺度非均匀性对偏微分方程解的影响。具有多个空间和时间尺度（微观、中观或宏观尺度）结构的系统在工业（复合材料、微流体）、生物学（组织、细胞膜、大脑）、生物物理学（海底）、气象学（云）、流体力学（湍流）和物理学（颗粒材料、物质结构）中无处不在。数学研究的一般精神是弄清楚如何将这些小尺度集成到渐近简化模型中。此外，这项工作的重点是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，能量转移，不稳定性和平衡动力学。这项基础研究具有深远的影响。这项工作的基础设计的新的数值方法，旨在证明数值格式的精度，并能够提高其效率。这个建议的重点是研究边界行为的解决方案和分析方程和系统的低正则性，无论是在系数，或在边界。许多现有的偏微分方程理论，即使是椭圆问题，已经发展为方程，在光滑域，常数或光滑系数，对称。类似地，渐近模型的推导通常依赖于强结构假设，例如周期性。新的应用使得放宽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开放问题。主要目标是（i）开发非对称椭圆方程和非常系数系统的工具，（ii）研究高度振荡的边界条件，（iii）放松与振荡边界有关的问题中的结构假设，（iv）在无限能量空间中的稳态线性或非线性系统的分析中取得进展，（v）提供易于处理的结果数值均匀化和（vi）证明严格的一些渐近模型在海洋学和粘弹性流体理论。这一领域的研究目前非常活跃，提出的问题也很重要。PI和他的合作者在最近的作品中阐述了处理这些问题的新方法。进一步发展这些工具不仅有助于解决问题（i）-（vi），而且还将为偏微分方程的许多领域带来新的想法：均匀化，调和分析，椭圆方程和系统，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.