CAREER: Statics and Dynamics of Singularities In Some Models From Material Science

职业：材料科学某些模型中奇点的静力学和动力学

• 批准号: 0239121
• 负责人: Sylvia Serfaty
• 金额: $ 40万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2008-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0239121&HistoricalAwards=false
• 关键词: CAREER; Statics; Dynamics; Singularities; Some

PI: Sylvia Serfaty, New York UniversityDMS-0239121This project, in the field of analysis and PDEs, is concerned with the analysis of statics and dynamics of singularities in some models from physics/material science. We have been and will continue to be particularly interested in two specific models: the Ginzburg-Landau model of superconductivity, and micromagnetics. In the area of micromagnetics, we studied previously some simplified two-dimensional model in the asymptotic regime corresponding to sharp "domain-wall" transitions. We established some optimal energy estimates, and exhibited optimal patterns for the transition profile, which fit very well with the "cross-tie wall" observations, and gave a first example of a non one-dimensional optimal profile in a vector-valued phase-transition problem. We intend to pursue this analysis towards more physically relevant models. In the area of the Ginzburg-Landau model, we previously focused on understanding the apparition, structure, and location, of vortices. We established a Gamma-convergence result deriving a limiting free-boundary problem, and proved convergence to some limiting vortex-densities, which we characterized, for minimizers as well as critical points. The project is to pursue further the analysis in the regime of high-applied fields, and turn towards the study of the associated dynamical models. We hope to establish the limiting dynamical laws obeyed by the vortices, first for a finite number of them, then for an infinite number of them, via some new estimates and an energetic approach to gradient-flow. Beyond that, we will be interested in understanding better the convergence of gradient flows for general Gamma-converging energies.This project, in the field of analysis and PDEs, is concerned with the analysis of statics and dynamics in some models from physics, in particular the Ginzburg-Landau model of superconductivity and microagnetics. In both cases the focus is on understanding, via rigorous mathematical proofs, the qualitative behavior of solutions to such models, and in particular explain the structure and the dynamics of singularities arising in some asymptotic regimes.The purpose of such research is thus two-fold. First to shed light on the understanding of the specific physical problems themselves: by obtaining asymptotic expansions and explicit formulas, one can explain physical experiments, confirm or disprove the validity of models, and by rigorousanalysis one can one also derive reduced simplified models (for example reducing the dimension) that are easier to work and compute/simulate with. The second purpose is more mathematical. As mentioned, the underlying philosophy is to manage to reduce the original asymptotic problems to simpler limiting problems (of lower dimension) on which the core phenomena (here singularities) can be tracked down. This is the main philosophy of ``Gamma-convergence''. In order to perform such analysis, one needs to develop appropriate mathematical tools and provide the right settings to understand the phenomena. One also wishes to understand how much of the behavior is particular, and how much can be extended and understood as a more general mathematical phenomenon, with the hope that what is understood for one specific model may in turn help to understand others.

Pi：Sylvia Serfaty，纽约大学DMS-0239121这个项目，在分析和偏微分方程领域，关注物理/材料科学中一些模型中奇点的静力学和动力学分析。我们一直并将继续对两个特定的模型特别感兴趣：超导的金兹堡-朗道模型和微磁学。在微磁学领域，我们以前研究了一些简化的二维模型，在渐近区域对应于尖锐的“畴-壁”相变。我们建立了一些最优的能量估计，给出了跃迁轮廓的最优模式，这与“十字搭桥”的观测结果非常吻合，并给出了向量值相变问题中非一维最优轮廓的第一个例子。我们打算对更多与身体相关的模型进行这一分析。在Ginzburg-Landau模型方面，我们以前的重点是了解涡旋的显现、结构和位置。我们建立了一个关于极限自由边界问题的Gamma收敛结果，并证明了它对于极小点和临界点都收敛到我们所刻画的极限涡旋密度。该项目将进一步在高应用性领域中进行分析，并转向相关动力学模型的研究。我们希望通过一些新的估计和一种关于梯度流的能量方法，首先建立有限数量的涡旋所遵循的极限动力学定律，然后再建立无限数量的涡旋遵守的极限动力学定律。除此之外，我们将有兴趣更好地了解一般伽马收敛能量的梯度流的收敛。这个项目，在分析和偏微分方程组领域，从物理上关注一些模型的静力学和动力学分析，特别是超导和微磁学的Ginzburg-Landau模型。在这两种情况下，重点都是通过严格的数学证明来理解这类模型解的定性行为，特别是解释在某些渐近状态下出现的奇点的结构和动力学。因此，这种研究的目的是双重的。首先阐明对具体物理问题本身的理解：通过获得渐近展开和显式公式，人们可以解释物理实验，确认或反驳模型的有效性，通过严格的分析，人们还可以推导出简化的简化模型(例如，降低维度)，这些模型更容易工作和计算/模拟。第二个目的更多的是数学化。正如前面提到的，基本的哲学是设法将原始的渐近问题简化为更简单的极限问题(较低维度的)，在这些问题上可以追踪到核心现象(这里是奇点)。这是“伽玛趋同”的主要哲学。为了进行这样的分析，人们需要开发适当的数学工具，并提供正确的环境来理解现象。人们还希望了解这种行为有多少是特殊的，有多少可以扩展和理解为更一般的数学现象，希望对一个特定模型的理解反过来可能有助于理解其他模型。