Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics

这个项目旨在了解当波在一个粗糙的开放系统中传播时，它的局部能量衰减的速度有多快。这个项目将通过改进Carleman估计、分离变量、b向量场分析和准模构造等工具，在微局部分析和偏微分方程组的数学子领域产生新的知识。该项目的预期结果是对弱于Lipschitz正则性的系统中的波传播进行一种新颖而全面的数学处理。这项研究应该会提供重大的好处，如提供关于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.