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

主要研究者：Sylvia Serfaty，纽约大学DMS-0239121本项目在分析和偏微分方程领域，涉及物理/材料科学中某些模型中奇点的静态和动态分析。我们一直并将继续对两个特定的模型特别感兴趣：超导的金斯堡-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制度对应的尖锐的“畴壁”转变。我们建立了一些最佳的能量估计，并表现出最佳模式的过渡轮廓，这与“交叉领带墙”的意见，并给出了第一个例子的非一维的最佳配置文件中的向量值相变问题。我们打算继续这种分析，以更物理相关的模型。在Ginzburg-Landau模型领域，我们以前专注于理解旋涡的出现、结构和位置。我们建立了一个Gamma收敛结果，导出了一个极限自由边界问题，并证明了收敛到一些极限涡密度，我们的特点，极小以及临界点。该项目将进一步开展高应用领域的分析，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对梯度流的一种能量方法，首先对有限数量的涡旋，然后对无限数量的涡旋，建立涡旋所服从的极限动力学定律。除此之外，我们将有兴趣更好地理解一般伽玛收敛能量的梯度流的收敛性。这个项目在分析和偏微分方程领域，涉及一些物理模型的静态和动态分析，特别是超导和微磁学的金斯堡-朗道模型。在这两种情况下，重点是理解，通过严格的数学证明，这种模型的解决方案的定性行为，特别是解释的结构和动力学的奇点出现在一些渐近制度。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，证实或反驳模型的有效性，并且通过严格的分析，人们还可以推导出更容易工作和计算/模拟的简化模型（例如减少维度）。第二个目的是更多的数学。如前所述，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），在此基础上可以追踪核心现象（这里是奇点）。这就是“伽玛收敛”的主要哲学。为了进行这种分析，需要开发适当的数学工具，并提供正确的设置来理解现象。人们还希望了解有多少行为是特殊的，有多少可以扩展和理解为更普遍的数学现象，希望对一个特定模型的理解反过来有助于理解其他模型。

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.