Mathematical Sciences: Macroscopic Structures in Nonlinear Partial Differential Equations

杰拉德·杰拉德将研究三类问题。其中两个是杰拉德与H·M·索纳合作开展的联合工作。Jerrad和Soner考察了在平面取值的Ginzburg-Landau(GL)系统解的拓扑缺陷的动力学。结果表明，在适当的标度极限下，对于适当的初始数据，解的能量密度集中在余维2流形附近，并刻画了这些流形的演化。在平面区域中，这些奇异流形是通过常微分方程组演化的简单奇点；在高维区域中，人们得到了通过余维2平均曲率流演化的极限流形。Jerrard现在建议研究关于GL型发展方程解的点亏动力学的一些问题，这与他早期的工作有关。特别是，他想要研究GL薛定谔方程和高维广义GL系统中点缺陷的动力学。其次，他计划研究与线缺陷的演化有关的问题，或者更一般地，维度大于1且余维至少为2的奇异流形。这将包括GL薛定谔方程中关于线缺陷演化的问题。第三类完全不同的问题涉及开发一个框架，在该框架内，人们可以理解违反某些关于系数的标准单调性假设的水平集方程。启发式地，这样的等式将指定每个水平集按某种几何规则进化，但不同的水平集按不同的规则进化。这类方程是混合抛物型/双曲型的。弱溶液的概念需要将粘性溶液技术与熵条件相结合，还可能需要一些其他成分。上述问题来源于两个领域：超导体和超流体的物理，以及图像处理。金兹堡-朗道系统是某些类型的超导体和超流体的一个广泛使用的数学模型，它们似乎与实验非常吻合。某些类型的超导材料，包括许多高温超导体，在较大的超导矩阵中表现出正常的非超导行为的小岛。这些岛屿被称为涡旋，它们可能会对超导体的性能产生负面影响。这一命题的一个主要焦点是试图从数学角度理解这些金兹堡-朗道涡旋的行为。在超流体中也可以看到类似的漩涡，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，同时保留带有一些重要信息的对比度。许多图像处理方案相当于求解某个微分方程解，它将给定的图像作为输入，并产生处理后的图像作为解。在这个框架中，尝试为方程建立一个完善的数学理论是有用的，这些方程具有平滑给定图像中的微小不规则性，同时锐化主要对比度的效果。这是杰拉德计划开展工作的另一个主要领域。

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.