對以負曲率流形作背景空間之納維-斯托克斯方程之解之大範圍特性之探索

Abstract

此研究計畫是關於以單連通且帶負曲率流形作背景空間之對定常或不定常納維-斯托克斯方程之解之大範圍特性之探索。

本研究計畫之主要目的在於研究一個具負曲率的背景空間之曲率，藉由何種方式 來影響定常或不定常納維-斯托克斯方程之解之分析、幾何或拓樸等諸種特性。在對本 研究計畫開展研究工作之過程中，其中最主要且有用的一個想法乃是嘗試藉賴著一個 負曲率流形之指數式的空間膨脹以得出定常或不定常納維-斯托克斯流於大範圍的較 佳的衰減特性。

This research project is about the investigation of global properties of solutions to Navier-Stokes equation or stationary Navier-Stokes equation on a negatively curved simply-connected Riemannian manifold of dimension 2 or 3. The main purpose of this project is to study the various possible ways in which the curvature of the negatively curved background manifold affects the analytical, geometric or topological properties of solutions to the Navier-Stokes equation on the negatively curved background manifold. In the process of carrying out this research project, one of the main ideas is to make use of the exponential expansion of the far range spatial structure of a negatively curved manifold to obtain better decay properties of Navier-Stokes flows in the far range.