Ginzburg-Landau type problems in superconductivity and cell motility

这个项目是数学、物理学和生物物理学的交汇点。它涉及Ginzburg-Landau(GL)型方程的理论和应用。这些方程是最基本的非线性偏微分方程组之一，因为它们描述了自然界中从相变、超导、超流到液晶和生物系统中的运动性的广泛现象。该项目由两部分组成，GL型方程的数学是统一的主题。第一部分的主要动机是寻求高能效材料，这些材料构成了新一代基于超导的微电子技术的基础。在实际应用中，流经超导体的电流产生磁涡流，磁涡流运动，能量耗散，从而导致功率损失。这就产生了固定旋涡的问题，称为旋涡钉扎问题。第二部分的动机是真核细胞在底物上的运动性问题。几个世纪以来，细胞运动性的研究一直是生物学中的经典课题，这可以追溯到范·列文虎克的著名发现。细胞运动是许多生物过程的重要组成部分，例如，上皮细胞爬过开放的伤口有助于伤口的愈合。研究人员开发数学技术来解决物理学和生物学中的这些基本问题。研究生作为该项目的一部分进行培训。该项目由两部分组成，统一于GL型偏微分方程组的多尺度分析思想和渐近方法。首先，研究人员研究了超导体中的钉扎和涡旋相分离，重点研究了一种新的梯形涡旋结构，这种结构是由于物理参数(如磁场的大小、钉扎位置的大小和超导样品的大小)的极值之间的微妙相互作用而出现的。考虑了一个双参数渐近问题，该问题模拟了含有大量极小柱状缺陷的超导体中的涡旋相分离现象。在此基础上，发展了适用于涡旋引起的能量集中和缺陷引起的强空间变化之间相互作用的Gamma收敛的均匀化技术。其次，研究人员在严格的渐近框架下研究了描述细胞运动的具有体积约束的相场模型。相场函数求解具有附加项的标量GL PDE(Allen-Cahn)，该附加项提供该场与满足矢量抛物线PDE的另一个控制场的梯度耦合。研究了在体积守恒约束下的尖锐界面极限问题。该系统的一个特点是梯度耦合，它导致了由曲率驱动的界面运动方程和一个新的非线性项。由于这种耦合，以前发展的经典粘性解技术和最新的伽马收敛技术不能用于梯度流，因此发展了新的方法。

This project is at the interface of mathematics, physics and biophysics. It is concerned with the theory and applications of Ginzburg-Landau (GL) type equations. These equations are among the most fundamental nonlinear partial differential equations because they describe a wide range of phenomena in nature from phase transitions, superconductivity, and superfluidity to liquid crystals and motility in biosystems. The project consists of two parts, with mathematics of GL type equations being the unifying theme. The first part is primarily motivated by the quest for energy-efficient materials that comprise a foundation for a new generation of superconductivity-based microelectronics. In practical applications the current passing through a superconductor creates magnetic vortices that move, dissipate the energy, and thus result in power loss. This poses the problem of immobilizing the vortices, known as the problem of vortex pinning. The second part is motivated by the problem of motility of eukaryotic cells on substrates. The study of cell motility has been a classical subject in biology for several centuries, dating back to the celebrated discovery by van Leeuwenhoek. Cell motility is an important ingredient of many biological processes, e.g., epithelial cells crawling across open wounds aid in their healing. The investigator develops mathematical techniques to address these fundamental problems in physics and biology. Graduate students are trained as part of this project. The project consists of two parts unified by the ideas and asymptotic methods of multiscale analysis applied to GL type PDEs. First, the investigator studies pinning and vortex phase separation in superconductors, with focus on the study of a novel terraced vortex structure that appears due to the subtle interplay between extremal values of physical parameters (such as the magnitude of the magnetic field, size of pinning sites, and size of the superconducting sample). A two-parameter asymptotic problem that models the vortex phase separation phenomenon in superconductors with a large number of very small columnar defects is considered. Then the homogenization techniques of Gamma-convergence suitable for the interplay between the energy concentrations due to vortices and strong spatial variations due to defects is developed. Second, the investigator studies in a rigorous asymptotic framework a phase field model with volume constraints that describes cell motility. The phase field function solves a scalar GL PDE (Allen-Cahn) with an additional term that provides a gradient coupling of this field with another governing field that satisfies a vectorial parabolic PDE. The sharp interface limit in the presence of volume conservation constraints is studied. A feature of this system is the gradient coupling that leads to an equation of interface motion driven by curvature and a novel nonlinear term. Due to this coupling, previously developed classical viscosity solution techniques and recent Gamma-convergence techniques for gradient flows cannot be applied, so new methods are developed.