Ginzburg-Landau type problems in superconductivity and cell motility

这个项目是在数学，物理学和生物物理学的接口。 它是关于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之一，因为它们描述了自然界中的各种现象，从相变，超导性和超流性到液晶和生物系统中的运动。 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节能材料，这些材料是新一代基于超导的微电子技术的基础。 在实际应用中，通过超导体的电流会产生磁涡流，这些磁涡流会移动，耗散能量，从而导致功率损耗。 这就提出了固定涡的问题，称为涡钉扎问题。 第二部分是由真核细胞在基质上的运动性问题引起的。 几个世纪以来，细胞运动的研究一直是生物学中的经典课题，可以追溯到货车列文虎克的著名发现。 细胞运动性是许多生物过程的重要成分，例如，上皮细胞在开放性伤口上爬行有助于伤口愈合。 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接受培训。 本课题由两部分组成，统一了GL型偏微分方程多尺度分析的思想和渐近方法。 首先，研究人员研究了超导体中的钉扎和涡旋相分离，重点研究了一种新型的阶梯涡旋结构，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微小柱状缺陷的超导体中的涡旋相分离现象。 然后，伽玛收敛的均匀化技术适合于由于涡流和强烈的空间变化，由于缺陷的能量集中之间的相互作用。 其次，研究人员在一个严格的渐近框架的相场模型与体积约束，描述细胞运动。 相场函数解决了标量GL PDE（艾伦-卡恩）与一个额外的条款，提供了一个梯度耦合的这个字段与另一个管理领域，满足矢量抛物型PDE。 研究了体积守恒约束下的锐界面极限。 该系统的一个特点是梯度耦合，导致界面运动方程驱动的曲率和一个新的非线性项。 由于这种耦合，以前开发的经典粘性溶液技术和最近的伽玛收敛技术的梯度流不能应用，所以新的方法被开发。

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.