High frequency scattering with applications to fluids

该项目是主要导师关于光谱和散射理论的更广泛研究计划的一部分。本项目主要研究波动型方程的高频解。该学生的工作将集中于研究存在诱捕的散射波。自Burq的工作以来，人们已经知道，相对于非捕获情况，对这种波的估计必须遭受损失。然而，这些损失发生的精确能量还不完全清楚。这个项目旨在提高我们对这一现象的理解。在进行这个项目时，学生将被训练使用几个重要的数学理论。学生将在数学散射理论以及微局部分析和泛函分析方面获得坚实的基础。这些工具最近在各种各样的问题中得到了应用，包括动力系统的研究，流体中的内强迫波，自由边界问题的动力学，广义相对论等等。在非线性偏微分方程的背景下，对线性动力学的精确理解是特别重要的。在项目的过程中，学生将被训练寻找和实施这些工具，特别是在数学流体动力学的背景下；二级主管的专业领域。研究方向：数学分析

The proposed project is part of the primary supervisor's broader research program on spectral and scattering theory. This project focuses on the study of high frequency solutions to wave-type equations. The student's work will focus on the study of scattered waves in the presence of trapping. It has been known since the work of Burq that estimates for such waves must suffer losses relative to non-trapping situations. However, the precise energies at which these losses occur are not fully understood. This project will aim to improve our understanding of this phenomena. While undertaking the project, the student will be trained to use several important theories in mathematics. The student will acquire a strong foundation in mathematical scattering theory as well as in microlocal analysis and functional analysis. These tools have recently found applications in an enormous variety of problems including the study of dynamical systems, internal forced waves in fluids, dynamics of free boundary problems, and general relativity to name a few. The precise understanding of the linearised dynamics is of particular importance in the context of nonlinear PDE. In the course of the project, the student will be trained to look for and implement these tools, particularly in the context of mathematical fluid dynamics; an area of expertise of the secondary supervisor.Research Area: Mathematical analysis