Nonlinear Hyperbolic Equations

该项目致力于两个相关的主题，即非线性双曲方程和具有非光滑系数的偏微分方程。我们考虑的双曲方程要么是半线性的（如临界波图和杨米尔斯方程），人们希望得到局部和全局的结果，要么是非线性的，其目的是改进局部理论。与非光滑系数偏微分方程相关的问题大多受到非线性偏微分方程的启发；这包括各种Strichartz和Carleman型估计，以及双曲边值问题。本项目所关注的方程可以粗略地描述为类波现象的模型，如广义相对论和非线性弹性。虽然许多物理模型是真正的非线性，但它们的数学分析极其困难，往往超出了当前的数学理论；因此，在大多数情况下，我们使用的简化模型是线性的或只是轻度非线性的。这项提议的工作是一个更大项目的一部分，其目的是理解这种完全非线性方程。该策略是研究一系列越来越难的模型问题，这些问题越来越接近相关的物理模型，以便在每一步都了解一些新的非线性特征。

The project is devoted to two related topics, namely nonlinear hyperbolicequations and partial differential equations with nonsmooth coefficients.The hyperbolic equations we consider are either semilinear (such asthe critical Wave-Maps and Yang Mills equations), for which one hopes toobtainboth local and global results, or nonlinear, for which the aim is toimprovethe local theory. The problems related to partial differential equationswith nonsmooth coefficients are mostly inspired by nonlinear pde's; thisincludes various Strichartz and Carleman type estimates, as well ashyperbolicboundary value problems.The equations of interest for this project can be loosely described asmodels for wave-like phenomena, such as generalized relativity andnonlinear ellasticity. While many physical models are genuinely nonlinear, their mathematical analysisis extremely difficult, often beyond the current mathemathical theory;thus,in most cases we contend ourselves with simplified models which are linearor only mildly nonlinear. The proposed work is part of a larger programwhose aim is to understand such fully nonlinear equations. The strategy isto study a sequence of increasingly difficult model problems which getcloser andcloser to the relevant physical model, so that at each step some newnonlinear feature is understood.