Limiting Theories in Material Science: Mathematical derivation and Analysis

最近的技术进步已经允许材料的工程化以不断减小的规模走向广泛的应用，薄膜就是其中的一个例子。不同长度尺度的存在往往使得材料科学中模型的数值研究过于昂贵。而不是对待他们的数值一个首先研究他们的分析，以获得一些了解他们的解决方案，然后使用所获得的洞察力铺平道路，发展更有效的数值方法。我们的目标是严格分析一些这样的问题。在第一部分中，我们研究了压缩弹性薄板中的条纹图案。在某些情况下，斜纹可能是不均匀的，实际图案显示出分支。为了理解退火微观结构，我们考虑变分的观点，并确定和分析研究的下一阶展开的能量（次优势能量）的极限消失片厚度。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系统，并将其应用于描述非均匀线弹性材料的模型。虽然微观行为可能相当复杂，但由于随机取消，宏观行为应该更简单和确定性，这一过程称为均匀化。我们将使用偏微分方程的方法来研究定量方面的随机均匀化椭圆系统。研究的最后一个领域涉及可压缩粘性流体在粗糙边界区域中的行为。而不是在一个粗糙的域中研究的问题，提出了一个光滑的域中的原始边界的粗糙度降低到一个有效的边界法的问题。利用相对能量不等式的概念，耗散解的Navier-Stokes系统，我们的目的是严格推导这些有效的边界条件，并分析错误的人采取这种方法。

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.