Hyperbolic systems with multiplicities

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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.