Analysis of Ginzburg-Landau models

Ginzburg-Landau 模型分析

• 批准号: 185065-2013
• 负责人: Bronsard, Lia
• 金额: $ 1.38万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635641
• 关键词: Analysis; Ginzburg; Landau; models

The goal of the proposed research is the rigorous analysis of nonlinear partial differential equations (PDE) and variational problems arising from models of the physical world. While the Ginzburg-Landau (GL) model was initially introduced to study phase transitions in superconductors, over time it has become ubiquitous in the mathematical description of multi-phase systems in physics and materials science and has lead to a wealth of deep mathematical results and new methods. In a Ginzburg-Landau model the state of a physical system is described as a critical point of a given energy functional, and solves a system of nonlinear elliptic PDE. Heuristic analysis suggests that in certain parameter limits, the solutions may develop geometrical singularities (such as vortices, monopoles, or domain walls,) which give the most salient features of the system and characterize its fundamental interactions and dynamics.This research proposal concerns vector-valued Ginzburg-Landau models (as opposed to scalar-valued) introduced in a variety of physical contexts such as high-temperature superconductivity, Bose-Einstein condensation (BEC), nematic liquid crystals, superfluidity, or ferromagnetism, as well as in the study of microstructures in alloys. Each one is a singular perturbation problem, leading to geometrical singularities (either point vortices or transition hypersurfaces,) and we ask: what type of singularities may be observed; how many are there and where; what is the profile of the minimizers near a singularity?An essential ingredient in this study will be the development of Liouville-type theorems, which classify the entire solutions to the appropriate blowup equations in all of space. Very few Liouville theorems are known for systems of elliptic PDE and original methods must be developed to study these systems. The analytical results obtained will give a more complete and reliable understanding of these models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.

拟议的研究的目标是严格分析非线性偏微分方程（PDE）和变分问题所产生的模型的物理世界。虽然Ginzburg-Landau（GL）模型最初是为了研究超导体中的相变而引入的，但随着时间的推移，它在物理学和材料科学中的多相系统的数学描述中变得无处不在，并导致了丰富的深刻的数学结果和新方法。在Ginzburg-Landau模型中，物理系统的状态被描述为给定能量泛函的临界点，并求解非线性椭圆偏微分方程组。启发式分析表明，在一定的参数限制，解决方案可能会发展的几何奇异性（如涡旋、磁单极或畴壁），它们给出了系统最显著的特征，并表征了系统的基本相互作用和动力学。本研究建议涉及向量值Ginzburg-Landau模型（与标量值相反）在各种物理环境中引入，例如高温超导性，玻色-爱因斯坦凝聚（BEC），非线性液晶，超流性，或铁磁性，以及在合金的微观结构的研究。每一个都是一个奇异摄动问题，导致几何奇点（点涡或过渡超曲面），我们问：什么类型的奇点可能会被观察到;有多少，在哪里;什么是在奇点附近的极小的轮廓？在这项研究中的一个重要组成部分将是刘维尔型定理的发展，其中分类的整体解决方案，以适当的爆破方程在所有的空间。对于椭圆型偏微分方程组，已知的刘维定理很少，必须开发原始方法来研究这些系统。所获得的分析结果将使我们对这些模型及其所描述的现象有一个更完整和可靠的理解，同时为分析、几何和物理之间的丰富相互作用提供了新的视角。