Hyperbolic problems with discontinuous coefficients

线性和非线性双曲偏微分方程出现在所有科学（物理，化学，医学，工程，天文学等）。特别是在物理学中，他们模拟了几种重要的现象，从波在介质中的传播（例如地震期间地震波的传播）到晶体中的折射和气体动力学。当模拟波在多层介质中的传播时，例如地震时的地基，利用不连续函数在物理上是有意义的。该项目希望研究最大可能的双曲方程和系统：具有可变多重性和不连续系数（取决于时间和空间）。这是众所周知的一个非常困难的问题，由于存在的多重性和低正则性的系数。这将需要制定新的分析方法，首先在规则性假设下采用（项目第一部分），然后逐步适用于不太规则的系数（项目第二部分）。为了提供一个统一的方法来解决具有间断系数的双曲问题，我们将用数值方法来检验我们新的分析方法的强度。这将在两种不同的双曲偏微分方程方法（分析和数值）之间建立一座桥梁，这座桥梁基于新思想的分析，比较和实施。

Linear and nonlinear hyperbolic PDEs arise in all sciences (physics, chemistry, medicine, engineering, astronomy, etc). In particular, in physics they model several important phenomena, from propagation of waves in a medium (for instance propagation of seismic waves during an earthquake) to refraction in crystals and gas-dynamics. When modelling wave propagation trough a multi-layered medium, for instance the subsoil during an earthquake, it is physically meaningful to make use of discontinuous functions. This project wants to study the largest possible class of hyperbolic equations and systems: with variable multiplicities and discontinuous coefficients (depending on time and space). This is notoriously a very difficult problem due to the presence of multiplicities and the low-regularity of the coefficients. It will require the development of new analytical methods which will be first introduced under assumptions of regularity (first part of the project) and then gradually adapted to less regular coefficients (second part of the project). In order to provide a unified approach to hyperbolic problems with discontinuous coefficients, we will test the strength of our new analytical methods numerically. This will build a bridge between two different approaches to hyperbolic PDEs (analytical and numerical), a bridge based on analysis, comparison and implementation of new ideas.