复标量场中量子化涡丝的动力学研究

This project mainly studies the dynamic behaviors of vortex filaments in some problems of quantum physics, and attempts to obtain strictly the law of vortex motion in mathematics. Specifically, we are interested in the Gross-Pitaevskii quantum vortex dynamics behavior in three dimension, and dynamic behavior of vortex filament in nonlinear theory，including the superconducting Ginzburg-Landau model. The equations in question arise from physical models. The energy concentration sets are often naturally interpreted as “quantized vortex filaments”, and also the zero level set of a solution. Physicists have much interests in the dynamics behavior of this kind of problems. There are a large number of results about elliptic and parabolic equations, although some significant open problems remain, and less is known about hyperbolic and (especially) dispersive equations. The study of this subject can further deepen the understanding the relationship of theoretical physics and geometry, and also promote the application of geometric measure theory in partial differential equations.

该项目主要研究量子物理的几个问题中涡丝的动力学行为，试图从数学上严格导出其动力学规律。具体地有三维Gross-Pitaevskii量子涡旋动力学行为，以及包含超导Ginzburg-Landau模型在内的非线性场论中涡丝的动力学行为。这类问题产生于具体的物理模型，能量集中集通常被理解为“量子涡丝”，该集合也是解的零水平集。物理学家对这类问题的动力学行为十分感兴趣。迄今为止，关于椭圆和抛物型的问题，已有一些进展，一些重要的问题仍没有被解决，而关于双曲型，特别是色散型，结果甚少。该课题的研究可进一步加深理论物理和几何关系的了解，促进几何测度论在方程中的应用。

本项目主要以分数次趋化（-流体）模型为对象，研究细胞在趋化剂的作用下向着更有利的环境以列维飞行的方式定向移动的动力学行为；借助于具有自由边界的反应扩散方程描述入侵种群的扩张行为；研究复杂网络上的反应扩散系统的种群生态学。利用科学的理论分析解释细胞的动力学行为，展现了一些重要的自然现象。这些结果将有助于我们揭示肿瘤的形成与转移规律；采取有效措施控制生物入侵及其危害；并提供了一种新视角、新方法进行复杂性研究。同时促进非线性偏微分方程理论在研究趋化（-流体）动力学模型方面的应用。.项目组所有人员所获得的研究成果以学术论文形式公开发表，其中标注该项目号的学术论文有22篇被SCI收录，这些论文在Web of Science上已经被引用近30篇次。在该项目的资助下承办了多次学术会议。