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.
|Effective start/end date||9/1/14 → 8/31/18|
- National Science Foundation: $284,000.00