1.1. From the Preface.

The aim of our work is to provide a proof of the nonlinear gravitational stability of Minkowski space-time. More precisely, we accomplish the following goals: (1) We provide a constructive proof of global, smooth, nontrivial, solutions to the Einstein vacuum equations which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. (2) We provide a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions, and give a rigorous derivation of the laws of gravitational radiation proposed by Bondi. (3) We obtain these solutions as dynamic developments of all initial data sets which are close, in a precise sense, to the initial data set of the Minkowski space-time, and thus establish the global dynamic stability of the latter. (4) Though our results are established only for developments of initial data sets which are uniformly close to the trivial one, they should in fact be valid in the complement of the domain of influence of a sufficiently large compact subset of the initial manifold of any 'strongly asymptotically flat' initial data set. We plan in fact to prove such a theorem in the future.

2.1. From the Publisher.

The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.

2.2. From the Introduction.

The aim of this book is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, our work accomplishes the following goals:

1. It provides a constructive proof of global, smooth, nontrivial solutions to the Einstein-Vacuum equations. which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities.

2. It provides a detailed description of the sense in which these solutions are close to the Minkowski space-time in all directions and gives a rigorous derivation of the laws of gravitational radiation proposed by Bondi. It also describes our new results concerning the behaviour of the gravitational field at null infinity.

3. It obtains these solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the initial data set of the Minkowski space-time, and thus it establishes the global dynamic stability of the latter.

4. Though our results are established only for developments of initial data sets which are uniformly close to the trivial one, they are in fact valid in the complement of the domain of influence of a sufficiently large compact subset of the initial manifold of any "strongly asymptotically flat" initial data set.

2.3. Review by: Alan D Rendall.
Mathematical Reviews MR1316662 (95k:83006).

This book presents the authors' theorem on the stability of Minkowski space, a landmark in the development of mathematical relativity. ... The main statement of the theorem is, informally, that, given any initial data set for the vacuum Einstein equations which is sufficiently close to the initial data induced on a hyperplane in Minkowski space, there exists a corresponding solution which is global in the sense of being geodesically complete, and whose asymptotic structure resembles that of Minkowski space. In the book there are three statements of versions of the main theorem which are increasingly precise (and technical). The first two are contained in Chapter 1 (the introduction) while the third is contained in Chapter 10, where the highest level steps of the proof are carried out. The book is not easy to read, due to the very technical nature of its contents, but under the circumstances the quality of the exposition is excellent.

2.4. Review by: Volker Perlick.
General Relativity and Gravitation 32 (4) (2000), 761-763.

For Einstein's vacuum field equation, it is a difficult task to investigate the existence of solutions with prescribed global properties. A very interesting result on that score is the topic of the book under review. The authors prove the existence of globally hyperbolic, geodesically complete, and asymptotically flat solutions that are close to (but different from) Minkowski space. These solutions are constructed by solving the initial value problem associated with Einstein's vacuum field equation. More precisely, the main theorem of the book says that any initial data, given on R3, that are asymptotically flat and sufficiently close to the data for Minkowski space give rise to a solution with the desired properties. In physical terms, these solutions can be interpreted as spacetimes filled with source-free gravitational radiation. Geodesic completeness means that there are no singularities. At first sight, this theorem might appear intuitively obvious and the enormous amount of work necessary for the proof might come as a surprise. The following two facts, however, should caution everyone against such an attitude. First, it is known that there are nonlinear hyperbolic partial differential equations (e.g., the equation of motion for waves in non-linear elastic media) for which even arbitrarily small localised initial data lead to singularities. Second, all earlier attempts to find geodesically complete and asymptotically flat solutions of Einstein's vacuum equation other than Minkowski space had failed. In the class of spherically symmetric spacetimes and in the class of static spacetimes the existence of such solutions is even excluded by classical theorems. These facts indicate that the theorem is, indeed, highly non-trivial. Yet even in the light of these facts it is still amazing that the proof of the theorem fills a book of about 500 pages. To a large part, the methods needed for the proof are rather elementary; abstract methods from functional analysis are used only insofar as a lot of L2 norms have to be estimated. What makes the proof involved and difficult to follow is that the authors introduce many special mathematical constructions, involving long calculations, without giving a clear idea of how these building-blocks will go together to eventually prove the theorem. The introduction, almost 30 pages long, is of little help in this respect. Whereas giving a good idea of the problems to be faced and of the basic tools necessary to overcome each problem, the introduction sheds no light on the line of thought along which the proof will proceed. For this reason the reader is likely very soon to get lost in a jungle of mathematical details without seeing the thread of the story. This is exactly what happened to the reviewer.

3.1. From the Publisher.

This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Demetrios Christodoulou is Professor of Mathematics at Princeton University. He has been awarded a John and Catherine MacArthur Fellowship, as well as a John Simon Guggenheim Fellowship. His previous book The Global Nonlinear Stability of the Minkowski Space (Princeton), cowritten with Sergiu Klainerman, won the Böcher Memorial Prize of the American Mathematical Society.

3.2. From the General Introduction.

The principle of stationary action, in the form in which it arose in classical mechanics in the work of Lagrange, led to the discovery of symplectic geometry and the development of Hamiltonian methods which are indispensable in the study of the equations of motion.

In classical mechanics the domain of the unknowns is the real line of time, the action is a single integral and the Euler-Lagrange equations are ordinary differential equations. The principle of stationary action was subsequently generalised to the case that the domain of the unknowns, the manifold of independent variables, is multi-dimensional, the action is a multiple integral, and the Euler-Lagrange equations are partial differential equations.

This generalisation originates in Lagrange's derivation of the partial differential equation which a function of two variables must satisfy so that its graph is a surface of least area in Euclidean space. Dirichlet's principle provided a simpler example of an action as a multiple integral, which stimulated the development of direct methods of the calculus of variations for several independent variables, beginning with the work of Hilbert. This development led to the solution of the problem of surfaces of least area by Douglas and Radó.

In the mean time Hilbert had extended the principle of stationary action to the case that the domain manifold is the four dimensional manifold of space-time and derived from a particular form of this principle Maxwell's equations for the electromagnetic field. Hilbert's work in this direction culminated with his discovery of the action principle which governs the geometry of the space-time manifold itself, leading to Einstein's equations of general relativity. The general form of the stationary action principle as envisioned by Hilbert, is at the present time a central unifying theme in theoretical physics.

Now, a great deal of deep work has been done in the calculus of variations in the case that the Euler-Lagrange equations are of elliptic type, in developing direct methods and in working out the regularity theory of solutions. Among the milestones in this development are Morrey's solution of the harmonic map problem from the unit disc to a Riemannian manifold, the first variational problem to be solved where the Euler-Lagrange equations constitute a non-linear system of partial differential equations, leading him directly to the solution of the problem of surfaces of least area in a Riemannian manifold. Also, the breakthrough in the regularity theory of more than two independent variables by De Giorgi and Nash, in the case of a single unknown function, which led, in particular, to the solution of the problem of hypersurfaces of minimal volume in higher dimensional spaces.

However, despite the remarkable progress, just outlined, in the elliptic case, the aspects of the action principle which are relevant to the case where the Euler-Lagrange equations are of hyperbolic type, the case occurring in physics, have been left largely undeveloped, with the notable exception of the principle connecting symmetries to conserved quantities propounded by Noether. The purpose of the present book is to contribute toward remedying this situation. Thus our aim is to introduce concepts and prove theorems which will be found useful in developing the theory of non-linear systems of hyperbolic type. For this reason we have completely left out all subject matter which pertains exclusively to the elliptic case. Nevertheless, since there is a number of concepts and theorems of a general nature, which apply equally well to the elliptic and hyperbolic cases in particular, we expound these in the first four chapters of the book.

After the introductory first chapter, the main developments expounded in this book are the following. First, in Chapter 2, symplectic geometry in the case of a multi-dimensional domain manifold is explored. In analogy with classical hydrodynamics, a theory of flows is developed. A flow corresponds to an n-parameter family of solutions of the Euler-Lagrange equations (n is the number of dependent variables), and the theory is an extension of the Hamilton-Jacobi theory of classical mechanics. In Chapter 3, a general theory of integral identities is developed, the theory of compatible currents, which extends the work of Noether. Whereas other methods, such as the maximum principle, are available for the treatment of elliptic equations, integral identities provide the only known general basis for approaching hyperbolic equations. In the development of the theory of compatible currents the great gulf between the case of two independent variables and the case of more than two independent variables becomes apparent. Chapter 4 deals with the case that the unknown is a section of a vector bundle over the domain manifold, rather than a mapping of the domain manifold into a target manifold. This is necessary for the developments of the last two chapters. Chapter 5 begins with our notion of hyperbolicity, which represents a significant departure from notions in the existing literature and suggests new methods for the solution of problems. We show how the new notion overcomes the difficulties associated with singularities of the characteristic variety. The causal structure on the domain manifold defined by a hyperbolic Lagrangian is then studied and the domain of dependence property of solutions is established. As is usual in this subject, the methods of the domain of dependence theorem lead readily to a local existence theorem, for given initial data. The results apply in particular to the theory of non-linear elasticity. We should note here that Leray's pioneering theory of strictly hyperbolic systems is applicable to non-linear elasticity only under certain restrictions. In fact, Fritz John has found a physical example to which Leray's theory does not apply. Moreover, Tahvildar-Zadeh has recently shown that Fritz John's example is stable within the framework of non-linear elasticity, thereby reinforcing its importance. In contrast to Leray's theory, our results apply without restrictions to the general framework of the theory of elasticity.

Finally, the last chapter deals with electromagnetic theory, the electrodynamics of a general non-linear continuous medium. Although electromagnetic theory may be considered to be a theory of sections of the cotangent bundle of the domain manifold, it cannot be reduced to the general theory of sections of vector bundles, for then the Legendre transformation which takes us from the Lagrangian to the Hamiltonian picture would be singular. This is a direct consequence of the requirement of gauge invariance and requires a reworking of all constructions. We have thus devoted a separate chapter to this theory, in view also of its physical significance. We establish results analogous to those of Chapters 3 and 5 (including the domain of dependence theorem) in the framework of non-linear electrodynamics. The results in this chapter are the first general results in this framework going beyond the linear approximation.

3.3. Review by: James Isenberg.
Mathematical Reviews MR1739321 (2003a:58001).

The use of an action principle as an alternative and more or less equivalent approach to working with a given system of partial differential equations is very familiar both to physicists working with classical and quantum field theories such as Maxwell's theory of electromagnetism or Einstein's theory of gravity, and to mathematicians working with geometrically motivated problems such as minimal surfaces or harmonic maps. The importance of this approach is familiar as well: action principles play a central part in relating symmetries and conservation laws ("Noether's theorem"), they play a key role in the proof of geometric results such as the Yamabe theorem, and they are indispensable for understanding how quantum field theories such as quantum electrodynamics are conceptualised and used.

This new book by Christodoulou is not the text I would recommend to a neophyte seeking a pedagogical introduction to action principles and their use in math and physics. Rather, it is a book of excursions into relatively unfamiliar and unexplored areas of their nature and application, led by a very experienced guide who doesn't tell you everything you might want to learn about, but instead uncovers and discusses with keen insight a number of very interesting aspects of action principles. It is not a book which rewards browsing. It is a book which requires careful study, and does reward hard work.

Many of the more important mathematical successes of the use of action principles to date have involved elliptic PDE systems, such as those noted above. While this book comments on such applications, the emphasis is more on their use with hyperbolic PDEs. This is not surprising, since the author is a leading expert in that field.

3.4. Review by: Piotr T Chrusciel.
General Relativity and Gravitation 35 (3) (2003), 499-501.

Most - if not all - important partial differential equations in physics derive from an action principle. This has useful consequences for the understanding of the structure of the theory: on a formal level this leads to various structures such as the symplectic one; on another, this allows one to obtain conserved currents via Noether's theorem. The key question addressed in the book by Demetrios Christodoulou is the following: can one infer existence and uniqueness of solutions of a set of variational equations by inspection the associated Lagrange function? The author announces a positive answer to this question in two cases:

1. a theory of maps between two manifolds - here the relevant physical problem is that of elastodynamics;

2. non-linear electrodynamics.

These are, in my view, the two most important results in this book.

The precise references to those results are the following: in Section 5.2 Christodoulou gives a definition of regularly hyperbolic Lagrangian. In Theorem 5.10 he proves, for such Lagrangians, uniqueness of solutions within appropriately defined domains of dependence. The resulting class of equations includes equations which were not covered by previous notions of hyperbolicity. Similarly in Definition 6.1 he defines a regularly hyperbolic Lagrangian in the context of non-linear electrodynamics. Theorem 6.1 provides the relevant uniqueness result.

One can only regret that no existence proofs have been given; in his introduction the author notices that this follows by standard arguments. However, there are several standard arguments proving existence for hyperbolic PDE's, not all of them applying simultaneously to all notions of hyperbolicity. Some indication how to proceed would dispel the worries about the associated technicalities which might arise in the mind of the reader.

The book also addresses, and (essentially) answers, the following important question: find all divergence identities which are implied by a given set of variational equations. Such identities provide a major tool in the studies of solutions, leading to local and global existence results, information about decay or blow- up, and so on. One such classical identity is the energy conservation law; two other examples are provided by the Pohozaev identity for the Laplace equation, and by its hyperbolic counterpart due to Morawetz. In Theorem 3.6 the author proves that for generic Lagrangians all such identities are the ones provided by Noether's theorem. This is an important no-go result, which will save many hours of work to those attempting to find such new identities. The theorem raises the following interesting question: do there exist non-generic Lagrangians which do possess other non-trivial divergence identities? The resulting equations, if any, would certainly be of interest both from a physical and mathematical point of view.

One can also regret that in the introductory material the author ignores the elegant symplectic framework of Kijowski and Tulczyjew, which has several advantages over the approach presented. As an example of simplifications which can be achieved we note the following: in his formalism Christodoulou uses tangent bundles to carry the information about the derivatives of the field. This leads to the need to introduce connections and, subsequently, to the need of proving that the constructions which have been performed are connection-independent. This problem does not even appear in the Kijowski-Tulczyjew formalism, where jet bundles are used instead. ...

Summarising, the book introduces new classes of well posed evolution equations. The new notion of hyperbolicity is tailored to fit variational equations, and is bound to change our thinking about evolution problems in physics when complete proofs of the theorems announced will become available. The book provides further new insights into the structure of hyperbolic variational problems. It raises interesting questions. Reading the book is hard work, but the rewards are there at the end of the road. It can be strongly recommended to experts interested in partial differential equations in mathematical physics or, more generally, to researchers interested in hyperbolic evolution problems.

4.1. From the Publisher.

The equations describing the motion of a perfect fluid were first formulated by Euler in 1752. These equations were among the first partial differential equations to be written down, but, after a lapse of two and a half centuries, we are still far from adequately understanding the observed phenomena which are supposed to lie within their domain of validity. These phenomena include the formation and evolution of shocks in compressible fluids, the subject of the present monograph. The first work on shock formation was done by Riemann in 1858. However, his analysis was limited to the simplified case of one space dimension. Since then, several deep physical insights have been attained and new methods of mathematical analysis invented. Nevertheless, the theory of the formation and evolution of shocks in real three-dimensional fluids has remained up to this day fundamentally incomplete. This monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. The author considers initial data for these equations which outside a sphere coincide with the data corresponding to a constant state. Under suitable restriction on the size of the initial departure from the constant state, he establishes theorems that give a complete description of the maximal classical development. In particular, it is shown that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signalling shock formation. The theorems give a detailed description of the geometry of this singular boundary and a detailed analysis of the behaviour of the solution there. A complete picture of shock formation in three-dimensional fluids is thereby obtained. The approach is geometric, the central concept being that of the acoustical spacetime manifold.

4.2. From the Prologue and Summary.

The equations describing the motion of a perfect fluid were first formulated by Euler in 1752, based, in part, on the earlier work of D Bernoulli. These equations were among the first partial differential equations to be written down, preceded, it seems, only by D'Alembert's 1749 formulation of the one-dimensional wave equation describing the motion of a vibrating string in the linear approximation. In contrast to D'Alembert's equation however, we are still, after the lapse of two and a half centuries, far from having achieved an adequate understanding of the observed phenomena which are supposed to lie within the domain of validity of the Euler equations.

The phenomena displayed in the interior of a fluid fall into two broad classes, the phenomena of sound, the linear theory of which is acoustics, and the phenomena of vortex motion. The sound phenomena depend on the compressibility of a fluid, while the vortex phenomena occur even in a regime where fluid may be considered to be incompressible. The formation of shocks, the subject of the present monograph, belongs to the class of sound phenomena, but lies in the nonlinear regime, beyond the range covered by linear acoustics. The phenomena of vortex motion include the chaotic form called turbulence, the understanding of which is one of the great challenges of science.

Let us make a short review of the history of the study of the phenomena of sound in fluids, in particular the phenomena of the formation and evolution of shocks in the nonlinear regime. At the time when the equations of fluid mechanics were first formulated, thermodynamics was in its infancy, however it was already clear that the local state of a fluid as seen by a comoving observer is determined by two thermodynamic variables, say pressure and temperature. Of these, only pressure entered the equations of motion, while the equations involve also the density of the fluid. Density was already known to be a function of pressure and temperature for a given type of fluid. However in the absence of an additional equation, the system of equations at the time of Euler, which consisted of the momentum equations together with the equation of continuity, was underdetermined, except in the incompressible limit. The additional equation was supplied by Laplace in 1816 in the form of what was later to be called the adiabatic condition, and allowed him to make the first correct calculation of the speed of sound.

The first work on the formation of shocks was done by Riemann in 1858. Riemann considered the case of isentropic flow with plane symmetry, where the equations of fluid mechanics reduce to a system of conservation laws for two unknowns and with two independent variables, a single space coordinate and time. He introduced for such systems the so-called Riemann invariants, and with the help of these showed that solutions which arise from smooth initial conditions develop infinite gradients in finite time. Riemann also realised that the solutions can be continued further as discontinuous solutions, but here there was a problem. Up to this time the energy equation was considered to be simply a consequence of the laws of motion, not a fundamental law in its own right. On the other hand, the adiabatic condition was considered by Riemann to be part of the main framework. Now as long as the solutions remain smooth it does not matter which of the two equations we take to be the fundamental law, for each is a consequence of the other, modulo the remaining laws. However this is no longer the case once discontinuities develop, so one must make a choice as to which of the two equations to regard as fundamental and therefore remains valid thereafter.

Here Riemann made the wrong choice. For, only during the previous decade, in 1847, had the first law of thermodynamics been formulated by Helmholtz, based in part on the experimental work of Joule on the mechanical equivalence of heat, and the general validity of the energy principle had thereby been shown.

In 1865 the concept of entropy was introduced into theoretical physics by Clausius, and the adiabatic condition was understood to be the requirement that the entropy of each fluid element remains constant during its evolution. The second law of thermodynamics, involving the increase of entropy in irreversible processes, had first been formulated in 1850 by Clausius without explicit reference to the entropy concept. After these developments the right choice in Riemann's dilemma became clear. The energy equation must remain at all times a fundamental law, but the entropy of a fluid element must jump upward when the element crosses a hypersurface of discontinuity, The formulation of the correct jump conditions that must be satisfied by the thermodynamic variables and the fluid velocity across a hypersurface of discontinuity was begun by Rankine in 1870 and completed by Hugoniot in 1889.

With Einstein's discovery of the special theory of relativity in 1905, and its final formulation by Minkowski in 1908 through the introduction of the concept of spacetime with its geometry, the domain of geometry being thereby extended to include time, a unity was revealed in physical concepts which had been hidden up to this point. In particular, the concepts of energy density, momentum density or energy flux, and stress, were unified into the concept of the energy-momentum-stress tensor and energy and momentum were likewise unified into a single concept, the energy-momentum vector. Thus, when the Euler equations were extended to become compatible with special relativity, it was obvious from the start that it made no sense to consider the momentum equations without considering also the energy equation, for these two were parts of a single tensorial law, the energy-momentum conservation law. This law together with the particle conservation law (the equation of continuity of the non-relativistic theory), constitute the laws of motion of a perfect fluid in the relativistic theory. The adiabatic condition is then a consequence for smooth solutions.

A new basic physical insight on the shock development problem was reached first, it seems, by Landau in 1944. This was the discovery that the condition that the entropy jump be positive as a hypersurface of discontinuity is traversed from the past to the future, should be equivalent to the condition that the flow is evolutionary, that is, that conditions in the past determine the fluid state in the future. More precisely, what was shown by Landau was that the condition of determinism is equivalent, at the linearised level, to the condition that the tangent hyperplane at a point on the hypersurface of discontinuity, is on one hand contained in the exterior of the sound cone at this point corresponding to the state before the discontinuity, while on the other hand intersects the sound cone at the same point corresponding to the state after the discontinuity, and that this latter condition is equivalent to the positivity of the entropy jump.

This is interesting from a general philosophical point of view, because it shows that irreversibility can arise, even though the laws are all time-reversible, once the solution ceases to be regular. To a given state at a given time there always corresponds a unique state at any given later time. If the evolution is regular in the associated time interval, then the reverse is also true: to a given state at a later time there corresponds a unique state at any given earlier time, the laws being time-reversible. This reverse statement is however false if there is a shock during the time interval in question. Thus determinism in the presence of hypersurfaces of discontinuity selects a direction of time and the requirement of determinism coincides, modulo the other laws, with what is dictated by the second law of thermodynamics which is in its nature irreversible. This recalls the interpretation of entropy, first discovered by Boltzmann in 1877, as a measure of disorder at the microscopic level. An increase of entropy was thus understood to be associated to an increase in disorder or to loss of information, and determinism can only be expected in the time direction in which information is lost, not gained.

An important mathematical development with direct application to the equations of fluid mechanics in the physical case of three space dimensions, was the introduction by Friedrichs of the concept of a symmetric hyperbolic system in 1954 and his development of the theory of such systems. It is through this theory that the local existence and domain of dependence property of solutions of the initial value problem associated to the equations of fluid mechanics are established. Another development in connection to this was the general investigation by Friedrichs and Lax in 1971 of nonlinear first order systems of conservation laws which for smooth solutions have as a consequence an additional conservation law. This is the case for the system of conservation laws of fluid mechanics, which consists of the particle and energy-momentum conservation laws, which for smooth solutions imply the conservation law associated to the entropy current. It was then shown that if the additional conserved quantity is a convex function of the original quantities, the original system can be put into symmetric hyperbolic form. Moreover, for discontinuous solutions satisfying the jump conditions implied by the integral form of the original conservation laws, an inequality for the generalised entropy was derived.

The problem of shock formation for the equations of fluid mechanics in one space dimension, and more generally for systems of conservation laws in one space dimension, was studied by Lax in 1964, and 1973, and John in 1974. The approach of these works was analytic, the strategy being to deduce an ordinary differential inequality for a quantity constructed from the first derivatives of the solution, which showed that this quantity must blow up in finite time, under a certain structural assumption on the system called genuine nonlinearity and suitable conditions on the initial data. The genuine nonlinearity assumption is in particular satisfied by the non-relativistic compressible Euler equations in one space dimension provided that the pressure is a strictly convex function of the specific volume.

A more geometric approach in the case of systems with two unknowns was developed by Majda in 1984 based in part on ideas introduced by Keller and Ting in 1966. In this approach, which is closer in spirit to the present monograph, one considers the evolution of the inverse density of the characteristic curves of each family and shows that under appropriate conditions this inverse density must somewhere vanish within finite time. In this way, not only were the earlier blow-up results reproduced, but, more importantly, insight was gained into the nature of the breakdown. Moreover Majda's approach also covered the case where the genuine nonlinearity assumption does not hold. but we have linear degeneracy instead. He showed that in this case, global-in-time smooth solutions exist for any smooth initial data.

The problem of the global-in-time existence of solutions of the equations of fluid mechanics in one space dimension was treated by Glimm in 1965 through an approximation scheme involving at each step the local solution of an initial value problem with piecewise constant initial data. The convergence of the approximation scheme then produced a solution in the class of functions of bounded variation. Now, by the previously established results on shock formation, a class of functions in which global existence holds must necessarily include functions with discontinuities, and the class of functions of bounded variation is the simplest class having this property. Thus, the treatment based on the total variation, the norm in this function space, in itself an admirable investigation, would be insuperable if the development of the one-dimensional theory was the goal of scientific effort in the field of fluid mechanics. However that goal can only be the mathematical description of phenomena in real three-dimensional space and one must ultimately face the fact that methods based on the total variation do not generalise to more than one space dimension. In fact it is clear from the study of the linearised theory, acoustics, which involves the wave equation, that in more than one space dimension only methods based on the energy concept are appropriate.

The first and thus far the only general result on the formation of shocks in three-dimensional fluids was obtained by Sideris in 1985. Sideris considered the compressible Euler equations in the case of a classical ideal gas with adiabatic index $\gamma > 1$ and with initial data which coincide with those of a constant state outside a ball The assumptions of his theorem on the initial data were that there is an annular region bounded by the sphere outside which the constant state holds, and a concentric sphere in its interior, such that a certain integral in this annular region of $\rho - \rho_{0}$, the departure of the density ρ from its value $\rho_{0}$ in the constant state, is positive, while another integral in the same region of $\rho \nu^{r}$, the radial momentum density, is non-negative. These integrals involve kernels which are functions of the distance from the centre. It is also assumed that everywhere in the annular region the specific entropy $s$ is not less than its value $s_{0}$ in the constant state. The conclusion of the theorem is that the maximal time interval of existence of a smooth solution is finite. The chief drawback of this theorem is that it tells us nothing about the nature of the breakdown. Also the method relies on the strict convexity of the pressure as a function of the density displayed by the equation of state of an ideal gas. and does not extend to more general equations of state.
...
The present monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We consider regular initial data on a space-like hyperplane $\Sigma _{0}$ in Minkowski spacetime which outside a sphere coincide with the data corresponding to a constant state. We consider the restriction of the initial data to the exterior of a concentric sphere in $\Sigma _{0}$ and we consider the maximal classical development of this data. Then, under a suitable restriction on the size of the departure of the initial data from those of the constant state, we prove certain theorems which give a complete description of the maximal classical development, which we call maximal solution. In particular, the theorems give a detailed description of the geometry of the boundary of the domain of the maximal solution and a detailed analysis of the behaviour of the solution at this boundary. A complete picture of shock formation in three-dimensional fluids is thereby obtained. Also, sharp sufficient conditions on the initial data for the formation of a shock in the evolution are established and sharp lower and upper bounds for the temporal extent of the domain of the maximal solution are derived.

The reason why we consider only the maximal development of the restriction of the initial data to the exterior of a sphere is in order to avoid having to treat the long time evolution of the portion of the fluid which is initially contained in the interior of this sphere. For, we have no method at present to control the long time behaviour of the point-wise magnitude of the vorticity of a fluid portion, the vorticity satisfying a transport equation along the fluid flow lines. ...

4.3. Review by: Philippe G LeFloch.
Mathematical Reviews MR2284927 (2008e:76104).

In this monograph the author studies the maximally defined, smooth solutions to the relativistic Euler equations of motion for a perfect fluid in Minkowski spacetime $M^{3+1}$.

The discussion begins with a review of earlier works, including pioneering work on shock formation by Riemann on isentropic fluid flows with plane symmetry and, more generally, on nonlinear hyperbolic systems of two conservation laws in one space variable: smooth solutions develop singularities in finite time. The formulation of the physically correct jump relations was later found by Rankine and Hugoniot. Further fundamental work was done by Friedrichs and Lax, and the general problem of shock formation for hyperbolic systems of conservation laws in one space dimension was solved by Lax in 1964 (for genuinely nonlinear systems) and John in 1974 (for general systems).

The strategy in the above works was to deduce an ordinary differential inequality for a quantity constructed from the first-order derivatives of the solution, and to show that this quantity must blow-up in finite time, at least under certain assumptions on the structure of the hyperbolic system.

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

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

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

5.1. From the Publisher.

General Relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. It is thus fundamentally rooted in classical mechanics. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss' intrinsic theory of curved surfaces in Euclidean space. The domain of application of the theory is astronomical systems.

One of the mathematical methods analysed and exploited in the present volume is an extension of Noether's fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations. Another method is the study and systematic use of foliations by characteristic (null) hypersurfaces, and is in the spirit of the approach of Roger Penrose in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media.

The book is intended for advanced students and researchers seeking an introduction into the methods and applications of general relativity.

5.2. From the General Introduction.

General Relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. The theory's roots extend over almost the entire previous history of physics and mathematics.

Its immediate predecessor, Special Relativity, established in its final form by Minkowski in 1908, accomplished the unification of space and time in the geometry of a 4-dimensional affine manifold, a geometry of simplicity and perfection on par with that of the Euclidean geometry of space. The root of Special Relativity is Electromagnetic Theory, in particular Maxwell's incorporation of Optics, the theory of light, into Electrodynamics.

General Relativity is based on and extends Newton's theory of Gravitation as well as Newton's equations of motion. It is thus fundamentally rooted in Classical Mechanics.

Perhaps the most fundamental aspect of General Relativity however, is its geometric nature. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss' intrinsic theory of curved surfaces in Euclidean space. The connection between gravitation and Riemannian geometry arose in Einstein's mind in his effort to uncover the meaning of what in Newtonian theory is the fortuitous equality of the inertial and the gravitational mass. Identification, via the equivalence principle, of the gravitational tidal force with spacetime curvature at once gave a physical interpretation of curvature of the spacetime manifold and also revealed the geometrical meaning of gravitation.

One sees here that descent to a deeper level of understanding of physical reality is connected with ascent to a higher level of mathematics. General Relativity constitutes a triumph of the geometric approach to physical science.

But there is more to General Relativity than merely a physical interpretation of a variant of Riemannian Geometry. For, the theory contains physical laws in the form of equations - Einstein's equations - imposed on the geometric structure. This gives a tightness which makes the resulting mathematical structure one of surpassing subtlety and beauty. An analogous situation is found by comparing the theory of differentiable functions of two real variables with the theory of differentiable functions of one complex variable. The latter gains, by the imposition of the Cauchy-Riemann equations, a tighter structure which leads to a greater richness of results.

The domain of application of General Relativity, beyond that of Newtonian theory, is astronomical systems, stellar or galactic, where the gravitational field is so strong that it implies the potential presence of velocities which are not negligible in comparison with the velocity of light. The ultimate domain of application is the study of the structure and evolution of the universe as a whole.

General Relativity has perhaps the most satisfying structure of all physical theories from the mathematical point of view. It is a wonderful research field for a mathematician. Here, results obtained by purely mathematical means have direct physical consequences.

One example of this is the incompleteness theorem of R Penrose and its extensions due to Hawking and Penrose known as the "singularity theorems". This result is relevant to the study of the phenomenon of gravitational collapse. It shall be covered in the second volume of the present work. The methods used to establish the result are purely geometrical - the theory of conjugate points. In fact, part of the main argument is already present in the theory of focal points in the Euclidean framework, a theory developed in antiquity.

Another example is the positive energy theorem, the first proof of which, due to R Schoen and S T Yau, is based on the theory of minimal surfaces and is covered in the present volume. In this example a combination of geometric and analytic methods are employed.

A last example is the theory of gravitational radiation, a main theme for both volumes of this work. Here also we have a combination of geometric and analytic methods. A particular result in the theory of gravitational radiation is the so-called memory effect, which is due to the non-linear character of the asymptotic laws at future null infinity and has direct bearing on experiments planned for the near future. This result will also be covered in our second volume.

The laws of General Relativity, Einstein's equations, constitute, when written in any system of local coordinates, a non-linear system of partial differential equations for the metric components. Because of the compatibility conditions of the metric with the underlying manifold, when piecing together local solutions to obtain the global picture, it is the geometric manifold, namely the pair consisting of the manifold itself together with its metric, which is the real unknown in General Relativity.

The Einstein equations are of hyperbolic character, as is explained in detail in this first volume. As a consequence, the initial value problem is the natural mathematical problem for these equations. This conclusion, reached mathematically, agrees with what one expects physically. For, the initial value problem is the problem of determining the evolution of a system from given initial conditions, as in the prototype example of Newton's equations of motion. The initial conditions for Einstein's equations, the analogues of initial position and velocity of Newtonian mechanics, are the intrinsic geometry of the initial space-like hypersurface and its rate of change under a virtual normal displacement, the second fundamental form. In contrast to the case of Newtonian mechanics however, these initial conditions are, by virtue of the Einstein equations themselves, subject to constraints, and it is part of the initial value problem in General Relativity - a preliminary part - to analyse these constraints. Important results can be obtained on the basis of this analysis alone and the positive energy theorem is an example of such a result.

An important notion in physics is that of an isolated system. In the context of the theory of gravitation, examples of such systems are a planet with its moons, a star with its planetary system, a binary or multiple star, a cluster of stars, a galaxy, a pair or multiplet of interacting galaxies, or, as an extreme example, a cluster of galaxies - but not the universe as a whole. What is common in these examples is that each of these systems can be thought of as having an asymptotic region in which conditions are trivial. Within General Relativity the trivial case is the flat Minkowski spacetime of Special Relativity. Thus the desire to describe isolated gravitating systems in General Relativity leads us to consider spacetimes with asymptotically Minkowskian regions. However it is important to remember at this point the point of view of the initial value problem: a spacetime is determined as a solution of the Einstein equations from its initial data. Consequently, we are not free to impose our own requirements on a spacetime. We are only free to impose requirements on the initial data - to the extent that the requirements are consistent with the constraint equations. Thus the correct notion of an isolated system in the context of General Relativity is a spacetime arising from asymptotically flat initial conditions, namely an intrinsic geometry which is asymptotically Euclidean and is a second fundamental form which tends to zero at infinity in an appropriate way. This is discussed in detail in this volume.

Trivial initial data for the Einstein equations consists of Euclidean intrinsic geometry and a vanishing second fundamental form. Trivial initial data gives rise to the trivial solution, namely the Minkowski spacetime. A natural question in the context of the initial value problem for the vacuum Einstein equations is whether or not every asymptotically flat initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. This question was answered in the affirmative in the joint work of the present author with Sergiu Klainerman, which appeared in the monograph The Global Nonlinear Stability of the Minkowski Space (1990). One of the aims of the present work is to present the methods which went into that work in a more general context, so that the reader may more fully understand their origin and development as well as be able to apply them to other problems. In fact, problems coming from fields other than General Relativity are also treated in the present work. These fields are Continuum Mechanics, Electrodynamics of Continuous Media and Classical Gauge Theories (such as arise in the mesoscopic description of superfluidity and superconductivity). What is common to all these problems from our perspective is the mathematical methods involved.

One of the main mathematical methods analysed and exploited in the present work is the general method of constructing a set of quantities whose growth can be controlled in terms of the quantities themselves. This method is an extension of the celebrated theorem of Noether, a theorem in the framework of the action principle, which associates a conserved quantity to each 1-parameter group of symmetries of the action. This extension is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations, as discussed in detail in this first volume. In fact we may say that such a system is hyperbolic at a particular background solution if linear perturbations about this solution possess positive energy in the high frequency limit.

The application of Noether's Principle to General Relativity requires the introduction of a background vacuum solution possessing a non-trivial isometry group, as is explained in this first volume. Taking Minkowski spacetime as the background, we have the symmetries of time translations, space translations, rotations and boosts, which give rise to the conservation laws of energy, linear momentum, angular momentum and centre of mass integrals, respectively. However, as is explained in this first volume, these quantities have geometric significance only for spacetimes which are asymptotic at infinity to the background Minkowski spacetime, so that the symmetries are in fact asymptotic symmetries of the actual spacetime.

The other main mathematical method analysed and exploited in the present work is the systematic use of characteristic (null) hypersurfaces. The geometry of null hypersurfaces has already been employed by R Penrose in his incompleteness theorem mentioned above. What is involved in that theorem is the study of a neighbourhood of a given null geodesic generator of such a hypersurface. On the other hand, in the work on the stability of Minkowski spacetime, the global geometry of a characteristic hypersurface comes into play. In addition, the properties of a foliation of spacetime by such hypersurfaces, also come into play. This method is used in conjunction with the first method, for, such characteristic foliations are used to define the actions of groups in spacetime which may be called quasi-conformal isometries, as they are globally as close as possible to conformal isometries and tend as rapidly as possible to conformal isometries at infinity. The method is introduced in this first volume and will be treated much more fully in the second volume. It has applications beyond General Relativity to problems in Fluid Mechanics and, more generally, to the Mechanics and Electrodynamics of Continuous Media.

This book is based on Nachdiplom Lectures held at the Eidgenössische Technische Hochschule Zurich during the Winter Semester 2002/2003. The author wishes to thank his former student Lydia Bieri for taking the notes of this lecture, from which a first draft was written, and for making the illustrations.

5.3. Review by: Alan D Rendall.
Mathematical Reviews MR2391586 (2008m:83008).

This book is based on lecture notes of the author on general relativity. It has three main parts. Chapter 2 introduces some basic ideas about general relativity and the Cauchy problem for the Einstein equations. Conservation laws for isolated systems are discussed in Chapter 3. Finally, Chapter 4 is an account on a somewhat informal and intuitive level of the proof of the nonlinear stability of Minkowski space due to the author and S Klainerman. Many of the ideas in this book can be found in other texts, but a large number cannot. This review will concentrate on the latter. In addition to the points mentioned specifically below, the text contains many insights on diverse issues, and I believe that even experienced researchers in general relativity will find something new and valuable in the book.
...
As far as the reviewer is aware, nothing comparable is available in the literature. Other accounts are written in a physicist's style which is hard for mathematicians to understand, or assume more background on the Lagrangian or Hamiltonian formalism than many potentially interested mathematicians have. Thus the book makes an important contribution here in providing a transparent exposition. The chapter also contains a largely self-contained proof of the positive mass theorem. The fourth chapter gives a panoramic view of the ideas used in the original proof of the stability of Minkowski space. This book is a valuable addition to the literature on mathematical relativity.

6.1. From the Publisher.

In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must come to an end. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity.

A major challenge since that time has been to find out how trapped surfaces actually form, by analysing the dynamics of gravitational collapse. The present monograph achieves this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of gravitational waves.

The theorems proved in the present monograph constitute the first foray into the long-time dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitable neighbourhood of trivial data. The main new method, the short pulse method, applies to general systems of Euler-Lagrange equations of hyperbolic type, and provides the means to tackle problems which have hitherto seemed unapproachable.

This monograph will be of interest to people working in general relativity, geometric analysis, and partial differential equations.

6.2. From the Prologue.

The story of the black hole begins with Schwarzschild's discovery of the Schwarzschild solution in 1916, soon after Einstein's foundation of the general theory of relativity and his final formulation of the field equations of gravitation, the Einstein equations, in 1915. The Schwarzschild solution is a solution of the vacuum Einstein equations which is spherically symmetric and depends on a positive parameter M, the mass. With r such that the area of the spheres, which are the orbits of the rotation group, is $4 \pi r^{2}$, the solution in the coordinate system in which it was originally discovered had a singularity at $r = 2M$. For this reason only the part which corresponds to $r > 2M$ was originally thought to make sense. This part is static and represents the gravitational field outside a static, spherically symmetric body with surface area corresponding to some $r_{0} > 2M$.

However, the understanding of Schwarzschild's solution gradually changed. First, in 1923 Birkoff proved a theorem which shows that the Schwarzschild solution is the only spherically symmetric solution of the vacuum Einstein equations. One does not therefore need to assume that the solution is static. Thus, Schwarzschild's solution represents the gravitational field outside any spherically symmetric body, evolving in any manner whatever, for example undergoing gravitational collapse.

Eddington, in 1924, made a coordinate change which transformed the Schwarzschild metric into a form which is not singular at $r = 2M$, however he failed to take proper notice of this. Only in 1933, with Lemaître's work, was it realised that the singularity at $r = 2M$ is not a true singularity but rather a failure of the original coordinate system. Eddington's transformation was rediscovered by Finkelstein in 1958, who realised that the hypersurface $r = 2M$ is an event horizon, the boundary of the region of spacetime which is causally connected to infinity, and recognised the dynamic nature of the region $r < 2M$. Now, Schwarzschild's solution is symmetric under time reversal, and one part of it, the one containing the future event horizon, the boundary of the region of spacetime which can send signals to infinity, is covered by one type of Eddington-Finkelstein coordinates, while the other part, the one containing the past event horizon, the boundary of the region of spacetime which can receive signals from infinity, is covered by the other type of Eddington-Finkelstein coordinates. Actually, only the first part is physically relevant, because only future event horizons can form dynamically, in gravitational collapse. Systems of coordinates that cover the complete analytic extension of the Schwarzschild solution had been provided earlier (in 1950) by Synge, and a single most convenient system that covers the complete analytic extension was discovered independently by Kruskal and Szekeres in 1960.

Meanwhile in 1939, Oppenheimer and Snyder had studied the gravitational collapse of a pressure-free fluid ball of uniform density, a uniform density "ball of dust". Even though this is a highly idealised model problem, their work was very significant, being the first work on relativistic gravitational collapse. As mentioned above, the space-time geometry in the vacuum region outside the ball is given by the Schwarzschild metric. Oppenheimer and Snyder analysed the causal structure of the solution. They considered in particular an observer on the surface of the dust ball sending signals to a faraway stationary observer at regularly spaced intervals as judged by his own clock. They discovered that the spacing between the arrival times of these signals to the faraway observer becomes progressively longer, tending to infinity as the radius $r_{0}$ corresponding to the surface of the ball approaches $2M$. This effect has since been called the infinite redshift effect. The observer on the surface of the dust ball may keep sending signals after $r_{0}$ has become less than $2M$, but these signals proceed to ever smaller values of $r$ until, within a finite affine parameter interval, they reach a true singularity at $r = 0$. The observer on the surface of the ball reaches this singular state himself within a finite time interval as judged by his own clock. The concept of a future event horizon, and hence of a region of spacetime bounded by this horizon from which no signals can be sent which reach arbitrarily large distances, was thus already implicit in the Oppenheimer-Snyder work.

The 1964 work of Penrose introduced the concept of null infinity, which made possible the precise general definition of a future event horizon as the boundary of the causal past of future null infinity. A turning point was reached in 1965 with the introduction by Penrose of the concept of a closed trapped surface and his proof of the first singularity theorem, or, more precisely, incompleteness theorem. Penrose defined a trapped surface as being a space-like surface in spacetime, such that an infinitesimal virtual displacement of the surface along either family of future-directed null geodesic normals to the surface leads to a point-wise decrease of the area element. ...
...
Once the notions of null infinity and of a closed trapped surface were introduced, it did not take long to show that a spacetime with a complete future null infinity which contains a closed trapped surface must contain a future event horizon, the interior of which contains the trapped surface. For the ideas and methods which go into Penrose's theorem the reader may consult, besides the monograph by Hawking and Ellis just mentioned, an article by Penrose in as well as his monograph. Further singularity theorems, which also cover cosmological situations, were subsequently established by Hawking and Penrose, but it is the original singularity theorem quoted above which is of interest in the present context, as it concerns gravitational collapse. We should also mention that the term black hole for the interior of the future event horizon was introduced by Wheeler in 1967.
...
Returning to our review of the historical development of the black hole concept, a very significant development took place in 1963, shortly before the work of Penrose. This was the discovery by Kerr of a two-parameter family of axially symmetric solutions of the vacuum Einstein equations, with an event horizon, the exterior of which is a regular asymptotically flat region.
...
The fascinating properties of the Kerr solution were revealed in the decade following its discovery.

6.3. Review by: Piotr T Chrusciel.
Mathematical Reviews MR2488976 (2009k:83010).

In Minkowski space-time, outgoing light fronts emanating from a round sphere increase in area, while ingoing ones decrease in area. Clearly something similar will hold for weak gravitational fields. A trapped surface, as defined by R Penrose, will have radically different properties: roughly speaking, it is a surface from which both ingoing and outgoing light fronts decrease in area.

Penrose has shown that, under natural hypotheses, existence of trapped surfaces leads to geodesic incompleteness. Hence the interest in understanding the formation of such surfaces. Furthermore, existence of a trapped surface will sometimes signal the presence of a black hole, but this requires many supplementary conditions.

The whole book under review is devoted to the proof of the theorem, that sufficiently focused "short pulse" initial data on a light-cone lead to the formation of a trapped surface within their Cauchy development.

7.1. From the Publisher.

Examines classical compressible Euler Equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. Under suitable restriction on the size of the initial departure from the constant state, the authors establish theorems which give a complete description of the maximal development.

7.2. From the Introduction.

Note. The Introduction is very similar to 4.2 above. We given below the last two paragraphs which explain how the present book extends the previous one.

The most recent and complete results on the formation of shocks in three dimensional fluids were obtained by Christodoulou in 2007 [The Formation of Shocks in 3-Dimensional Fluids]. Christodoulou considered the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. He considered the regular initial data on a space-like hyperplane $\Sigma _{0}$ in Minkowski spacetime which outside a sphere coincide with the data corresponding to a constant state. He considered the restriction of the initial data to the exterior of a concentric sphere in $\Sigma _{0}$ and the maximal classical development of this data. Under suitable restriction on the size of the departure of the initial data from those of the constant state, he proved certain theorems which give a complete description of the maximal classical development. In particular, theorems give a detailed description of the geometry of the boundary of the domain of the maximal classical solution and a detailed analysis of the behaviour of the solution at this boundary.

The aim of the present monograph is to derive analogous results for the classical, non-relativistic, compressible Euler's equations taking the data to be irrotational and isentropic, and to give a proof of these results which is considerably simpler and completely self-contained. The present monograph in fact not only gives simpler proofs but also sharpens some of the results. In addition the present monograph explains in depth the ideas on which the approach is based. Finally certain geometric aspects which pertain only to the non-relativistic theory are discussed.

7.3. Review by: Michael Dreher.
Mathematical Reviews MR3288725.

This monograph investigates classical solutions to the compressible Euler equations of fluid dynamics in three-dimensional space ...

It is well known that for most smooth initial data, the solutions to the Euler system develop singularities in finite time. The aim of the monograph is to present a complete description of the behaviour of the classical solution before the breakdown. In particular, a geometrical characterisation of the boundary of the maximal classical solution is presented, and the behaviour of the solution near that boundary is presented in detail.

The main tools of the approach are a partial hodograph transform that translates from the usual rectangular coordinates to the acoustic coordinates, so-called variation fields that play the same role in the three-dimensional case as the well-known Riemann invariants in the one-dimensional case, and a collection of commutation vector fields. A sophisticated application of these commutation vector fields and a careful choice of multiplier vector fields then leads to a family of energy estimates that subsequently prove the main result. Then a characterisation of the behaviour of the solution before the formation of shocks is established.

The results in this monograph all refer to the non-relativistic case, and they are companions to similar results in [D Christodoulou, The formation of shocks in 3-dimensional fluids, 2007], where the formation of shocks was studied in the relativistic situation, in three dimensions.

The exposition of the results and their proofs is fully self-contained, and several proofs have been simplified. The ideas behind the geometric approach are explained very well, and the book should be of interest to researchers in the field of formation of singularities for hyperbolic systems.

8.1. From the Publisher.

This monograph addresses the problem of the development of shocks in the context of the Eulerian equations of the mechanics of compressible fluids. The mathematical problem is that of an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions.

The free boundary is the shock hypersurface and the boundary conditions are jump conditions relative to a prior solution, conditions following from the integral form of the mass, momentum and energy conservation laws. The prior solution is provided by the author's previous work which studies the maximal classical development of smooth initial data. New geometric and analytic methods are introduced to solve the problem. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularisation of the problem is achieved.

Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bivariational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities. Shocks are an ubiquitous phenomenon and also occur in magnetohydrodynamics, nonlinear elasticity, and the electrodynamics of nonlinear media. The methods developed in this monograph are likely to be found relevant in these fields as well.

8.2. From the Prologue.

The subject of this monograph is the shock development problem in fluid mechanics. This problem is formulated in the framework of the Eulerian equations of a compressible perfect fluid as completed by the laws of thermodynamics. These equations express the differential conservation laws of mass, momentum, and energy and constitute a quasilinear hyperbolic 1st-order system for the physical variables, that is, the fluid velocity and the two positive quantities corresponding to a local thermodynamic equilibrium state. Smooth initial data for this system of equations lead to the formation of a surface in spacetime where the derivatives of the physical quantities with respect to the standard rectangular coordinates blow up.