奇摄动变分问题的理论与应用

The main purpose of this project (conducted by Pan Xingbin) is to study the concentration phenomena of the Ginzburg-Landau equations with a large parameter, to examine the surface nucleation of superconductivity, and to explore the effect of the geometry of a superconductor on its surface superconductivity. The results.obtained in this project are presented in 9 papars in international journals, and 12 invited talks given at international conferences/workshops. We obtained an estimate of the third critical field for cylindrical superconductors, and proved that, as the applied magnetic field decreases from the third critical field, superconductivity nucleates first at the surface of the cylinder where the curvature is maximal; and also gave a mathematical theory to describe the surface superconducting state. For.bounded 3 dimensional superconductors with smooth surface, we showed that.superconductivity nucleates first at the surface where the applied field is tangential. We also showed that the value of the third critical field is higher for a superconductor with edges and conners. Our research indicates that the geometry of domains has.important effects on the behavior of solutions of partial differential equations. Some other problems related to the vortex nucleation of superconductivity and the phase.transitions of liquid crystals have also been investigated in the project..The other topic of this project (conducted by Fang Daoyuan) is to study the existence time of solutions of semilinear Klein-Gordon system with different speed in one space dimension for weakly decaying small Cauchy data in certain circumstances of nonlinearity, and study the nonlinear type singularities for nonlinear wave equations.

本项目以超导与液晶的数学理论为中心，研究各种奇摄动变分问题和偏微分方程，及解的各种奇异性态和凝聚现象。重点研究超导材料在外场中的性态，表面成核现象，磁通涡旋的分布和运动规律，几何性质对超电性的影响。研究液晶的奇性、相变过程及边界层现象。在这些研究的基础上发展奇摄动变分问题和非线性凝聚现象的数学理论，并应用于其它学科。