ON THE EXISTENCE OF SOLUTIONS AND CAUSALITY FOR RELATIVISTIC VISCOUS CONFORMAL FLUIDS

. We consider a stress-energy tensor describing a pure radiation viscous ﬂuid with conformal symmetry introduced in [3]. We show that the cor- responding equations of motions are causal in Minkowski background and also when coupled to Einstein’s equations, and solve the associated initial-value problem.


1.
Introduction. Consider the following stress-energy tensor for a relativistic fluid with viscosity: + λ(u α u µ ∇ µ u β + u β u µ ∇ µ u α ) + 1 3 χπ αβ ∇ µ u µ + χu α u β ∇ µ u µ + λ 4 (u α π µ β ∇ µ + u β π µ α ∇ µ ) + 3χ 4 u α u β u µ ∇ µ + χ 4 π αβ u µ ∇ µ . (1) Here, u is the four-velocity of fluid particles, normalized so that is the energy density of the fluid, g is a (Lorentzian) metric, ∇ is the Levi-Civita connection associated with g, π αβ = g αβ + u α u β , and η, λ, and χ are viscous transport coefficients -so that η = λ = χ = 0 corresponds to an ideal fluid. The transport coefficients are non-negative functions of . Coefficient η is the usual coefficient of shear viscosity, whereas λ and χ are related to relaxation times. More precisely, while λ and χ, differently than η, have no analogue in more familiar theories such as classical, non-relativistic Navier-Stokes, their physical meaning can be understood from the derivation of (1) from kinetic theory given in [3]. In that case, one may interpret λ/(sθ) and χ/(sθ), where s is the entropy density and θ the temperature, as relaxation times that restore causality (since intuitively causality says that the system needs some time to relax back to equilibrium after a perturbation). See [3] for details. Convention 1. We work in units where 8πG = c = 1, where G is Newton's constant and c is the speed of light in vacuum. Our signature for the metric is − + ++. Greek indices run from 0 to 3 and Latin indices from 1 to 3.
We shall couple (1) to Einstein's equations: where R αβ and R are, respectively, the Ricci and scalar curvature of the metric g, and Λ is a constant (the cosmological constant). We recall that in light of the Bianchi identities, a necessary condition for (3) to hold is that Naturally, equations (3)-(4) are defined in a four-dimensional differentiable manifold, the space-time.
We shall establish the following.
Main result. (see Theorems 2.2 and 2.3 for precise statements) Under appropriate conditions on the initial data and the transport coefficients, the system of Einstein's equations coupled to (1) is causal and admits a unique solution. Causality and uniqueness are here understood in the usual sense of general relativity. Existence, uniqueness, and causality remain true if we consider solely (4) in Minkowski spacetime.
The tensor (1) was introduced 1 in [3]. As discussed there, (1) is the first example in the literature of a stress-energy tensor for relativistic viscous fluids satisfying the following list of physical requirements: in Minkowski background, equations (4) are (i) linearly stable with respect to perturbations around homogeneous thermodynamic equilibrium, (ii) well-posed, and (iii) causal; (iv) Einstein's equations coupled to (1) are well-posed and causal; (v) equations (4) reduce to the standard Navier-Stokes equations in the non-relativistic limit; (vi) an out-of-equilibrium entropy can be defined so that solutions to (4) satisfy the (out of equilibrium) second law of thermodynamics; and (vii) T αβ can be derived from microscopic kinetic theory.
One reason for seeking a stress-energy tensor satisfying the above properties is that the traditional forms of the relativistic Navier-Stokes equations fail to be causal and stable [23,35], and attempts to construct a relativistic viscous theory satisfying (i)-(vi) have been limited so far 2 . See [12,15,16,37] for a discussion. In [3] it is also shown that T αβ yields a well-defined temperature in the test-case of the Gubser flow, in contrast to the traditional relativistic Navier-Stokes' equations that yield a negative temperature, and that a hydrodynamic attractor exists for the dynamics of the Bjorken flow.
Tensor (1) describes a conformal fluid. Loosely speaking, this means that (1) is well-behaved under conformal changes of the metric. More precisely, consider a conformal transformation g αβ = e −2φ g αβ , and the transformed quantities u α = e −φ u α , = e 4φ . Then the fluid is called conformal if T αβ is traceless and the corresponding transformed T αβ satisfies T αβ = e 2φ T αβ .
One can show [2,4] that under these conditions ∇ α (T ) α β = e 4φ ∇ α T α β , so in particular solutions are preserved by the above transformations. There exists a large literature on conformal fluids and their applications in physics, to which the reader is referred for a discussion (see, e.g., [11,20] and references therein; for the mathematical background for these references, see [19]). We restrict ourselves to mentioning that conformal fluids are of importance in the study of the quark-gluon plasma that forms in high-energy collisions of heavy-ions; the quark-gluon plasma at very high temperatures is the prototypical example of a relativistic viscous fluid with an equation of state of pure radiation.
The definition of conformal fluid, stated above, will play no direct role in this work per se. Rather, we shall use one of its main consequences, namely, that for such fluids we have where a 1 and a 2 are constants. Therefore all transport coefficients are determined once we are given η = η( ).
Our main result has previously appeared in [3], but the letter format of that manuscript and the fact that it was addressed primarily to a physical audience prevented us from presenting several details of the proof. In particular, the argument in [3] may not be entirely satisfactory for a mathematical audience. Definition 1.1. For the rest of the the paper, we shall refer to the system of equations (3), with T αβ given by (1) and u satisfying (2), as the viscous Einsteinconformal fluid (VECF) system.
2. Statement of the results. We now turn to the precise formulation of the Main Result. We begin by discussing the initial data for the VECF system. Definition 2.1. An initial data set for the VECF system consists of a threedimensional smooth manifold Σ, a Riemannian metric g 0 on Σ, a symmetric twotensor κ on Σ, two real-valued functions 0 and 1 defined on Σ, and two vector fields v 0 and v 1 on Σ, such that the Einstein constraint equations are satisfied.
We recall that the constraint equations are given by the following system of equations on Σ: where R g0 is the scalar curvature of g 0 , ∇ g0 , tr g0 , div g0 , and | · | g0 are the covariant derivative, trace, divergence, and norm with respect to g 0 . The quantities ρ and j are given by ρ = T (n, n) and j = T (n, ·), where n is the future-pointing unit normal to Σ inside a development of the initial data and T is the stress-energy tensor.
Because T αβ involves first derivatives of u and , initial conditions for their time derivatives have to be given, hence the necessity of two functions and two vector fields. Even though u is a four-vector, it suffices to specify vector fields on Σ, with initial conditions for the non-tangential components of u derived from (2) (see section 3.2). It is well-known that initial data for Einstein's equations cannot be prescribed arbitrarily, having to satisfy the associated constraint equations, see, e.g., [21], for details.
We can now state our main result. The definition of spaces G s and G m,s is recalled in Appendix A.1. We refer the reader to the general relativity literature (e.g., [7,21,25,38,40]) for the terminology employed in Theorem 2.2.
2) M is causal, in the following sense. Let (g, , u) be a solution to the VECF system provided by the globally hyperbolic development M . For any p ∈ M in the future of Σ, (g(p), u(p), (p)) depends only on I| i(Σ)∩J − (p) , where J − (p) is the causal past of p and i : Σ → M is the embedding associated with the globally hyperbolic development M .
We note that, in the standard PDE language, Theorem 2.2 is local in time. But as usual in general relativity, solutions to Einstein's equations are geometric (a solution to Einstein's equations is a Lorentzian manifold) and, in particular, coordinate independent, whereas a statement like "there exists a T > 0..." (as is usual in local in time results) requires the introduction of coordinates. This is why the theorem is better stated as the existence of a globally hyperbolic development 3 . We assumed that Σ is compact for simplicity, otherwise asymptotic conditions would have to be prescribed. The type of asymptotic conditions one would impose had Σ been non-compact depends on the type of questions one is investigating. For instance, it is customary to require g 0 to be asymptotically flat, but other conditions, such as asymptotically hyperbolic, are often used. As for the matter variables, several choices are possible. One can require v 0 and 0 to approach zero, a constant, or some other specified profile at infinity. The literature on Einstein's equations with non-compact Σ is vast, and a discussion of asymptotic conditions can be found, e.g., [7,8] and references therein. The assumption 0 > 0 in Theorem 2.2 (which implies a uniform bound from below away from zero by the compactness of Σ), however, is crucial. This is apparent from expression (1), but it is worth mentioning that allowing 0 to vanish leads to severe technical difficulties even in the better studied case of the Einstein-Euler system (see [18,24,36] for the known results and [13] for a discussion; in fact, the difficulties with vanishing density are present already in the non-relativistic case, see the discussion in [14,31]). In particular, if we were dealing with a non-compact Σ and had chosen an asymptotic condition where 0 approaches zero, the techniques here employed would not directly apply. 3 We recall that a globally hyperbolic development is, roughly speaking, a Lorentzian manifold where Einstein's equations are satisfied and in which Σ embeds isometrically as a Cauchy surface taking the correct data. We also recall that once a globally hyperbolic development is shown to exist, one can prove the existence of the "largest" possible global hyperbolic development, i.e., the maximal globally hyperbolic development of the initial data, which is (geometrically) unique. See [25,38] for details.
The assumptions a 1 = 4 and a 2 ≥ 4 are technical 4 , but they are consistent with conditions that guarantee the previously mentioned linear stability of (1). Note that while our proof is restricted to the Gevrey class, our result guarantees that causality will be automatically satisfied in any function space where uniqueness can be established. This is relevant in view of the difficulties of constructing causal theories of relativistic viscous fluids.
Next, we consider the case of a Minkowski background.
While formally Theorem 2.3 can not be derived as a corollary of Theorem 2.2, its validity should come as no surprise once we know the latter to be true. In fact, the proof of Theorem 2.3 will be essentially contained in that of Theorem 2.2, as we shall see. It is nonetheless useful to state Theorem 2.3 given the importance of viscous fluids in Minkowski background for applications. Remark 1. The difference between s > 1 in Theorem 2.2 and s ≥ 1 in Theorem 2.3 comes from the fact that in the proof of Theorem 2.2 we work in local coordinates and employ bump functions, which cannot be analytic (case s = 1). In Minkowski space, however, we can use global coordinates and analyticity is not prevented.
3. Proof of Theorem 2.2. In this section we prove Theorem 2.2, thus we henceforth assume its hypotheses. We will always denote by s a number in (1, 17 16 ), as in the statement of the theorem. The proof will be split in several parts. Some of the arguments parallel well-known constructions in general relativity in the smooth setting, but we present them because some additional steps are required in the Gevrey class.
3.1. The equations of motion. Here we write the VECF in coordinates and in a more explicit form. At this point, we are only interested in writing the equations in a suitable form, thus we assume the validity of (2) and (3) (and consequently (4)), and derive relations of interest.
As is customary, we shall write (3) in trace-reversed form and in wave coordinates. More precisely, we consider the reduced Einstein equations given by where above and henceforth we adopt the following: Notation 1. We shall employ the letters B and B, with indices attached when appropriate, to denote a general expression depending on at most the number of derivatives indicated in its argument. For instance, in (6), B αβ represents an expression depending on at most first derivatives of , first derivatives of u, and first derivatives of g. As another example, B( , ∂u, ∂ 2 g) denotes an expression depending on at most zero derivatives of , one derivative of u, and two derivatives of g. B and B can vary from expression to expression. It can be easily verified that B and B will always be an analytic function (typically involving only products and quotients) of its arguments.
Equations (4) become 5 The term B β (∂u, g)∂ 2 g, which is linear in ∂ 2 g, comes from derivatives of the Christoffel symbols, after expanding the second covariant derivatives of u. This term is of the form B β (∂u, g, ∂ 2 g) according to Notation 1, but we wrote it as B β (∂u, g)∂ 2 g to emphasize that we shall consider it as a second order quasi-linear operator on g. The particular form of this operator will not be needed, but it is important that it be included in the principal part of the system for the derivative counting employed below. Applying u α u µ ∇ α ∇ µ to (2) produces u λ u α u µ ∂ 2 αµ u λ + B(∂u, g)∂ 2 g = B(∂u, ∂g).

3.2.
Initial data. We now investigate the appropriate initial conditions for (9). We remind the reader that the geometric data in the assumptions of Theorem 2.2 are intrinsic to Σ, thus they do not determine full data for the system 6 . Hence, we need to complete the given data to a full set of initial data.
Assume that I is given as in the statement of Theorem 2.2. Embed Σ into R × Σ and consider p ∈ {0} × Σ. We shall initially obtain a solution in a neighborhood of p, hence we prescribe initial data locally.
Take coordinates {x α } 3 α=0 in a neighborhood U of p such that {x i } 3 i=1 are coordinates on Σ, which we assume to be normal coordinates for g 0 centered at p. We remark that in these coordinates the initial data will be in G (s) ({x 0 = 0} ∩ U). For, by our assumption on I, there exist local coordinates {y i } 3 i=1 in a neighborhood Y ⊆ Σ of p such that, in these coordinates, the initial data is Gevrey regular. One obtains (short-time) geodesics starting at p by solving the geodesic equation, which will be an ODE with Gevrey data in the {y i } coordinates. Since we can equip Gevrey spaces with a norm, the usual Picard iteration can be applied to solve the geodesic equation, and hence we obtain solutions that are Gevrey regular and vary within the Gevrey class with the initial data. Therefore, the exponential map and, as a consequence, the coordinates {x i } are Gevrey regular in Y with respect to the {y i } coordinates. Expressing the initial data now in {x i } coordinates, we conclude from standard properties of composition and products of Gevrey maps (see, e.g., [32]) that the initial data is in We prescribe the following initial conditions for g αβ on {x 0 = 0} ∩ U: 6 For example, g 0 is a metric on Σ which is a three-manifold; thus, g 0 contains only nine (six independent) components locally, whereas there are sixteen (ten independent) components in the full space-time metric. Similarly, κ does not determine all transversal derivatives of g on Σ, and v 0 and v 1 determine only the initial three-velocity and its transversal derivatives, whereas we need the four-velocity u and its tranversal derivatives initially. These mismatches are, as it is well-known, related to the gauge freedom of Einstein's equations. See, e.g., [7] for more discussion. and ∂ 0 g 0α (0, ·) is chosen such that {x α } are wave coordinates for g at x 0 = 0 (which is well-known to always be possible).
For u β , we prescribe (Note that the radicands are non-negative because g 0 is a Riemannian metric.) The initial conditions for u 0 and ∂ 0 u 0 have been derived from (2) and the above initial conditions for g αβ . Finally,

3.3.
Initial conditions for the system in R 4 . Consider the local coordinates introduced in section 3.2. Via these coordinates and identifying p with the origin, we can regard system (9) as defined in an open set U of R 4 containing the origin, with the initial conditions prescribed on {x 0 = 0} ∩ U. Note that we can also take (9) as a system of equations on the whole of R 4 , and we therefore do so. We seek to extend the initial data to the whole hypersurface {x 0 = 0}, thus determining initial conditions for the system in R 4 . Let V be compactly contained in {x 0 = 0} ∩ U and W be compactly contained in V. Let ϕ : {x 0 = 0} → R be a function in G (s) (R 3 ) such that 0 ≤ ϕ ≤ 1, ϕ = 1 in W, and ϕ = 0 in the complement of V. Denote by h the Minkowski metric and set, on {x 0 = 0}, These will be initial conditions for g αβ (for equations (9) in R 4 ), with an usual abuse of notation to denote the initial conditions involving ∂ 0 . As our coordinates have been chosen with {x i } normal coordinates for g 0 centered at p, we have that g ij (0) = h ij and the deviations ofg ij from the Minkowski metric restricted to {x 0 = 0} ∩ U are quadratic on the coordinates away from the origin. Writing we see that, shrinking U if necessary and taking into account our choice forg 0α , g αβ is a perturbation of the Minkowsi metric restricted to {x 0 = 0}. Therefore,g αβ defines a Lorentzian metric.
Next, we introduceů i = ϕv i 0 , ∂ 0ů i = ϕv i 1 , with the initial conditions forů 0 and ∂ 0ů 0 obtained by the same formulas as in section (3.2), with the appropriate replacements byů i andg on the right-hand sides. Finally, set˚ By the compactness of Σ and the assumption 0 > 0, it follows that 0 ≥ C for some constant C > 0, thus˚ for some constant C . The initial data for (9) in R 4 described in this section will be denoted byŮ .
3.4. Solving the system in R 4 . In this section, we solve system (9) with the initial conditions described in section 3.3 (see Proposition 1 below). We shall employ the techniques, terminology, and notation of Leray-Ohya systems reviewed in the appendix.
It is understood that we have one index m I for each unknown of the fifteen unknowns and one index n J for each one of the fifteen equations in (9). For instance, by m 1 = m 1 (u β , ) = 2 we mean m(u 0 ) = m(u 1 ) = m(u 2 ) = m(u 3 ) = m( ) = 2, and so on. One readily verifies that with this choice of indices, (9) has the structure of a Leray system. Indeed, we list below for each row J in (9) or, equivalently, for each equation in the system (6), (7), and (8), the value of n J ; the highest derivatives of each unknown entering in the coefficients and on the right-hand side of the equation; and the difference m I − n J : rows 1-4 ≡ eq. (7) : n 1 = 0; ∂u, ∂ , ∂g; and rows 6-15 ≡ eq. (6) : n 2 = 0; ∂u, ∂ , ∂g; For example, in equations (7), for which n 1 = 0, we have that the left-hand side consists of differential operators of order 2 acting on (u β , ) (m(u β , ) − n 1 = 2) and differential operators of order 2 acting on (g αβ ) (m(g αβ )−n 1 = 2), whose coefficients depend on at most first derivatives of the unknowns (∂u, ∂ , ∂g, i.e., m(u β , )−n 1 −1 and m(g αβ )−n 1 −1); the right-hand side of (7), as the coefficients of the differential operators, depends on at most first derivatives of the unknowns. Assumption 1. We henceforth make explicit use of (5), with a 1 = 4 and a 2 ≥ 4, in accordance with the assumptions of Theorem 2.2.
For the proof of the next proposition, the reader is reminded of the Definition A.9 of A s (Σ, Y ), which consists of the space of functions sufficiently near the Cauchy data.
and this solution is unique in this class.
Proof. We fix the initial dataŮ as constructed in section 3.3 and consider U = ( u α , , g αβ ) ∈ A s (Σ, Y ). Shrinking Y if necessary, we can assume that g αβ is a Lorentzian metric, that > 0, and that u is time-like for g αβ , since these properties hold forŮ . Because the coefficients of the matrix of differential operators M(U, ∂) depend on at most first derivatives of the unknowns, we can evaluate these coefficients on U . Denote the corresponding operator by M( U , ∂). The characteristic determinant P ( U , ξ) of (9), evaluated at U , is where 7 and and the contractions in these expressions are done with respect to the metric g αβ . The computation of P ( U , ξ), and the corresponding factorization in the above polynomials, is done through a lengthy and tedious algebraic calculation, part of which was done with the help of the software Mathematica 8 . Note that the block diagonal form of M(U, ∂) allowed us to compute the characteristic determinant without providing the specific form of the operators B β (∂u, g)∂ 2 g and B(∂u, g)∂ 2 g.
It is easy to see that the polynomials u µ ξ µ and ξ µ ξ µ are hyperbolic polynomials as long as g αβ is a Lorentzian metric and u is time-like with respect to g αβ . Both conditions are satisfied in view of the constructions in section 3.3. Therefore, p 1 (ξ) is the product of four hyperbolic polynomials (recall that > 0 and η( ) > 0), and p 4 (ξ) is the product of ten hyperbolic polynomials. We now move to analyze p 2 (ξ) and p 3 (ξ).
Write p 2 (ξ) = ( p 2 (ξ)) 2 , where p 2 (ξ) is the second-degree polynomial between brackets in the definition of p 2 (ξ). We claim that p 2 (ξ) is a hyperbolic polynomial. To show this, we need to investigate the roots ξ 0 = ξ 0 (ξ 1 , ξ 2 , ξ 3 ) of the equation p 2 (ξ) = 0. Consider first the case where p 2 (ξ) is evaluated at the origin, i.e., p 2 (ξ) = p 2 ( U (0), ξ), and assume for a moment that g αβ (0) is the Minkowski metric and that u µ u µ = −1. In this case, the roots are where , and · is the Euclidean inner product. We see that if ξ = 0, then ξ 0,± = 0, and hence ξ = 0. Thus, we can assume ξ = 0. The Cauchy-Schwarz inequality gives u 2 ξ 2 − ( u · ξ) 2 ≥ 0, hence ξ 0,+ and ξ 0,− are real and distinct for a 2 ≥ 4. We conclude that p 2 (ξ) is a hyperbolic polynomial at the origin. Since the roots of a polynomial vary continuously with the polynomial coefficient, p 2 (ξ) will have two distinct real roots at any point on {x 0 = 0} if g αβ is sufficiently close to the Minkowski metric and u µ u µ sufficiently close to −1. We know from section 3.3 that these last conditions are fulfilled upon taking U and Y sufficiently small (recall thatg αβ (0) equals the Minkowski metric.). Therefore, p 2 (ξ) is a hyperbolic polynomial, and p 2 (ξ) is the product of two hyperbolic polynomials.
We conclude that P ( U , ξ) is the product of four degree one (i.e., p 1 (ξ)), two degree two (i.e., p 2 (ξ)), one degree two (i.e., p 3 (ξ)), and ten degree two (i.e., p 4 (ξ)) hyperbolic polynomials. The Gevrey index of (9) is thus 17 16 (see Remark 15). Recall that 1 < s < 17 16 by assumption. Since m I − n J = 2 for all I, J, and I m I − J n J ≥ 2, we have verified the conditions of Theorem A.14 in the appendix. Hence we obtain the diagonalized where M(U, ∂) is a diagonal matrix whose entries are differential operators of order 30 (the order of the characteristic determinant, see the appendix) whose coefficients depend on at most 29 derivatives of U , and q(U ) contains all the lower order terms. We want to invoke Theorem A.10 to solve (17). To do so, we need to provide initial conditions for (17). Since our goal is to obtain a solution to (9) out of a solution to (17), such initial conditions need to be compatible with solutions to (9). We shall show that all derivatives of U , restricted to {x 0 = 0}, can be formally computed from (9) and written in terms of the initial data. In particular, initial conditions to (17) compatible with (9) can be determined. As usual in these situations, it suffices to show that we can inductively compute ∂ k 0 U on {x 0 = 0} as the tangential derivatives ∂ i can always be computed.
From (6), we can determine ∂ 2 0 g αβ {x 0 =0} in terms of the initial dataŮ . Using the result into (7), we can write B β (∂u, g)∂ 2 g restricted to {x 0 = 0} in terms ofŮ . Equations (7) and (8) then give where b can be written in terms of the initial data on {x 0 = 0}, and the matrix a is the matrix of the coefficients of the terms ∂ 2 0 u β and ∂ 2 0 in equations (7) and (8). At the origin, whereg αβ (0) equals the Minkowski metric, the determinant of a is which is never zero for a 2 ≥ 4 (recall that 0 > 0 and η( 0 ) > 0). Invoking once more the fact thatg αβ is a perturbation of the Minkowski metric, we conclude that det(a)| {x 0 =0} never vanishes. We can thus invert a and write ∂ 2 0 u β and ∂ 2 0 at x 0 = 0 in terms ofŮ .
The initial data for (17), denoted˚ U , consists of the original initial dataŮ for (9), and the values of ∂ k 0 U {x 0 =0} determined by the above procedure for k = 2, . . . , 29.
Remark 2. The above procedure determines all derivatives of U , evaluated at x 0 = 0, in terms of the initial conditionsŮ . It follows that if the initial dataŮ is analytic, a well-known argument using power series can be employed to construct an analytic solution to (9) in a neighborhood of {x 0 = 0}. These techniques for construction of analytic solutions, however, say nothing about causality.
(We note that in invoking Theorem A.10, we are using that the intersections of the cones determined by the polynomials p i (ξ) have non-empty interiors (recall definition A.4). This follows from the above expressions, but it can also be verified from the explicit computations in section 3.5.) The conclusions that > 0 and g is a Lorentzian metric follow by continuity in the x 0 variable, since these conditions are true at x 0 = 0. Now we move to obtain a solution to (9) in R 4 . The argument is similar to the one in [30], thus we shall go over it briefly.
Let {Ů k } ∞ k=1 be a sequence of analytic initial conditions for the system (9) converging in G (s) ({x 0 = 0}) toŮ . For each k, let (u k , k , g k ) be the analytic solution to (9), defined in a neighborhood of {x 0 = 0}, and taking on the initial dataŮ k (see Remark 2). Let˚ U k be the initial data for (17) obtained fromŮ k and compatible with (9), i.e., the one derived by the inductive procedure previously described. Then, In light of the compatibility of˚ U k , and because (17) was derived from (9) via diagonalization, the solutions (u k , k , g k ) also satisfy (17). Furthermore, this solution to (17) also agrees with the one given by Theorem A.10 (since this theorem also applies for analytic data, i.e., s = 1). The energy-type of estimates proved by Leray and Ohya [28] guarantee then that (u k , k , g k ) → (u, , g) in G (s) and that (u, , g) satisfy the original system (9). By construction, (u, , g) take on the initial dataŮ .
Remark 3. The initial conditions for the VECF system have to satisfy the Einstein constraint equations (recall Definition 2.1). The initial conditionsŮ satisfy the constraints in the region W in light of the way thatŮ was constructed out of I| U . This is, naturally, necessary for the eventual construction of a full solution to the VECF system. However, purely from the point of view of (9) in R 4 , initial condition can be prescribed freely, i.e., they do not have to satisfy any constraints. Therefore, the existence of the analytic initial dataŮ k follows simply by the density of analytic functions in G (s) . Also by density, we can guarantee that the components (˚ 0 ) k and (g αβ ) k inŮ k satisfy (˚ 0 ) k > 0 and that (g αβ ) k is a Lorentzian metric.
Remark 4. The above calculations involving (a 2 2 − 2a 2 − 8) ≥ 0 show why we have the technical assumption a 2 ≥ 4. As our calculations were presented already with a 1 = 4 in place, they do not reveal the reason for this assumption, which as follows. Computing the characteristic determinant with general a 1 produces a very complicated expression with some terms proportional to a 1 − 4. These terms vanish when a 1 = 4, and the corresponding expression simplifies to (11). This can be seen explicitly in Appendix C.
3.5. Causality. Having obtained solutions, we now investigate the causality of equations (9). As in section 3.4, we use results and terminology recalled in the appendix. Proof. Fix x ∈ [0, T ) × R 3 . The characteristic determinant of (9) at x is given by (11), with the obvious replacement of U by U and evaluated at x; the polynomials p i (U (x), ξ) ≡ p i (x, ξ), i = 1, . . . , 4, are given by expressions (12) to (15), again with the obvious replacement by U (x). By the same argument used in section 3.4 to prove that the p i (ξ)'s are hyperbolic polynomials on {x 0 = 0}, namely, that g αβ is near the Minkowski metric, we know that the polynomials p i (x, ξ) are hyperbolic (perhaps after shrinking T if necessary).
Assume for a moment that g is the Minkowski metric at x and that u λ u λ = −1 (note that we have not proved yet that u remains normalized for x 0 > 0). The roots of {p 2 (x, ξ) = 0} are given by (16), changing u by u, which we can write as where θ is the angle between u and ξ in R 3 , we used u · ξ = u 2 ξ 2 cos θ, and we omitted the dependence of u and θ on x for simplicity. Equation (18) determines the two halves of the characteristic cone V 2 (x) in the cotangent space at x. We will have that K * ,± (x) ⊆ Γ * ,± 2 (x) if the slopes s ± satisfy −1 < s ± (u, θ) < 1 for each u and θ. To see that this is the case, compute s ± (u, 0) = s ± (u, 2π) = − ± √ a 2 + (a 2 − 1) u 2 (1 + u 2 ) 1 + (a 2 − 1)(1 + u 2 ) , and observe that this expression is always between −1 and 1 for a 2 ≥ 4. We seek the maxima and minima of s ± (u, θ) for 0 < θ < 2π. Computing the derivative with respect to θ and solving for sin θ, we find sin θ = 0, i.e., θ = π. We readily verify that −1 < s ± (u, π) < 1, thus −1 < s ± (u, θ) < 1. Since this last condition is open, the result remains true when g is sufficiently close to the Minkowski metric and u sufficiently close to unitary, which is the case if T is taken sufficiently small. The same argument shows that K * ,± (x) ⊆ Γ * ,± 3 (x), where again one uses the condition a 2 ≥ 4.
We conclude that for any , and the result now follows from Theorem A.11 and Remark 12.
Remark 5. The characteristics associated with p 1 (ξ) and p 4 (ξ) are of course those of the flow lines and gravitational waves. The characteristics associated with p 3 (ξ) and p 2 (ξ) are interpreted, respectively, as sound waves and shear waves. The latter is sometimes called a second sound wave and is present also in the Müller-Israel-Stewart theory [22]. It is useful to compare these characteristics to those of the ideal fluid. In the latter case we have the flow lines and the sound cone (i.e., the characteristics of the sound waves; see [17] for a detailed discussion of the role of the sound cone in the relativistic Euler equations). Here it is as if the sound cone had "split" into two sound-type characteristics. This resembles what happens in magnetohydrodynamics: there two different characteristics are present for the magnetoacoustic waves, namely, the so-called fast and slow magnetoacoustic waves (see [1] for details).
3.6. Existence and causality for the system in R × Σ. Here we show how the solution found in section 3.4 can be used to construct a causal solution in a region of R × Σ, thus effectively proving Theorem 2.2. Recall that we embedded Σ into R × Σ. Remark 6. Consider the solution U = (u, , g) to (9) obtained in Proposition 1. Let p be a point on {x 0 = 0} × Σ and W be as in section 3.3. Let D + g (W) ⊆ [0, T ) × R 3 be the future domain of dependence of W in the metric g, where replacing W with a smaller set if necessary, we can assume that x 0 < T for every (x 0 , x 1 , x 2 , x 3 ) ∈ D + g (W). In the coordinates on D + g (W) induced from the coordinates on [0, T ) × W, the solution U is in G (2,s) The solution will remain in G (2,s) upon coordinate changes that are Gevrey regular [32]. Note that there are plenty of such coordinate changes in that a smooth manifold always admits a maximal compatible analytic atlas.
Proof. The vector field u satisfies (8), whose explicit form is This can be written as This is an equation for the scalar u λ u λ . The operator u α u µ ∇ α ∇ µ satisfies the assumptions of Theorem A.10. Therefore, u α u α = −1 in D + g (W) if this condition is satisfied initially, which is the case by construction.
Lemma 3.4. For every q ∈ Σ there exists a neighborhood Z q ⊆ Σ of q in Σ and a globally hyperbolic development M q of I| Zq , where M q ⊆ [0, T q ) × Σ for some T q > 0.
Proof. Let p be a point on {x 0 = 0} × Σ and W be as in section 3.3. Since the initial conditionsŮ (whereŮ is as in section 3.3) agree on W with those from the initial data I, in view of Lemma 3.2, we conclude that U is a solution to the reduced Einstein equations within D + g (W). It is well-known that a solution to the reduced equations within D + g (W) is also a solution to the full Einstein's equations if and only if the constraints are satisfied, which is the case by the definition of I. Because p was an arbitrary point, the result is proven.
We now glue the different M q 's in order to obtain a global (in space) solution.
Proposition 2. Let q, r ∈ Σ, Z q and Z r be neighborhoods of q and r as in lemma 3.4, with globally hyperbolic developments M q and M r of I| Zq and I| Zr , respectively, and corresponding solutions U q = (u q , q , g q ) and U r = (u r , r , g r ) of the VECF equations. Assume that Z q ∩ Z r = ∅. Then, for any w ∈ Z q ∩ Z r , there exist neighborhoods U q and U r of w in M q and M r , respectively, and a diffeomorphism ψ : U q → U r such that U q = ψ * (U r ).
Proof. We shall construct harmonic coordinates for g q in a neighborhood of w in M q as follows. Identifying (a portion of) Σ with its embedding in M q , take normal coordinates (V, {y i }) for g 0 on Σ centered at w, where g 0 comes from the initial data I. Note that the initial data is Gevrey regular in the {y i } coordinates (see the argument in section 3.2). We can thus assume that U q is in G (2,s) (see Remark 6) On [0, T q ) × V , where T q > 0 is some small number such that U q is defined on [0, T q ) × V , we introduce coordinates {y α }, y 0 ∈ [0, ∞). Consider family of initial-value problems parametrized by α: where ∇ is the covariant derivative in the metric g q . This problem has a Gevrey regular solution in a neighborhood of w in [0, T q ) × V , and a standard implicit function type of argument shows that the functions x α ≡ f (α) define (harmonic) coordinates near w. We now consider the change of coordinates x = x(y) where V is a neighborhood of w in V , T > 0 is determined by the foregoing conditions guaranteeing the existence of the coordinates {x α }, and W is an open set containing the origin. Pulling U q back to W via x −1 , it follows from these constructions that (x −1 ) * (U q ) satisfies the reduced Einstein equations in W . Since U q originally satisfied (2) and (4) as well, we conclude that it is a solution to (9) in W .
We can repeat the above argument to obtain wave coordinates {z α } for g r . Because (V, {y i }) is intrinsically determined by g 0 , and M q and M r induce on Z q ∩ Z r the same initial data, the map z agrees with x on {0}×V (in the region where both are defined). From these facts, we conclude that (x −1 ) * (U q ) and (z −1 ) * (U r ) (i) are solutions to (9) in some domain [0, t) × Y ⊆ [0, ∞) × R 3 containing the origin, and (ii) take the same initial data on {0} × Y .
Using Proposition 2, we can now identify overlapping globally hyperbolic developments, thus obtaining a globally hyperbolic development of I as stated in Theorem 2.2. Causality follows essentially from Lemma 3.2: by the foregoing, we can assume that M is diffeomorphic to [0, T ) × Σ for some T > 0. Shrinking T if necessary, we reduce the problem to local coordinates, in which case we can employ wave coordinates. Causality, as stated in Theorem 2.2, is preserved by diffeomorphisms, thus the result follows from the causality of the reduced system guaranteed by Lemma 3.2. This finishes the proof of Theorem 2.2.

4.
Proof of Theorem 2.3. The proof of Theorem 2.3 is essentially contained in the above. In the case of a Minkowski background, the system reduces to where m is as in (10), U = (u β , ) and q(U ) is as in (9) with the appropriate changes for this 5 × 5 system. The system can be analyzed as in section 3.4. We can do this directly in R 4 , without the complications of constructing the initial dataŮ . The characteristic determinant is given by p 1 (ξ)p 2 (ξ)p 3 (ξ), where these polynomials are as before, with the simplification that now we need not carry out any near-Minkowski arguments. Without the matrix g µν ∂ 2 µν coming from Einstein's equations, the Gevrey index of the system is 7 6 , and analogues of Proposition 1 and Lemma 3.2 establish the result.
Appendix A. Tools of weakly hyperbolic systems. For the reader's convenience, we state in this appendix the results about Leray-Ohya systems (sometimes called weakly hyperbolic systems) that are used in the proof of Theorem 2.2. These results have been established by Leray and Ohya in [27,28] for the case of systems with diagonal principal part, and extended by Choquet-Bruhat in [6] to more general systems. These works build upon the classical work of Leray on hyperbolic differential equations [26]. The reader can consult these references for the proofs of the results stated below. Further discussion can be found (without proofs) in [7,10,12]. Related results can also be found in [34].
We start by recalling some standard notions and fixing the notation that will be used throughout. Given T > 0, let X = [0, T ] × R n . By ∂ k we shall denote any k th order derivative. We shall denote coordinates on X by {x α } n α=0 , thinking of x 0 ≡ t as the time-variable. We use the multi-index notation to write where |α| = α 0 + α 1 + α 2 + · · · + α n .
A.1. Gevrey spaces. In this section we review the definition of Gevrey spaces. Roughly speaking, a function is of Gevrey class if it obeys inequalities similar, albeit weaker, than those satisfied by analytic functions. One of the crucial properties of Gevrey spaces for their use in general relativity is that they admit compactly supported functions.
We say that f : R n → C belongs to the Gevrey space Let K ⊂ R n be the cube of unit side. We say that f belongs to the local Gevrey space G (s) where sup K is taken over all side one cubes K in R n .
We note that the case s = 1, i.e., G (1) (R n ), corresponds to the space of analytic functions.
We next introduce the space of maps defined on X whose derivatives up to order m belong to G (s) ({x 0 = t}), 0 ≤ t ≤ T .
Definition A.2. On X, denote S t = {x 0 = t}. Let s ≥ 1, and let m ≥ 0 be an integer. We denote by α a multi-index α = (α 0 , . . . , α n ) for which α 0 = 0. We define G m,(s) (X) as the set of maps f : X → C such that sup α, |β|≤m, 0≤t≤T Let Y be an open set of R d . We define G m,(s) (X ×Y ) as the set of maps f : X ×Y → C such that sup α, γ, |β|≤m, 0≤t≤T Let K t ⊂ S t be the cube whose sides have unit length. The spaces G m,(s) loc (X) and G m,(s) loc (X × Y ) are defined as the set of maps f : X → C and f : X × Y → C, respectively, such that sup α, |β|≤m, 0≤t≤T and sup α, γ, |β|≤m, 0≤t≤T where sup Kt is taken over all cubes of side one within S t . Remark 7. Definitions A.1 and A.2 are easily generalized to vector and tensor fields in R n and X, and to open subsets of R n and X. In particular, replacing R n by an open set Ω and X by [0, T ] × Ω in the above definitions we obtain the corresponding spaces for Ω. This allows one to define Gevrey spaces on manifolds. If M is a differentiable manifold, we say that f : M → C belongs to G (s) (M ) if for every p ∈ M there exists a coordinate chart (x, U ) about p such that f • x −1 ∈ G (s) (Ω), where Ω = x(U ). This definition generalizes for vector and tensor fields.
Remark 8. The reason to treat X and Y differently in definitions of G (s) (X × Y ) and G m,(s) (X × Y ) is that, in the theorems of section A.2, we need to distinguish between the regularity with respect to the space-time X and the regularity with respect to the parametrization of the initial data.
Remark 9. We could similarly define for manifolds the analog of the other Gevrey spaces introduce above. However, this can be somewhat cumbersome and not always natural. In particular, the spaces G m,(s) require a distinguished coordinate that plays the role of time. This can always be done locally, and it can be done for globally hyperbolic manifolds if we fix a particular foliation in terms of space-like slices (as done, e.g., in [10,12]), although it is debatable how canonical this is. Here we prefer to avoid extra complications, i.e., we in fact only need the definition of G (s) (Σ), which is used for the construction of appropriate local coordinates and the construction of the initial data for the system in R 4 (sections 3.2 and 3.3) and in the results of section 3.6. The bulk of the proofs are carried out for the system in R 4 , where all the different Gevrey spaces play a role. It follows that the solution in R 4 is in particular smooth, giving rise to a smooth globally hyperbolic development. Note that for the conclusion of Theorem 2.2 it is not needed to assert that the full solution enjoys certain Gevrey regularity.
For more about Gevrey spaces, see, e.g., [28,39]. We remark that the terminology "local" and the notation G loc are not standard.
Clearly, p(x, ξ) is a homogeneous polynomial of degree k. It is called the characteristic polynomial (at x) of the operator a.
The cone V x (p) of p in T * x X is defined by the equation p(x, ξ) = 0.
Definition A.3. With the above notation, p(x, ξ) is called a hyperbolic polynomial (at x) if there exists ζ ∈ T * x X such that every straight line through ζ that does not contain the origin intersects the cone V x (p) at k real distinct points. The differential operator a(x, ∂ k ) is called a hyperbolic operator (at x) if p(x, ξ) is hyperbolic.
Leray proved in [26] that (if X is at least three-dimensional) if p(x, ξ) is hyperbolic at x, then the set of points ζ satisfying the condition of Definition A.3 forms the interior of two opposite half-cones Γ * ,+ x (a), Γ * ,− x (a), with Γ * ,± x (a) non-empty, with boundaries that belong to V x (p) .
Remark 10. Another way of stating Definition A.3 is as follows. Given ζ ∈ T x X, consider a non-zero vector θ that is not parallel to ζ and form the line λζ + θ, where λ ∈ R is a parameter. We then require this line to intersect the cone V x (p) at k distinct real points. An equivalent definition of hyperbolic polynomials is as follows [9]: p(x, ξ) is hyperbolic at x if for each non-zero ξ = (ξ 0 , . . . , ξ n ) ∈ T * x X, the equation p(x, ξ) = 0 has k distinct real roots ξ 0 = ξ 0 (ξ 1 , . . . , ξ n ).
With applications to systems in mind, we next consider the N × N diagonal linear differential operator matrix Each a J (x, ∂ k J ), J = 1, . . . , N is a linear differential operator of order k J .
Definition A.4. The operator A(x, ∂) is called Leray-Ohya hyperbolic (at x) if: (i) The characteristic polynomial p J (x, ξ) of each a J (x, ∂ k J ) is a product of hyperbolic polynomials, i.e.
Remark 11. When the above hyperbolicity properties hold for every x, we call the corresponding operators hyperbolic (we can also talk about hyperbolicity in an open set, a certain region, etc.). When we say that an operator is Leray-Ohya hyperbolic on the whole space (or in an open set, etc.), this means not only that Definition A.4 applies for every x, but also that the numbers r J and the degree of the polynomials p J,q (x, ξ), q = 1, . . . , r J , J = 1, . . . , N , do not change with x.
Definition A.5. We define the dual convex half-cone C + x (A) at T x X as the set of v ∈ T x X such that ξ(v) ≥ 0 for every ξ ∈ Γ * ,+ x (A); C − x (A) is analogously defined, and we set C x (A) = C + x (A) ∪ C − x (A). If the convex cones C + x (A) and C − x (A) can be continuously distinguished with respect to x ∈ X, then X is called time-oriented (with respect to the hyperbolic form provided by the operator A). A path in X is called future (past) time-like with respect to A if its tangent at each point x ∈ X belongs to C + x (A) (C − x (A)), and future (past) causal if its tangent at each point x ∈ X belongs or is tangent to for each x ∈ Σ. It follows that for a time-oriented X, the concepts of causal past, future, domains of dependence and influence of a set can be defined in the same way one does when the manifold is endowed with a Lorentzian metric. We refer the reader to [26] for details. Here we need only the following: the causal past J − (x) of a point x ∈ X is the set of points that can be joined to x by a past causal curve.
Remark 12. The definitions in Definition A.5 endow X with a causal structure provided by the operator A. Despite the similar terminology, however, it should be noticed that all of the above definitions depend only on the structure of the operator A, and do not require an a priori Lorentzian metric on X. The case of interest in general relativity, however, is when the causal structure of the spacetime is connected with that of A. In this regard, the following observation is useful. Suppose that X has a Lorentzian metric g. For causal solutions of the systems of equations here described (see Theorem A.11 below) to be causal in the sense of general relativity, one needs that, for all x ∈ X, C ± x (A) ⊆ K ± x , where K ± x are the two halves of the light-cone {g µν ξ µ ξ ν ≤ 0}. By duality, this is equivalent to saying that in the cotangent spaces we have K * ,± x ⊆ Γ * ,+ x (A), where K * ,± x are the two halves of the dual light-cone {g µν ξ µ ξ ν ≤ 0}.
Next, we consider the following quasi-linear system of differential equations where Definition A.7. A Leray-Ohya system (with diagonal principal part) is a Leray system where the matrix A is Leray-Ohya hyperbolic. In the quasi-linear case, since the operators a depend on U , we need to specify a function U that is plugged into A(x, U, ∂) in order to compute the characteristic polynomials. In this case we talk about a Leray-Ohya system for the function U . The primary case of interest is when U assumes the values of the given Cauchy data.
When considering a quasi-linear system, we write p(x, U, ξ) and similar expressions to indicate the dependence on U .
We now formulate the Cauchy problem for Leray systems.
Definition A.8. Let Σ be a regular hypersurface in X, which we assume for simplicity to be given by {x 0 = 0}. The Cauchy data on Σ for a Leray system in X consists of the values of U = (u I ) and their derivatives up to order m I − 1 on Σ, i.e., ∂ α u I Σ , |α| ≤ m I − 1, I = 1, . . . , N . The Cauchy data is required to satisfy the following compatibility conditions. If V = (v I ) is an extension of the Cauchy data defined in a neighborhood of Σ, i.e. ∂ α v I Σ = ∂ α u I Σ , |α| ≤ m I − 1, I = 1, . . . , N , then the difference a J (x, V, ∂)U − b J (x, V ) and its derivatives of order less than n J vanish on Σ, for J = 1, . . . , N . When to a Leray system A(x, U, ∂)U = B(x, U ) we prescribe initial data satisfying these conditions, we say that we have a Cauchy problem for A(x, U, ∂)U = B(x, U ).
Notice that by definition, the Cauchy data for a Leray system satisfies the aforementioned compatibility conditions. We also introduce the following notions related to the Cauchy problem for a Leray system. It is convenient to consider A(x, U, ∂) as a differential operator defined over X × Y , as follows. We shall assume that there exists a differential operator A(x, y, ∂) defined over X × Y with the following property. If (x, y) ∈ X × Y and V = (v J ) is a sufficiently regular function on Σ such that y = (∂ max I m I −n J v J (x)) J=1,...,N , then A(x, V (x), ∂) = A(x, y, ∂). We shall write A(x, y, ∂) for A(x, y, ∂). We are now ready to state the results of this appendix. We use the above notation and definitions in the statement of the theorems below.
Theorem A.10 (Existence and uniqueness). Consider the Cauchy problem for (19). Suppose that the Cauchy data is in G (s) (Σ), and that Suppose that for any V ∈ A s (Σ, Y ) the system is Leray-Ohya hyperbolic with indices m I and n J ; thus for all x ∈ Σ, each p J (x, V, ξ) is the product of r J hyperbolic polynomials, Suppose that each p J,q+1 (x, V, ξ), q = 0, . . . , r J − 1, depends on at most m I − m J,q − r I + q derivatives of v I , I = 1, . . . , N , where m J,q = n J + deg(p J,1 ) + · · · + deg(p J,q ), m J,r J = m J , m J,0 = n J .
Above, deg(p J,q ) is the degree, in ξ, of the polynomial p J,q (x, V, ξ).
Let 0 ≤ g I ≤ r I be the smallest integers such that a J (x, V, ∂ m J −n J ) and b J (x, V ) depend on at most m I − n J − r I + g I derivatives of v I , I = 1, . . . , N , J = 1, . . . , N . Finally, assume that 1 ≤ s ≤ r J g J and n 2 + r J < n J , J = 1, . . . , N.
Then, there exists a T > 0 and a solution U = (u I ) to the Cauchy problem for (19) and defined on [0, T ) × R n ⊆ X. The solution satisfies Furthermore, the solution is unique in this regularity class.
Theorem A.11 (Causality). Assume the same hypotheses of Theorem A.10, and suppose further that Let T and U be as in the conclusion of Theorem A.10. Then, if T is sufficiently small, the operator A(x, U, ∂) is Leray-Ohya hyperbolic (thus the causal past of a point is well-defined), and for each Remark 14. Theorem A.10 assumes that the system is Leray-Ohya hyperbolic for V ∈ A(Σ, Y ), which is essentially the space of values near the initial data. (Naturally, it would not make sense to require the system to be Leray-Ohya hyperbolic for the yet to be proven to exist solution U .) Once U is constructed, one can then ask whether the system is Leray-Ohya hyperbolic for U . This will be the case if T is small, since in this case the values of U will be close to those of the initial data by continuity, guaranteeing that U (x) ∈ A(Σ, Y ).
Theorems A.10 and A.11 are proven in [28] (reprinted in [29]). We now consider a system whose principal part is not necessarily diagonal. The definition of a Leray system depends only on the existence of the indices m I and n J with the stated properties, and thus can be extended to non-diagonal systems.
Definition A.12. Consider a system of N partial differential equations and N unknowns in X, and denote the unknown as U = (u I ), I = 1, . . . , N . The system is a (not necessarily diagonal in the principal part) Leray system if it is possible to attach to each unknown u I a non-negative integer m I and to each equation a non-negative integer n J , such that the system reads is a homogeneous differential operator of order m I − n J (which can be zero), whose coefficients depend on at most m K − n J − 1 derivatives of u K , K = 1, . . . N , and there is a sum over I in h J I (·)u I . The remaining terms, b J (x, ∂ m K −n J −1 u K ), also depend on at most m K − n J − 1 derivatives of u K , K = 1, . . . N . As before, these indices are defined only up to an overall additive integer.
As done above, for a given sufficiently regular U , h J I (x, ∂ m K −n J −1 U K , ∂ m I −n J ) are well-defined linear operators, and we can ask about their hyperbolicity properties. The case of interest will be, again, when we evaluate these operators at some given Cauchy data.
Definition A. 13. The characteristic determinant of (21) at x ∈ X and for a given U is the polynomial p(x, ξ) in the co-tangent space T * x X, ξ ∈ T * x X, given by p(x, U, ξ) = det (H(x, U, ξ)).
Note that p is a homogeneous polynomial of degree Under appropriate conditions, (21) can be transformed into a Leray-Ohya system of the form (19), i.e., with diagonal principal part. More precisely, we have the following.
Let d q be the degree of p q (x, U, ξ), q = 1, . . . , Q, and suppose that Then, there exists a N × N matrix C(x, U, ∂) of differential operators whose coefficients depend on U , such that where I is the N ×N identity matrix, p(x, U, ∂) is the differential operator associated with p(x, U, ξ), and B 1 (x, U ) and B 2 (x, U ) depend on at most − 1 derivatives of U , as do the coefficients of the operator p(x, U, ξ). Furthermore, there is a choice of indices that makes the system into a Leray system. In particular, if the intersections ∩ q Γ * ,+ x (a q ) and ∩ q Γ * ,− x (a q ), where Γ * ,± x (a q ) are the half-cones associated with the hyperbolic polynomials p q (x, U, ξ), have non-empty interiors, then (23) is a Leray-Ohya system with diagonal principal part in the sense of definition A.7.
Definition A. 15. Under the hypotheses of Theorem A.14, the number Q Q−1 is called the Gevrey index of the system. Remark 15. Suppose that (23) forms a Leray-Ohya system in the sense of definition A.7, i.e., the half-cones have non-empty interiors as stated in Theorem A.14. It can then be shown [6] that a value of s sufficient to apply Theorems A.10 and A.11 is 1 ≤ s < Q Q−1 . Let us make a brief comment about the proofs of the above results. Theorem A.10 is proven as follows. First, one solves the associated linear problem. This is done by a method of majorants reminiscent of the Cauchy-Kowalevskaya theorem. One uses the fact that Gevrey functions admit a formal series expansion that provides a consistent way of constructing successive approximating solutions to the problem. The non-linear problem is then treated via a fixed point argument, upon solving successive linear problems. Theorem A.11 is obtained by a Holmgren type of argument. We remark that the assumption that p J,q+1 (x, V, ξ), q = 0, . . . , r J − 1, depends on at most m I − m J,q − r I + q derivatives of v I , I = 1, . . . , N , ensures that the coefficients of the associated differential operators a J,q+1 (x, U, ∂) do not depend on too many derivatives of U , as it should be in the treatment of quasi-linear equations.
Theorem A.14 is based on the following identity: where a is an N × N invertible matrix and c T the transpose of the co-factor matrix. At the level of differential operators, this identity produces the lower order terms B 1 .
One then needs to match the order of the resulting differential operators and lower order terms with appropriate indices satisfying the definition of a Leray system. This is possible under the conditions on d q and stated in the theorem.