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In this post, I review the book Partial Differential Equations in General Relativity, recently published by A.D. Rendall, in: Graduate Texts in Mathematics, vol. 16, Oxford University Press, 2008. I recommend it as an excellent source of inspiration for problems arising in general relativity. The text below has appeared in the “Gazette des Mathématiciens” of the SMF (Sociéte Française de Mathématiques) and is, therefore,… in French.

Ce livre est consacré aux aspects mathématiques de la relativité générale et en présente les développements les plus récents de manière concise. Il porte une attention particulière à la formulation des équations d’Einstein en un système hyperbolique symétrique avec contraintes couplé avec les équations d’évolution de la matière. Il est destiné aux étudiants de Mastère, tant mathématiciens que physiciens, mais sera aussi apprécié des spécialistes de la relativité générale.

Rappelons que l’inconnue principale de la théorie est une variété lorentzienne de dimension quatre, ${(M,g)}$, satisfaisant aux équations

$\displaystyle G_{\alpha \beta} + \Lambda \, g_{\alpha \beta} = 8 \pi \, T_{\alpha\beta},$

reliant le tenseur de courbure d’Einstein ${G_{\alpha \beta}}$ au tenseur de moment-énergie de la matière ${T_{\alpha\beta}}$. Le scalaire ${\Lambda}$ représente la constante cosmologique de l’espace-temps, et les indices ${\alpha, \beta}$ varient de ${0}$ à ${3}$. (Par exemple, dans le vide le tenseur de courbure de Ricci d’une telle variété est identiquement nul.) Pour formuler le problème, on se donne une variété riemannienne de dimension trois ${(N,h)}$ munie d’un champ de ${2}$-tenseurs symétriques ${K}$. On cherche alors une variété lorentzienne (un développement maximal) satisfaisant aux équations d’Einstein avec la contrainte que ${(N,h)}$ est isométriquement plongée dans ${(M,g)}$ et admet ${K}$ comme deuxième forme fondamentale.

Les deux premiers chapitres présentent les bases incontournables de la relativité générale d’un point de vue à la fois physique et mathématique: notions de géométrie lorentzienne, théorèmes d’incomplétude de Penrose et de Hawking, feuilletages ${3+1}$, et décompositions des équations d’Einstein.

L’auteur entre dans le vif de son sujet en mettant en parallèle différents modèles de matière: équations d’ondes pour les champs scalaires (à valeurs réelles ou à valeurs dans une variété), équations de Maxwell de l’électro\-magné\-tisme, équations de Yang-Mills, équations d’Euler des fluides compressibles et équation de Vlasov de la théorie cinétique des fluides raréfiés. Pour chacun de ces modèles, un Lagrangien détermine l’expression du tenseur de moment-énergie ${T_{\alpha \beta}}$ en fonction de la métrique lorentzienne et des variables physiques décrivant l’état de la matière.

Peu de résultats sont disponibles dans la littérature à ce niveau de généralité, et, le plus souvent, des hypothèses sur les symétries de l’espace-temps sont nécessaires. L’auteur introduit ici les espaces-temps admettant un ou plusieurs champs de Killing: statiques, stationnaires, spatialement homogènes, etc.

Il consacre ensuite un chapitre à l’étude des espaces spatialement homogènes pour lesquels les équations d’Einstein se réduisent à des équations différentielles non-linéaires. En dépit de leur simplicité apparente, ces modèles sont intéressants pour l’interprétation physique de la théorie. Par ailleurs, leur étude mathé\-matique est très délicate et fait appel à toutes les facettes de la théorie des équations différentielles: variété centrale, systèmes dynamiques, théorie des bifurcations, etc.

La deuxième moitié de l’ouvrage traite directement la résolution du problème de Cauchy pour les équations d’Einstein. L’auteur présente d’abord les résultats principaux d’existence sans hypothèse de symétrie; ceux-ci sont centrés sur le théorème de Christodoulou et de Klainerman (stabilité de l’espace-temps de Minkowski). Il explique aussi les techniques d’analyse (harmonique) intervenant dans les démonstrations. Les derniers chapitres concernent les modèles possédant deux champs de Killing et l’auteur étudie en détail l’existence globale des solutions des équations d’Einstein et la nature géométrique de leurs singularités, ce qui lui permet de déterminer le comportement asymptotique des espace-temps construits.

En conclusion, il s’agit d’un ouvrage particulièrement bien organisé et documenté, dont la lecture est vivement recommandéee et permet d’accéder à l’état de l’art sur le sujet.

Philippe G. LeFloch (Paris).

In this post, I want to review the book The formation of shocks in 3-dimensional ßuids” by D. Christodoulou, EMS Monographs in Mathematics, 2007.

The author studies here the maximally defined, smooth solutions to the relativistic Euler equations of motion for a perfect fluid in Minkowsky spacetime ${M^{3+1}}$. The discussion begins with a review of earlier works, including pioneering work on shock formation by Riemann on isentropic fluid flows with plane symmetry and, more generally, on nonlinear hyperbolic systems of two conservation laws in one space variable: smooth solutions develop singularities in finite time. The formulation of the physically correct jump relations was later found by Rankine and Hugoniot. Further fundamental work was done by Friedrichs and Lax, and the general problem of shock formation for hyperbolic systems of conservation laws in one space dimension was solved by Lax in 1964 (genuinely nonlinear systems) and John in 1974 (general systems). The strategy in the above works was to deduce an ordinary differential inequality for a quantity constructed from the first-order derivatives of the solution, and to show that this quantity must blow-up in finite time, at least under certain assumptions on the structure of the hyperbolic system.

More recently, for the Euler equations of perfect compressible fluids, an entirely different approach was introduced by Sideris in 1985 which, instead, used integral quantities associated with the solution. The main drawback of this method is that it tells us nothing about the nature of the breakdown. Moreover, it requires the pressure of the fluid to be strictly convex in terms of the density. In another direction, in 1983, Majda began an ambitious program on the stability of shock fronts for nonlinear hyperbolic systems in several space dimensions; this was continued and expanded by Gues, Metivier, and followers.

In the present work, the author considers the relativistic Euler equations for a perfect fluid with an arbitrary equation of state. Initial data are imposed on a given spacelike hyperplane and are constant outside a compact set. Attention is restricted to the evolution of the solution within a region limited by two concentric spheres. Given a smooth solution to the Euler equations, the main objective of the author is to investigate the geometry of the boundary of its domain of definition, that is, the locus where shock waves may form.

At the end of this book, under certain smallness assumptions on the size of the initial data, a remarkable and complete picture of the formation of shock waves in three dimensions is obtained. In addition, sharp sufficient conditions on the initial data for the formation of shocks in the evolution are established, and sharp lower and upper bounds for the time of existence of a smooth solution are derived.

The main strategy proposed in this book is as follows. Given an arbitrary initial data that is constant outside a sphere and under suitable smallness conditions on this initial data, the author controls first the solution for a time interval of order ${1/\eta}$, where ${\eta}$ is a reference sound speed. He shows that at the end ${T}$ of this time interval the flow is irrotational and isentropic within an annular region limited by two concentric spheres. Then, he proceeds by studying the maximal development of the restriction of the data at the time ${T}$ to the exterior of the inner sphere. Next, he relies on the property that for irrotational and isentropic flows, the exists a function ${\phi}$ which suffices to characterize the fluid and satisfies a wave equation which is the main equation studied here.

This book represents an amazing “tour de force” by the author. The analysis relies heavily on differential geometric concepts and methods; one key unknown of the Euler equations is the one-form velocity field, denoted here by ${\beta}$, suitably multiplied by the relativistic enthalpy of the fluid; in the irrotational case, ${\beta}$ coincides with the exterior differential of the potential ${\phi}$ mentioned earlier. The estimates derived in this work are based on the natural action principle associated with the fluid equations and on the construction of vector fields adapted to the geometry of the solution. High-order energy-type estimates are derived which yield a sharp control of a geometric foliation of the solution.

The book is structured as follows. The first four chapters provide notation and set up the general framework. Chapters 5 to 13 restrict attention to irrotational and isentropic fluids and culminate with the shock formation result in Theorem 13.1. Chapter 5 contains the fundamental energy estimate. Chapter 6 contains a discussion of the properties of several vector fields of interest. Chapter 7 deals with source-terms arising in higher-order estimates and present a recursion formula for these terms. Chapters 8 and 9 contain the crucial technical part of the present work, and establish the higher-order estimates. In particular, in Chapter 9 the evolution of the second fundamental form of the leaves of the foliation is discussed and the key structure of the problem (elliptic equations on two-dimensional submanifolds of the foliation, ordinary differential equations along the generators of the foliation) is uncovered.

Theorem 14.1 provides sharp sufficient conditions on the initial data for shock wave to form during the evolution. The rest of Chapter 14, is devoted to the general problem of shock formation for flow that need be irrotational and isentropic. The connection is made here with Theorem 14.1, since the author finds conditions at the beginning of the time interval of definition of the solution which guarantee that the solution at the end of the time interval will satisfy the assumptions of Theorem 14.1. This analysis leads to the main result stated in Theorem 14.2.

Next, Chapter 15 is devoted to the investigation of the geometry of the boundary of the domain of definition of the solution. Another main result of this book is that the boundary of the domain of definition of a solution consists of a singular part ${S= \partial_- UH \bigcup H}$ and a regular part ${C}$. Here, ${\partial_- H}$ denotes the past boundary of ${H}$. Each component of ${C}$ is an incoming characteristic hypersurface having a singular past boundary, while ${S}$ is the locus where the inverse density vanishes.

In so-called acoustical coordinates (associated with a metric taking into account the acoustic part of the relativistic Euler equations), the solution extends smoothly up to the boundary, but a particular function associated with the solution and denoted by ${\mu}$ vanishes on the singular part ${S}$. On the other hand, the function ${\mu}$ is positive on the regular part of the boundary, and the solution is smooth in this part even in the original coordinates. In addition, the author shows that each connected component of the boundary ${\partial_- H}$ is a smooth two-dimensional embedded submanifold in Minkowsky spacetime hich is spacelike with respect to the acoustical metric. On the other hand, the corresponding component of ${H}$ is a smooth embedded three-dimensional submanifold ruled by invariant curves of vanishing arc length with respect to the acoustical metric, having past end points on the component of ${\partial_- H}$. The corresponding component of ${C}$ is precisely the incoming null hypersurface associated with the component of ${\partial_- H}$. It is ruled by incoming null geodesics of the acoustical metric with past end points on the component of ${\partial_- H}$.

The author also points out that the limit toward the non-relativistic Euler equation does not involve any singular behavior and, therefore, from his results one can deduce similar results about shock formation in non-relativistic fluids.

Finally, the author discusses the physical continuation of the solution. He observes that the standard notion of maximal development is not appropriate up to ${H}$. In order to determine the physically correct hypersurface of discontinuity (denoted by ${K}$ below), the author then defines a shock development problem, as follows,

Given a component ${A}$ of ${\partial_- H}$ and the corresponding components ${C^A}$ and ${H^A}$ of ${C}$ and ${H}$, respectively, the physically relevant problem associated with ${A}$ is as follows: Find a hypersurface ${K \subset M^{3+1}}$ lying in the past of ${H^A}$ and with the same past boundary (namely ${A}$) and the same tangent hyperplane at each point along ${A}$, together with a solution of the Euler equations defined on the domain of ${M^{3+1}}$ bounded in the past by ${C}$ and ${K}$ and such that on the regular part ${C^A}$ the solution coincides with the data induced by the maximal solution; on the other hand, across ${K}$, the solution must suffer a jump with respect to the prescribed data (given the maximal solution).

Of course, the jump relations under consideration are those associated with the integral form of the Euler equations. Moreover, the hypersurface ${K}$ must be spacelike for the acoustical metric induced by the maximal solution (in the past of ${K}$), but timelike with respect to the new solution (in the future of ${K}$). The author then proceeds with a rigorous derivation of the jump conditions and justifies that his construction is consistent with the second law of thermodynamics and provides the physically relevant discontinuous solution.

Following questions raised at a (very interesting) conference I attended at Bilbao in September 2009 on hyperbolic systems with source-terms and non-conservative form, I want to emphasize here a fact of importance in numerical computations for such problems.

Consider the issue of computing shock waves to nonconservative hyperbolic systems, as well as to hyperbolic conservation laws generated by zero diffusive-dispersive limits,  In these two instance, an additional selection criterion beyond an entropy inequality is necessary, that is, a kinetic relation for the latter and a family of paths (or one or several kinetic functions) for the former. The shocks under consideration turn out to strongly depend on small scales that are represented by higher-order terms (viscosity, heat conduction, capillarity, etc) that must be included in an associated augmented model. No universal selection criterion should be sought for, but rather a different theory of shock waves is necessary in each physical context.

The key point to stress is that no finite difference schemes (say expressed as smooth algebraic expressions of the coefficients of the conservative or nonconservative equations) has never been found to converge to correct weak solutions ! This difficulty was observed in Hou and LeFloch in 1993 for nonconservative systems and in Hayes and LeFloch in 1994 for diffusive-dispersive limits.

The point is that in both the continuous model and the discrete scheme, small scale features are critical to the selection of shocks. The balance between diusive and dispersive features determines which shocks are selected. In nonconservative systems, the competition takes place between the (hyperbolic) propagation part and the (viscous) regularization. These small scale features cannot be quite the same at the continuous and at the discrete levels, since a continuous dynamical system of ordinary dierential equations cannot be exactly represented by a discrete dynamical system of ﬁnite dierence equations. Strictly speaking, finite difference schemes do not converge to the correct weak solution when small-scaled are the driving factor for the selection of shock waves.

Still, the good news is that schemes can be developed that approach the exact solutions as much as one requests. The design of suitable schemes is achieved via an analysis of the equivalent equation associated with a scheme by formal Taylor expanding its coefficients. The accuracy of the numerical solution improves as the equivalent equation coincides with the augmented model at a higher and higher order of approximation. For recent developments, we refer the interested reader to the following two papers.

P.G. LeFloch and M. Mohamadian, Why many shock wave theories are necessary. Fourth-order models, kinetic functions, and equivalent equations, J. Comput. Phys. 227 (2008), 4162–4189.

Abstract. We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit under-compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by high-order ﬁnite diﬀerence schemes, and uncover several new features of the kinetic function associated with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a ﬁnite diﬀerence scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model – especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid ﬁlm model and the generalized Camassa–Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals ﬂuids, whose ﬂux-function admits two inﬂection points.

M.J. Castro, P.G. LeFloch, M.L. Munoz-Ruiz, and C. Pares, Why many theories of shock waves are necessary. Convergence error in formally path-consistent schemes, J. Comput. Phys. 227 (2008), 8107–8129.

Abstract. We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics,and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat’s theory, a shock wave theory for a given nonconservative system requires prescribing a~priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. We first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an {\sl convergence error source-term} which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually “small”. In the special case that the scheme converges in the sense of graphs —a rather strong convergence property often violated in practice— then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as we demonstrate, plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme.