Hyperbolic problems with discontinuous coefficients

线性和非线性双曲偏微分方程组存在于所有的科学领域(物理、化学、医学、工程、天文学等)。特别是，在物理学中，他们模拟了几个重要的现象，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。在模拟地震过程中多层介质(例如地基)中的波传播时，使用不连续函数是有物理意义的。这个项目想要研究可能最大的一类双曲型方程和系统：具有可变的重数和不连续的系数(取决于时间和空间)。众所周知，这是一个非常困难的问题，因为存在多重性和系数的低规则性。这将需要开发新的分析方法，这些方法将首先在规律性假设下引入(项目的第一部分)，然后逐渐适应较不规则的系数(项目的第二部分)。为了给具有间断系数的双曲型问题提供一个统一的方法，我们将用数值方法来检验我们新的分析方法的能力。这将在两种不同的双曲偏微分方程(解析和数值)方法之间架起一座桥梁，一座基于新想法的分析、比较和实施的桥梁。

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.