Long -time limit for the Ginzburg -Landau system with pinning
Abstract
Superconductivity invokes more and more attention by physicists and mathematicians. Much work has been produced to explore the properties of various aspects of superconductors. Unfortunately, there are only a few papers related to inhomogeneous superconductors, and even fewer touch the dynamics of superconductivity. In my work, I set forth to address these issues, and cover dynamic properties of inhomogeneous superconductors. In my thesis, the inhomogeneous superconductor is analyzed within the framework of Ginzburg-Landau theory, where the state of the superconductor is described by the Gor'kov-Eliashberg system (TDGL system) with a parameter ε and a penalty function a(x). We present a novel way to find long-time properties for the superconductor through the analysis of the gauge transformation equation, and show that the solution of the TDGL system converges to a static state under various gauges. By setting an appropriate gauge, the convergence is in C 2,α globally. When a(x) has zero points in the domain, and the parameter ε is small enough, it is shown that the degree properties of the solution for the TDGL system resemble its initial value in a certain sense; moreover, the degree of the limit is the same as the degree of the initial value around a zero point of a(x), which gives another way to analyze the degree properties for the static solution of the superconductor. My numerical simulations of the TDGL system provide an auxiliary yet vivid verification of the analysis.
Degree
Ph.D.
Advisors
Bauman, Purdue University.
Subject Area
Mathematics
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