Mathematical Sciences: Variable Coefficient Fourier Analysis

Sogge DMS-9424418 Sogge将继续他在经典分析和与波动方程有关的PDE问题上的工作。他将尝试证明关于区域上的傅立叶分析的尖锐结果，这将应用于有界域上的非线性波动方程的问题。他还将研究欧氏空间中半线性波动方程的小数据存在定理。这将需要对涉及非平凡角模的波动方程的解进行新类型的混合范数估计。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

Sogge DMS-9424418 Sogge will continue his work on problems in classical analysis and pde's that relate to the wave equation. He will attempt to prove sharp results concerning Fourier analysis in domains which will have applications to questions about nonlinear wave equations in bounded domains. He will also investigate small-data existence theorems for semilinear wave equations in Euclidean space. This will require new types of mixed-norm estimates for solutions of the wave equation involving non-trivial angular norms. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.