Hyperbolic systems with multiplicities

双曲方程模拟了物理学中的不同现象：从波在介质中的传播(例如地震时穿过地球各层)，到晶体中的锥形折射，从气体动力学到信号传输。高阶方程通常通过降阶为一阶系统来研究，因此双曲标量方程的分析可以看作是线性双曲组的分析。存在两类系统：没有多重性(严格双曲)的系统和有多重性(弱双曲)的系统。我们对严格双曲系统有很好的理解，但当出现多重性时，情况就完全不同了。这个项目致力于研究具有多重性的双曲型系统，这是偏微分方程界出了名的一个难题。这一研究领域的复杂性被迄今为止所取得的支离破碎的结果和众多的公开问题所证明。这个项目承诺开发一种新的方法来研究具有多重性的双曲型系统，这将解决该领域长期存在的开放问题。注意，非线性系统的研究通常从线性化过程开始，在双曲型的情况下导致系统的多重性，因此本项目的进展将对非线性双曲型方程和系统的研究同样有用。在项目的第一部分，我将集中于系数仅依赖于时间(t依赖)的线性弱双曲型系统。作为第一步，我将把系统简化为一种特殊形式：块西尔维斯特形式(目标一)。这个约化是非常重要的，因为这将使我更容易找到合适的能量，并证明相应的柯西问题的适定性。此外，我还将证明，对于不一定是线性的双曲系统，可以简化为块西尔维斯特形式，因此客观I也将与非线性系统的分析相关。在这一预备部分之后，第一部分将转到考虑具有t依赖正则系数的弱双曲组(目标II)。在这里，Regular的意思是流畅的或有分析能力的。通过使用迄今仅用于标量方程而不用于系统(准对称)的技巧，我将证明相应的柯西问题在每个Gevrey类(解析函数和光滑函数之间的中间类)或更多类中一般在光滑函数和/或分布空间中的适定性。这将需要关于低阶项的精确条件(Levi条件)，其最优性仍需理解。最具挑战性的目标将是描述较低阶项水平的适当性。作为一个自然的目标III，然后我会问自己，当系数的规律性合理地降低时会发生什么。现有的结果总是假设至少Hölder正则性，并且是以Gevrey适定的形式表示的。我打算取消这一规律性限制。这就需要开发新的方法和技术。其主要思想是研究一个正则化问题，其中的系数已通过与软化器卷积而正则化。这种正则化不改变系统的性质，但提供了一族更规则的系统(取决于参数趋向于0)，可以根据目标III进行研究。然后，正则化问题的解的网络(广义解)将被渐近分析，并最终通过极限程序得到经典解。项目的最后部分(目标IV)将致力于具有(t，x)依赖系数的弱双曲型系统，并将采用与第一部分完全不同的技术(半群)。更准确地说，目标四是目标三的雄心勃勃的(t，x)版本，旨在去掉t和x中的规律性假设。

Hyperbolic equations model different phenomena in physics: from propagation of waves in a medium (for instance through the Earth layers during an earthquake), to conical refraction in crystals, from gas dynamics to signal transmission. Higher order equations are usually studied via reduction to a first order system, so the analysis of hyperbolic scalar equations can be regarded as analysis of linear hyperbolic systems. There exist two classes of systems: systems without multiplicities (strictly hyperbolic) and systems with multiplicities (weakly hyperbolic). We have a very good understanding of strictly hyperbolic systems but the situation is completely different when multiplicities appear. This project is devoted to hyperbolic systems with multiplicities, a notoriously difficult topic in the field of partial differential equations. The complex nature of this research area is testified by the fragmented results obtained so far and by the numerous open problems.This project promises to develop a new approach to hyperbolic systems with multiplicities which will solve long-standing open problems in the field.Note that, the studying of nonlinear systems often starts with a linearisation process, which in the hyperbolic case leads to systems with multiplicties, so the advances in this project will be useful for the research on non-linear hyperbolic equations and systems as well.In the first part of the project I will concentrate on linear weakly hyperbolic systems with coefficients depending only on time (t-dependent). As a first step I will work out a reduction of the system to a special form: a block Sylvester form (Objective I). This reduction is very important because will allow me to find more easily a suitable energy and to prove well-posedness for the corresponding Cauchy problem. In addition, I will prove that a reduction to block Sylvester form can be done on hyperbolic systems which are not necessarily linear, so Objective I will be relevant for the analysis of nonlinear systems as well. After this preliminary part I will pass to consider weakly hyperbolic systems with t-dependent regular coefficients (Objective II). Here regular means smooth or analytic. By using techniques so far employed only for scalar equations and not for systems (quasi-symmetriser) I will prove well-posedness of the corresponding Cauchy problem in every Gevrey class (intermediate classes between analytic functions and smooth functions) or more in general in the space of smooth functions and/or distributions. This will require precise conditions on the lower order terms (Levi conditions) whose optimality still has to be understood. The ultimate challenging goal will be a characterisation of well-posedness at the lower order terms level. As a natural Objective III, I will then ask myself what happens when the regularity of the coefficients is sensibly reduced. The existing results always assume at least Hölder regularity and are formulated in terms of Gevrey well-posedness. It is my intention to drop this regularity restriction. This requires the development of new methodologies and techniques. The main idea is to work on a regularised problem, where the coefficients have been regularised by convolution with a mollifier. Such a regularisation does not change the nature of the system but provides a family of more regular systems (depending on a parameter tending to 0) which can be studied thanks to Objective III. The net of solutions of the regularised problem (generalised solution) will then be analysed asymptotically and eventually lead to a classical solution via limit procedure.The final part of the project (Objective IV) will be devoted to weakly hyperbolic systems with (t,x)-dependent coefficients and will employ techniques (semigroups) completely different from the ones of the first part. More precisely, Objective IV is the ambitious (t,x)-version of Objective III, aiming to drop regularity assumptions in both t and x.