Mathematical Sciences: Macroscopic Structures in Nonlinear Partial Differential Equations

数学科学：非线性偏微分方程的宏观结构

• 批准号: 9600080
• 负责人: Robert Jerrard
• 金额: $ 6.73万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-05-15 至 1999-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9600080&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Macroscopic; Structures; Nonlinear

Abstract Jerrard Jerrard will work on three classes of problems. Two of these grow out of joint work carried out by Jerrard in collaboration with H.M Soner. Jerrad and Soner examined the dynamics of topological defects in solutions of Ginzburg-Landau (GL) systems taking values in the plane. They showed that, under an appropriate scaling limit and for appropriate initial data, the energy density of solutions concentrates around codimension 2 manifolds, and they characterized the evolution of these manifolds. In planar domains, these singular manifolds are simply singular points which evolve via a system of ordinary differential equations; in higher dimensional domains, one obtains in the limit manifolds which evolve via codimension 2 mean curvature flow. Jerrard now proposes to study a number of questions about dynamics of point defects of solutions of evolution equations of GL type, related to his earlier work. In particular, he wants to look at dynamics of point defects in GL Schroedinger equations and in generalized GL systems in higher dimensions. Secondly, he plans to look at problems concerning the evolution of line defects or, more generally, singular manifolds of dimension greater than 1 and codimension at least 2. These would include questions about evolution of line defects in GL Schroedinger equations. A third, completely different class of problems has to do with developing a framework within which one can make sense of level set equations in which certain standard monotonicity assumptions on the coefficients are violated. Heuristically, such an equation would specify that every level set evolves by some geometric rule, but that different level sets evolve by different rules. Such equations are of mixed parabolic/hyperbolic type. A notion of weak solution would need to combine viscosity solution techniques with entropy conditions, and would probably also require some other ingredients. The problems described above are drawn from two areas: physics of superconductors and sup erfluids, and image progessing. Ginzburg-Landau systems are a widely-used mathematical model for certain kinds of superconductors and superfluids, and they seem to agree quite well with experiment. Some classes of superconducting materials, including many high temperature superconductors, exhibit small islands of normal, nonsuperconducting behavior within a larger superconducting matrix. These islands are known as vortices, and they can have a negative impact on the performance of superconductors. A major focus of this propsal is to attempt to understand, from a mathematical viewpoint, the behavior of these Ginzburg-Landau vortices. Similar vortices are seen in superfluids, such as liquid helium. Finally, a major problem in image processing is to eliminate "meaningless" noise, while retaining contrasts that carry some significant information. Many image processing schemes amount to finding a solution of some differential equation, which takes a given image as the input and produces the processed image as a solution. In this framework, it is useful to try to have a well-developed mathematical theory for equations which have the effect of smoothing out small irregularities in the given image, while sharpening major contrasts. This is the other main area in which Jerrard proposes to work.

抽象的杰勒德 杰勒德将研究三类问题。其中两个是由杰勒德与H.M Soner合作进行的联合工作。Jerrad和Soner研究了在平面上取值的Ginzburg-Landau（GL）系统的解中拓扑缺陷的动力学。他们表明，在适当的尺度限制和适当的初始数据下，解的能量密度集中在余维2流形周围，他们表征了这些流形的演化。在平面域中，这些奇异流形是通过常微分方程系统演化的简单奇点;在高维域中，人们在通过余维2平均曲率流演化的极限流形中获得。杰勒德现在提出研究一些问题的动态点缺陷的解决方案的发展方程的GL型，有关他的早期工作。 特别是，他想看看动力学的点缺陷在GL薛定谔方程和广义GL系统在更高的维度。其次，他计划看看问题的演变线缺陷，或更一般地说，奇异流形的维度大于1和余维至少2。这些问题包括GL薛定谔方程中的线缺陷的演化。 第三类完全不同的问题是建立一个框架，在这个框架内，人们可以理解水平集方程，在这个方程中，对系数的某些标准单调性假设被违反了。启发式地，这样的方程将指定每个水平集按照某种几何规则演化，但是不同的水平集按照不同的规则演化。这类方程是混合抛物型/双曲型方程。弱解的概念需要将联合收割机粘性解技术与熵条件结合起来，而且可能还需要一些其他的成分。 上述问题来自两个领域：超导和超流体物理和图像处理。Ginzburg-Landau系统是一种广泛用于某些类型超导体和超流体的数学模型，它们 看起来和实验很吻合。某些类别的超导材料，包括许多高温超导体，在较大的超导基质中表现出正常的非超导行为的小岛。这些岛被称为涡流，它们会对超导体的性能产生负面影响。这个propsal的一个主要重点是试图从数学的角度来理解这些金兹伯格-朗道涡的行为。在超流体中也可以看到类似的涡旋，比如液氦。最后，图像处理中的一个主要问题是消除“无意义”的噪声，同时保留携带一些重要信息的对比度。 许多图像处理方案相当于找到一些问题的解决方案。 微分方程，它以给定的图像作为输入，并产生处理后的图像作为解决方案。在这个框架中，试图拥有一个完善的数学理论来处理方程是有用的，这些方程具有平滑给定图像中的小不规则性的效果，同时锐化主要对比度。这是杰勒德提议工作的另一个主要领域。