Limiting Theories in Material Science: Mathematical derivation and Analysis

最近的技术进步使材料工程化在不断缩小的规模下走向广泛的应用，超薄膜就是其中的一个例子。不同长度尺度的存在往往使材料科学中模型的数值研究变得昂贵得令人望而却步。人们不是用数值来处理它们，而是首先对它们进行解析研究，以获得对其解的一些理解，然后利用所获得的洞察力为开发更有效的数值方法铺平道路。我们的目标是严格分析几个这样的问题。在项目的第一部分中，我们研究了压缩弹性薄板的起皱规律。在某些情况下，褶皱可能是不均匀的，实际图案显示为分叉。为了理解起皱微结构，我们考虑了变分的观点，并识别和解析地研究了能量(次优势能)在消失薄板厚度极限内的下一级展开。在这个框架内，我们将研究几种物理情况，描述石墨烯纳米带的模型就是其中之一。在第二部分中，我们研究了具有随机和快速振荡系数的椭圆组，并将其应用到描述非均匀线弹性材料的模型中。尽管微观行为可能相当复杂，但由于随机抵消，宏观行为应该简单得多，而且是确定性的，这一过程被称为均匀化。我们将使用偏微分方程法研究椭圆型系统随机齐次化的定量方面。最后一个研究领域涉及具有粗糙边界的区域中可压缩粘性流体的行为。不是在粗糙域中研究问题，而是在光滑域中提出问题，在光滑域中原始边界的粗糙度被缩减为有效的边界律。利用Navier-Stokes系统耗散解的相对能量不等概念，我们的目的是严格地推导出这些有效边界条件，并分析采用这种方法所造成的误差。

Recent technological advances have allowed for engineering of materials at ever decreasing scales toward broad applications, ultrathin films being one of the examples. The presence of different length scales often makes the numerical study of models in material science prohibitively expensive. Instead of treating them numerically one first studies them analytically to obtain some understanding of their solutions and then uses the acquired insight to pave the way for development of more effective numerical methods. Our goal is to rigorously analyze a few such problems. In the first part of the project we study the wrinkling patterns in compressed thin elastic sheets. In some situations the wrinkling could be non-uniform and the actual pattern shows branching. To understand wrinkling microstructure we consider a variational viewpoint, and identify and analytically study the next-order expansion of the energy (the subdominant energy) in the limit of vanishing sheet thickness. Within this framework we will study several physical situations, a model describing graphene nanoribbons being one of them. In the second part we study elliptic systems with random and rapidly oscillating coefficients, with the application to the model describing heterogeneous linearly-elastic materials in mind. Though the microscopic behavior could be quite complicated, due to stochastic cancellations the macroscopic behavior should be much simpler and deterministic, a process called homogenization. We will use PDE methods to study quantitative aspects of the stochastic homogenization for elliptic systems. The last area of research concerns behavior of compressible viscous fluids in domains with rough boundaries. Rather than study the problem in a rough domain, one poses the problem in a smooth domain where the roughness of the original boundary is reduced to an effective boundary law. Using the concept of relative energy inequality for dissipative solutions to the Navier-Stokes system our aim is to rigorously derive these effective boundary conditions and analyze the error one makes by taking this approach.