Mathematical Sciences: Estimates for Linear and Nonlinear Wave Equations

这个项目将涉及非线性严格双曲 微分方程 我们将考虑以下问题： 线性方程的估计，解的衰减和扩展 的非常数系数的结果已知的 达兰伯特算子 这些研究将包括分析 非光滑表面上奇点的相互作用。 进一步构建表现出非线性的示例 将尝试奇点。 精确正则性 定理将发展，以表明，某些明确的 近似解与实际解不同。 技术的发展，如更好的雷达，X射线， 设备和扫描仪最终依赖于对 双曲方程在这个项目中考虑。

This project will involve nonlinear strictly hyperbolic differential equations. We will consider the following problems: estimates for linear equations, decay of solutions, and extension to nonconstant coefficients of results known for the D'Alambertian operator. The studies will include the analysis of the interaction of singularities across nonsmooth surfaces. Further construction of examples exhibiting nonlinear singularities will be attempted. Precise regularity theorems will be developed in order to show that certain explicit approximate solutions differ from the actual ones. Technological developments such as better radars, X-ray devices, and scanners ultimately rely on the understanding of the hyperbolic equations considered in this project.