Hyperbolic systems with multiplicities

双曲方程模拟了物理学中不同的现象：从波在介质中的传播（例如在地震期间穿过地层），到晶体中的锥形折射，从气体动力学到信号传输。高阶方程的研究通常通过还原为一阶系统来进行，因此双曲型标量方程的分析可以看作是线性双曲型系统的分析。存在两类系统：无多重性系统（严格双曲型）和有多重性系统（弱双曲型）。我们对严格意义上的双曲系统有很好的理解，但是当多重性出现时，情况就完全不同了。该项目致力于具有多重性的双曲系统，这是偏微分方程领域中一个众所周知的难题。这一研究领域的复杂性由迄今为止获得的碎片化结果和众多悬而未决的问题证明。该项目有望开发一种具有多重性的双曲系统的新方法，这将解决该领域长期存在的开放性问题。请注意，非线性系统的研究通常从线性化过程开始，在双曲情况下，线性化过程导致系统具有乘法，因此本项目的进展对非线性双曲方程和系统的研究也很有用。在项目的第一部分，我将专注于线性弱双曲系统，其系数仅依赖于时间（t相关）。作为第一步，我将把系统简化为一种特殊形式：块Sylvester形式（目标1）。这种简化是非常重要的，因为它可以让我更容易地找到一个合适的能量，并证明相应的柯西问题的适定性。此外，我将证明在不一定是线性的双曲系统上可以进行块Sylvester形式的约简，因此目标1也将与非线性系统的分析相关。在这个初步部分之后，我将转而考虑具有t相关正则系数的弱双曲系统（目标II）。这里正则的意思是光滑的或解析的。通过使用迄今为止仅用于标量方程而不用于系统（拟对称）的技术，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）或更一般地在光滑函数和/或分布空间中的适定性。这将需要低阶项（Levi条件）的精确条件，其最优性仍有待理解。最终具有挑战性的目标将是在低阶项水平上描述适位性。作为一个自然的目标III，我会问自己当系数的规律性被合理地减少时会发生什么。现有的结果总是假定至少Hölder有规律性，并且是根据格夫里适定性来表述的。我打算放弃这个规则限制。这就要求开发新的方法和技术。主要思想是处理一个正则化的问题，其中系数已经通过与缓和器的卷积进行了正则化。这样的正则化不会改变系统的性质，但会提供一组更规则的系统（取决于趋向于0的参数），这可以通过目标III进行研究。然后将渐近分析正则化问题（广义解）的解网，并最终通过极限过程得到经典解。项目的最后一部分（目标IV）将致力于具有（t,x）相关系数的弱双曲系统，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IV是目标III的雄心勃勃的（t,x）版本，旨在消除t和x中的正则性假设。

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.