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Hyperbolic systems with multiplicities

具有重数的双曲系统

基本信息

批准号：
EP/L026422/1
负责人：
Claudia Garetto
金额：
$ 12.69万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FL026422%2F1
关键词：
Hyperbolic systems multiplicities

项目摘要

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.

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（例如在地震期间通过地球层），到晶体中的锥形折射，从气体动力学到信号传输。高阶方程通常通过一阶方程组的分析来研究，因此双曲型标量方程的分析可以看作是线性双曲型方程组的分析。存在两类系统：无重性系统（严格双曲）和重性系统（弱双曲）。我们对严格双曲系统有很好的理解，但当多重性出现时，情况完全不同。这个项目致力于双曲型系统的多重性，一个臭名昭著的困难的主题在偏微分方程领域。该研究领域的复杂性由迄今为止所获得的零散结果和众多的开放问题所证明。该项目有望开发一种新的方法来研究具有多重性的双曲系统，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始于线性化过程，在双曲情况下，这会导致具有多重性的系统，因此，本项目的研究成果对非线性双曲方程和方程组的研究也有一定的参考价值。在本项目的第一部分中，我将主要研究系数仅依赖于时间（t-依赖）的线性弱双曲方程组。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I）。这种约化非常重要，因为它使我能够更容易地找到合适的能量并证明相应柯西问题的适定性。此外，我将证明，减少到块西尔维斯特形式可以做双曲系统，不一定是线性的，所以目标我将相关的非线性系统的分析。在这个初步的部分之后，我将考虑具有t依赖正则系数的弱双曲方程组（目标II）。这里，正则意味着平滑或解析。通过使用的技术，到目前为止只采用标量方程，而不是系统（拟对称）我将证明适定性的相应柯西问题在每一个Gevrey类（中间类之间的解析函数和光滑函数）或更一般的空间中的光滑函数和/或分布。这将需要精确的条件下的低阶项（列维条件），其最优性仍然需要理解。最终的挑战性目标将是在低阶项水平的适定性的特征。作为一个自然的目标III，我会问自己，当系数的正则性明显降低时会发生什么。现有的结果总是假设至少Hölder正则性，并制定Gevrey适定性。我打算放弃这种规律性的限制。这就需要开发新的方法和技术。主要思想是处理一个正则化问题，其中系数已经通过与缓和剂的卷积进行正则化。这样的正则化并不改变系统的性质，但提供了一个家庭的更正规的系统（取决于一个参数趋于0），可以研究感谢目标III。正则化问题（广义解）的解的网络将被渐近地分析，并最终通过极限方法得到经典解。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，旨在放弃t和x的正则性假设。