项目名称: 基于算子空间的微分流形及非线性偏微分方程的研究
项目编号: No.11501445
项目类型: 青年科学基金项目
立项/批准年度: 2016
项目学科: 数理科学和化学
项目作者: 方莉
作者单位: 西北大学
项目金额: 18万元
中文摘要: 本项目应用微分几何流形的思想,研究算子空间中的黎曼流形与格拉斯曼流形问题,通过讨论测地线、指数映射、曲率以及挠率的相关性质,得到流形上的测地线理论。另一方面,应用算子空间理论、希尔伯特空间理论,研究一类非线性偏微分方程,通过建立该类偏微分方程解的存在性,得到其解的适定性理论。预期研究成果不仅为算子论与算子代数本身开辟一些新的研究课题,而且拓宽算子论与算子代数在偏微分方程等其他学科分支中的应用前景。
中文关键词: 算子空间;黎曼流形;格拉斯曼流形;测地线理论;解的适定性
英文摘要: In this project, theories and methods in manifolds of differential geometry are used to study the geometric structure of Riemann manifold and Glassman manifold in operator space. Theory of geodesics on manifold is obtained with discussing some properties of geodesics curve, exponential mapping, curvature and torsion. On the other hand, theories of operator space and Hilbert space are applied to consider a class of non-linear partial differential equations. The well-posedness of solution to this non-linear partial differential equations is got by establishing the existence of solution to the partial differential equations. The anticipated research results not only open up some new research topics for the operator theory and operator algebras, and wide the application prospect of operator theory and operator algebras in the other branches such as partial differential equations.
英文关键词: operator space;Riemannian manifold;Grassmann manifold;theory of geodesics;the well-posedness of solution