This book presents a view of the state of the art in multi-dimensional hyperbolic partial differential equations, with a particular emphasis on problems in which modern tools of analysis have proved useful. Ordered in sections of gradually increasing degrees of difficulty, the text first covers linear Cauchy problems and linear initial boundary value problems, before moving on to nonlinear problems, including shock waves. The book finishes with a discussion of the application of hyperbolic PDEs to gas dynamics, culminating with the shock wave analysis for real fluids.

We present an n-population generalization of the Lighthill-Whitham and Richards traffic flow model. This model is analytically interesting because of several non-standard features. For instance, it leads to non-classical shocks and enjoys an unexpected stability in spite of the presence of umbilic points. Furthermore, while satisfying all the minimal 'common sense' requirements, it also allows for a description of phenomena often neglected by other models, such as overtaking.or more complicated relations, for instance making the distinction between light and congested traffic.One might question, a priori, the validity of (1.6). In general, density-speed relations may well be different from one class to the other. In a merely numerical study, the special form (1.6) of speed laws can obviously be abandoned. However, the system (1.3)-(1.4) is too general for any analytical study because even its hyperbolicity is unclear. To reach significant analytical results, we assume (1.6) throughout the paper.Concerning the specific choice of ψ, (1.7) has the advantage of making the flux of the system (1.3)-(1.6) quadratic. As we shall demonstrate below, this special and nevertheless meaningful feature leaves the possibility of a much richer mathematical analysis of (1.3)-(1.6). From the modelling point of view, this very specific choice is questionable, but it will not lead to any qualitative inconsistency.Anyway, we shall also point out properties of (1.3)-(1.6) that are valid for any ψ. For a discussion of numerical results with the other choice (1.8), we refer to Wong & Wong [30].Interestingly, both from the applied point of view and the mathematical point of view, we shall show that (1.3)-(1.6) is endowed with a far from standard structure. In particular, due to the presence of an umbilic point, system (1.3)-(1.6) does not fit into any of the recent theoretical results [4,8,26]. Besides, (1.3) comprehends an n × n analog of the Keyfitz-Kranzer system [17], and leads to the presence of over-compressive shocks as well.In spite of that, as the study below and several numerical experiments suggest, we expect it to be well-posed.The next section deals with the general n-population model, while § 3 specializes to the case n = 2. Some technical details are deferred to the final Appendix.

International audienceThe Euler-Korreweg model results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, which takes into account capillarity effects in regions where the density experiences large variations, typically across interfaces for fluids exhibiting phase changes. One of the main difficulties in the analysis of the Cauchy problem for this model, a third order system of conservation laws, is the absence of dissipative regularization, since viscosity is neglected. The Cauchy problem for isothermal fluids in one space dimension has been addressed by the authors in an earlier paper, using Lagrangian coordinates. Here the Cauchy problem is investigated in arbitrary space dimension N, still for isothermal fluids, and a variable capillarity coefficient. A local well-posedness result is obtained in Sobolev spaces as though the density gradient and the velocity field were solutions of a symmetrizable hyperbolic system. More precisely, well-posedness is shown for Hs+1 x H-s (s > N/2 + 1) perturbations of smooth global solutions, either constant states or traveling profiles. In addition, almost-global existence is proved for small enough perturbations, and a blow-up criterion is shown. Proofs rely on a suitable extended formulation of the system, which turns out to amount to a nonlinear degenerate Schrodinger equation coupled with a transport equation, and on a priori estimates without loss of derivatives for the extended system, which necessitate various 'gauge' functions to cancel out bad commutators

Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipative models are considered, and a most basic question is investigated, namely the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg-de Vries equation, and the less known Euler-Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water waves theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre [Comm. Partial Differential Equations limit. Then numerical investigations are carried out for the modulated equations of the Euler-Korteweg system with two types of 'pressure' laws, namely the quadratic law of shallow water equations, and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.