Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics

该项目旨在了解波在粗糙的开放系统中传播时，波的局部能量衰减的速度。该项目将产生新的知识，在微观局部分析和偏微分方程的数学子领域，通过细化工具，如Carleman估计，变量分离，b-向量场分析和准模结构。这个项目的预期成果是一个新的和全面的数学处理系统中的波传播弱于Lipschitz正则性。这项研究应该提供显着的好处，如通知粗糙系统中的波的预测，包括地震波的传播，并导致医学和地质成像的进步。

This project aims to understand how fast the local energy of a wave decays when it propagates in a rough, open system. This projects will generate new knowledge in the mathematical subfields of microlocal analysis and partial differential equations by refining tools such as Carleman estimates, separation of variables, b-vector field analysis, and quasimode constructions. The expected outcome of this project is a novel and comprehensive mathematical treatment of wave propagation in systems with weaker than Lipschitz regularity. This research should provide significant benefits such as informing predictions about waves in rough systems, including the propagation of seismic waves, and lead to advances in medical and geological imaging.