二维薛定谔映照的长时间行为的研究

Schr?dinger map is a partial differential equation with numerous applications both in differential geometry. This projection is about to study the long time behviour of the solueions to Schr?dinger map. There are three part in this projection: (1)On the Cauchy problem, we intoduce the concept of m-equivariant map combined with the Hasimoto transformation and Strichartz estimates to stability of the solutions; (2)On the initial-periodic boundary problem, we fix the boundary value as a constant vector, the initial value as a family of curves induced by the stereographic projection. Then we are going to probe the blow up property of the solutions using the well known results of the corresponding Schrodinger equations. (3)Under the initial-boundary value condition given by (2), Adaptive mesh method, which has been applied exentively in computational mathematics, will be used to investigate the shape of the solution near the blow up points.

薛定谔映照是一个有着丰富几何意义同时在物理学和材料学得到广泛应用的一个偏微分方程。本项目拟在二维情况下对其解的长时间行为展开研究。主要分以下三部分内容：（1）对Cauchy问题，我们引进m度等变映射（m-equivariant map）的概念，结合Hasimoto变换与Strichartz估计，研究解的稳定性问题；（2）对周期初边值问题，固定边值为常数向量，初值为一族由球极平面映射所诱导的曲线，结合对周期非线性Schr?dinger方程的研究，探讨解的爆破（blowup）性质； (3) 数值计算方面,，利用在计算数学中应用非常广泛的适配网格法（adaptive mesh method）探讨解在爆破点附近的形态。

薛定谔映照是一个有着丰富的微分几何意义同时在物理学和材料学中得到广泛应用的偏微分方程，本项目对薛定谔映照的解的稳定性问题，利用等变映射，将其转化为一个一维非线性的抛物性偏微分方程，考虑其解是否具有blow up的性质。我们的结论是：如果解的初值在中间大于\pi而在1处小于\pi，那么解会爆破；另一方面，如果解的初值在中间大于\pi而在1处小于\pi，那么解可能会blow up也可能具有稳定性。对带Gilbert项的Landau-Lifshitz方程，也就是薛定谔映照加了粘性项后的情况，我们在不同的条件下构造了爆破的精确解以及全局解。