Editorial note to: Existence theorem for the Einsteinian gravitational field equations in the non-analytic case, by Yvonne Fourès-Bruhat

1 Editorial note

by Demetrios Christodoulou

Yvonne Fourès-Bruhat, known to us of the succeeding generation as Mme. Yvonne Choquet-Bruhat, made her debut to the scientific community at the age of 26 with the short article “Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique” [1], presented to the Paris Academy in February 1950. The presenter was Jacques Hadamard (1865–1963) who was the greatest living mathematician at that time. Hadamard had proved in 1896 the prime number theorem, the most important theorem in number theory since the time of Gauss, conjectured by Gauss himself. (A proof was also given simultaneously and independently by de la Vallée Poussin.) Hadamard had made several other major contributions to mathematics and mathematical physics, such as the Hadamard inequality on determinants and Hadamard matrices in algebra, Hadamard manifolds and the geodesic flow on Riemannian manifolds of negative sectional curvature in differential geometry, and, most important in the present context, foundational work on partial differential equations, in particular equations of hyperbolic type. Hadamard was the first to formulate the notion of well-posedness of a problem in connection with the system of partial differential equations. He was also the first to emphasize that the Cauchy–Kowalewski theory of analytic solutions is inadequate because it is unable to capture the physical domain of dependence property, and that the development of a theory of non-analytic solutions is required. To Hadamard is also due the representation formula for the solution of a linear inhomogeneous hyperbolic partial differential equation of 2nd order, generalizing Kirchhoff’s formula for the classical wave equation and known as the Hadamard parametrix.

I note for the benefit of younger readers that the purpose in those days of Academy announcements like Yvonne’s short article was to draw attention of the community to an upcoming publication and to secure priority, given the fact that there was no ArXiv in those days and there was a two year wait for publication of a longer paper to a mathematical journal. The longer paper in fact came out two years later with the title “Théorème d’existence pour certains systèmes d’équations aux dérivées partielles nonlinéaires”, again under the name Yvonne Fourès-Bruhat, in Acta Mathematica 88, 141–225 (1952). This paper is found in English translation at the Max Planck Institute for the History of Science (Preprints No. 480) with the title “Existence theorem for certain systems of nonlinear partial differential equations” [2]. In this note, I shall refer to sections and page numbers as they appear in this translation.

For lack of sufficient space to explain things, the short article by itself is virtually unintelligible to the reader. One must resort to the longer paper to find adequate explanations of the concepts and quantities employed as well as for the layout of the approach and for the structure of the proof. For this reason I shall in this note take the short article as a point of departure and point out the relevant sections of the longer paper where the concepts and quantities introduced in the short article are explained and their use in the approach is made evident. Even the longer paper however, does not explain the motivation for the approach, which is very different from the approach to a local in time existence theorem that a student of partial differential equations is taught nowadays, and which became established not long after the publication of Yvonne’s longer paper. It is the purpose of the present note to explain the motivation of Yvonne’s approach, to outline the difficulties that she faced, to discuss the state of knowledge in this mathematical area at that time, showing the originality of her work, and also to explain why a different approach to the local in time existence theorem subsequently dominated. I shall finally discuss how Yvonne’s approach eventually bore fruit in addressing another important problem in the context of general relativity, finding the conditions under which solutions break down.

The physical motivation for Yvonne’s work, though not that of her mathematical approach, is expressed in the Introduction of the longer paper. Note in particular her statement on page 1: “One could also see the emergence of gravitational waves and gravitational rays, giving to the gravitational field the character of a propagation phenomenon, and one could see the identity of the propagation laws for light and for the gravitational field”. Moreover, the 3rd paragraph of page 2 in the Introduction makes it clear that her aim was not only to prove an existence theorem but also to provide a means by which approximate solutions can be calculated which approach the exact solution as much as desired. This paragraph reads: “It seemed to me that, for the problems considered by the theory of relativity, it would be interesting to obtain, under the minimal possible amount of assumptions, an existence theorem easy to use, enabling [one] to find properties of the solutions that can be compared with the classical properties of light waves and gravitational potentials, and to have formulas which can be an efficient method of calculating gravitational fields, at least approximately, that correspond to given initial conditions.” Coming back to general relativity in the last paragraph of the Introduction (page 3) she summarizes in the last sentence what she has achieved from a physical point of view: “I have also built an Einstein spacetime corresponding to nonanalytic initial data, assigned on a spacelike domain, in such a way that it highlights the propagation character which is peculiar to relativistic gravitation.”

The short article addresses systems of 2nd order hyperbolic partial differential equations of the form

$$\begin{aligned} A^{\lambda \mu }\frac{\partial ^2 W_s}{\partial x^\lambda \partial x^\mu }+f_s=0 \end{aligned}$$

(1)

in 4 spacetime dimensions, $W_s : s=1,...,n$ being the unknowns and $x^\lambda : \lambda =1,...,4$ the spacetime coordinates. Here the principal coefficients $A^{\lambda \mu }$ depend on the unknowns and their first derivatives as do the functions $f_s$. The application to general relativity is concerned only with the somewhat simpler case where the $A^{\lambda \mu }$ depend on the unknowns but not on their first derivatives. The more general case occurs for example when we study minimal 4-dimensional timelike surfaces in a $n+4$-dimensional Minkowski spacetime. These surfaces being described as graphs over the 4-dimensional Minkowski spacetime, there being n extra spatial coordinates $y^a$, we have $y^a=u^a(x^1,...,x^4)$, the unknowns are the functions $u^a$ and $A^{\lambda \mu }$ is the reciprocal of the induced metric on the graph:

$$\begin{aligned} \eta _{\lambda \mu }+\sum _{a=1}^n \frac{\partial u^{a}}{\partial x^\lambda } \frac{\partial u^a}{\partial x^\mu } \,, \end{aligned}$$

where $\eta _{\lambda \mu }$ is the Minkowski metric. The longer paper actually deals in detail only with the somewhat simpler case. The more general case is only briefly referred to in the Introduction, in a comment to the effect that the approach can be extended to that case. I should mention here that the more general case is not in fact the most general case, where the principal part is of the form

$$\begin{aligned} A^{\lambda \mu r}_{ \ \ s} \frac{\partial ^2 W_r}{\partial x^\lambda \partial x^\mu }. \end{aligned}$$

This most general case occurs in the theory of nonlinear elasticity. The theory of characteristic hypersurfaces such as characteristic conoids is in this case much more difficult.

The backbone of Yvonne’s work is pointwise estimates for the solutions of linear systems of the form

$$\begin{aligned} A^{\lambda \mu }\frac{\partial ^{2} u_s}{\partial x^\lambda \partial x^\mu }+ B^{r\lambda }_s\frac{\partial u_r}{\partial x^{\lambda }}+f_s=0. \end{aligned}$$

(2)

These are naturally associated to the nonlinear system (1) by taking derivatives. For example when one derivative is taken, and the $A^{\lambda \mu }$ depend only on the unknowns $W_s$ but not on their derivatives, we arrive at a system of the form (2) with $W_{s\nu }=\partial W_s/\partial x^\nu$ in the role of the unknowns $u_s$,

$$\begin{aligned} B^{r\lambda \mu }_{s\nu }=\frac{\partial A^{\lambda \mu }}{\partial W_t}W_{t\nu }\delta ^r_s +\frac{\partial f_s}{\partial W_{r\lambda }}\delta ^\mu _\nu \end{aligned}$$

in the role of the coefficients $B^{r\lambda }_s$ of the next-to-principal terms, and

$$\begin{aligned} f_{s\nu }=\frac{\partial f_{s}}{\partial W_{r}}W_{r\nu } \end{aligned}$$

in the role of the inhomogeneous terms $f_s$. Note that the next-to-principal coefficients $B^{r\lambda \mu }_{s\nu }$ depend only on the original unknowns $W_s$ and their first derivatives $W_{s\nu }$. Taking additional derivatives the form of the equation does not change with the next-to-principal coefficients depending again only on the original unknowns $W_s$ and their first derivatives $W_{s\nu }$, only the inhomogeneous terms now depend on up to the mth derivatives of the original unknowns if a total of m derivatives are taken so that $W_{s\nu _1...\nu _m}=\partial ^m W_s/\partial x^{\nu _1}\cdot \cdot \cdot \partial x^{\nu _m}$ constitute the new unknowns. For the linear theory, to be presently discussed, to be applied it is required that the principal coefficients $A^{\lambda \mu }$ be 4 times continuously differentiable and the next-to-principal B coefficients be 2 times continuously differentiable. This is why one must consider the derived system for the 4th derivatives $W_{s\alpha \beta \gamma \delta }$ of the original unknowns. In the somewhat more general case addressed in the short article, where the $A^{\lambda \mu }$ depend also on the first derivatives of the $W_s$, an additional derivative must be taken and the derived system for the 5th derivatives of the $W_s$ must be considered. In the following I shall focus on the simpler case. There are trivial equations connecting the lower order derivatives of the $W_s$ and $W_s$ themselves to the top order derivatives, and these equations, in integral form, constitute the 1st group of integral equations listed in the short article.

The main integral equation is that of the 3rd group. This is a generalized Kirchhoff formula for the solution $u_s$ of the linear system (2), applied to the case where the top derivatives $W_{s\alpha \beta \gamma \delta }$ of the solution $W_s$ of the nonlinear system (1) are placed in the role of $u_s$. The formula expresses $u_s$ at an arbitrary point $M_0$ in the spacetime domain in terms of an integral on $\Sigma _0$, the past null geodesic cone of $M_0$ truncated in the past by the initial hypersurface $x^4=0$. The formula expresses $u_s$ at the vertex $M_0$ as the sum of a volume integral on $\Sigma _0$ and a surface integral on the intersection of $\Sigma _0$ with the initial hypersurface $x^4=0$. The surface integral involves the initial data, while the volume integral involves, besides the inhomogeneous term $f_s$, also the solution $u_s$ itself, but, what is essential, not the first derivatives of the solution. This is essential because even in the case of the classical inhomogeneous wave equation $\square \phi +\rho =0$ it is impossible to estimate pointwise the 1st derivatives of $\phi$ on the basis of a bound on $\rho$ not involving the 1st derivatives of $\rho$. The fact that the integrand in the volume integral in the right hand side of the generalized Kirchhoff formula does not involve the 1st derivatives of $u_s$ is what allows an iteration method using pointwise estimates as in Yvonne’s paper to work, producing the solution as the limit of a sequence of iterates where each new iterate is defined by taking the right hand side in the generalized Kirchhoff formula to be that determined by the previous iterate. In view of the fact that the solution $u_s$ appears also on the right hand side, Yvonne’s generalized Kirchhoff formula is not really a representation formula for the solution of the linear system (2). An actual representation formula follows from the work of Hadamard, which proceeds by successive approximations requiring a progressively higher degree of differentiability of the coefficients $A^{\lambda \mu }$ and $B^{r\lambda }_s$ at each successive step of the approximation. The exact formula obtained in the limit also involves an integral in the interior of the past characteristic conoid, and requires infinite order differentiability of the coefficients, a fact which prohibits application to nonlinear problems. It seems that it was Sobolev who first proposed, in the simpler context of a single 2nd order hyperbolic equation, a formula of the kind described above, which only requires finite order differentiability of the coefficients, and Yvonne gives him credit in her short article as well as in her longer paper.

Since the generalized Kirchhoff formula involves integration on the past characteristic conoid and this is generated by the past-directed bicharacteristics, that is the past-directed null geodesics of the Lorentzian metric whose reciprocal is $A^{\lambda \mu }$, issuing from the vertex $M_0$, the equations obeyed by these null geodesics naturally enter the setup. These are the canonical equations

$$\begin{aligned} \frac{dx^\mu }{d\lambda }=\frac{\partial H}{\partial p_{\mu }},\quad \frac{dp_\mu }{d\lambda }=-\frac{\partial H}{\partial x^\mu } \end{aligned}$$

(3)

associated to the Hamiltonian

$$\begin{aligned} H=\frac{1}{2}A^{\mu \nu }p_\mu p_\nu \end{aligned}$$

(4)

and we are considering the Hamiltonian flow on the zero level set of the Hamiltonian, as we are considering null geodesics. The evolution parameter $\lambda$ is then an affine parameter. Yvonne finds it convenient, capitalizing on the fact that the Hamiltonian is homogeneous in the momenta, to introduce rescaled momenta ${\hat{p}}_\mu =p_\mu /p_4$ so that ${\hat{p}}_4=1$, and a rescaled Hamiltonian

$$\begin{aligned} {\hat{H}}=\frac{1}{2}A^{\mu \nu }{\hat{p}}_{\mu }{\hat{p}}_{\nu } \end{aligned}$$

(5)

Introducing also a rescaled evolution parameter ${\hat{\lambda }}$ by the condition $d{\hat{\lambda }}/d\lambda =p_4$ Eq. (3) take the form:

$$\begin{aligned} \frac{dx^\mu }{d{\hat{\lambda }}}=A^{\mu 4}+A^{\mu j}{\hat{p}}_j,\quad \frac{d{\hat{p}}_i}{d{\hat{\lambda }}}=-\frac{\partial {\hat{H}}}{\partial x^i}+ {\hat{p}}_i\frac{\partial {\hat{H}}}{\partial x^4} \end{aligned}$$

(6)

the Latin indices taking the values 1, 2, 3. The condition that we are on the zero level set of the Hamiltonian need only be imposed at the vertex $M_0$, where by a suitable linear transformation of the coordinates we can assume that the coordinate vectorfields $\partial /\partial x^\mu : \mu =1,...,4$ constitute at $M_0$ an orthonormal frame with $\partial /\partial x^4$ at $M_0$ being the timelike vector. Then the condition that ${\hat{H}}$ vanishes at $M_0$ reduces to:

$$\begin{aligned} \sum _{i=0}^3({\hat{p}}^{0}_{i})^2=1 \,, \end{aligned}$$

(7)

where ${\hat{p}}^{0}_{i}$ denote the momenta at $M_0$. These therefore define a point on the unit sphere, the direction of issue of the null geodesic, hence correspond to spherical angles $\lambda _2,\lambda _3$. In Yvonne’s work the ${\hat{p}}_i$ and ${\hat{p}}^{0}_{i}$ are denoted simply by $p_i$ and $p^0_i$ respectively and ${\hat{\lambda }}$ is denoted by $\lambda _1$. I shall follow Yvonne’s notation from this point on. Equation (6) imply equations for the partial derivatives with respect to the initial parameters $p^i_0$ of up to 3rd order, $\partial x^i/\partial p^0_j$, $\partial ^2 x^i/\partial p^0_j\partial p^0_k$, $\partial ^3 x^i/\partial p^0_j\partial p^0_k\partial p^0_l$, denoted $y^i_j$, $y^i_{jk}$, $y^i_{jkl}$ respectively, and $\partial p_i/\partial p^{0}_j$, $\partial ^2 p_i/\partial p^{0}_j\partial p^{0}_k$, $\partial ^P{3} p_i/\partial p^0_j\partial p^0_k\partial p^0_l$, denoted $z^i_j$, $z^i_{jk}$, $z^i_{jkl}$ respectively. The corresponding integral equations constitute the equations of part a of the 2nd group listed in the short article.

The above quantities enter the generalized Kirchhoff formula as we shall presently see. This formula is derived in Sects. 4–20 (pp. 6–24) of the longer paper, where also the properties of the functions entering this formula are derived. In the derivation of the formula the spatial coordinates $x^1,x^2,x^3$ are used in place of the parameters $\lambda _1,\lambda _2,\lambda _3$ as coordinates on $\Sigma _0$ and the restriction to $\Sigma _0$ of a function $\phi$ defined on the spacetime domain is denoted by $[\phi ]$. Denoting the left hand side of Eq. (2) by $E_s$ so that these equations read $E_s=0$, Yvonne considers $\sigma ^r_s[E_s]$ where the $\sigma ^r_s$ are certain auxiliary functions on $\Sigma _0$ to be chosen, and shows that, in general,

$$\begin{aligned} \sigma ^r_s[E_r]=\frac{\partial E^{i}_s}{\partial x^i}+[u_r]L^r_s+\sigma ^r_s f_r -\left[ \frac{\partial u_r}{\partial x^{4}}\right] D^r_s . \end{aligned}$$

(8)

Here

$$\begin{aligned} E^i_s&=[A^{ij}]\sigma ^r_s\frac{\partial [u_r]}{\partial x^{j}} -[u_r]\frac{\partial }{\partial x^j}([A^{ij}]\sigma ^r_s) +2\sigma ^r_s([A^{ij}]p_j\nonumber \\&\quad +[A^{i4}])\left[ \frac{\partial u_r}{\partial x^4}\right] +[B^{ti}_r][u_t]\sigma ^r_s \end{aligned}$$

(9)

are the components of vectorfields on $\Sigma _0$, so the first term on the right in (8) is a divergence. Note that the $E^i_s$ depend on the unknowns $u_s$ and their first derivatives, including their transversal derivatives $\partial u_s/\partial x^4$ on $\Sigma _0$. Moreover,

$$\begin{aligned} L^r_s=\frac{\partial ^2([A^{ij}]\sigma ^r_s)}{\partial x^i\partial x^j} -\frac{\partial }{\partial x^i}([B^{ri}_t]\sigma ^t_s) \end{aligned}$$

(10)

depend on up to the 2nd derivatives of the auxiliary functions $\sigma ^r_s$, and

$$\begin{aligned} D^r_s&=2([A^{ij}]p_j+[A^{i4}])\frac{\partial \sigma ^r_s}{\partial x^i} \nonumber \\&\quad +\left\{ 2\frac{\partial }{\partial x^i}([A^{ij}]p_j+A^{i4}]) -[A^{ij}]\frac{\partial p_j}{\partial x^i}\right\} \sigma ^r_s \nonumber \\&\quad -([B^{r4}_t]+[B^{ri}_t]p_i)\sigma ^t_s . \end{aligned}$$

(11)

Crucial for the construction is the choice of the auxiliary functions $\sigma ^r_s$ so as to satisfy

$$\begin{aligned} D^r_s=0 . \end{aligned}$$

(12)

Now, by the first of Eq. (6) the vectorfield

$$\begin{aligned} T^i\frac{\partial }{\partial x^i}:=([A^{ij}]p_j+[A^{i4}])\frac{\partial }{\partial x^i}=\frac{d}{d\lambda _1} \end{aligned}$$

(13)

is the tangent field to the null geodesic generators of $\Sigma _0$, thus Eq. (12) constitutes an ordinary differential equation for the auxiliary functions $\sigma ^r_s$ along the generators of $\Sigma _0$. This equation being satisfied, the solutions of Eq. (2) satisfy on $\Sigma _0$

$$\begin{aligned} \frac{\partial E^i_s}{\partial x^{i}}+[u_r]L^r_s+\sigma ^r_s f_r=0 \end{aligned}$$

(14)

the dependence on the first derivatives of the unknowns $u_s$, except through the divergence term, having been eliminated. The functions $\sigma ^r_s$ are factored into

$$\begin{aligned} \sigma ^r_s=\sigma \omega ^r_s, \end{aligned}$$

(15)

the function $\sigma$ is subjected to a simple but singular equation

$$\begin{aligned} \frac{d\sigma }{d\lambda _1}+\frac{1}{2}\sigma \frac{\partial T^i}{\partial x^i}=0 \end{aligned}$$

(16)

while the functions $\omega ^r_s$ are subjected to the more complicated but regular equations

$$\begin{aligned} \frac{d\omega ^{r}_s}{d\lambda _1}+\frac{1}{2}\omega ^r_s\left( p_j\frac{\partial [A^{ij}]}{\partial x^i} +\frac{\partial [A^{i4}]}{\partial x^i}\right) -\frac{1}{2}\omega ^t_s([B^{ri}_t]p_i+[B^{r4}_t])=0. \end{aligned}$$

(17)

The general solution of (16) is

$$\begin{aligned} \sigma =\frac{f(\lambda _2,\lambda _3)}{|\triangle |^{1/2}} \,, \end{aligned}$$

(18)

where $\triangle$ is the Jacobian determinant

$$\begin{aligned} \triangle =\frac{\partial (x^1,x^2,x^3)}{\partial (\lambda _1,\lambda _2,\lambda _3)}. \end{aligned}$$

(19)

Since $\triangle$ tends to zero as $\lambda _1\rightarrow 0$, that is as we approach the vertex $M_0$, $\sigma$ blows up as we approach $M_0$. Choosing $f(\lambda _1, \lambda _2)=\sin \lambda _2$ fixes $\sigma$. The auxiliary functions $\omega ^r_s$ are fixed by imposing the condition that $\omega ^r_s\rightarrow \delta ^r_s$ as $\lambda _1\rightarrow 0$. Equation (17) imply equations for the partial derivatives with respect to the initial parameters $p^0_i$ up to 2nd order $\partial \omega ^r_s/\partial p^0_i:=\omega ^r_{si}$, $\partial ^2\omega ^r_s/\partial p^0_i\partial p^0_j:=\omega ^r_{sij}$. The corresponding integral equations constitute the equations of part b of the 2nd group of equations listed in the short article.

Integrating (14) on the portion of $\Sigma _0$ below the hypersurface $x^4=x^4_0-\eta$ where $x^{4}_0$ is the value of $x^{4}$ at $M_0$ and then taking the limit $\eta \rightarrow 0$ in which the portion left out contracts to $M_0$, the generalized Kirchhoff formula results:

$$\begin{aligned} 4\pi u_s(M_0)&=\int \int \int _V ([u_r]L^r_s+\sigma ^r_s[f_r])\triangle d\lambda _1 \, d\lambda _2 \, d\lambda _3 \nonumber \\&\quad +\int _0^{2\pi }\int _0^\pi \left\{ \frac{E^{i}_s p_i\triangle }{|T|^{4}}\right\} _{x^{4}=0} d\lambda _1 \, d\lambda _2. \end{aligned}$$

(20)

Here V is the parameter domain corresponding to $\Sigma _0$, the surface integral is on the trace of $\Sigma _0$ on the initial hypersurface $x^4=0$, and |T| denotes the Euclidean magnitude of the vectorfield T. From (9) $\left\{ E^i_s\right\} _{x^4=0}$ depends on the initial data for $u_s$, that is the restriction of $u_s$ to the initial hypersurface $x^4=0$, its 1st derivative, and the transversal derivative $\partial u_s/\partial x^4$ along the initial hypersurface. The formula (20) being applied placing the 4th derivatives $W_{s\alpha \beta \gamma \delta }$ of the original unknowns $W_s$ in the role of $u_s$, the surface integral involves the 5th derivatives of the functions $\phi _s$ and the 4th derivatives of the functions $\psi _s$, these functions being the initial data for the original system (1), namely the restriction of $W_s$ and of its transversal derivative $\partial W_s/\partial x^4$ to the initial hypersurface $x^4=0$. The assumptions of the existence theorem as stated in Sect. 31 of the longer paper accordingly require $\varphi _s$ and $\psi _s$ to possess continuous derivatives up to order five and four respectively and the conclusion for the solution $W_s$ is that it possesses continuous derivatives up to order four. Thus a loss of one order of differentiability is involved. I shall come back to this point below.

In regard to the application to general relativity, the short article gives only a brief discussion in the first paragraph, mentioning the role of harmonic coordinates (which are called isothermal in both articles) in reducing the Einstein equations to standard hyperbolic form, but as to how it is proved that a solution of the reduced equations satisfies the harmonic condition, hence is a solution of the original equations, only a hint is given in the statement that the uniqueness theorem allows this proof. The longer article devotes the last chapter to the application to general relativity, stating the crucial fact that modulo the reduced equations the transversal derivative of the harmonic condition on the initial hypersurface is equivalent to the constraint equations on the initial data, and giving a complete proof of the theorem that given initial data for the Einstein equations, that is first and second fundamental form of the initial hypersurface, satisfying the constraint equations, the data extends to data for the reduced equations in such a way that the harmonic condition is satisfied initially. Then the corresponding solution of the reduced equations satisfies, by the aforementioned equivalence, also the transversal derivative of the harmonic condition initially, hence by the uniqueness theorem hinted at in the short article, satisfies the harmonic condition throughout the domain of existence of the solution, therefore we have a solution of the actual Einstein equations.

I now come to the point about the loss of one degree of differentiability in evolution which I mentioned above, and which is not discussed even in the longer paper. Some members of the mathematical community might have considered this loss an unsatisfactory feature and would have considered Yvonne’s result as provisional, while acknowledging the importance of her work given the state of knowledge at that time. The loss of differentiability is in fact unavoidable and occurs even for the classical wave equation in more than one spatial dimension (more than two independent variables). This is explained in the Courant & Hilbert classical treatise Methods of Mathematical Physics (1962), Volume II, Chapter VI “Hyperbolic Differential Equations in More Than Two Independent Variables”. The relevant section is §10.4 “Focussing. Example of Nonpersistence of Differentiablity” (pp. 673–674). The point is that the $C^k$ classes of functions with continuous derivatives up to the kth order are not preserved in evolution, and what are preserved are the classes of functions with finite kth order energy norms, which correspond to functions with square integrable derivatives of up to the kth order. For this essential insight Courant & Hilbert gives credit to Kurt Otto Friedrichs and Hans Lewy in their paper “Über die Eindeutigkeit und das Abhängigkeitsgebeit der Lösungen beim Anfangswertproblem linearer hyperbolischer Differentialgleichungen” (“On the uniqueness and domain of dependence of the solutions to the initial value problem for linear hyperbolic differential equations”), Math. Ann. 98, 192–204 (1928). This is the reason why Yvonne’s approach, which is based on $C^k$ classes, despite the advantage of being more constructive, was abandoned soon after Jean Leray’s memoir “Lectures on Hyperbolic Equations” (Institute for Advanced Study, Princeton, 1952) appeared. It should be noted that this memoir was not a book but a bound set of typed notes that was available only in the leading mathematical libraries. This could be circulated immediately after typing and was not subject to the usual waiting time in mathematics of two years either for a journal article or for a monograph. In his memoir Leray addressed the most general quasilinear hyperbolic system (although he had to introduce a condition called “strict hyperbolicity” which ruled out most physical applications) and established local existence by the method of energy estimates. Yvonne attended the lectures given by Leray in Princeton, having, after her thesis, been offered by Leray himself a position as his assistant, as recounted in the accompanying biography, for the academic years 1951–1952 and 1952–1953.

Energy estimates were in fact first employed in 1935 by Juliusz Schauder, an outstanding Jewish mathematician from Poland and friend of Leray who was executed by the Nazis during World War II, to establish the local existence theorem for a quasilinear hyperbolic equation in more than two independent variables. His paper “Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichung zweiter Ordnung in beliebiger Anzahl von unabhängigen Veränderlichen” (“The initial value problem for a quasilinear hyperbolic equation of second order in any number of independent variables”), Fundam. Math. 24, 213–246 (1935), is acknowledged as a major contribution by Courant & Hilbert (p. 675).

However, Schauder’s work concerned a single equation rather than a system of equations, and the intervening period until 1950 saw no major developments in this topic, possibly because of the war, so when Yvonne composed her work there was no mathematical work in existence that could address the Einstein equations. This gives the measure of originality of her work.

There is however more to say: while her method was superseded as an approach to proving local existence of solutions, it eventually bore fruit in the discovery of a breakdown criterion for the Einstein equations by Sergiu Klainerman and Igor Rodnianski in the paper “On a breakdown criterion in general relativity”, J. Amer. Math. Soc. 23(2), 345–382 (2010), where they make essential use of her method. This paper considers a spacetime, solution of the Einstein vacuum equations, which is given as a smooth foliation $\cup _{t\in [0,t_*)}H_t$ (note that the value $t_*$ is not included) into maximal spacelike hypersurfaces $H_t$ such that with k being the 2nd fundamental form of the $H_t$, $\mathrm{tr}k=0$ expressing the maximality condition, and ${\overline{d}}\log \alpha$ the differential of the restriction to the $H_t$ of the logarithm of the lapse function $\alpha$ of the foliation, $\Vert k\Vert _{L^\infty (H_t)}$ and $\Vert {\overline{d}}\log \alpha \Vert _{L^\infty (H_t)}$, functions of t, are bounded on $[0,t_*)$, and shows that the spacetime extends smoothly as a solution of the Einstein vacuum equations continuing the maximal foliation to $t\in [t_*, t_*+\varepsilon ]$ for some $\varepsilon >0$. This means that the solution cannot break down unless one or both of $\Vert k\Vert _{L^\infty (H_t)}$, $\Vert {\overline{d}}\log \alpha \Vert _{L^\infty (H_t)}$ blow up as t approaches $t_*$. The crucial step in the proof is a pointwise bound on the spacetime curvature obtained using Yvonne’s method.

2 Biography of Yvonne Choquet-Bruhat (formerly Fourès-Bruhat)

by Richard Kerner

The renowned mathematician and relativist Yvonne Suzanne Marie-Louise Choquet-Bruhat was born in Lille, on 29 December 1923, to Georges Bruhat and his wife Berthe née Hubert. Georges Bruhat was a respected university professor who wrote a universally praised multi-volume physics treatise, in use until today; her mother was a high-school teacher of literature. In 1932, Georges Bruhat became professor at the Sorbonne University, and Yvonne’s family – the parents and their three children, Yvonne, her older sister Jeanne and younger brother François, moved to Paris.

The war which France and Great Britain declared on Germany started officially on 3 September 1939, two days after Germany invaded Poland, but no military response followed. Nevertheless the Bruhat family moved to Bordeaux, then to Poitiers, where Yvonne continued high school; finally, after the French surrender the next year, they returned to Paris, where Georges Bruhat took on the directorship of the École Normale Supérieure de la rue d’Ulm. During the following years he protected numerous Jewish students, arranging their transfer to the unoccupied zone, and covered others who went to England to join the Allied armed forces.

Yvonne entered the École Normale de Sèvres, a sister establishment of the École Normale of Paris in 1943 reserved for young women, where she became one of the most brilliant students. She also followed lectures given both in the Institut Henri Poincaré, and at Sèvres, by prominent teachers, including the great Elie Cartan, and gained a taste for high level mathematics.

Tragedy struck the Bruhat family in 1944, at the very end of the occupation: German police (the “Gestapo”) were looking for several École Normale students who were involved in the French Resistance, and ordered Georges Bruhat to reveal their whereabouts, which he refused to do. He was deported to the sinister camp in Buchenwald, then to Sachsenhausen, where he perished due to the inhuman treatment; even his burying place is unknown. The dramatic loss of her father who served as guide and example, marked Yvonne for life; she vowed to be worthy of his memory.

One of her father’s students, Léonce Fourès, who was among the ones involved in the resistance movement at the École Normale, became her close friend and lover; they married in 1946 and spent their honeymoon in Chamonix in the French Alps. Both were following the lectures by Jean Leray, the great specialist in partial differential equations of hyperbolic type. At the same time, Yvonne read André Lichnerowicz’ books: “Tensor calculus” and “Global problems of relativistic mechanics”, and decided to write her doctoral thesis under Lichnerowicz’s direction, thus becoming his first PhD student. There were many open problems in her Master’s work, which she brilliantly solved. Her first publication was a short note in the Comptes Rendus de l’Académie des Sciences, “Théorème de Gauss en Relativité Générale” (1948); three others followed, two co-authored by Lichnerowicz, all developing the global approach to Einstein’s equations. Global existence and uniqueness theorems became the principal subject of Yvonne’s thesis, in which she developed a new approach to Einstein’s equations, symbolically called 3 + 1, considering them as a hyperbolic system describing time development of 3-dimensional geometries, long before the similar approach by Arnowitt, Deser and Misner. Yvonne’s results were important enough to deserve publication in extenso [1, 2]. Her work in this area is the subject of this editorial note.

In December 1950, Yvonne gave birth to her first daughter Michelle who became an ecologist. Soon after, she and her husband brilliantly defended their doctoral theses and got teaching positions in Paris. Jean Leray, who was a universally respected mathematician, received a proposal for a permanent position in Princeton. He declined the permanent job, but obtained instead a temporary position for the next five years, spending only a semester in Princeton each year. He also had the privilege of choosing his teaching assistants, and proposed Yvonne and Léonce to follow him. They eagerly agreed, and spent the following academic years (1951–1952; 1952–1953) in Princeton.

Yvonne was presented to Einstein, who was happy to learn about her results and encouraged her to continue in the chosen direction. She had several unforgettable discussions with the great man. The two years’ long stay in Princeton profoundly influenced Yvonne’s career. She met several great scientists like Robert Oppenheimer and John von Neumann.

Back in France after two years spent (with two months of vacations in France) in Princeton, Yvonne and Léonce were offered associate professorships in Marseille, and moved with their daughter there. Life continued, with important teaching duties, but also with successful research and participation at many scientific congresses where Yvonne’s results were presented and appreciated by her peers and renowned scientists, many of whom befriended her. After the first gathering in Bern in 1955, there followed in 1957 the first great International Congress on relativity organized at the Chapel Hill University by Bryce and Cécile De Witt. Yvonne was invited along with her mentor Lichnerowicz and Mme. Tonnelat, the former student of Louis de Broglie. She met many outstanding relativists, including Einstein’s assistant Peter Bergmann, John Archibald Wheeler and his student Charles Misner, and Stanley Deser – all of whom became her friends from then on.

Other international travels followed, in particular the first Congress of Romance language speaking mathematicians held in 1957 in Bucharest. However, Léonce and Yvonne’s interests in everyday life and career started to diverge and the marriage was becoming unstable. At the same time their common friend and mentor, Gustave Choquet, developed a closer relation with Yvonne. Both divorced, not without difficulty for Choquet, but at the end of 1961 the marriage with Choquet, already a renowned mathematician, could be celebrated.

The newlyweds bought a house in the southern suburb of Paris, Parc de Sceaux, Antony, which became their family residence for the next 50 years. Two children were born from this union, a son Daniel who is now a well-known biologist, head of an important research laboratory in Bordeaux and a member of the Academy of Sciences, and daughter Geneviève who is a successful medical doctor in Lyon.

In 1965, Yvonne got a professorship at the Mechanics Department of the Sorbonne University, where she remained until her retirement in 1989. With thus stabilized family and professional life, Yvonne could devote herself to her beloved science with doubled enthusiasm. An important milestone was participation in the Battelle Rencontres, organized by Bryce and Cécile De Witt, where she presented new results obtained by applying global hyperbolicity concepts elaborated by Jean Leray to the system of Einstein’s equations. The theorems of uniqueness and treatment of singularities generalized and clarified the heuristic constructions of Geroch, Penrose and Hawking, all present at the meeting, alongside the great geometers Bott and Helgason.

During the next decades Yvonne Choquet-Bruhat’s scientific interests constantly evolved, as well as the circle of her collaborators. In 1965, she applied the Leray–Ohya non-strict hyperbolicity definition to the analysis of the relativistic generalization of the Navier–Stokes equations. In the next few years, after meeting Joe Weber who promoted experimental detection of gravitational waves, her attention was drawn to this phenomenon: a Taylor expansion of Einstein’s equations in vacuo was performed with respect to the small parameter which was the inverse of frequency. This method enabled Yvonne to find approximate plane wave solutions describing propagation of high frequency gravitational waves. The stability of gravity wave solutions was also investigated.

The analysis of the mathematical description of relativistic fluids led quite naturally to the relativistic generalization of Boltzmann’s equation, to which Yvonne devoted a series of articles written in collaboration with her PhD students. Another bunch of physical theories to which rigorous methods of functional analysis could be applied were the Maxwell–Einstein system and Yang–Mills fields, which were treated by Yvonne with her usual invention and mathematical precision. The analysis of conserved constraints was of particular interest for the subsequent developments and applications of Einstein’s gravitation.

From 1968 on, she headed the university research group which was also affiliated to the CNRS, the “Laboratoire de Mécanique Relativiste”, whose members were mathematicians and physicists, including André Avez, Charles-Michel Marle, Philippe Droz-Vincent, Lise Lamoureux-Brousse, Richard Kerner, Carlos Moreno, Bartolomé Coll, and occasionally post-doctoral visitors and PhD students. The two faithful secretaries, Marie-Pierre Delègue and Denise Goudmand, ensured smooth and efficient functioning and cohesion. The weekly seminar led in the Collège de France jointly with André Lichnerowicz became a focal point of General Relativity and Mathematical Physics, reinforced by the annual meetings “Journées Relativistes”.

After Yvonne’s official retirement the Laboratoire was dissolved in 1991: several members including Yvonne, who became Emeritus Professor, migrated to the “Laboratoire de Gravitation et Cosmologie Relativistes” headed by Ph. Tourrenc and from 1991 on by me, until its dismantling in 2001. The ultimate scientific haven for Yvonne was provided by Thibault Damour in the prestigious Institut des Hautes Études Scientifiques (IHES) in Bures, easily accessible from Yvonne’s home in Antony by suburban train. After her husband Gustave Choquet died in 2006 she continued to live in their house in Antony, but in 2016 she moved to her eldest daughter Michelle’s house in central France, and in 2018 to a home for the elderly.

In 1979 she was elected to the French Academy of Sciences, becoming the first female full member of this venerable Institution. She was awarded the most important French distinction, the Légion d’Honneur, promoted “Grand Commandeur” (the highest rank) in 2012. From 1980 to 1983 she was President of the International Society on General Relativity and Gravitation.

Having become a renowned teacher, Yvonne had numerous students and pupils. The list of Yvonne’s students and collaborators impresses by diversity and quality. Among her former students we find Alice Chaljub-Simon, Lise Lamoureux-Brousse, Daniel Bancel, Norbert Noutchegeme and Richard Kerner; the post-docs Piotr Chruściel and Pawel Nurowski. Close collaborators and co-authors of many important works comprise James W. York, James Isenberg, Arthur Fischer, Vincent Moncrief and Jerrold Marsden from the USA; Antonio Greco and Tommaso Ruggeri from Italy; Demetrios Christodoulou and Spiros Cotsakis from Greece, and many others. Her influence which has imbued General Relativity with rigorous methods of modern mathematics is still visible in the works of the present generation, e.g. in the excellent contributions by Sergiu Klainerman of Princeton.

A tireless traveler, Yvonne Choquet-Bruhat visited universities and research centers all over the world, on five continents, and had not only excellent collaborators, but also good and genuine friends. To enumerate them all would take several pages; let us mention Abraham Taub in Berkeley, Jules Géhéniau and Michel Cahen in Brussels, Giorgio Ferrarese and Remo Ruffini in Rome, Walter Thirring and Peter Aichelburg in Vienna, Andrzej Trautman in Warsaw, Jürgen Ehlers in München, Irving Segal in MIT, Olga Oleinik and Nail Ibragimov in Moscow, Gu Chao Hao and Hu Hesheng in Shanghai, Roy McLenaghan in Waterloo, Canada, Robert Bartnik in Canberra, and many others.

How highly Yvonne was appreciated by her peers can be measured by the number of workshops and symposia organized in her honor: after retirement in 1991 a full size symposium was held in the Collège de France, with her friends delivering excellent talks which appeared in the proceedings [3]. Her 80th birthday was celebrated in 2002 and 2003 by a workshop organized by Jean-Pierre Bourguignon and Thibault Damour at the IHES, and by her Italian friends on the island of Elba. The last workshop in her honor took place in IHES at her 90th anniversary in 2013.

Her autobiographical book [4] was written and published when she was more than 90 years old, and is available in English [5].