Celebratio Mathematica

[1] Y. Fourès-Bruhat: “Sur une expression intrinsèque du théorème de Gauss en relativité générale,” C. R. Acad. Sci. Paris 226 (1948), pp. 218–220. MR 23640 Zbl 0030.38402 article

[2] Y. Fourès-Bruhat and A. Lichnerowicz: “Sur un théorème global de réduction des \( ds^2 \) statiques généraux d’Einstein,” C. R. Acad. Sci. Paris 226 (1948), pp. 775–777. MR 24704 Zbl 0031.23902 article

Racine a donné pour les équations du champ gravitationnel stationnaire une forme simple en introduisant le repère orthogonal mobile tel que la métrique prenne la forme \[ ds^2 = V^2 (dx^0)^2-\sum (\omega_i)^2 \] (\( i=1,2,3 \)) où \( \omega_i = \bar{\omega}+ \lambda_i dx^0 \), les \( \bar{\omega}_i \) étant des formes purement spatiales. L’auteur utilise ces équations pour obtenir une expression simple du vecteur \( h \) qui intervient dans le théorème de Gauss donné par le rapporteur et dont la divergence est égale à \( R^0_0 \). \[ Vh_i=\partial_2 V-\sum_j\lambda_j P_{ji} \] (\( i,j=1,2,3 \)), les \( P_{ji} \) étant coefficients de la seconde forme quadratique fondamentale des sections d’espace.

[3] Y. Fourès-Bruhat: “Sur l’intégration du problème des conditions initiales en mécanique relativiste,” C. R. Acad. Sci. Paris 226 (1948), pp. 1071–1073. MR 25310 Zbl 0030.38401 article

On sait que le problème de l’intégration des équations d’Einstein peut être partagé en deux problèmes: le problème de la recherche de conditions initiales (valeurs des \( g_{ij} \) et de leurs dérivées premières sur une hypersurface \( S \) orientée dans l’espace) et le problème de l’évolution dans le temps de potentiels. Les conditions initiales sont astreintes à satisfaire à 4 des équations d’Einstein. Les équations d’Einstein sont écrites par l’auteur sous une forme nouvelle par l’introduction de la méthode du repère mobile d’E. Cartan; l’intégration des 4 équations du problème des conditions initiales est etudiée par réduction à un système de trois équations linéaires dérivées partielles et à une équation non linéaire du second ordre de type elliptique à une seule fonction inconnue.

[4] A. Lichnerowicz and Y. Fourès-Bruhat: “Théorème global sur les \( ds^2 \) extérieurs généraux d’Einstein,” C. R. Acad. Sci. Paris 226 (1948), pp. 2119–2120. MR 26456 Zbl 0031.23903 article

[5] Y. Fourès-Bruhat: “Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique,” C. R. Acad. Sci. Paris 230 (1950), pp. 618–620. MR 34135 Zbl 0041.56303 article

Je me propose de résoudre le problème de Cauchy pour le système \( R_{\alpha\beta} = 0 \) dans le cas non analytique: on sait qu’un théorème d’unicité valable dans le même cas a été donné par Stellmacher. La surface \( (S) \) portant les données de Cauchy sera supposée orientée dans l’espace. Si \( (S) \) est représentée par \( x^4=0 \), je suppose de plus que les données de Cauchy satisfont sur \( (S) \) aux conditions d’isothermie: \begin{align*} & \frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^{\lambda}}(\sqrt{-g}\, g^{\lambda\mu}) = 0, \\ & \frac{\partial}{\partial x^4} \biggl[\frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^{\lambda}} (\sqrt{-g}\, g^{\lambda\mu})\biggr] = 0 \quad \text{pour } x^4=0 \end{align*} et je recherche les solutions du système d’Einstein écrit en coordonnées isothermes: le théorème d’unicité permet de montrer que de telles solutions satisfont effectivement aux conditions d’isothermie, donc au système \( R_{\alpha\beta} = 0 \).

[6] Y. Fourès-Bruhat: “Un théorème d’existence sur les systèmes d’équations aux dérivées partielles quasi linéaires,” C. R. Acad. Sci. Paris 231 (1950), pp. 318–320. MR 36927 Zbl 0038.25503 article

[7] Y. Fourès-Bruhat: “Théorèmes d’existence et d’unicité pour les équations de la théorie unitaire de Jordan–Thiry,” C. R. Acad. Sci. Paris 232 (1951), pp. 1800–1802. MR 42224 Zbl 0044.31702 article

L’auteur a développé récemment des méthodes donnant des théorèmes d’existence et d’unicité, dans le cas non analytique, pour les septimes d’équations aux dérivées partielles régissant les phénomènes physiques se propageant par ondes (de type totalement hyperbolique). Les preuves détaillées seront publiées prochainement dans les Acta Math.. L’auteur applique ici ses méthodes aux équations relativistes de la gravitation et de l’électromagnétisme; elle envisage deux cas, celui des équations de la théorie de Jordan–Thiry à quinze variables de champ et celui des équations “primitives” d’Einstein–Maxwell. Dans les deux cas, des théorèmes d’existence et d’unicité sont obtenues pour le problème de Cauchy, en supposant les données de Cauchy sur \( x^4=0 \) quatre ou cinq fois différentiables et en introduisant des coordonnées isothermes. La comparaison des résultats concernant les deux théories est intéressante; la théorie “primitive” doit être une première approximation de la théorie “unitaire” et cela se doit constater sur les solutions correspondant à une même valuer intiale du facteur de gravitation \( V \). L’auteur établit qu’il en est bien ainsi et même que \( V \) reste constant s’il est stationnaire, c’est-à-dire si \( V = \text{const.} \) et \( \partial_4 V=0 \) pour \( x^4=0 \).

[8] Y. Fourès-Bruhat: “Théorème d’existence pour des systèmes d’équations aux dérivées partielles à quatre variables,” C. R. Acad. Sci. Paris 234 (1952), pp. 500–502. MR 45287 Zbl 0046.09903 article

[9] Y. Fourès-Bruhat: “Solution du problème de Cauchy pour des systèmes d’équations hyperboliques du second ordre non linéaires,” C. R. Acad. Sci. Paris 234 (1952), pp. 585–587. MR 45288 Zbl 0046.10001 article

L’auteur considère d’abord un système d’équations aux dérivées partielles du second ordre linéaires à \( n \) variables \( x^{\alpha} \) et \( N \) fonctions inconnues \( u_s \) du type \[ A^{\lambda\mu} \frac{\partial^2 u_s}{\partial x^{\lambda} \partial x^u} + B^{t \lambda}_s \frac{\partial u_t}{\partial x^{\lambda}} + f_s = 0, \] (\( \lambda,\mu=1,2,\dots,n \); \( s=1,2,\dots,N \)) où \( A^{\lambda\mu}_1 B^{t\lambda}_s, f_s \) sont des fonctions des \( x^x \) possédant des dérivées partielle continues et bornées jusqu’aux ordres \( n,n-2, n/2-1 \). On suppose que \( A^{nn} \) est positif et que pour \( x_n=0 \), la forme quadratique \( A^{\lambda\mu} x_{\lambda} x_{\mu} \) se réduit à une forme définie négative.

En utilisant une méthode voisine de celle qu’elle a employée pour \( n=4 \), l’auteur ramène le problème de Cauchy à l’étude d’un système d’équations integrales.

L’auteur considère ensuite le système non linéaire à \( n \) variables et à \( N \) fonctions inconnues \( W_s \) \begin{equation} A^{\lambda\mu} (W_T,W_T^x, x^x) \frac{\partial^2 W_s}{\partial x^{\lambda} \partial x^{\mu}} + F_s (W_T, W_{Tx}, x^x)=0.\tag{1} \end{equation} Sous des conditions convenables de réguarité imposées à \( A^{\lambda\mu}, f_s \) et aux données de Cauchy sur \( x^n=0 \), le problème de Cauchy pour (1) admet une solution \( n+1 \) fois différentiable. Les résultats précédents généralisent à un nombre quelcoque de variables les formules de résolution et les théorèmes d’existence donnés précédemment par l’auteur pour \( n=4 \).

[10] Y. Fourès-Bruhat: “Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires,” Acta Math. 88 (1952), pp. 141–225. MR 53338 Zbl 0049.19201 article

Je me suis posé le problème de Cauchy pour les équations aux dérivées partielles hyperboliques non linéaires à propos des équations de la gravitation d’Einstein. Ces équations se présentent en effet eomme un système de dix équations du second ordre, linéaires, à quatre variables (espace et temps) et dix fonctions inconnues, les potentiels de gravitation. Ces équations sont du types hyperbolique normal dans un système de coordonnées spatio-temporelles régulier. Le problème du déterminisme se pose, dans la théorie d’Einstein, sous forme du problème de Cauchy, les données étant portées par une variété orientée dans l’espace, relativement à ce système d’équations.

[11] Y. Fourès-Bruhat: “Les distributions sur les multiplicités,” C. R. Acad. Sci. Paris 236 (1953), pp. 2201–2202. MR 56168 Zbl 0050.33902 article

[12] Y. Fourès-Bruhat: “Résolution du problème de Cauchy pour des équations hyperboliques du second ordre non linéaires,” Bull. Soc. Math. France 81 (1953), pp. 225–288. MR 62929 Zbl 0053.25101 article

Le but de ce Mémoire est la résolution du problème de Cauchy pour les équations hyperboliques du second ordre à un nombre quelconque de variables, non linéaires, du type suivant: \[ A^{\lambda \mu}\frac{\partial^2u}{\partial x^l \partial x^{\mu}}+f = 0 \] (\( \lambda,\mu=1,2,\dots,n \)). Les coefficients \( A^{\lambda\mu} \) et \( f \) sont des fonctions des inconnues \( \mu \), de leurs dérivées partielles premières et des variables \( x^x \).

Les résultats s’appliquent aux systèmes d’équations de la forme \[ A^{\lambda\mu} \frac{\partial^2 u_s}{\partial x^{\lambda} \partial x^{\mu}}+ f_s=0 \] (\( s=1,2,\dots, N \)), où les coefficients \( A^{\lambda\mu} \) et \( f_s \) sont les fonctions des \( N \) inconnues \( u_t \), de leurs dérivées partielles premières et des variables \( x^x \) , les coefficients \( A^{\lambda\mu} \) étant les mêmes pour les \( N \) équations.

[13] Y. Fourès: “Résolution du problème de Cauchy pour des équations hyperboliques du second ordre non linéaires,” pp. 25–33 in Premier colloque sur les équations aux dérivées partielles (Louvain, 1953). Georges Thone; Masson & Cie (Liège; Paris), 1954. An earlier version of this paper was published in Bull. Soc. Math. France 81, 225–288 (1953). MR 62930 incollection

[14] Y. Fourès and I. E. Segal: “Causality and analyticity,” Trans. Amer. Math. Soc. 78 (1955), pp. 385–405. MR 69401 Zbl 0064.36805 article

We treat an abstract version of a linear mechanical system, demonstrating a connection between causality in the system and analyticity of its gain function. This function is in general merely Lebesgue measurable; it is suitably analytic if and only if the system is causal in the sense that the future cannot influence the past. This result is for the “continuous” case in which the output is globally a continuous function of the input. Systems described by partial differential equations with constant coefficients will not in general be of this type, and we are led thereby to an examination of the influence of causality in the “discontinuous” case.

[15] Y. Fourès-Bruhat: “Solution élémentaire d’équations ultrahyperboliques,” C. R. Acad. Sci. Paris 240 (1955), pp. 395–396. MR 70840 Zbl 0064.09406 article

[16] Y. Fourès-Bruhat: “Solution élémentaire d’équations ultrahyperboliques à coefficients variables,” C. R. Acad. Sci. Paris 242 (1956), pp. 1566–1568. MR 77774 Zbl 0075.10003 article

[17] Y. Fourès-Bruhat: “Solution élémentaire d’équations ultrahyperboliques,” J. Math. Pures Appl. (9) 35 (1956), pp. 277–288. An earlier version of this paper was published as “Solutions élémentaire d’équations ultrahyperboliques,” C.R. Acad. Sci. Paris 240, 395–396 (1955). MR 83667 article

Construction d’une solution élémentaire de l’opérateur ultra-hyperbolique \[\square = \sum^{n_1}_{i=1} \frac{\partial^2}{(\partial x^i)^2} - \sum^{n_2}_{j=1} \frac{\partial^2}{(\partial x^j)^2}. \] Si \( n_1 \) et \( n_2 \) sont impairs, l’auteur détermine cette solution élémentaire sous la forme d’une distribution \( E \) satisfaisant à l’équation \( \square E = \delta \) dans \( R^{n_1+n_2} \) (\( \delta \): function de Dirac) et consituée par une somme finie de dérivées transversales de distributions portées par le cône caractéristique \( \sum \) de l’operateur \( \square \), d’équation \[ \sum^{n_1}_{i=1} (x^i)^2 - \sum^{n_2}_{i=1} (x^j)^2 = 0. \] Ces distributions sont en fait de simples puissances d’un paramètre variant sur une génératrice de \( \sum \) et sont obtenues comme solutions d’un système d’équations différentielles ordinaires. L’auteur passe aux autres valeurs de \( n_1 \) et \( n_2 \) par la méthode de descente.

[18] Y. Fourès-Bruhat: “Sur l’intégration des équations de la relativité générale,” J. Rational Mech. Anal. 5 (1956), pp. 951–966. MR 85123 Zbl 0075.21602 article

It is well-known from the investigations of Cartan, Darmois, and Lichnerowicz that the integration of the field equations of the general relativity theory consists firstly in finding the initial conditions on a certain space-like hypersurface, say \( x^4=0 \), and then in solving the corresponding Cauchy problem. The author develops here a method for integrating the system of four initial conditions \( S^4_{\alpha} =0 \), \( S_{\alpha\beta} \) being the energy-momentum tensor. For this purpose she transcribes the field equations of Einstein into a suitable system of coordinates. Thereafter she reduces one elliptic equation with the help of the Green’s function and three hyperbolic equations into a system of integral equations. If the domain of integration is sufficiently small, this system can be integrated by means of successive approximations. In certain special cases there exist solutions which hold without any restriction of the size of the domain of integration. The developed method can be applied for the equations describing the gravitational field in the empty space (i.e., \( S_{\alpha\beta} = 0 \)) as well as for the case when the space is filled with matter (\( S^{\alpha\beta}_{\alpha\beta} \neq 0 \)).

[19] Y. Fourès-Bruhat: “Problème des conditions initiales,” C. R. Acad. Sci. Paris 245 (1957), pp. 1384–1386. MR 90448 Zbl 0089.44403 article

[20] Y. Fourès-Bruhat: “Le problème de l’évolution dans le cas matière pure,” C. R. Acad. Sci. Paris 246 (1958), pp. 1809–1812. MR 105292 Zbl 0091.21206 article

[21] Y. Fourès-Bruhat: “Équations d’Helmholtz: Détermination des vitesses à partir des tourbillons. Problème d’évolution,” C. R. Acad. Sci. Paris 246 (1958), pp. 3319–3322. MR 105293 Zbl 0092.42603 article

Es werden die relativistischen Helmhotzschen Gleichungen für ideale Flüssigkeiten für den Fall abgeleitet, daß sich der den Erhaltungsgesetzen \[ \nabla_{\alpha}T^{\alpha\beta} = 0 \] genügende Energieimpulstensor in der Form \[ T_{\alpha\beta} = (\rho + p)u_{\alpha} u_{\beta} - pg_{\alpha\beta} \] schreiben läßt, wo \( \rho = \varphi(p) \) die Flüssigkeitsdichte, \( p \) den Druck und \( u_{\alpha} \) den Geschwindigkeitsvektor bedeuten. Dann werden einige Relationen zwischen den Wirbeln und den Geschwindigkeiten angegeben, weiter wird die Fortplflanzung der hydrodynamischen Wellen untersucht. Endlich wird darauf hingewiesen, daß die Fl”ussigkeitsbewegung durch gewisse Anfangsbedingungen für die Geschwindigkeit und den Druck eindeutig bestimmt ist.

[22] Y. Fourès-Bruhat: “Théorèmes d’existence en mécanique des fluides relativistes,” Bull. Soc. Math. France 86 (1958), pp. 155–175. MR 105294 Zbl 0092.20804 article

[23] Y. Fourès-Bruhat: “Fluides chargés de conductivité infinie,” C. R. Acad. Sci. Paris 248 (1959), pp. 2558–2560. MR 103026 Zbl 0085.21203 article

[24] Y. Fourès-Bruhat: “Conditions de continuité et équations de choc,” C. R. Acad. Sci. Paris 248 (1959), pp. 1782–1784. MR 104497 Zbl 0085.42605 article

[25] Y. Fourès-Bruhat: “Mécanique des fluides relativistes,” Cahiers de Phys. 13 (1959), pp. 463–468. MR 109720 article

[26] Y. Fourès-Bruhat: “Propagateurs et solutions d’équations homogènes hyperboliques,” C. R. Acad. Sci. Paris 251 (1960), pp. 29–31. MR 113042 Zbl 0147.08501 article

[27] Y. Bruhat: “Système elliptique pour le problème des conditions initiales,” C. R. Acad. Sci. Paris 252 (1961), pp. 3411–3413. MR 129941 Zbl 0100.40502 article

[28] Y. Bruhat: “The Cauchy problem,” pp. 130–168 in Gravitation: An introduction to current research. Wiley (New York), 1962. MR 143626 incollection

[29] Y. Fourès-Bruhat: “Les fluides chargés en relativité générale,” pp. 157–163 in Les théories relativistes de la gravitation (Royaumont, 1959). Éditions du Centre National de la Recherche Scientifique (CNRS) (Paris), 1962. MR 164729 incollection

[30] A. Lichnerowicz and Y. Fourès-Bruhat: “Problèmes mathématiques en relativité,” pp. 73–87 in Recent developments in general relativity. Pergamon; PWN (Polish Scientific Publishers) (Oxford; Warsaw), 1962. MR 164751 incollection

[31] Y. Choquet-Bruhat: Recueil de problèmes de mathématiques à l’usage des physiciens. Masson & Cie (Paris), 1963. MR 157517 Zbl 0114.05901 book

[32] Y. Bruhat: “Un théorème d’unicité de solutions faibles d’équations hyperboliques,” C. R. Acad. Sci. Paris 258 (1964), pp. 3949–3951. MR 168927 Zbl 0178.45404 article

[33] Y. Bruhat: “Sur la théorie des propagateurs,” Ann. Mat. Pura Appl. (4) 64 (1964), pp. 191–228. MR 169659 Zbl 0173.36501 article

[34] Y. Bruhat: “Ondes asymptotiques pour certaines équations aux dérivées partielles nonlinéaires,” C. R. Acad. Sci. Paris 258 (1964), pp. 3809–3812. MR 201788 Zbl 0124.30704 article

[35] Y. Bruhat: “Fluides chargés inductifs de conductivité finie,” C. R. Acad. Sci. Paris 261 (1965), pp. 3545–3548. MR 187560 article

[36] Y. Bruhat and P. M. Quan: “Thermodynamique des fluides chargés,” C. R. Acad. Sci. Paris 261 (1965), pp. 3987–3990. MR 189406 article

[37] Y. Bruhat: “Uniformisation de la solution d’un problème de Cauchy analytique non linéaire,” C. R. Acad. Sci. Paris 261 (1965), pp. 2056–2059. MR 190560 Zbl 0135.15003 article

The author extends the results of [Gårding et al. 1964] on linear systems to non linear systems. Let \( X \) be a complex analytic manifold of complex dimension \( l \) and \( S \) be an analytic sub-manifold of \( X \) which is defined by \( S \): \( \xi - s(x) = 0 \) (\( \operatorname{grad} s(x) \neq 0 \), \( \xi \in C \)). The author considers the Cauchy problem: \begin{equation} a_{j,k}(x,\xi,D^{m_k - n_j-1}, u^h, D^{m_k-n_j})u^k + b_j(x,\xi,d^{m_h- n_j-1}u^h = 0) \tag{1} \end{equation} (\( j,k=1,\dots,N \)) with initial conditions \[ \Bigl(\frac{\partial}{\partial x}\Bigr)^{\beta} (u^k(\xi,x)- w^k(\xi,x))= 0 \] (\( |\beta|\leq m_k-1 \)) on \( S \), where \( a_{j,k} \) and \( w^k (x,\xi) \) are holomorphic functions and \( S \) admits characteristic points of the equation (1). He proves that there exists a uniformisation \( \xi(t,x) \), \( \xi(0,x) = s(x) \) [depending on the Cauchy data \( w^k(x,\xi) \)] for the solutions \( u^k(\xi,x) \) up to order \( m_k-1 \), i.e., \[ \Bigl(\frac{\partial}{\partial x}\Bigr)^{\beta} u^k (\xi(t,x),x) \] (\( |\beta| \leq m_k-1 \)) are holomorphic for sufficiently small \( t \).

[38] Y. Bruhat: “Diagonalisation des systèmes quasi linéaires et hyperbolicité non stricte: Applications à la magnétohydrodynamique relativiste,” C. R. Acad. Sci. Paris 261 (1965), pp. 4315–4317. MR 203260 Zbl 0134.08302 article

[39] Y. Bruhat: “Etude des équations des fluides chargés relativistes inductifs et conducteurs,” Comm. Math. Phys. 3 : 5 (1966), pp. 334–357. MR 1552502 Zbl 0171.46605 article

In the first part a system of equations for an inductive charged relativistic fluid with finite conductivity is written in a space time with given metric, taking into account thermodynamic phenomena. Speeds of propagation of various types of waves are determined under a restrictive hypothesis concerning the heat current \( q \): that \( q \) depends only on the thermodynamical quantities and the gradient of one function of these quantities.

In the second part it is shown, by a detailed study of the characteristic polynomial and of its irreducible factors, that, when \( q \) is negligible, the proposed system is non-strictly hyperbolic in the sense of J. Leray and Y. Ohya and existence and uniqueness theorems of a certain Gevrey class are verified the relativistic causality principle is satisfied under some physically reasonable assumptions on the thermodynamical quantities. The system becomes strictly hyperbolic (existence and uniqueness theorems obtain in classes of functions with a finite number of derivatives) when the fluid is both non inductive and of zero electrical conductivity.

In the third part we show briefly, by the methods of the second part, that the equations of relativistic fluids, with an infinite electrical conductivity is also non-strictly hyperbolic. The linearized equations (in the neighborhood of constant values) are strictly hyperbolic.

[40] Y. Choquet-Bruhat: “Diagonalisation des systèmes quasi-linéaires et hyperbolicité non stricte,” J. Math. Pures Appl. (9) 45 (1966), pp. 371–386. An earlier version of this paper appeared in C.R. Acad. Sci. 261, 4315–4317 (1965). MR 216131 Zbl 0147.08505 article

[41] Y. Choquet-Bruhat: “Uniformisation de la solution d’un problème de Cauchy non linéaire à données holomorphes,” Bull. Soc. Math. France 94 (1966), pp. 25–48. MR 217423 Zbl 0147.08201 article

Nous nous proposons d’étendre certains résultats de Jean Leray [Leray 1957] et de L. Gårding, T. Kotaké et J. Leray [Gårding et al. 1964] sur la solution d’un problème de Cauchy linéaire à donnés holomorphes au cas d’un système d’équations aux dérivées partielles non linéaires. Nous montrons que pour un système d’équations aux dérivées partielles quasi-linéaire, à coefficients holomorphes, régulier au sens de Cauchy–Kovalevski, généralisé par Leray–Gårding, il existe une application holomorphe uniformisant la solution du problème de Cauchy donné sur une variété analytique \( S \) pouvant admettre des points caractéristiques. En général, \( u \) est algébroïde et le support \( K \) de ses singularités est un ensemble analytique de codimension 1.

[42] Y. Choquet-Bruhat: Problems and solutions in mathematical physics. Holden-Day (San Francisco, CA–London–Amsterdam), 1967. Translation of Recueil de problèmes de mathématiques à l’usage des physiciens. MR 213076 Zbl 0148.18303 book

[43] Y. Choquet-Bruhat: “Ondes asymptotiques pour un système d’équations aux dérivées partielles non linéaires,” C. R. Acad. Sci. Paris Sér. A-B 264 (1967), pp. A625–A628. MR 218720 Zbl 0146.33201 article

[44] Y. Choquet-Bruhat: Géométrie différentielle et systèmes extérieurs. Monographies Universitaires de Mathématiques 28. Dunod (Paris), 1968. MR 236824 Zbl 0164.22001 book

[45] Y. Choquet-Bruhat: “Hyperbolic partial differential equations on a manifold,” pp. 84–106 in Battelle Rencontres: 1967 Lectures in Mathematics and Physics. Edited by C. M. DeWitt and J. A. Wheeler. Benjamin (New York), 1968. MR 239299 incollection

[46] Y. Choquet-Bruhat: “Théorème global d’unicité pour les solutions des équations d’Einstein,” C. R. Acad. Sci. Paris Sér. A-B 266 (1968), pp. A182–A184. MR 232588 Zbl 0162.29702 article

Nous nous proposons, après avoir donné les définitions nécessaires, de démontrer le théorème suivant:

Théorème global d’unicité:
Hypothèses:

\( (V_4,g) \) et \( (V^{\prime}_4,g^{\prime}) \) sont desux espaces-temps einsteiniens, réguliers et globalement hyperboliques; \( (V^{\prime}_4,g^{\prime}) \) est complet vers le futur;
\( P \) (respectivement \( P^{\prime} \)) étant un ensemble compact vers le passé de \( (V_4,g) \) (respectivement \( (V^{\prime}_4,g^{\prime}) \)) tel que \( P=\mathcal{E}^+_P \) (respectivement \( P^{\prime}=\mathcal{E}^+_{P^{\prime}} \)) il existe une isométrie \( f \) de \( \complement P \) sur \( \complement^{\prime} P^{\prime} \).

Conclusion: Il existe une isométrie de \( (V_4,g) \) sur \( (D^{\prime},g^{\prime}) \) où \( D^{\prime} \) est un ouvert de \( V^{\prime}_4 \).

[47] Y. Choquet-Bruhat: “Espaces-temps einsteiniens généraux, chocs gravitationnels,” Ann. Inst. H. Poincaré Sect. A (N.S.) 8 (1968), pp. 327–338. MR 233576 Zbl 0162.29703 article

Nous donnons dans cet article une définition très large de ce qu’est un espace-temps, aussi large qu’il semble possible pour que les principes de base de la Relativité Générale soient respectés d’une part, pour que les équations d’Einstein aient un sens dans le cadre de la théorie des distributions d’autre part. Nous donnons des conditions dites “de régularité forte” pour que le problème de Cauchy (local) ait une solution et une seule.

Un cas particulier des espaces-temps ainsi définis est celui des espaces-temps présentant des chocs gravitationnels. Nous écrivons les équations de ces chocs sous forme intrinsèque, à la traversée d’une hypersurface S que nous supposons seulement lipshitzienne. Nous montrons que les quatre conditions vérifiées lors d’un tel choc à travers une variété caractéristique expriment la continuité des conditions d’harmonicité.

Ce dernier résultat nous permet de montrer l’unicité du prolongement d’un espace-temps à travers un conoïde caractéristique, même si ce dernier peut être surface d’onde de choc gravitationnel.

[48] Y. Choquet-Bruhat: “Construction de solutions radiatives approchées des équations d’Einstein,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 44 (1968), pp. 649–652. A paper of the same title appears in Comm. Math. Phys., 12, 16–35 (1969). MR 242456 article

[49] Y. Choquet-Bruhat: “Ondes asymptotiques pour un système d’équations aux dérivées partielles à caractéristiques multiples,” C. R. Acad. Sci. Paris Sér. A-B 267 (1968), pp. A596–A598. MR 252823 Zbl 0159.38602 article

[50] Y. Choquet-Bruhat: “Construction de solutions radiatives approchées des équations d’Einstein,” Comm. Math. Phys. 12 (1969), pp. 16–35. This paper seems to be related to an earlier paper of the same title, published in 1968 in Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 44. MR 241087 article

In this paper we construct rigorously, without any averaging scheme, an hyperbolic metric \begin{equation} g_{\alpha,\beta}(x, \omega\, \varphi) \,\overset{0}g_{\alpha,\beta}(x) + \frac{1}{\omega} \overset{1}{g}_{\alpha,\beta}(x, \omega\, \varphi) + \frac{1}{\omega^2} \overset{2}{g}_{\alpha,\beta} (x,\omega\,\varphi) \tag{1} \end{equation} where \( x \) is a point of the space-time \( V_4 \), \( \varphi \) a scalar function on \( V_4 \) (the “phase”) and \( \omega \) a great parameter (the “frequency”). This metric is an approximate solution of Einstein’s equations: it verifies \begin{equation} R_{\alpha,\beta} = O(\omega^{-1}), \tag{2} \end{equation} \( \overset{0}{g}_{\alpha,\beta_1} \) and its derivatives, \( \overset{1}{g}_{\alpha,\beta} \) and \( \overset{2}{g}_{\alpha,\beta} \) being bounded, the first and second derivatives of \( \overset{1}{g}_{\alpha,\beta} \), \( \overset{2}{g}_{\alpha,\beta} \) or order respectively \( O(\omega) \) and \( O(\omega^2) \).

We first show that the Ricci tensor of (1) stays bounded (when \( \omega \) increases), with a perturbation \( \overset{1}{g}_{\alpha,\beta} \) physically significant, if and only if \( \varphi \) verifies the characteristic equation of the back-ground metric and \( \overset{1}{g}_{\alpha,\beta} \) four algebraic, linear relations, (5.7) and (5.8) in radiative coordinates \( x^0 = \varphi \).

We show afterwards that (1) satisfies (2) if \( \overset{1}{g}_{\alpha,\beta} \) satisfies ordinary differential first order equations (which take a very simple form in radiative coordinates) along the rays of the background, the preceding algebraic relations can be considered as “initial conditions” if the Ricci tensor of the background is of the radiative form \[ R_{\alpha,\beta}(\overset{0}{g}_{\lambda,\mu}) = \tau \,\partial_\alpha \varphi \,\partial_\beta \varphi. \] It is possible to find \( \overset{1}{g}_{\alpha,\beta} \) and \( \overset{2}{g}_{\alpha,\beta} \) with the required conditions of boundedness only if \( \tau > 0 \). We apply the results to the Vaydya metric (13.1) and show that, by an oscillatory perturbation of this metric one can satisfy Einstein’s equations (to the order \( \omega^{-1} \)) if the coefficient \( m \) usually interpreted as the mass is a decreasing function of \( \varphi = u \), which gives, in this context, a proof of the loss of mass by gravitational radiation.

[51] Y. Choquet-Bruhat: “Problème de Cauchy oscillatoire pour un système de deux équations quasi linéaires à deux inconnues,” C. R. Acad. Sci. Paris Sér. A-B 268 (1969), pp. A1560–A1563. MR 244632 Zbl 0175.39501 article

[52] Y. Choquet-Bruhat and R. Geroch: “Global aspects of the Cauchy problem in general relativity,” Comm. Math. Phys. 14 (1969), pp. 329–335. MR 250640 Zbl 0182.59901 article

[53] Y. Choquet-Bruhat and R. Geroch: “Problème de Cauchy intrinsèque en relativité générale,” C. R. Acad. Sci. Paris Sér. A-B 269 (1969), pp. A746–A748. A version of this paper appeared as “Problème de Cauchy global en relativité générale,” Séminaire Jean Leray, no. 2, 1–7 (1969–1970) and a related paper was published in English as “Global aspects of the Cauchy problem in general reliativity,” Commun. Math. Phys. 14, 329–335 (1969). MR 255214 article

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	AUTHOR = {Choquet-Bruhat, Yvonne and Geroch, Robert},
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	JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. A-B},
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[54] Y. Choquet-Bruhat: “Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivées partielles non linéaires,” J. Math. Pures Appl. (9) 48 (1969), pp. 117–158. This paper appears to be related to two other versions by the same or similar title, one earlier and one contemporaneous with the present entry. They are, respectively: (1) “Ondes asymptotiques et approchées pour un système d’équations aux dérivées partielles non linéaires,” C.R. Aad. Sc. 264, 625–638 (1967); and (2) “Ondes asymptotiques et approchées pour un système d’équations aux dérivées partielles non linéaires,” Séminaire Jean Leray 3, 1–10 (1969). MR 255964 Zbl 0177.36404 article

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	TITLE = {Ondes asymptotiques et approch\'{e}es
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		 aux d\'{e}riv\'{e}es partielles non
		 lin\'{e}aires},
	JOURNAL = {J. Math. Pures Appl. (9)},
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		 partielles non lin\'{e}aires,'' \textit{S\'eminaire
		 Jean Leray} \textbf{3}, 1--10 (1969).
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[55] Y. Choquet-Bruhat: “Ondes asymptotiques et approchées pour un système d’équations aux dérivées partielles non linéaires,” Séminaire Jean Leray 3 (1969), pp. 1–10. article

Nous nous proposons de construire des ondes asymptotiques et des ondes approchées, pour un système d’équations aux dérivées partielles non linéaires sur une variété différentiable \( X \) en utilisant la méthode générale donnée par J. Leray [Leray 1961] et Gårding–Kotaké–Leray [Gårding et al. 1964] pour les systèmes linéaires à caractéristiques simples. Nous effectuons cette construction au voisinage d’une solution donnée du système et nous obtenons les termes successifs du développement par des quadratures sur des “rayons”, bicaractéristiques correspondant à la solution donnée, commes dans le cas linéaire. Nous voyons cependant apparaître ici un phénomène nouveau pour les ondes authentiquement non linéaires (ondes non exceptionelles au sens de Lax) qui est une déformation des signaux (“raidissement” conduisant aux phénomènes de choc). Les ondes asymptotiques obtenues ne sont, d’autre part, en général, dans le cas non linéaire, des ondes approchées qu’en se limitant aux deux premiers terms du dévloppement.

[56] Y. Choquet-Bruhat: “Problème de Cauchy global en relativité générale,” Séminaire Jean Leray 2 (1969–1970), pp. 1–7. A related paper was published in English as “Global aspects of the Cauchy problem in general reliativity,” Commun. Math. Phys. 14, 329–335 (1969). article

[57] Y. Choquet-Bruhat: “Solutions oscillatoires approchées des équations de la mécanique des fluides,” pp. 145–155 in La magnétohydrodynamique classique et relativiste (Lille, 1969). Éditions du Centre National de la Recherche Scientifique (CNRS) (Paris), 1970. MR 280163 Zbl 0267.76088 incollection

Author studies the approximate solutions of the system of nonlinear partial differential equations of relativistic fluid mechanics. The method is shown to work for the equations describing perfect (non-viscous, non-heat-conducting) fluids. The method is a kind of generalized WKB procedure and permits to find approximations to the solution proceeding according to the inverse powers of a parameter the value of which is greater than unity. It can be applied to any system of quasilinear partial differential equations admitting real simple or multiple characteristics for which the dimension of the characteristic proper space is equal to the multiplicity of the characteristic, and gives approximate waves. If this condition does not apply the equations admit discontinuities.

[58] Y. Choquet-Bruhat: “Un théorème d’unicité pour les équations d’Einstein–Liouville,” C. R. Acad. Sci. Paris Sér. A-B 270 (1970), pp. A728–A730. MR 258418 Zbl 0189.27601 article

[59] Y. Choquet-Bruhat: “Theorem of uniqueness and local stability for Liouville–Einstein equations,” J. Mathematical Phys. 11 (1970), pp. 3238–3243. MR 269263 article

[60] Y. Choquet-Bruhat: “Un théorème d’existence pour le système intégrodifférentiel d’Einstein–Liouville,” C. R. Acad. Sci. Paris Sér. A-B 271 (1970), pp. A625–A628. MR 270699 article

[61] Y. Choquet-Bruhat: “New elliptic system and global solutions for the constraints equations in general relativity,” Comm. Mat. Phys. 21 (1971), pp. 211–218. MR 295724 Zbl 0212.26802 article

[62] Y. Choquet-Bruhat: “Solutions \( C^{\infty } \) d’équations hyperboliques non linéaires,” C. R. Acad. Sci. Paris Sér. A-B 272 (1971), pp. A386–A388. MR 279444 Zbl 0216.12902 article

La question a été soulevée de savoir si un système d’équations aux dérivées partielles hyperboliques quasi linéaires à \( n \) variables avait une solution \( C^{\infty} \), pour des données de Cauchy \( C^{\infty} \): les méthodes usuelles de démonstration de l’existence d’une solution du problème de Cauchy (cf. [Leray 1953], [Dionne 1962]) pour de telles équations utilisent des espaces de Sobolev \( H_{\mu} (\Omega) \), où \( \Omega \) est un voisinage de la variété initiale portant les données de Cauchy, qui dépend de \( \mu \) et qui, d’après sa construction, tend vers zéro quand \( \mu \) augmente indéfiniment, de sorte que l’on est assuré, pour des données \( C^{\infty} \), d’une-solution \( C^p \), quel que soit \( p \), mais dont le domaine d’existence dépend de \( p \) et peut tendre vers zéro quand \( p \) tend vers l’infini. Nous allons montrer par une méthode très simple, pour un système à coefficients \( C^{\infty} \) , et des données de Cauchy \( C^{\infty} \), que la solution est \( C^{\infty} \) dans tout le domaine déterminé par la classe de Sobolev minimum nécessaire pour assurer son existence, si les coefficients des dérivées d’ordre le plus élevé, \( \nu,m \), ne contiennent que les dérivées d’ordre au plus \( m- 2 \) des inconnues par la classe suivante dans les autres cas.

[63] Y. Choquet-Bruhat: “Quelques résultats récents sur les équations aux dérivées partielles d’Einstein,” Math. Balkanica 1 (1971), pp. 27–31. MR 288709 Zbl 0221.35066 article

This paper contains two parts: (1) a short review of the recent results of the author about the evolution problem for the coupled Einstein–Liouville system. (2) A new result about the global solution of the constraints equations in empty space. This system has been changed into a simpler form than in a previous work of the author, and it is shown that the fundamental equation \[ \Delta \varphi - R\varphi + M\varphi^{-7} = 0 \] has for each \( R\geq0 \), \( M\geq0 \), one and only one solution \( \varphi = 0 \), on a given Riemannian manifold with boundary, taking given positive values on the boundary: this result has since been extended, for \( R > 0 \), \( M > 0 \), to closed manifolds and the interior case has been studied.

[64] Y. Choquet-Bruhat: “Problème de Cauchy pour le système intégro différentiel d’Einstein–Liouville,” Ann. Inst. Fourier (Grenoble) 21 : 3 (1971), pp. 181–201. MR 337248 Zbl 0208.14303 article

[65] Y. Choquet-Bruhat: “\( C^{\infty } \)solutions of hyperbolic non linear equations applications in G.R.G,” Gen. Relativity Gravitation 2 (1971), pp. 359–362. MR 395674 Zbl 0331.35041 article

The question has been raised by S. Hawking (colloquium on “The Bearings of Topology upon General Relativity” Bern, May 1970, cf. [Hawking 1971]) whether, for \( C^{\infty} \) Cauchy datas on an initial 3-manifold \( S \), Einstein’s equations had a \( C^{\infty} \) solution: he had remarked that in the usual proofs of existence of a solution of the Cauchy problem, using for instance Sobolev spaces \( H_{\mu} \), this existence is proved in a neighbourhood \( \Omega_{\mu} \) of \( S \), which depends on \( \mu \), and which appears to tend to zero as \( \mu \) increases towards infinity (cf. Fig. 3 in [Hawking 1971]): so we are assured, for \( C^{\infty} \) initial datas, of a solution \( C^p \)for every \( p \), but its domain of existence might tend to zero as \( p \) tends to infinity.

We will in this paper prove that this does not happen.

[66] Y. Choquet-Bruhat: “The bearings of global hyperbolicity on existence and uniqueness theorems in general relativity,” Gen. Relativity Gravitation 2 (1971), pp. 1–6. MR 411553 Zbl 0336.53022 article

The main result of this talk will be a global, geometric, uniqueness theorem for the solutions of Einstein equations, which has been proved jointly by Robert Geroch and myself. I had obtained previously a weaker result, on global uniqueness for space-times complete towards the future, but the theorem we have now is much more interesting since it expresses for the Einsteinian space-times a uniqueness analogous to the one obtained for the solutions of ordinary differential equations, i.e., the existence of a unique maximal extension.

I will first recall some definitions, with a few comments, then I will state the intrinsic Cauchy problem, and sketch the proof of the semi local, geometric, existence and uniqueness theorems of its solutions relative to Einstein equations using classical local theorems. I will then show the existence of a geometrically unique, maximal space-tie, solution of this Cauchy problem, in the class of globally hyperbolic space-times.

[67] Y. Choquet-Bruhat: “Existence and uniqueness for the Einstein–Maxwell–Liouville system,” pp. 23–39, 356 in Gravitation: Problems, prospects (dedicated to the memory of A. Z. Petrov). Edited by V. P. Šelest, A. E. Levašov, M. F. Širokov, and K. A. Piragas. Izdat. “Naukova Dumka” (Kiev), 1972. MR 334801 Zbl 0271.35063 incollection

[68] Y. Choquet-Bruhat: “Stabilité de solutions d’équations hyperboliques non linéaires: Application à l’espace-temps de Minkovski en relativité générale,” C. R. Acad. Sci. Paris Sér. A-B 274 (1972), pp. A843–A845. MR 291607 Zbl 0227.35068 article

Nous montrons que, pour une équation hyperbolique quasi-linéaire d’ordre \( m \), sur \( \mathbf{R}^l \), admettant une solution globale (régulière) \( u_0 \) correspondant à des données de Cauchy \( \varphi_0 \) sur \( \mathbf{R}^{l-1} \) (hypersurface spatiale de \( \mathbf{R}^l \)), il existe, quel que soit \( T > 0 \), un voisinage \( V_T(\varphi_0) \) de ces données de Cauchy tel que l’équation ait une solution (régulière) sur \[ \mathbf{R}^{l-1} \times [-T,T] \] pour toutes les données de Cauchy \( \varphi \in V_T (\varphi_0) \). Ce théoème, joint à un théorème précédemment démontré pour les équations des contraintes en Relativité générale, démontre l’existence sur toute bande \[ \mathbf{R}^3\times[-T,T] \] de solutions voisines de la solution de Minkovski des équations d’Einstein. La démonstration utilise la théorie de Leray des équations hyperboliques dans les espaces de Sobolev \( H_\mu \) et le théorème des fonctions implicites.

[69] Y. Choquet-Bruhat and J. Leray: “Sur le problème de Dirichlet, quasilinéaire, d’ordre 2,” C. R. Acad. Sci. Paris Sér. A-B 274 (1972), pp. A81–A85. MR 291623 Zbl 0227.35045 article

Let \( V \) be a differentiable manifold of \( \mathbb{R}^n \) with or without boundary. Suppose that \( V \) and \( \partial V \) (when not empty) are smooth enough, \( \bar{V} = V\cup\partial V \) is compact, and \( \beta_0 \), \( \gamma_0:\bar{V}\to \mathbb{R} \), \( \varphi: \partial V\to \mathbb{R} \) are given functions such that \begin{align*} & -\infty\leq\beta_0(x) \lt \gamma_0(x) \leq +\infty \quad\text{ for all } x\in \bar{V};\\ & \beta_0(x)\lt \varphi(x)\lt \gamma_0(x) \quad\text{for all }x\in \varphi V. \end{align*} The authors consider the following Dirichlet problem (\( \pi_0 \)) \begin{align*} & \operatorname{div} A(x,\partial u, u) = a(x,\partial u, u) \quad\text{in } V,\\ & u|_{\partial V} = \varphi, \, \beta_0(x)\lt u(x) \lt \gamma_0(x) \quad\text{in } \bar{V}, \end{align*} where \( A= (A^1,\dots,A^n) \) and \( a \) are vectorial respectively scalar real-valued functions, \[ \partial u = \Bigr(\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\Bigr). \] Suppose that \( \operatorname{div} A \) defines an elliptical differential operator for \( x\in V \), \( u(x)\in(\beta_0(x), \gamma_0(x)) \). The authors introduce the notion “over (sub)solution” as a real-valued function for which \( \operatorname{div} A(x,\partial u, u)\lt a(x,\partial u, u) \) in \( V \), \( u|_{\partial V}\lt \varphi \) (\( \operatorname{div} A > a \), \( u|_{\partial V} > \varphi \)), \( \beta_0(x)\lt u(x)\lt \gamma_0(x) \) in \( \bar{V} \). The second problem which is studied in this paper is the following: \( (\pi) \) Being given an oversolution \( \gamma \) and a subsolution \( \beta \), find a solution of \( \pi_0 \) for which \( \beta(x) \leq u(x) \leq\gamma(x) \). The authors give sufficient conditions for the existence and uniqueness of the solution to the problem \( \pi \) and conditions for the non-existence of a solution to the problem \( \pi_0 \). The conditions are given by some inequalities and the proofs use the maximum principle and Leray–Schauder’s fixed point theorem.

[70] Y. Choquet-Bruhat: “Problème des contraintes sur une variété compacte,” C. R. Acad. Sci. Paris Sér. A-B 274 (1972), pp. A682–A684. MR 293959 Zbl 0228.53044 article

It is known, if the initial 3-geometry is given up to a conformal factor \( \varphi \), the system of constraints in general relativity in empty space can be split in two subsystems, a linear one of the type: \( \nabla_i A^i_j=0 \), \( A^i_i = 0 \) an dear non linear second-order equation for the gauge factor \( \varphi \): \begin{equation} 8\Delta\phi - R\varphi +M\varphi^{-7}=0 \tag{1}, \end{equation} \( M=A^i_j A^j_i\geq 0 \). This paper proves the existence, and uniqueness of a \( \varphi > 0 \) on any closed manifold with \( R > 0 \), if \( M > 0 \). It also proves some theorems of existence, uniqueness and non existence, for the equation corresponding to (1) in the exterior case: \[ 8\Delta \varphi - R\varphi + M\varphi^{-7} = 2T^0_0\varphi^5 ,\] if \( R \), \( M \) and \( T_0^0 \) satisfy some inequalities. Large classes of initial datas compatible with the constraints on closed manifolds are thus obtained.

[71] Y. Choquet-Bruhat and S. Deser: “Stabilité initiale de l’espace temps de Minkowski,” C. R. Acad. Sci. Paris Sér. A-B 275 (1972), pp. A1019–A1021. MR 327250 article

[72] Y. Choquet-Bruhat: “Global solutions of the equations of constraints in general relativity on closed manifolds,” pp. 317–325 in Symposia Mathematica, Vol. XII (Convegno di Relativita, INDAM) (Rome, 1972). Academic Press (London), 1973. MR 353926 Zbl 0272.35013 incollection

In this paper the author studies the problem of existence and uniqueness of a global solution of the Lichnerowicz equation \[ 8\Delta\varphi - R\varphi + M\varphi^{-7} + Q\varphi^{5}=0 \] on a closed manifold and this is done using the Leray-Schauder theory of topological degree. This fundamental equation appears in the problem of constraints in general relativity. It is shown that on a closed manifold the Lichnerowicz equation has, in the case \( Q=0 \), one and only one solution \( \varphi > 0 \) for each \( M > 0 \) if the scalar curvature \( R \) of the manifold is also positive. There is no solution \( \varphi > 0 \) if \( M\geq 0 \), \( R\leq 0 \) and not both identically zero; there is no positive solution either f \( M=0 \) and \( R \), not identically zero, does not change sign. In the case of \( Q\not\equiv 0 \) (sources of energy are present) relations between \( R \), \( M \) and \( Q \) are found. If these relations are verified, there exists a positive solution of the Lichnerowicz equation. Finally, conditions on \( R \), \( M \), \( Q \) are given under which the equation has no global positive solution on a closed manifold.

[73] Y. Choquet-Bruhat: Distributions: Théorie et problèmes. Masson et Cie (Paris), 1973. MR 482020 Zbl 0249.46018 book

[74] Y. Choquet-Bruhat: “Un théorème d’instabilité pour certaines équations hyperboliques non linéaires,” C. R. Acad. Sci. Paris Sér. A-B 276 (1973), pp. A281–A284. In French. MR 315300 Zbl 0245.35058 article

[75] Y. Choquet-Bruhat and L. Lamoureux-Brousse: “Sur les équations de l’élasticité relativiste,” C. R. Acad. Sci. Paris Sér. A-B 276 (1973), pp. A1317–A1320. MR 317624 Zbl 0286.73019 article

[76] Y. Choquet-Bruhat and S. Deser: “On the stability of flat space,” Ann. Physics 81 (1973), pp. 165–178. MR 341358 Zbl 0265.35059 article

[77] D. Bancel and Y. Choquet-Bruhat: “Existence, uniqueness, and local stability for the Einstein–Maxwell–Boltzman system,” Comm. Math. Phys. 33 (1973), pp. 83–96. MR 356790 Zbl 0283.76080 article

[78] I. Šoke-Brjua: “Stability of solutions of nonlinear hyperbolic equations on a manifold,” Uspehi Mat. Nauk 29 : 2 (176) (1974), pp. 314–322. In Russian. Dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973). MR 407906 article

[79] Y. Choquet-Bruhat: “Couplage d’ondes gravitationnelles et électromagnétiques à haute fréquence,” pp. 85–100 in Ondes et radiations gravitationnelles (Paris, 1973). Colloq. Internat. CNRS 220. Centre Nat. Recherche Sci. (Paris), 1974. MR 496351 incollection

[80] Y. Choquet-Bruhat: “Global solutions of the constraints equations on open and closed manifolds,” Gen. Relativity Gravitation 5 : 1 (1974), pp. 49–60. MR 406309 Zbl 0332.35030 article

[81] Y. Choquet-Bruhat and C. Gilain: “Difféomorphismes harmoniques et unicité,” C. R. Acad. Sci. Paris Sér. A 279 (1974), pp. 827–830. MR 410791 Zbl 0292.53021 article

[82] Y. Choquet-Bruhat: “Sous-variétés maximales, ou à courbure constante, de variétés lorentziennes,” C. R. Acad. Sci. Paris Sér. A-B 280 (1975), pp. Aiv, A169–A171. In French. MR 371346 Zbl 0296.53043 article

[83] Y. Choquet-Bruhat: “Quelques propriétés des sous-variétés maximales d’une variété lorentzienne,” C. R. Acad. Sci. Paris Sér. A-B 281 : 14 (1975), pp. Aii, A577–A580. MR 643821 Zbl 0324.53046 article

[84] Y. Choquet-Bruhat: “The problem of constraints in general relativity: Solution of the Lichnerowicz equation,” pp. 225–235 in Differential geometry and relativity. Edited by M. Cahen and M. Flato. Mathematical Phys. and Appl. Math. 3. Reidel (Dordrecht), 1976. MR 462479 Zbl 0355.53013 incollection

[85] Y. Choquet-Bruhat and J. E. Marsden: “Sur la positivité de la masse,” C. R. Acad. Sci. Paris Sér. A-B 282 : 11 (1976), pp. Aii, A609–A612. MR 397785 Zbl 0326.35055 article

[86] Y. Choquet-Bruhat: “Maximal submanifolds and submanifolds with constant mean extrinsic curvature of a Lorentzian manifold,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 : 3 (1976), pp. 361–376. MR 423405 Zbl 0332.53035 article

[87] Y. Choquet-Bruhat and J. E. Marsden: “Solution of the local mass problem in general relativity,” Comm. Math. Phys. 51 : 3 (1976), pp. 283–296. MR 478215 Zbl 0364.58013 article

[88] Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick: Analysis, manifolds and physics. North-Holland (Amsterdam), 1977. MR 467779 Zbl 0385.58001 book

Cécile M. DeWitt-Morette	Related
Margaret Dillard-Bleick	Related

[89] Y. Choquet-Bruhat, A. Fisher, and J. Marsden: “Équations des contraintes sur une variété non compacte,” C. R. Acad. Sci. Paris Sér. A-B 284 : 16 (1977), pp. A975–A978. MR 433498 Zbl 0364.58010 article

The basic data one needs for the “initial-value problem” in general relativity are: a three-dimensional manifold \( S \) with a Riemannian metric \( g \) and a symmetric rank two tensor field \( k \) which satisfy certain constraint equations. Theorems from p.d.e. state that \( g \) and \( k \) will generate a four-dimensional space-time which satisfies Einstein field equations and contains \( S \) as a spacelike hypersurface with second fundamental form \( k \). The constraint equations are said to be linearization stable at a given solution if every nonzero solution of the linearized constraint equations is tangent to a curve of solutions of the nonlinear constraint equations. This article gives a new simple method for proving linearization stability of the constraint equations in the case where \( S \) is not necessarily compact, but trace \( (k) \) is constant.

[90] Y. Choquet-Bruhat: “Properties of maximal submanifolds in space-times with compact or open space-like sections,” Rep. Mathematical Phys. 12 : 1 (1977), pp. 9–17. MR 461389 Zbl 0363.53029 article

[91] Y. Choquet-Bruhat and A. H. Taub: “High-frequency, self-gravitating, charged scalar fields,” Gen. Relativity Gravitation 8 : 8 (1977), pp. 561–571. MR 471867 article

In this paper we shall apply the Wentzel–Kramers–Brillouin (WKB) method of expansion (the two-timing method) discussed in [Choquet-Bruhat 1969] to the equations governing the behavior of a charged scalar field \( \psi \), the vector potential \( A_{\mu} \) of an electromagnetic field that satisfies the Maxwell equations with a current vector derived from \( \psi \), and the metric tensor \( g_{\mu\nu} \) that satisfies the Einstein field equations with a stress energy tensor \( T_{\mu\nu} \) due to the electromagnetic and scalar fields. This method, in contrast to the averaged Lagrangian method given in [MacCallum and Taub 1973], allows one to determine propagation equations for various functions as well as equations involving values of various variables averaged over a period of the perturbations that are introduced.

[92] Y. Choquet-Bruhat: “Compactification de variétés asymptotiquement euclidiennes: Applications,” C. R. Acad. Sci. Paris Sér. A-B 285 : 16 (1977), pp. A1061–A1064. MR 517104 Zbl 0383.35016 article

[93] Y. Choquet-Bruhat: “Maximal submanifolds and the positivity of the gravitational mass,” pp. 9–15 in Proceedings of the International Symposium on Relativity and Unified Field Theory (Calcutta, 1975–1976). S. N. Bose Inst. Phys. Sci. (Calcutta), 1978. MR 524255 inproceedings

[94] A. Chaljub-Simon and Y. Choquet-Bruhat: “Solutions asymptotiquement euclidiennes de l’équation de Lichnerowicz,” C. R. Acad. Sci. Paris Sér. A-B 286 : 20 (1978), pp. A917–A920. This paper is related to a later paper published in English by the same authors: “Global solutions of the Lichnerowicz equation in {G}eneral {R}elativity on an asymptotically {E}uclidean complete manifold,” General Relativity and Gravity 12 (1980). MR 498871 Zbl 0388.35006 article

[95] G. Choquet and Y. Choquet-Bruhat: “Sur un problème lié à la stabilité des données initiales en relativité générale,” C. R. Acad. Sci. Paris Sér. A-B 287 : 15 (1978), pp. A1047–A1049. MR 519238 article

[96] Y. Choquet-Bruhat, D. Christodoulou, and M. Francaviglia: “Cauchy data on a manifold,” Ann. Inst. H. Poincaré Sect. A (N.S.) 29 : 3 (1978), pp. 241–255. MR 519694 Zbl 0412.35018 article

Mauro Francaviglia	Related
Demetrios Christodoulou	Related

[97] Y. Choquet-Bruhat, D. Christodoulou, and M. Francaviglia: “Problème de Cauchy sur une variété,” C. R. Acad. Sci. Paris Sér. A-B 287 : 5 (1978), pp. A373–A375. MR 524042 Zbl 0412.35017 article

Mauro Francaviglia	Related
Demetrios Christodoulou	Related

[98] A. Chaljub-Simon and Y. Choquet-Bruhat: “Problèmes elliptiques du second ordre sur une variété euclidienne à l’infini,” Ann. Fac. Sci. Toulouse Math. (5) 1 : 1 (1979), pp. 9–25. MR 533596 Zbl 0411.35044 article

[99] A. Chaljub-Simon and Y. Choquet-Bruhat: “Solution de l’équation de Lichnerowicz sur une variété asymptotiquement euclidienne,” C. R. Acad. Sci. Paris Sér. A-B 288 : 16 (1979), pp. A779–A784. MR 535811 Zbl 0407.58038 article

[100] Y. Choquet-Bruhat, D. Christodoulou, and M. Francaviglia: “On the wave equation in curved spacetime,” Ann. Inst. H. Poincaré Sect. A (N.S.) 31 : 4 (1979), pp. 399–414. MR 574143 Zbl 0454.58016 article

Mauro Francaviglia	Related
Demetrios Christodoulou	Related

[101] A. Chaljub-Simon and Y. Choquet-Bruhat: “Global solutions of the Lichnerowicz equation in general relativity on an asymptotically Euclidean complete manifold,” Gen. Relativity Gravitation 12 : 2 (1980), pp. 175–185. MR 575238 article

We prove some existence and uniqueness theorems for the solution of the Liehnerowicz equation: \[ 8\Delta_g \phi - R\phi + M\phi^{-7} + \mathcal{Q}\phi^{-3} + \tau \phi^{5} = 0 ,\] on an asymptotically Euclidian manifold. This equation governs the conformal factor of a metric solution of the constraints in General Relativity. In the first part we prove existence and uniqueness under the simple assumption \( R\geq 0 \), \( M\geq 0 \), \( \mathcal{Q} \geq 0 \), \( \tau \geq 0 \), which insures the monotony of the non-differentiated part. In the second part we obtain an existence theorem under more general hypothesis on the co-efficients, by use of the Leray–Schauder degree theory. The results of this paper have been announced in [Chaljub-Simon and Choquet-Bruhat 1978].

[102] Y. Choquet-Bruhat and D. Christodoulou: “Systèmes elliptiques sur une variété euclidienne à l’infini,” C. R. Acad. Sci. Paris Sér. A-B 290 : 17 (1980), pp. A781–A785. MR 580565 Zbl 0453.58021 article

[103] Y. Choquet-Bruhat and J. W. York, Jr.: “The Cauchy problem,” pp. 99–172 in General relativity and gravitation, vol. 1. Edited by A. Held. Plenum (New York–London), 1980. MR 583716 Zbl Zbl 0537.00011 incollection

[104] Y. Choquet-Bruhat and D. Christodoulou: “Elliptic systems in \( H_{s,\delta } \) spaces on manifolds which are Euclidean at infinity,” Acta Math. 146 : 1–2 (1981), pp. 129–150. MR 594629 article

In this paper we study elliptic differential systems of order \( m \) on non-compact manifolds which are euclidean at infinity, in weighted Sobolev spaces \( H_{s,\delta} \). Such a study has been done in weighted Hölder spaces \( C^{1,\alpha}_{\beta} \), for equations of order 2 in [Chaljub-Simon and Choquet-Bruhat 1978]. On the other hand, M. Cantor has proved [Cantor 1979] closed range and isomorphism theorems for elliptic operators of order \( m \) in \( \mathbf{R}^n \), in weighted Sobolev spaces \( W^{p}_{s,\delta} \), where \( p > n/(n-m) \). His paper is based on a work by L. Nirenberg and H. Walker [Nirenberg and Walker 1973] on the null spaces of such operators with continuous coefficients. In the present article we show that this restriction on \( p \) is unnecessary. Although we shall treat explicitly only the case \( p = 2 \) which is of special interest since \( W^2_{s,\delta}=H_{s,\delta} \) is a Hilbert space, the results extend trivially to any \( p > 1 \). The hypotheses on the coefficients which we make, permit the study of nonlinear systems in the same framework.

[105] Y. Choquet-Bruhat: “Solutions globales des équations de Maxwell–Dirac–Klein–Gordon (masses nulles),” C. R. Acad. Sci. Paris Sér. I Math. 292 : 2 (1981), pp. 153–158. MR 610307 Zbl 0498.35053 article

A system of field equations (the Maxwell, Dirac and Klein–Gordon ones) is considered, the Dirac field \( \psi \) and the complex scalar field \( U \) contributing to the current density on the right hand side of the Maxwell equations (a further extension of this analysis is proposed with the interaction terms \( U^{\ast}U \psi \) and \( U \bar{\psi}{\psi} \) also in the Dirac and the Klein–Gordon equations respectively). Starting with the local existence theorem for the Cauchy problem concerning this system, the author uses the conformal invariance of the system (when \( psi \) and \( U \) are massless fields, the Klein–Gordon equation being written with the term \( RU/6 \)) which enables her to transfer the complete Minkowski space-time to an open bounded region of the Einstein universe (Penrose’s lemma), so that the global Cauchy problem for the system under consideration is reduced to a local one, though in a curved manifold. This helps the author also to overcome the difficulty caused by the nonpositive-definiteness of the Dirac field energy density which prevented direct a priori majorizations. As a result, the global existence of solutions of the Cauchy problem in the Minkowski space-time is proved for the Cauchy data with zero total charge.

[106] Y. Choquet-Bruhat and D. Christodoulou: “Existence de solutions globales des équations classiques des théories de jauge,” C. R. Acad. Sci. Paris Sér. I Math. 293 : 3 (1981), pp. 195–199. MR 635980 Zbl 0478.58027 article

[107] Y. Choquet-Bruhat and D. Christodoulou: “Existence of global solutions of the Yang–Mills, Higgs and spinor field equations in \( 3+1 \) dimensions,” Ann. Sci. École Norm. Sup. (4) 14 : 4 (1981), pp. 481–506. MR 654209 Zbl 0499.35076 article

[108] Y. Choquet-Bruhat and I. E. Segal: “Solution globale des équations de Yang–Mills sur l’univers d’Einstein,” C. R. Acad. Sci. Paris Sér. I Math. 294 : 6 (1982), pp. 225–230. MR 654042 Zbl 0501.58023 article

[109] Y. Choquet-Bruhat and T. Ruggeri: “Hyperbolicité du système \( 3+1 \) des équations d’Einstein,” C. R. Acad. Sci. Paris Sér. I Math. 294 : 12 (1982), pp. 425–429. An English version of the French original appeared in Comm. Math. Phys. 89: 2 (1983). MR 659737 article

[110] Y. Choquet-Bruhat: “Solution globale des équations de Maxwell–Dirac–Klein–Gordon,” Rend. Circ. Mat. Palermo (2) 31 : 2 (1982), pp. 267–288. MR 670401 Zbl 0497.35078 article

[111] Y. Choquet-Bruhat: “Recent results on the Cauchy problem for gravitation and Yang–Mills fields,” pp. 167–178 in Proceedings of the Second Marcel Grossmann Meeting on General Relativity: Part A, B (Trieste, 1979). Edited by R. Ruffini. North-Holland (Amsterdam; New York), 1982. MR 678940 inproceedings

[112] Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick: Analysis, manifolds and physics, 2nd edition. North-Holland (Amsterdam–New York), 1982. The first edition included the authorship of Margaret Dillard-Bleick and was published in 1977 under the same title. The second edition was revised and updated without Dillard-Bleick’s participation. MR 685274 Zbl Zbl 0492.58001 book

Cécile M. DeWitt-Morette	Related
Margaret Dillard-Bleick	Related

[113] Y. Choquet-Bruhat: “Global solutions of Yang–Mills field equations,” Rend. Sem. Mat. Fis. Milano 52 (1982), pp. 247–259. MR 802944 Zbl 0571.35087 article

[114] Y. Choquet-Bruhat: Propos sur la cosmologie: Séance publique annuelle des cinq académies. Eight-page booklet, Institut de France (Paris), 1982. techreport

[115] Y. Choquet-Bruhat and A. Greco: “High frequency asymptotic solutions of Yang–Mills and associated fields,” J. Math. Phys. 24 : 2 (1983), pp. 377–379. MR 692317 Zbl 0557.70020 article

[116] Y. Choquet-Bruhat and T. Ruggeri: “Hyperbolicity of the \( 3+1 \) system of Einstein equations,” Comm. Math. Phys. 89 : 2 (1983), pp. 269–275. This appears to be an English version of the French article of the same title published a year earlier in C. R. Acad. Sci. Paris Sér. I Math. 294:12 (1982). MR 709467 Zbl 0521.53034 article

[117] F. Cagnac and Y. Choquet-Bruhat: “Solution globale d’une équation non linéaire sur une variété hyperbolique,” C. R. Acad. Sci. Paris Sér. I Math. 296 : 20 (1983), pp. 845–849. A paper by the same title was subsequently published in J. Math. Pures Appl. 63:4 377–390 (1984). MR 712755 Zbl 0529.58032 article

The authors prove the existence of a global, unique solution for the equation \[ \square_g u + au + bu^3=0 \] on a globally hyperbolic 4-dimensional manifold \( (V,g) \) with given Cauchy data on a space-like section \( S \).The case when \( a \) and \( b \) are given positive functions on \( V \) is considered. First, the definitions and the hypothesis used in the work are given. The space \( (V,g) \) is organized as a global hyperbolic manifold and some regularity hypotheses are introduced. Secondly, a local theorem regarding the existence of a solution for the linear equation \[ \square_g u + au = f \] is given. The existence of the solution is proved using the energy majoring inequalities. Then it is shown that the equation \[ \square_g u + au + bu^3 = 0 \] admits a unique solution with the Cauchy data and the hypotheses imposed. Finally, the existence of a global solution for the equation \[ \square_g u + au + bu^3 = 0 \] is proved. The existence theorem is proved using an a priori condition of majoring and also the lemmas and theorems proven previously in the work.

[118] Y. Choquet-Bruhat: “Supergravité classique: Le problème de Cauchy,” C. R. Acad. Sci. Paris Sér. I Math. 297 : 1 (1983), pp. 71–76. MR 719951 Zbl 0535.53030 article

[119] Y. Choquet-Bruhat, S. M. Paneitz, and I. E. Segal: “The Yang–Mills equations on the universal cosmos,” J. Funct. Anal. 53 : 2 (1983), pp. 112–150. MR 722506 Zbl 0535.58022 article

Global existence and regularity of solutions for the Yang–Mills equations on the universal cosmos \( \tilde{\mathbf{M}} \), which has the form \( R^1 \times S^3 \) for each of an 8-parameter continuum of factorizations of \( \tilde{\mathbf{M}} \) as \( \textit{time}\times\textit{space} \) are treated by general methods. The Cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the \( \textit{field}\oplus\textit{potential} \) in \[ L_{2,r}(S^3)\oplus L_{2,r+1}(S^3) ,\] where \( r \) is an integer \( > \) 1 and \( L_{2,r} \) denotes the class of sections whose first \( r \) derivatives are square-integrable; if \( r= 1 \), the problem is soluble locally in time. When \( r \) is 3 or more the solution is identifiable with a classical one; if infinite, the solution is in \( C^{\infty}(\tilde{\mathbf{M}}) \). These results extend earlier work and approaches. Solutions of the equations on Minkowski space-time \( M \), extend canonically (modulo gauge transformations) to solutions on \( \tilde{\mathbf{M}} \) provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on \( \mathbf{M}_0 \), follow, and in turn imply various decay estimates. Thus the energy in regions uniformly bounded in direction away from the light cone is \( O(|x_0|^{-5}) \), where \( x_0 \) is the Minkowski time coordinate; analysis solely in \( \mathbf{M}_0 \), earlier yielded the estimate \( O(|x_o|^{-2}) \) applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang–Mills equuions in \( \mathbf{M}_0 \), is finite, in fact absolutely convergent.

Stephen Mark Paneitz	Related
Irving Ezra Segal	Related

[120] Y. Choquet-Bruhat: “The Cauchy problem in classical supergravity,” Lett. Math. Phys. 7 : 6 (1983), pp. 459–467. MR 728641 Zbl 0529.58039 article

[121] Y. Choquet-Bruhat: “Classical supergravity with Weyl spinors,” Proc. Einstein Found. Internat. 1 : 1 (1983), pp. 43–53. MR 731352 article

[122] Y. Choquet-Bruhat: “Global solutions of hyperbolic equations of gauge theories,” pp. 108–134 in Relativity, cosmology, topological mass and supergravity (Caracas, 1982). Edited by C. Aragone. World Sci. Publ. (Singapore), 1983. MR 743271 incollection

[123] Y. Choquet-Bruhat and D. Christodoulou: “Cauchy problem at past infinity for nonlinear equations in curved spacetime,” pp. 73–91 in Studies in applied mathematics. Edited by V. Guillemin. Adv. Math. Suppl. Stud. 8. Academic Press (New York), 1983. MR 759906 Zbl 0517.53028 incollection

The authors consider the Cauchy problem at past infinity for both linear and non-linear wave equations on a globally hyperbolic curved spacetime whose metric tends to be stationary at past infinity. The first five sections consider the problem of a linear wave equation with a source term, and the final two sections consider weakly coupled systems of non-linear wave equations. The last section applies these results to a general Yang–Mills system in the Lorentz gauge with a source on a curved spacetime. Contents include sections on definitions and hypotheses, function spaces, energy estimates, weighted function spaces, existence theorems for linear equations, quasi-linear wave equations, existence theorems.

[124] Y. Choquet-Bruhat: “Causality of classical supergravity,” pp. 61–84 in Asymptotic behavior of mass and spacetime geometry (Corvallis, Ore., 1983). Edited by F. J. Flaherty. Lecture Notes in Phys. 202. Springer, 1984. MR 771854 Zbl 0546.53047 incollection

[125] Y. Choquet-Bruhat: “Supergravities,” pp. 88–106 in Gravitation, geometry and relativistic physics (Aussois, 1984). Lecture Notes in Phys. 212. Springer, 1984. MR 780219 Zbl 0557.53050 incollection

A survey of the various supergravity theories is given. The mathematical and physical motivations of supergravity (simple or extended) are discussed and then a general formulation of the supergravity in a space- time V of \( d = 4 + n \) dimensions is presented. One considers on V a connection \( \omega \), metric but with torsion S; in addition to the metric g appears, as auxiliary unknown, the orthonormal moving frame e.

The contorsion tensor and the curvature are defined on V and then a Lagrangian of supergravity (i.e., for a spinor valued 1-form \( \psi = (\psi_M ) \), \( M = 0, 1 \), …, \( d-1 \), on V) is written. The simple supergravity \( (d = 4) \) and the extended supergravities \( (d = 4 + N ) \) are considered in detail. In particular, the Kaluza–Klein supergravity is analysed and some remarks about the \( d = 11 \) supergravity are given.

[126] F. Cagnac and Y. Choquet-Bruhat: “Solution globale d’une équation non linéaire sur une variété hyperbolique,” J. Math. Pures Appl. (9) 63 : 4 (1984), pp. 377–390. MR 789558 Zbl 0554.58049 article

[127] Y. Choquet-Bruhat: “Positive-energy theorems,” pp. 739–785 in Relativity, groups and topology, II (Les Houches, 1983). Edited by B. S. DeWitt and R. Stora. North-Holland (Amsterdam), 1984. MR 830248 Zbl 0593.53055 incollection

General relativity deals with space-time models consisting of a metric of Lorentz signature over a four-manifold satisfying the Einstein equations. These require that the Einstein tensor equal the matter tensor, which in turn must satisfy certain conditions of physical reasonableness. Under certain circumstances, it is possible to define a total energy including gravitational contributions, for such a space-time. This problem requires either asymptotic flatness, which must be defined, or a comparison manifold. Further, the definition must be physically reasonable, and result in a constant total energy for an isolated system. In physics, energy is intimately tied up with a definition of time. In general relativity, time, as a coordinate on a manifold, may be defined only locally. These and other considerations have made the definition of total energy for a gravitational system a non-trivial problem in mathematical physics.

This paper begins with a review of these questions and the standard approaches to them and proceeds to the main issue which is the so-called “positive energy” theorems. These assert that such quantities must be positive definite for isolated systems. Only recently theorems have been proven establishing such results rigorously and under various conditions. This paper presents a very thorough and easily followed discussion of this problem and the known theorems.

[128] D. Bao, Y. Choquet-Bruhat, J. Isenberg, and P. B. Yasskin: “The well-posedness of (\( N=1 \)) classical supergravity,” J. Math. Phys. 26 : 2 (1985), pp. 329–333. MR 776502 Zbl 0563.53060 article

In this paper the authors investigate whether classical (\( N=1 \)) supergravity has a well-posed locally causal Cauchy problem. One defines well-posedness to mean that any choice of initial data (from an appropriate function space) which satisfies the supergravity constraint equations and a set of gauge conditions can be continuously developed into a space-time solution of the supergravity field equations around the initial surface. Local causally means that the domains of dependence of the evolution equations coincide with those determined by the light cones. They show that when the fields of classical supergravity are treated as formal objects, the field equations are (under certain gauge conditions) equivalent to a coupled system of quasilinear nondiagonal second-order partial differential equations which is formally nonstrictly hyperbolic (in the sense of Leray-Ohy). Hence, if the fields were numerical valued, there would be an applicable existence theorem leading to well-posedness.

James Allen Isenberg	Related
David Dai-Wai Bao	Related
Philip Barry Yasskin	Related

[129] Y. Choquet-Bruhat: “The Cauchy problem in extended supergravity, \( N=1 \), \( d=11 \),” Comm. Math. Phys. 97 : 4 (1985), pp. 541–552. MR 787117 Zbl 0593.53062 article

[130] I. Shoke-Bryua: “Mathematical problems in general relativity,” Uspekhi Mat. Nauk 40 : 6(246) (1985), pp. 3–39. In Russian; translated from the English by A. P. Veselov. MR 815488 article

[131] Y. Choquet-Bruhat: “Supermanifolds and supergravities,” pp. 25–42 in Geometrodynamics proceedings, 1985 (Cosenza, 1985). Edited by A. Pràstavo. World Sci. Publ. (Singapore), 1985. MR 825786 Zbl 0652.58038 incollection

[132] Y. Choquet-Bruhat: “Einstein–Cartan theory with spin \( {3/2} \) source and classical supergravity,” pp. 27–56 in On relativity theory: Proceedings of the Sir Arthur Eddington Centenary Symposium, II (Nagpur, 1984). Edited by Y. Choquet-Bruhat and T. M. Karade. World Sci. Publ. (Singapore), 1985. MR 827562 inproceedings

[133] Y. Choquet-Bruhat: “Causalité des théories de supergravité,” pp. 79–93 in Élie Cartan et les mathématiques d’aujourd’hui [The mathematical heritage of Élie Cartan] (Lyon, 1984). Astérisque (numéro hors série). 1985. MR 837195 Zbl 0604.53047 incollection

[134] Y. Choquet-Bruhat: “Kaluza–Klein theories,” pp. 155–169 in Proceedings of the conference commemorating the 1st Centennial of the Circolo Matematico di Palermo (Palermo, 1984), published as Rend. Circ. Mat. Palermo 8 : suppl. (1985). MR 881396 Zbl 0615.53073 incollection

[135] Y. Choquet-Bruhat: “Supergravities and Kaluza–Klein theories,” pp. 31–48 in Topological properties and global structure of space-time (Erice, 1985). Edited by P. G. Bergmann and V. D. Sabbata. NATO Adv. Sci. Inst. Ser. B: Phys. 138. Plenum (New York), 1986. MR 1102937 Zbl 0687.53072 incollection

[136] Y. Choquet-Bruhat: “Applications harmoniques hyperboliques,” C. R. Acad. Sci. Paris Sér. I Math. 303 : 4 (1986), pp. 109–113. MR 853598 Zbl 0607.58011 article

[137] Y. Choquet-Bruhat and N. Noutchegueme: “Système hyperbolique pour les équations d’Einstein avec sources,” C. R. Acad. Sci. Paris Sér. I Math. 303 : 6 (1986), pp. 259–263. MR 860831 article

[138] Y. Choquet-Bruhat: “The Cauchy problem in higher-dimensional gravity and Kaluza–Klein theories,” pp. 1223–1231 in Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity, Part A, B (Rome, 1985). Edited by R. Ruffini. North-Holland (Amsterdam), 1986. MR 879816 inproceedings

[139] Y. Choquet-Bruhat: “Global existence theorems for hyperbolic harmonic maps,” Ann. Inst. H. Poincaré Phys. Théor. 46 : 1 (1987), pp. 97–111. MR 877997 Zbl 0608.58018 article

Consider a hyperbolic manifold \( (M,g) \), \( M = S \times \mathbb{R} \), of dimension \( n + 1 \) and a Riemannian manifold \( (N,h) \) embedded in Euclidean space. A smooth function \( f : M \to N \) is a (hyperbolic) harmonic map if and only if \[ \operatorname{tr}_g \nabla^2 f = 0 .\] Imposing some (geometric) regularity conditions on \( M \) and \( N \) the author shows local existence of a solution \( u \) defined on \( S \times (-\ell, \ell) \) for some \( \ell > 0 \) of the Cauchy problem with initial data \[ u|_{S\times\{0\}} = \phi, \qquad \partial_0 u|_{S\times \{0\}} = \psi \] in suitable Sobolev spaces. The proof is based on standard existence results for second order hyperbolic systems. In the special case \( n = 1 \) the local existence theorem can be sharpened (using energy estimates) to give global existence of a solution.

[140] F. Cagnac, Y. Choquet-Bruhat, and N. Noutchegueme: “Solutions of the Einstein equations with data at past infinity,” pp. 35–54 in Seventh Italian conference on general relativity and gravitational physics (Rapallo, 1986). Edited by U. Bruzzo, R. Cianci, and E. Massa. World Sci. Publ. (Singapore), 1987. MR 892547 incollection

Francis Cagnac	Related
Norbert Noutchegueme	Related

[141] Y. Choquet-Bruhat: “Mathematics for classical supergravities,” pp. 73–90 in Differential geometric methods in mathematical physics (Salamanca, 1985). Edited by P. L. García and A. Pérez-Rendón. Lecture Notes in Math. 1251. Springer, 1987. MR 897114 Zbl 0619.53051 incollection

[142] Y. Choquet-Bruhat: “Mathematical problems,” pp. 331–335 in General relativity and gravitation: Proceedings of the 11th International Conference on General Relativity and Gravitation (Stockholm, 1986). Edited by M. A. H. MacCallum. Cambridge Univ. Press, 1987. MR 915367 incollection

[143] Y. Choquet-Bruhat: “Non existence de solutions globales de certaines équations d’onde non linéaires sur une variété,” C. R. Acad. Sci. Paris Sér. I Math. 305 : 19 (1987), pp. 817–821. In French. MR 923206 article

[144] Y. Choquet-Bruhat: “Spin \( {1/2} \) fields in arbitrary dimensions and the Einstein–Cartan theory,” pp. 83–106 in Gravitation and geometry. Edited by W. Rindler and A. Trautman. Monogr. Textbooks Phys. Sci. 4. Bibliopolis (Naples), 1987. MR 938534 incollection

[145] Y. Choquet-Bruhat and M. Novello: “Système conforme régulier pour les équations d’Einstein,” C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 305 : 3 (1987), pp. 155–160. MR 979834 Zbl 0633.53070 article

[146] Y. Choquet-Bruhat: “Global existence for nonlinear \( \sigma \)-models,” pp. 65–86 in Nonlinear hyperbolic equations in applied sciences, published as Rend. Sem. Mat. Univ. Politec. Torino special issue (1988). MR 1007367 Zbl 0678.58015 incollection

[147] Y. Choquet-Bruhat: “Cas d’existence globale de solutions de l’équation \( \square u=A|u|^p \),” C. R. Acad. Sci. Paris Sér. I Math. 306 : 8 (1988), pp. 359–364. MR 934619 Zbl 0673.35076 article

[148] Y. Choquet-Bruhat: “Problème de Cauchy pour les modèles gravitationnels avec termes de Gauss–Bonnet,” C. R. Acad. Sci. Paris Sér. I Math. 306 : 10 (1988), pp. 445–450. MR 937983 Zbl 0646.53074 article

[149] Y. Choquet-Bruhat: “Global problems in general relativity,” pp. 3–27 in SILARG, VI (Rio de Janeiro, 1987). Edited by M. Novello. World Sci. Publ. (Singapore), 1988. MR 948977 Zbl 0658.53075 incollection

This paper concentrates its attention on a review of two topics in the global structure of general relativity; the Cauchy problem and the strong and weak cosmic censorship hypotheses. After an initial discussion comparing the study of the Cauchy problem for field theories on a Minkowski background with a similar but more difficult study for Einstein’s equations of general relativity, a brief mention is made of the as yet unsolved problem regarding the stability of Minkowski space of dimension 4. The author then reviews the Klainerman–Christodoulou theorems which solve this problem for \( n\geq 5 \). The first part of the paper is concluded with a detailed discussion of more general types of stability problems concerning Einstein’s equations with particular emphasis of the role of harmonic coordinates and the structure of Einstein’s equations in these coordinates.

The second part of the paper starts with a discussion of the weak and strong cosmic censorship hypotheses of Penrose and their relation to global hyperbolicity. A survey is then given of examples of violation of the cosmic censorship hypothesis (but the author admits that the examples given contain sources which are not physically reasonable). They include certain types of collapsing dust balls and other forms of fluid balls and a detailed analysis is given of a Tolman-Bondi type solution in this context. The paper ends with a detailed discussion of an example of Christodoulou which satisfies the Einstein-scalar field equations. This example should, according to the author, provide a good indication of the results one might expect for more realistic sources in general relativity — and the results are, apparently, in good agreement generally with cosmic censorship conjectures.

[150] Y. Choquet-Bruhat: “The Cauchy problem for stringy gravity,” J. Math. Phys. 29 : 8 (1988), pp. 1891–1895. MR 955194 Zbl 0658.53078 article

[151] Y. Choquet-Bruhat: “Les travaux de André Lichnerowicz en relativité générale,” pp. 1–10 in Physique quantique et géométrie: Colloq. Géom. Phys. (Paris, 1986). Edited by D. Bernard and Y. Choquet-Bruhat. Travaux en Cours 32. Hermann (Paris), 1988. MR 955859 Zbl 0648.01004 incollection

In recognition of Lichnerowicz’s contribution to the General Theory of Relativity one of his pupils gives a survey. From the beginning onwards Lichnerowicz, provided with a geometric mind, was interested in manifolds with a hyperbolic metric in correlation with the Einstein equations. Lichnerowicz was able to prove that gravitational solitons do not exist. Proceeding from this result Lichnerowicz worked on the integration of the Einstein equations. He was one of the first to find out that gravitational radiation appears in form of a curvature tensor. The curved space-time also was within Lichnerowicz’s main interests. Another subject of his work were the boson-fields which led him to investigate on the theory of spinors in pseudo-Riemannian manifolds. During his whole carrier Lichnerowicz dealt with the relativistic mechanics of continua. Astrophysicists recognized well Lichnerowicz’s article on shock-waves in magnetohydrodynamics, which had appeared in 1975. Four appendices enriched by references allow to verify the mathematical details.

[152] Y. Choquet-Bruhat: “Ondes à haute fréquence pour la gravitation avec termes de Gauss–Bonnet,” C. R. Acad. Sci. Paris Sér. I Math. 307 : 12 (1988), pp. 693–696. MR 967815 Zbl 0648.53046 article

[153] Y. Choquet-Bruhat: “Global existence theorems by the conformal method,” pp. 16–37 in Recent developments in hyperbolic equations (Pisa, 1987). Edited by L. Cattabriga, F. Colombini, M. K. V. Murthy, and S. Spagnolo. Pitman Res. Notes Math. Ser. 183. Longman Sci. Tech. (Harlow), 1988. MR 984357 Zbl 0734.35142 incollection

This paper is a brief review of previous results. The author recalles the conformal mapping from the Minkowski space time \( M_{n+1} \) onto a bounded subset of \[ \Sigma_{n+1}= S^n \times \mathbb{R} .\] Then she shows how it has been used by [Christodoulou 1986] to prove the global existence of solutions of the Cauchy problem for general second order equations of the form \[ \square u = f (u, \partial u, \partial^2 u) \] where \( \square \) is the usual wave operator on \( M_{n+1} \). Finally, she indicates how these theorems can be extended to some asymptotically flat manifolds and notes that global existence theorems, obtained by the conformal method, include decay estimates of the solutions [Choquet-Bruhat, Paneitz and Segal 1983] and [Choquet-Bruhat and Christodoulou 1981].

[154] Y. Choquet-Bruhat and C. DeWitt-Morette: Analysis, manifolds and physics, Part II: 92 applications. North-Holland Publishing Co. (Amsterdam), 1989. A companion volume to the revised 1982 edition. MR 1016603 Zbl 0682.58002 book

[155] Y. Choquet-Bruhat: “Gravitation with Gauss–Bonnet terms,” pp. 53–72 in Conference on Mathematical Relativity (Canberra, 1988). Edited by R. Bartnik. Proc. Centre Math. Anal. Austral. Nat. Univ. 19. Austral. Nat. Univ. (Canberra), 1989. MR 1020790 Zbl 0684.53061 incollection

In the theory of gravity with Gauss–Bonnet terms, the Lagrangian consists of the scalar curvature plus a particular polynomial combination of the Riemann tensor such that the resulting field equations are of the second order. In the present paper the author studies the initial value problem of that theory. She shows that, as in Einstein’s gravity, the equations split into constraint and evolution equations. For analytical data it is shown, as a simple application of the Cauchy–Kowalewski theorem, that the evolution equations preserve the constraints. It is found that the characteristics of the equations are not necessarily tangent to the light cones. In addition, high frequency waves of the Gauss–Bonnet theory are obtained and analyzed.

[156] Y. Choquet-Bruhat: “Global existence for solutions of \( \square u=A|u|^p \),” J. Differential Equations 82 : 1 (1989), pp. 98–108. MR 1023303 Zbl 0694.58040 article

[157] Y. Choquet-Bruhat: Graded bundles and supermanifolds. Monographs and Textbooks in Physical Science. Lecture Notes 12. Bibliopolis (Naples), 1989. MR 1026098 Zbl 0707.58006 book

[158] Y. Choquet-Bruhat: “Global solutions of Yang–Mills equations on anti-de Sitter spacetime,” Classical Quantum Gravity 6 : 12 (1989), pp. 1781–1789. MR 1028380 Zbl 0698.53040 article

[159] Y. Choquet-Bruhat: “High frequency waves for stringy gravity,” pp. 349–361 in Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity, Part A, B (Perth, 1988). Edited by D. G. Blair, M. J. Buckingham, and R. Ruffini. World Sci. Publ. (Teaneck, NJ), 1989. MR 1056882 inproceedings

[160] Y. Choquet-Bruhat and C. H. Gu: “Existence globale d’applications harmoniques sur l’espace-temps de Minkowski \( M_3 \),” C. R. Acad. Sci. Paris Sér. I Math. 308 : 6 (1989), pp. 167–170. In French. MR 984915 Zbl 0661.53043 article

[161] Y. Choquet-Bruhat: “Solutions globales d’équations d’ondes sur l’espace-temps anti de Sitter,” C. R. Acad. Sci. Paris Sér. I Math. 308 : 11 (1989), pp. 323–327. MR 989897 Zbl 0662.53068 article

[162] Relativistic fluid dynamics: Lectures given at the first 1987 session of the Centro Internazionale Matematico Estivo (C.I.M.E.) (Noto, Italy, 1987). Edited by A. M. Anile and Y. Choquet-Bruhat. Lect. Notes Math. 1385. Springer (Cham), 1989. book

[163] Y. Choquet-Bruhat and R. McLenaghan: “Noyau de diffusion en coordonnées de Bondi de l’opérateur des ondes sur une variété lorentzienne,” C. R. Acad. Sci. Paris Sér. I Math. 311 : 8 (1990), pp. 483–486. MR 1076477 Zbl 0717.35044 article

[164] Y. Choquet-Bruhat and N. Noutchegueme: “Solution des systèmes de Yang–Mills–Vlasov: Théorèmes d’existence locale [Solutions of the Yang–Mills–Vlasov systems: Local existence theorems],” C. R. Acad. Sci. Paris Sér. I Math. 311 : 10 (1990), pp. 611–616. MR 1081417 Zbl 0717.35072 article

[165] Y. Choquet-Bruhat and N. Noutchegueme: “Solutions globales du système de Yang–Mills–Vlasov (masse nulle)” [Global solutions of the Yang–Mills–Vlasov system with zero rest mass], C. R. Acad. Sci. Paris Sér. I Math. 311 : 12 (1990), pp. 785–788. MR 1082633 Zbl 0715.53046 article

In a previous Note [Choquet-Bruhat 1990] we have considered the Yang–Mills–Vlasov system on a pseudo-Riemannian manifold \( (V,g) \) of hyperbolic signature \begin{align} & \hat{\nabla}_a F^{\alpha,\beta} = J^{\beta}, \quad J^{\beta}(x)= \int_{\mathcal{P}_x} f(x,p,q)p^{\beta}q\omega_p \omega_q, \tag{1}\\ & \mathcal{L}_Y f \equiv p^\alpha \frac{\partial f}{\partial x^{\alpha}} + (-\Gamma^{\alpha}_{\lambda\mu}p^{\lambda} p^{\mu} + q\cdot F^{\alpha}_{\lambda} p^{\lambda})\frac{\partial f}{\partial p^{\alpha}} - p^{\alpha}[A_{\alpha},q]\cdot \frac{\partial f}{\partial q} = 0 \tag{2} \end{align} where \( F = (F_{\alpha,\beta}) \) is a Yang–Mills field with potential \( A = (A_{\alpha}) \), \[ \hat{\nabla} = \nabla + [A,\,] \] is the gauge covariant derivative, \( [\ ,\,] \) the Lie bracket in the algebra \( \mathcal{G} \) where the fields take their values and a dot denotes an Ad invariant positive definite scalar product; \( J = (J_{\beta}) \) is the \( \mathcal{G} \) valued current generated by the distribution function f of particles with energy momentum p and Yang–Mills charge q: in the absence of collisions f satisfies the Vlasov equation (2). A possible physical situation modeled by (1), (2) is a plasma of quarks and gluons. In the above cited paper we have proved that when \( (V,g) \) is regularly hyperbolic, when the particles have a given rest mass m and their charges a given norm the Cauchy problem for (1), (2) has a local in time solution if:

The Cauchy data are in appropriate weighted Sobolev spaces.
These data satisfy the constraint \( \hat{\nabla}_{\alpha}F^{\alpha 0} = J^0 \).
In the case \( m = 0 \) the initial value \( \phi \) of f satisfies in addition Support \( \phi \subset \{p^0 > c > 0\} \).

In this Note we use this result together with the conformal invariance of the system in the case \( m = 0 \) to prove a global existence theorem on Minkowski space time \( M_4 \), for small initial data.

[166] Y. Choquet-Bruhat: “Yang–Mills fields on Lorentzian manifolds,” pp. 289–313 in Mechanics, analysis and geometry: 200 years after Lagrange. Edited by M. Francaviglia. North-Holland Delta. North-Holland (Amsterdam), 1991. MR 1098521 Zbl 0717.53066 incollection

[167] Y. Choquet-Bruhat: “Yang–Mills–Higgs fields in three space time dimensions,” pp. 73–97 in Analyse globale et physique mathématique (colloque à la mémoire d’Edmond Combet) (Lyon, 1989). Mém. Soc. Math. France 46. 1991. MR 1125837 Zbl 0773.53036 incollection

This paper is concerned with the problem of the global existence of solutions to the Y-M-H system, i.e. to the Yang–Mills equations coupled with the Higgs equations for a scalar multiplet. When \( M_{n+1} \) is the Minkowski space-time, existence theorems for global solutions are already known. They have been proved for \( n=1 \) or 2 by Ginibre and Velo and for \( n=3 \) by Eardley and Moncrief. Ginibre and Velo use a priori estimates in temporal gauge which do not allow to complete the proof for \( n=3 \). On the other hand, the author and Christodoulou proved the global existence only for small Cauchy data, but including also spinor sources. Their proof extends only to Lorentzian manifolds which are asymptotically Minkowskian at infinity. In this paper, the author shows that the proof of Ginibre and Velo extends to a general regularly (and then globally) hyperbolic manifold of dimension 2 or 3, even if the Yang–Mills bundle is not trivial. She uses the local existence theorem proved for such manifolds by her, Paneitz and Segal, and the Sobolev and Gagliardo–Nirenberg inequalities for \( n \lt 4 \), to obtain the required a priori estimates.

[168] Y. Choquet-Bruhat: “Fluides chargés non abéliens,” C. R. Acad. Sci. Paris Sér. I Math. 313 : 8 (1991), pp. 551–555. MR 1131874 Zbl 0733.76098 article

[169] Y. Choquet-Bruhat and N. Noutchegueme: “Système de Yang–Mills–Vlasov en jauge temporelle,” Ann. Inst. H. Poincaré Phys. Théor. 55 : 3 (1991), pp. 759–787. MR 1141713 Zbl 0767.58045 article

[170] Y. Choquet-Bruhat and G. Pichon: “Plasmas with discrete velocities,” pp. 85–96 in Discrete models of fluid dynamics (Figueira da Foz, 1990). Edited by A. S. Alves. Ser. Adv. Math. Appl. Sci. 2. World Sci. Publ. (River Edge, NJ), 1991. MR 1192797 Zbl 0828.76002 incollection

[171] Y. Choquet-Bruhat: “High frequency Gauss–Bonnet gravity,” pp. 57–73 in Classical mechanics and relativity: Relationship and consistency (papers from the international conference in memory of Carlo Cattaneo) (Elba, 1989). Edited by G. Ferrarese. Monogr. Textbooks Phys. Sci. Lecture Notes 20. Bibliopolis (Naples), 1991. MR 1204438 Zbl 0934.83003 incollection

[172] Y. Choquet-Bruhat: “Fluides chargés non abéliens de conductivité infinie,” C. R. Acad. Sci. Paris Sér. I Math. 314 : 1 (1992), pp. 87–91. MR 1149646 Zbl 0800.76552 article

[173] Y. Choquet-Bruhat: “Cosmological Yang–Mills hydrodynamics,” J. Math. Phys. 33 : 5 (1992), pp. 1782–1785. MR 1158999 Zbl 0753.76201 article

[174] Y. Choquet-Bruhat and N. Noutchegueme: “Yang–Mills Vlasov systems,” pp. 52–71 in Nonlinear hyperbolic equations and field theory: Papers from a workshop on nonlinear hyperbolic equations (Lake Como, 1990). Edited by M. K. V. Murthy and S. Spagnolo. Pitman Res. Notes Math. Ser. 253. Longman Sci. Tech.; Wiley (Harlow, UK; New York), 1992. MR 1175201 Zbl 0799.35188 incollection

[175] Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief: “Solutions of constraints for Einstein equations,” C. R. Acad. Sci. Paris Sér. I Math. 315 : 3 (1992), pp. 349–355. MR 1179734 Zbl 0796.35161 article

[176] Y. Choquet-Bruhat: “Cosmological Yang–Mills fluids,” pp. 685–687 in On recent developments in theoretical and experimental general relativity, gravitation and relativistic field theories: The Sixth Marcel Grossmann Meeting, Part A, B (Kyoto, 1991). Edited by H. Sato and T. Nakamura. World Sci. Publ. (River Edge, NJ), 1992. MR 1236927 incollection

[177] Y. Choquet-Bruhat: “Solution of the Einstein constraints on nonmaximal submanifolds,” pp. S205–S206 in Les Journées Relativistes (Amsterdam, 1992), published as Classical Quantum Gravity 10 : suppl. (1993). MR 1212813 Zbl 0788.53082 incollection

It has long been known that, by use of the “conformal method”, the Einstein constraints on an initial manifold split into a linear elliptic system and a nonlinear elliptic (Lichnerowicz) equation in the case where the manifold has constant mean extrinsic curvature. However there are spacetimes which do not admit such submanifolds (cf. Brill, Bartnik). In a recent paper, in collaboration with J. Isenberg and V. Moncrief, the author proved existence theorems for the coupled system of constraints, on an initial compact manifold with non-constant mean extrinsic curvature. Here the author considers the non-compact, asymptotically Euclidean case of non-maximal initial submanifolds.

[178] Y. Choquet-Bruhat: “Solution des contraintes pour les équations d’Einstein sur une variété asymptotiquement euclidienne non maximale” [Solution of constraints for Einstein equations on non maximal asymptotically Euclidean manifolds], C. R. Acad. Sci. Paris Sér. I Math. 317 : 1 (1993), pp. 109–114. MR 1228975 Zbl 0783.53047 article

[179] Y. Choquet-Bruhat: “Applications of generalized functions to shocks and discrete models,” pp. 37–49 in Generalized functions and their applications: Proceedings of the international symposium (Varanasi, 1991). Edited by R. S. Pathak. Plenum (New York), 1993. MR 1240062 Zbl 0849.46052 incollection

Generalized functions, also called distributions, have proved useful in many problems related to physics even before their rigorous and general definition. Generalized functions are also a powerful tool to formulate in a unified manner fundamental physical laws. We shall give here two general examples of such an application of generalized functions. The first is to the writing of the equations satisfied by shock waves: we establish in particular the algebraic conditions satisfied across a gravitational shock wave and the propagation of the corresponding discontinuities. In the second example we use a representation of the distribution function of kinetic theory by a sum of discrete measures to find the Boltzmann equation for a model with a finite number of velocities, together with propagation equations satisfied by these velocities.

[180] Y. Choquet-Bruhat: “Einstein equations with 1 parameter spacelike isometry group,” pp. 137–145 in Modern group analysis: Advanced analytical and computational methods in mathematical physics (Acireale, 1992). Edited by N. H. Ibragimov, M. Torrisi, and A. Valenti. Kluwer Acad. (Dordrecht), 1993. MR 1259548 Zbl 0793.53066 incollection

In a recent series of interesting articles V. Moncrief has obtained reduced Hamiltonians for vacuum Einstein, or Einstein–Maxwell–Higgs equations. We show how one can obtain by direct methods, without making use of a Hamiltonian formalism, the splitting of vacuum Einstein equations into a system elliptic on each spacelike slice on the one hand, and on the other hand a hyperbolic system which is an harmonic map equation from a pseudo-Riemannian manifold, whose metric depends on the solution of the elliptic system, into a fixed 2-dimensional symmetric Riemannian space. We discuss, following with some variants ideas of Cameron and Moncrief, the general solution of the system, depending on the topology of space.

[181] Y. Choquet-Bruhat: “Solution of Einstein equations,” Math. Today 11 (1993), pp. 3–12. MR 1311850 Zbl 0789.53044 article

In this lecture I shall not try to enter into the mystery of the action of gravity through eventual graviton exchange, neither in its quantization. Following J. V. Narlikar in his second Vaidya–Raychauduri Endowment Lecuture I shall quote from Newton’s letter to Bently “…cause of gravity is what I do not pretend to know and therefore would take more time to consider of it.” I shall only try to give a short review of what mathematicians know today about the general solutions of Einstein field equations of classical General Relativity, comparing it with the stronger results obtained in recent years for the so called Yang–Mills equations which govern the three other fundamental fields: the electromagnetic, weak and strong interactions.

[182] Y. Choquet-Bruhat: “Solution of the coupled Einstein constraints on asymptotically Euclidean manifolds,” pp. 83–96 in Directions in general relativity: Proceedings of the international symposium (University of Maryland, College Park, MD, 1993). Cambridge University Press (Cambridge), 1993. Papers in honor of Charles Misner on the occasion of his 60th birthday. Zbl 0850.83012 incollection

[183] Y. Choquet-Bruhat: “Cosmological Yang–Mills magnetohydrodynamics,” C. R. Acad. Sci. Paris Sér. I Math. 318 : 8 (1994), pp. 775–782. MR 1272347 Zbl 0802.76099 article

[184] Y. Choquet-Bruhat and V. Moncrief: “An existence theorem for the reduced Einstein equation,” C. R. Acad. Sci. Paris Sér. I Math. 319 : 2 (1994), pp. 153–159. MR 1288395 Zbl 0813.35060 article

[185] Y. Choquet-Bruhat: “Théorèmes d’existence globaux pour des fluides ultra relativistes,” C. R. Acad. Sci. Paris Sér. I Math. 319 : 12 (1994), pp. 1337–1342. MR 1310683 Zbl 0834.76100 article

[186] Y. Choquet-Bruhat: “Hydrodynamics and magnetohydrodynamics of Yang–Mills fluids,” pp. 54–68 in Waves and stability in continuous media (Bologna, 1993). Edited by S. Rionero and T. Ruggeri. Ser. Adv. Math. Appl. Sci. 23. World Sci. Publ. (River Edge, NJ), 1994. MR 1320064 incollection

[187] A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr.: “Einstein and Yang–Mills theories in hyperbolic form without gauge fixing,” Phys. Rev. Lett. 75 : 19 (1995), pp. 3377–3381. MR 1356171 Zbl 1020.83503 article

Andrew M. Abrahams	Related
Arlen Anderson	Related
James W. York	Related

[188] Y. Choquet-Bruhat and J. W. York: “Geometrical well posed systems for the Einstein equations,” C. R. Acad. Sci. Paris Sér. I Math. 321 : 8 (1995), pp. 1089–1095. MR 1360579 Zbl 0839.53063 article

[189] Y. Choquet-Bruhat and V. Moncrief: “Existence theorem for solutions of Einstein’s equations with 1 parameter spacelike isometry groups,” pp. 67–80 in Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994). Edited by W. Arveson, T. Branson, and I. Segal. Proc. Sympos. Pure Math. 59. Amer. Math. Soc. (Providence, RI), 1996. MR 1392984 Zbl 0861.53063 incollection

We shall first show in this paper how one can obtain by direct methods, without making use of a Hamiltonian formalism, the splitting of part of the vacuum Einstein equations with 1 parameter isometry group into a system elliptic on each spacelike slice on the one hand, and on the other hand a hyperbolic system which is essentially a harmonic map equation from a pseudo-Riemannian manifold, whose metric depends on the solution of the elliptic system, into the 2-dimensional symmetric Riemannian space \( \mathcal{H} \).

We then prove global in space, local in time existence and uniqueness of solutions of the Cauchy problem for the hyperbolic-elliptic system so obtained, when \( \Sigma = S^2 \), in appropriate functional spaces. We show that in that case, the obtained solution satisfies the original Einstein equations. We finally discuss the general solution of the system, in the case where \( \Sigma \) is not simply connected. We show that when the genus of \( \Sigma \) is greater than 1, we can determine gt so that it is conformal to a metric at which remains in some chosen cross section of a bundle over the Teiehmuller space of \( \Sigma \) (cf. an analogous method in [Moncrief 1989] for the 3-dimensional Einstein vacuum equations). Here the evolution of at is determined by requiring the mixed elliptic-hyperbolic system obtained previously to be a solution of the original, 4-dimensional, Einstein equations

[190] Y. Choquet-Bruhat: “Yang–Mills plasmas,” pp. 3–31 in Global structure and evolution in general relativity: Proceedings of the First Samos Meeting on Cosmology, Geometry and Relativity (Karlovassi, 1994). Edited by S. Cotsakis and G. W. Gibbons. Lecture Notes in Phys. 460. Springer (Berlin), 1996. MR 1398907 Zbl 0844.53053 incollection

The Yang–Mills plasmas generalize to the case of a Yang–Mills type, non-abelian, charge the usual electrically charged plasmas. It has been proved recently that at very high temperature deconfined quarks and their gluon field can be effectively modeled by such a plasma. Yang–Mills type plasmas, with more “elementary” particles could also have occurred in the early stages of the universe. We consider in the first part the kinetic model of a Yang–Mills plasma: the particles have a charge which takes its values in a Lie algebra. They follow between collisions the flow lines corresponding to the average Yang-Mills field they generate through their distribution function. This function satisfies a Liouville–Vlasov equation if collisions are neglected, a Boltzmann equation if they are taken into account.

In the second part we study fluid models, with finite or infinite conductivity.

[191] Y. Choquet Bruhat: “Global existence theorems for Einstein equations in high dimensions,” pp. 19–28 in Gravity, particles and space-time. Edited by P. Pronin and G. Sardanashvily. World Sci. Publ. (River Edge, NJ), 1996. MR 1450873 incollection

Classical General Relativity is the best available theory of the gravitational field. It is tested by the most recent astronomical observations with a great accuracy against alternative theories, through very sophisticated experimental as well as mathematical devices. Its fundamental unknown, the gravitational potential, is a pseudo riemannian metric \( g \) of lorentzian signature on a four dimensional manifold. It satisfies the Einstein field equations. Analogous equations on higher dimensional manifolds have been introduced long ago. Kaluza and Klein used a metric in five dimensions, with a one parameter isometry group, to unify the gravitational and electro magnetic fields. Later it has been proved by Kerner that \( 4 + N \) dimensional Einstein equations with cosmological constant can be used to obtain the Einstein Yang–Mills system on the classical space time and an N dimensional Yang Mills group. In recent physical theories which may be pertinent at microscopic scales, one considers anew space-times with dimension greater than 4, for instance 10 in some superstrings models. We will give in this article a general study of Einstein equations in arbitrary dimensions and prove a global existence theorem for small data in dimensions greater than six.

[192] Y. Choquet-Bruhat and A. Greco: “Interaction of gravitational and fluid waves,” pp. 111–123 in Proceedings of the Eighth International Conference on Waves and Stability in Continuous Media, Part I (Palermo, 1995), published as Rend. Circ. Mat. Palermo 45 : suppl. (1996). MR 1461067 Zbl 0893.76098 inproceedings

[193] A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York: “Un système hyperbolique non strict pour les équations d’Einstein,” C. R. Acad. Sci., Paris, Sér. II, Fasc. b 323 : 12 (1996), pp. 835–841. Zbl 0916.35126 article

Andrew M. Abrahams	Related
Arlen Anderson	Related
James W. York	Related

[194] Y. Choquet-Bruhat and A. Greco: “Ondes gravitationnelles à haute fréquence et interaction avec la matière,” C. R. Acad. Sci., Paris, Sér. II, Fasc. b 323 : 2 (1996), pp. 117–124. Zbl 0936.83014 article

[195] Y. Choquet-Bruhat and J. W. York, Jr.: “Well posed reduced systems for the Einstein equations,” pp. 119–131 in Mathematics of gravitation, Part I (Warsaw, 1996). Edited by P. T. Chruściel. Banach Center Publ. 41. Polish Acad. Sci. Inst. Math. (Warsaw), 1997. MR 1466512 Zbl 0896.53052 incollection

[196] A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr.: “Einstein–Bianchi hyperbolic system for general relativity,” Topol. Methods Nonlinear Anal. 10 : 2 (1997), pp. 353–373. MR 1634577 Zbl 0917.35145 article

Arlen Anderson	Related
James W. York	Related

[197] Y. Choquet-Bruhat: “A global existence theorem for ultra relativistic fluids on Minkowski space time,” pp. 14–22 in Collection of papers on geometry, analysis and mathematical physics. Edited by T.-T. Li. World Sci. Publ. (River Edge, NJ), 1997. In honor of Professor Gu Chaohao. MR 1635656 Zbl 1021.35132 incollection

[198] A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr.: “Geometrical hyperbolic systems for general relativity and gauge theories,” Classical Quantum Gravity 14 : 1A (1997), pp. A9–A22. MR 1691883 Zbl 0866.58059 article

The evolution equations of Einstein’s theory and of Maxwell’s theory — the latter used as a simple model to illustrate the former — are written in gauge-covariant first-order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom (the spatial shift vector \( \beta^i(t, x^j) \) and the spatial scalar potential \( \phi(t, x^j) \), respectively) are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of \( (t, x^j) \) from subsidiary equations that are not part of the evolution equations. Propagation of certain (“radiative”) dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by

taking a further time derivative of the equation of motion of the canonical momentum, and
adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier \( \beta^i \)) or of the Gauss law constraint of electromagnetism (Lagrange multiplier \( \phi \)).

General relativity also requires a harmonic time-slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first-order symmetric form. The dynamically propagating gravity fields straightforwardly determine the “electric” or “tidal” parts of the Riemann tensor

Andrew M. Abrahams	Related
Arlen Anderson	Related
James W. York	Related

[199] Y. Choquet-Bruhat: “Applications d’ondes sur un univers en expansion,” C. R. Acad. Sci. Paris Sér. I Math. 326 : 10 (1998), pp. 1175–1180. MR 1650238 Zbl 0908.53047 article

[200] Y. Choquet-Bruhat: “Global existence of wave maps,” pp. 143–152 in Proceedings of the Ninth International Conference on Waves and Stability in Continuous Media (Bari, 1997), published as Rend. Circ. Mat. Palermo 57 : suppl. Issue edited by M. Maiellaro and S. Rionero. 1998. MR 1708504 Zbl 0933.58028 inproceedings

[201] Y. Choquet-Bruhat: “Wave maps in general relativity,” pp. 147–169 in On Einstein’s path (New York, 1996). Edited by A. Harvey. Springer, 1999. MR 1658871 Zbl 0981.83010 incollection

[202] M. Berger, J.-P. Bourguignon, Y. Choquet-Bruhat, C.-M. Marle, and A. Revuz: “André Lichnerowicz (1915–1998),” Notices Amer. Math. Soc. 46 : 11 (1999), pp. 1387–1396. MR 1723248 Zbl 0932.01060 article

Marcel Berger	Related
Jean Pierre Bourguignon	Related
Charles-Michel Marle	Related

[203] Y. Choquet-Bruhat: “Lichnérowicz et la relativité générale,” Gaz. Math. 82 (1999), pp. 99–101. MR 1730680 Zbl 1388.01019 article

[204] Y. Choquet-Bruhat, J. W. York, Jr., and A. Anderson: “Curvature-based hyperbolic systems for general relativity,” pp. 112–121 in The Eighth Marcel Grossmann Meeting, Part A, B (Jerusalem, 1997). Edited by T. Piran. World Sci. Publ. (River Edge, NJ), 1999. MR 1891864 Zbl 0970.83005 incollection

Arlen Anderson	Related
James W. York	Related

[205] Y. Choquet-Bruhat: “The null condition and asymptotic expansions for the Einstein equations,” Ann. Phys. 9 : 3–5 (2000), pp. 258–266. MR 1770106 Zbl 0973.83018 article

The global existence of solutions of non linear wave equations on the Minkowski spacetime of dimension 4 is linked with the non linearities satisfying the “null condition” discovered by Klainerman and Christodoulou. This null condition also improves local existence results, lowering the sufficient regularity of the Cauchy data. The non linear stability of the Minkowski spacetime among solutions of the vacuum Einstein equations relies on estimates for the Weyl tensor probably linked with some kind of null condition. It had been proved long ago that the significant part of an high frequency gravitational wave in General Relativity obeys a transport law along the rays which is linear, but inflicts a “back reaction” on the background metric. These two properties reflect the fact that the Einstein equations satisfy almost, but not quite, a “polarized” null condition. We will explicit this result, also for associated field equations, and explain its relation with the properties of high frequency waves.

[206] Y. Choquet-Bruhat: “Jean Leray, souvenirs,” Gaz. Math. 84 : suppl. (2000), pp. 7–9. MR 1775585 article

[207] Y. Choquet-Bruhat: “Global wave maps on Robertson–Walker spacetimes,” Nonlinear Dynam. 22 : 1 (2000), pp. 39–47. MR 1776183 Zbl 0997.83116 article

[208] Y. Choquet-Bruhat, J. Isenberg, and J. W. York, Jr.: “Einstein constraints on asymptotically Euclidean manifolds,” Phys. Rev. D (3) 61 : 8 (2000), pp. 084034, 20. MR 1791413 article

[209] Y. Choquet-Bruhat: “Global wave maps on curved space times,” pp. 1–29 in Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, Samos, 1998). Edited by S. Cotsakis and G. W. Gibbons. Lecture Notes in Phys. 537. Springer, 2000. MR 1843031 Zbl 0994.83006 incollection

Wave maps from a pseudoriemannian manifold of hyperbolic (Lorentzian) signature \( (V, g) \) into a pseudoriemannian manifold are the generalisation of the usual wave equations for scalar functions on \( (V, g) \). They are the counterpart in hyperbolic signature of the harmonic mappings between properly riemannian manifolds.

The wave map equations are an interesting model of geometric origin for the mathematician, in local coordinates they look like a quasilinear quasidiagonal system of second order partial differential equations which satisfy the [Christodoulou 1993] and [Klainerman 1993] null condition. They also appear in various areas of physics (cf. [Nutku 1974], [Misner 1978].)

[210] A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr.: “Einstein’s equations and equivalent hyperbolic dynamical systems,” pp. 30–54 in Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, Samos, 1998). Edited by S. Cotsakis and G. W. Gibbons. Lecture Notes in Phys. 537. Springer (Berlin), 2000. MR 1843032 Zbl 0987.83006 incollection

We discuss several explicitly causal hyperbolic formulations of Einstein’s dynamical \( 3 + 1 \) equations in a coherent way, emphasizing throughout the fundamental role of the “slicing function,” \( \alpha \) — the quantity that relates the lapse \( N \) to the determinant of the spatial metric \( \bar{g} \) through \( N = \bar{g}^{1/2}\alpha \). The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations. Specifically, the slicing function plays an essential role

in a clearer form of the canonical action principle and Hamiltonian dynamics for gravity, and leads to a recasting
of the Bianchi identities \( \nabla_{\beta} G^{\beta} \alpha \equiv 0 \) as a well-posed system for the evolution of the gravitational constraints in vacuum, and also
of \( \nabla_{\beta} T^{\beta}_{\alpha} \equiv 0 \) as a well-posed system for evolution of the energy and momentum components of the stress tensor in the presence of matter,
in an explicit rendering of four hyperbolic formulations of Einstein’s equations with only physical characteristics, and
in providing guidance to a new “conformal thin sandwich” form of the initial value constraints.

Arlen Anderson	Related
James W. York	Related

[211] Y. Choquet-Bruhat and C. DeWitt-Morette: Analysis, manifolds and physics. Part II., Revised and enl. edition. Elsevier/North-Holland (Amsterdam), 2000. A revised and enlarged version of the 1989 edition. book

[212] Y. Choquet-Bruhat and V. Moncrief: “Future complete Einsteinian space times with \( U(1) \) isometry group,” C. R. Acad. Sci. Paris Sér. I Math. 332 : 2 (2001), pp. 137–144. A version of this paper was published as “Future global in time Einsteinian spacetimes with \( U(1) \) isometry group,” in Ann. Henri Poincaré 6, 1007–1064 (2001). MR 1813771 Zbl 0976.83004 article

[213] Y. Choquet-Bruhat and V. Moncrief: “Future global in time Einsteinian spacetimes with \( U(1) \) isometry group,” Ann. Henri Poincaré 2 : 6 (2001), pp. 1007–1064. MR 1877233 Zbl 0998.83007 article

[214] Y. Choquet-Bruhat and J. W. York: “On H. Friedrich’s formulation of the Einstein equations with fluid sources,” Topol. Methods Nonlinear Anal. 18 : 2 (2001), pp. 321–335. MR 1911385 Zbl 1043.35141 article

[215] Y. Choquet-Bruhat and S. Cotsakis: “Global hyperbolicity and completeness,” J. Geom. Phys. 43 : 4 (2002), pp. 345–350. MR 1929912 Zbl 1022.83002 article

[216] Y. Choquet-Bruhat and J. W. York: “Bianchi–Euler system for relativistic fluids and Bel–Robinson type energy,” C. R. Math. Acad. Sci. Paris 335 : 8 (2002), pp. 711–716. MR 1941654 Zbl 1011.83010 article

[217] Y. Choquet-Bruhat: “Local and global results on the Cauchy problem for the Einstein equations,” pp. 7–27 in The Ninth Marcel Grossmann Meeting: On recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories (Rome, Italy, 2000), vol. 1. World Scientific (Singapore), 2002. Zbl 1032.83013 incollection

[218] Y. Choquet-Bruhat and J. W. York: “Constraints and evolution in cosmology,” pp. 29–58 in Cosmological crossroads. An advanced course in mathematical, physical and string cosmology. Lectures given at first Aegean summer school on cosmology (Samos Island, Greece, 2001). Lecture Notes in Physics 592. Springer (Berlin), 2002. Zbl 1043.83006 incollection

[219] Y. Choquet-Bruhat and V. Moncrief: “Nonlinear stability of an expanding universe with the \( S^1 \) isometry group,” pp. 57–71 in Partial differential equations and mathematical physics (Tokyo, 2001). Edited by K. Kajitani and J. Vaillant. Progr. Nonlinear Differential Equations Appl. 52. Birkhäuser (Boston), 2003. MR 1957625 Zbl 1062.35149 incollection

[220] Y. Choquet-Bruhat: “Future complete \( S^1 \) symmetric Einsteinian spacetimes, the unpolarized case,” C. R. Math. Acad. Sci. Paris 337 : 2 (2003), pp. 129–136. MR 1998845 Zbl 1027.83005 article

[221] Y. Choquet-Bruhat: “Causal evolution for Einsteinian gravitation,” pp. 129–144 in Hyperbolic differential operators and related problems. Edited by V. Ancona and J. Vaillant. Lecture Notes in Pure and Appl. Math. 233. Dekker (New York), 2003. MR 2004863 Zbl 1205.83016 incollection

[222] Y. Choquet-Bruhat: “Global wave maps on black holes,” pp. 469–482 in Jean Leray ’99 Conference Proceedings. Edited by M. de Gosson. Math. Phys. Stud. 24. Kluwer Acad. (Dordrecht), 2003. MR 2051505 inproceedings

[223] Y. Choquet-Bruhat: “Einstein constraints on compact \( n \)-dimensional manifolds,” Classical Quantum Gravity 21 : 3 (2004), pp. S127–S151. MR 2053003 Zbl 1040.83004 article

[224] Y. Choquet-Bruhat: “Future complete \( \mathrm{ U}(1) \) symmetric Einsteinian spacetimes, the unpolarized case,” pp. 251–298 in The Einstein equations and the large scale behavior of gravitational fields. Edited by P. T. Chruściel and H. Friedrich. Birkhäuser (Basel), 2004. MR 2098918 Zbl 1064.83005 incollection

In this paper I generalize the non linear stability theorem obtained in collaboration with V. Moncrief [Choquet-Bruhat and Moncrief 2001; Choquet-Bruhat and Moncrief 2003] for vacuum Einsteinian 4-manifolds \( (V, {}^{(4)}g) \) where \( V = M \times R \) with \( M \) a circle bundle over a compact, orient able surface \( \Sigma \) of genus greater than 1. The Lorentzian metric \( {}^{(4)}g \) admits a Killing symmetry along the (spacelike) circular fibers. I remove the so-called polarization condition, i.e., the orthogonality of the fibers to quotient 3-manifolds. The reduced field equations take now the form of a wave map equation, instead of a linear wave equation in the polarized case, coupled to \( 2 + 1 \) gravity. I use results on wave maps from curved manifolds obtained in [Choquet-Bruhat 1998a; Choquet-Bruhat 1998b]. Like in [Choquet-Bruhat and Moncrief 2003] we do not restrict the conformal geometry of \( \Sigma \) to avoid those regions of Teichmüller space for which the lowest positive eigenvalues of the scalar Laplacian lie, in our normalization, in the gap \( (0, 1/8] \). A consequence is that the asymptotic behavior of the wave map field does not exhibit a universal rate of decay but instead develops a decay rate which depends upon the asymptotic values of the lowest eigenvalue.

[225] Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief: “Topologically general U(1) symmetric vacuum space-times with AVTD behavior,” Nuovo Cimento Soc. Ital. Fis. B 119 : 7–9 (2004), pp. 625–638. MR 2136898 article

James Allen Isenberg	Related
Vincent Edward Moncrief	Related

[226] Y. Choquet-Bruhat: “Asymptotic solutions of non linear wave equations and polarized null conditions,” pp. 125–141 in Actes des Journées Mathématiques à la mémoire de Jean Leray. Edited by L. Guillopé and D. Robert. Sémin. Congr. 9. Soc. Math. France (Paris), 2004. MR 2145939 Zbl 1062.35148 incollection

La généralisation faite par Leray de la méthode WKB pour la construction de solutions asymptotiques à haute fréquence de systèmes arbitraires d’équations aux dérivées partielles linéaires a permis le traitement de systèmes quasilinéaires et l’apparition de propriétés nouvelles comme la distorsion des signaux. La non linéarité est aussi une obstruction à l’existence de solutions globales des systèmes d’évolution. On introduit une condition nulle polarisée, généralisation de la condition nulle de Christodoulou–Klainerman à des systèmes mal posés par suite de l’invariance de jauge. On montre qu’elle conduit à une équation de transport linéaire le long des rayons d’une solution asymptotique. Elle est satisfaite par le modèle standard, mais un terme résiduel dans le cas des équations d’Einstein conduit à une “réaction en retour” sur la métrique de base.

[227] Y. Choquet-Bruhat, J. Isenberg, and D. Pollack: “The Einstein-scalar field constraints on asymptotically Euclidean manifolds,” Chinese Ann. Math. Ser. B 27 : 1 (2006), pp. 31–52. MR 2209950 Zbl 1112.83008 article

[228] Y. Choquet-Bruhat and J. W. York: “Einstein–Bianchi system with sources,” pp. 59–70 in Nonlinear hyperbolic fields and waves: A tribute to Guy Boillat, published as Rend. Circ. Mat. Palermo 78 : suppl. (2006). MR 2210593 Zbl 1101.35047 incollection

[229] Y. Choquet-Bruhat: “From the big bang to future complete cosmologies,” pp. 110–121 in “WASCOM 2005”: 13th Conference on Waves and Stability in Continuous Media. Edited by R. Monaco, G. Mulone, S. Rionero, and T. Ruggeri. World Sci. Publ. (Hackensack, NJ), 2006. MR 2231273 Zbl 1345.83023 incollection

[230] Y. Choquet-Bruhat and J. Isenberg: “Half polarized \( \mathrm{ U}(1) \) symmetric vacuum spacetimes with AVTD behavior,” J. Geom. Phys. 56 : 8 (2006), pp. 1199–1214. MR 2234438 Zbl 1113.83006 article

[231] Y. Choquet-Bruhat and H. Friedrich: “Motion of isolated bodies,” Classical Quantum Gravity 23 : 20 (2006), pp. 5941–5949. MR 2270110 Zbl 1107.83011 article

[232] Y. Choquet-Bruhat, P. T. Chruściel, and J. Loizelet: “Global solutions of the Einstein–Maxwell equations in higher dimensions,” Classical Quantum Gravity 23 : 24 (2006), pp. 7383–7394. MR 2279722 Zbl 1117.83024 article

We consider the Einstein–Maxwell equations in space-dimension \( n \). We point out that the Lindblad–Rodnianski stability proof applies to those equations whatever the space-dimension \( n \geq 3 \). In even spacetime dimension \( n + 1 \geq 6 \), we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski spacetime and which are Schwarzschildian outside of a compact set lead to geodesically complete spacetimes, with a complete Scri, with a smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as \( r^{-(n-1)/2} \).

Piotr T. Chruściel	Related
Julien Loizelet	Related

[233] Y. Choquet-Bruhat, J. Isenberg, and D. Pollack: “Applications of theorems of Jean Leray to the Einstein-scalar field equations,” J. Fixed Point Theory Appl. 1 : 1 (2007), pp. 31–46. MR 2282342 Zbl 1169.83007 article

James Allen Isenberg	Related
Daniel Pollack	Related

[234] Y. Choquet-Bruhat, J. Isenberg, and D. Pollack: “The constraint equations for the Einstein-scalar field system on compact manifolds,” Classical Quantum Gravity 24 : 4 (2007), pp. 809–828. MR 2297268 Zbl 1111.83002 article

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.

[235] Y. Choquet-Bruhat: “Results and open problems in mathematical general relativity,” Milan J. Math. 75 (2007), pp. 273–289. MR 2371545 Zbl 1164.83001 article

[236] Y. Choquet-Bruhat: “Fuchsian partial differential equations,” pp. 153–161 in “WASCOM 2007”: 14th Conference on Waves and Stability in Continuous Media. Edited by N. Manganaro, R. Monaco, and S. Rionero. World Sci. Publ. (Hackensack, NJ), 2008. MR 2484759 Zbl 1184.35305 incollection

It is generally believed that our cosmos started with an initial singularity, the big bang. In fact all solutions of Einstein equations with compact spacelike sections present a singularity, at least in one time direction, except if the space is the flat three torus. The behavior near the singularity, chaotic or not, is a subject of active research. A case amenable to rigorous mathematical treatment is when the difference of the solution with a given spacetime metric satisfies a Fuchsian system of partial differential equations. In this article we give a more general definition than Kichenassamy and Rendall of a Fuchsian system and give a simpler proof of existence of solutions tending to zero at the singularity. Our generalization is interesting for Hamiltonian systems, as pointed out by Thibault Damour and Sophie de Buyl.

[237] Y. Choquet-Bruhat: General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press (Oxford, UK), 2009. MR 2473363 Zbl 1157.83002 book

[238] Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García: “The light-cone theorem,” Classical Quantum Gravity 26 : 13 (2009), pp. 135011, 22. MR 2515694 Zbl 1171.83001 article

Piotr T. Chruściel	Related
José María Martín-García	Related

[239] Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García: “A property of light-cones in Einstein’s gravity,” C. R. Math. Acad. Sci. Paris 347 : 15–16 (2009), pp. 971–977. MR 2542904 Zbl 1170.83005 article

Piotr T. Chruściel	Related
José María Martín-García	Related

[240] Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García: “Einstein constraints on a characteristic cone,” pp. 93–102 in Proceedings “WASCOM 2009” 15th Conference on Waves and Stability in Continuous Media. World Sci. Publ. (Hackensack, NJ), 2010. MR 2762005 Zbl 1243.83015 inproceedings

Piotr T. Chruściel	Related
José María Martín-García	Related

[241] Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García: “The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions,” Ann. Henri Poincaré 12 : 3 (2011), pp. 419–482. MR 2785136 Zbl 1215.83016 article

Piotr T. Chruściel	Related
José María Martín-García	Related

[242] Y. Choquet-Bruhat: “Positive gravitational energy in arbitrary dimensions,” C. R. Math. Acad. Sci. Paris 349 : 15–16 (2011), pp. 915–921. MR 2835903 Zbl 1227.83017 article

[243] Y. Choquet-Bruhat and J. M. Martín-García: “A geometric energy estimate for data on a characteristic cone,” pp. 17–25 in Advances in Lorentzian geometry. Edited by M. Plaue, A. Rendall, and M. Scherfner. AMS/IP Stud. Adv. Math. 49. Amer. Math. Soc. (Providence, RI), 2011. MR 2867849 Zbl 1243.83016 incollection

The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa, proved for a scalar wave equation, for initial data in functional spaces relevant for their proofs. It is difficult to check that the initial data that we have constructed as solutions of the Einstein wave-map gauge constraints satisfy the more general of the Cagnac-Dossa hypotheses which uses weighted energy estimates. In this paper we start a new study of energy estimates using on the cone coordinates adapted to its null structure which are precisely the coordinates used to solve the constraints, following work of Rendall who considered the Cauchy problem for Einstein equations with data on two intersecting characteristic surfaces.

[244] Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García: “An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations,” pp. 73–81 in Complex analysis and dynamical systems, IV: Part 2. Edited by M. Agranovsky, M. Ben-Artzi, G. Galloway, L. Karp, S. Reich, D. S. G. Weinstein, and L. Zalcman. Contemp. Math. 554. American Mathematical Society; Bar-Ilan University (Providence, RI; Ramat Gan), 2011. MR 2884395 Zbl 1235.83011 incollection

Piotr T. Chruściel	Related
José María Martín-García	Related

[245] Y. Choquet-Bruhat, P. T. Chruściel, and J. M. Martín-García: “An existence theorem for the Cauchy problem on the light-cone for the vacuum Einstein equations with near-round analytic data,” Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki 153 : 3 (2011), pp. 115–138. MR 3244416 Zbl 1344.83002 article

[246] Y. Choquet-Bruhat: Introduction to general relativity, black holes, and cosmology. Oxford University Press (Oxford, UK), 2015. MR 3379262 Zbl 1307.83001 book

[247] Y. Choquet-Bruhat: “Beginnings of the Cauchy problem for Einstein’s field equations,” pp. 1–16 in Surveys in differential geometry 2015: One hundred years of general relativity. Edited by L. Bieri and S.-T. Yau. Surv. Differ. Geom. 20. Int. Press (Boston), 2015. MR 3467361 Zbl 1339.83006 incollection

[248] Y. Choquet-Bruhat: Une mathématicienne dans cet étrange univers : mémoires. Odile Jacob (Paris), 2016. book

[249] Y. Choquet-Bruhat: A lady mathematician in this strange universe: memoirs. World Sci. Publ. (Hackensack, NJ), 2018. Translated from the 2016 French original. MR 3753699 Zbl 1387.83002 book

[250] Y. Fourès-Bruhat: “Republication of: ‘Existence theorem for the Einsteinian gravitational field equations in the non-analytic case’,” Gen. Relativity Gravitation 54 : 4 (2022), pp. Paper No. 35, 4. Reprint of 1950 original. Republication of the article “Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique,” originally published in Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, 230, 618–620 (1950). MR 4406810 Zbl 1490.83007 article

@article {key4406810m,
	AUTHOR = {Four\`es-Bruhat, Yvonne},
	TITLE = {Republication of: `{E}xistence theorem
		 for the {E}insteinian gravitational
		 field equations in the non-analytic
		 case'},
	JOURNAL = {Gen. Relativity Gravitation},
	FJOURNAL = {General Relativity and Gravitation},
	VOLUME = {54},
	NUMBER = {4},
	YEAR = {2022},
	PAGES = {Paper No. 35, 4},
	DOI = {10.1007/s10714-022-02917-4},
	NOTE = {Reprint of 1950 original. Republication
		 of the article ``Th\'{e}or\`eme d'existence
		 pour les \'{e}quations de la gravitation
		 einsteinienne dans le cas non analytique'',
		 originally published in \textit{Comptes
		 Rendus Hebdomadaires des S\'{e}ances
		 de l'Acad\'{e}mie des Sciences}, \textbf{230},
		 618--620 (1950). MR:4406810. Zbl:1490.83007.},
	ISSN = {0001-7701},
}

Yvonne Choquet-Bruhat

Complete Bibliography