High frequency scattering with applications to fluids

拟议的项目是主要主管的光谱和散射理论更广泛的研究计划的一部分。本计画主要研究波型方程式之高频解。学生的工作将集中在研究存在陷波时的散射波。自Burq的工作以来，人们已经知道，相对于非陷波情况，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没有完全了解。该项目旨在提高我们对这一现象的认识。在进行该项目时，学生将被训练使用数学中的几个重要理论。学生将获得数学散射理论以及微局部分析和泛函分析的坚实基础。这些工具最近发现应用在各种各样的问题，包括研究动力系统，内部强迫波的流体，动力学的自由边界问题，广义相对论仅举几例。在非线性偏微分方程的背景下，线性化动力学的精确理解是特别重要的。在项目的过程中，学生将接受培训，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。研究领域：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