Local Existence of Solutions of Self Gravitating Relativistic Perfect Fluids
aa r X i v : . [ m a t h . A P ] M a y LOCAL EXISTENCE OF SOLUTIONS OF SELF GRAVITATINGRELATIVISTIC PERFECT FLUIDS
UWE BRAUER AND LAVI KARP
Abstract.
This paper deals with the evolution of the Einstein gravitational fields whichare coupled to a perfect fluid. We consider the Einstein–Euler system in asymptoticallyflat spacestimes and therefore use the condition that the energy density might vanish ortend to zero at infinity, and that the pressure is a fractional power of the energy density. Inthis setting we prove a local in time existence, uniqueness and well posedness of classicalsolutions. The zero order term of our system contains an expression which might not bea C ∞ function and therefore causes an additional technical difficulty. In order to achieveour goals we use a certain type of weighted Sobolev space of fractional order. In [4] weconstructed an initial data set for these of systems in the same type of weighted Sobolevspaces.We obtain the same lower bound for the regularity as Hughes, Kato and Marsden [14]got for the vacuum Einstein equations. However, due to the presence of an equation ofstate with fractional power, the regularity is bounded from above. Introduction
This paper deals with the Cauchy problem for the Einstein–Euler system describing arelativistic self-gravitating perfect fluid, whose density either has compact support or fallsoff at infinity in an appropriate manner.The evolution of the gravitational field is described by the Einstein equations(1.1) G αβ = R αβ − g αβ R = 8 πT αβ , where g αβ is a semi Riemannian metric having a signature ( − , + , + , +), R αβ is the Riccicurvature tensor, and R is the scalar curvature. Both tensors are functions of the metric g αβ and its first and second order partial derivatives. The right-hand side of (1.1) consistsof the energy–momentum tensor T αβ , which in the case of a perfect fluid takes the form(1.2) T αβ = ( ǫ + p ) u α u β + pg αβ , where ǫ is the energy density, p is the pressure and u α is the four-velocity vector. The vector u α is a unit timelike vector, which means that it satisfies the normalization condition Mathematics Subject Classification.
Primary 35L45, 35Q75 ; Secondary 58J45, 83C05.
Key words and phrases.
Einstein-Euler systems, hyperbolic symmetric systems, energy estimates,Makino variable, weighted fractional Sobolev spaces.Research supported ORT Braude College’s Research Authority. (1.3) g αβ u α u β = − . The Euler equations describing the evolution of the fluid take the form(1.4) ∇ α T αβ = 0 , where ∇ denotes the covariant derivative associated with the metric g αβ . Equations (1.1)and (1.4) are not sufficient to determinate the structure uniquely, a functional relationbetween the pressure p and the energy density ǫ (equation of state) is also needed. Wechoose an equation of state that has been used in astrophysical problems. It is the analogueof the well known polytropic equation of state in the non-relativistic theory, given by(1.5) p = p ( ǫ ) = Kǫ γ , K, γ ∈ R + , < γ. The sound velocity is denoted by σ = dpdǫ , and the range of the energy density ǫ will be restricted so that the causality condition σ < g αβ , the velocity vector u α and the energy density ǫ . These are functions of t and x a , where x a ( a = 1 , ,
3) are theCartesian coordinates on R . The alternative notation x = t will also be used and Greekindices will take the values 0 , , , ǫ is expected to have compact support, or tendto zero at spatial infinity in an appropriate sense. It is well known that the usual sym-metrization of the Euler equations is badly behaved in cases where the density tends tozero somewhere. The coefficients of the system degenerate or become unbounded when ǫ approaches zero. It was observed by Makino [19] that this difficulty can be to some extendcircumvented in the case of a non-relativistic fluid by using a new matter variable w inplace of the mass density. For this reason we introduce the quantity(1.6) w = M ( ǫ ) = ǫ γ − , and we call it the Makino variable. A similar device was used by Gamblin [12] and Bezard[2] for the Euler-Poisson equations, and by Rendall [24] and Oliynyk [22] for the Einstein–Euler equations. The common method for solving the Cauchy problem for the Einsteinequations consists usually of the following steps.1. Initial data must satisfy the constraint equations, which are intrinsic to the initialhypersurface. Therefore, the first step is to construct solutions of these constraints.2. The second step is to use the harmonic coordinate condition and to solve the evo-lution equations with these initial data. OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 3
3. The last step is to prove that the harmonic coordinate condition and the solutionof the constraints propagate. That means if they held on a initial hypersurface,they hold for later times.The last step was treated in detail, for example in Fisher and Marsden[11]. The idea is towork out the condition ∇ α G αβ = 0. Since our energy–momentum satisfy (1.4), their resultcan be immediately generalised for our case. But for the sake of brevity we have omittedthe details.However, the presence of the equation of state (1.5) introduces an additional step: thecompatibility problem of the initial data for the fluid and the gravitational field (see (2.11)).There are three types of initial data for the Einstein–Euler system: • The gravitational data is a triple ( M , h, K ab ), where M is a space-like manifold, h is a proper Riemannian metric on M , and K ab is the second fundamental form on M (extrinsic curvature). The pair ( h, K ab ) must satisfy the constraint equations(2.10); • The matter variables, consisting of the energy density z and the momentum density j a , appear on the right hand side of the constraints (2.10); • The initial data for Makino variable w and the velocity vector u α of the perfectfluid.The only type of Sobolev spaces which are known to be useful for existence theorems for theconstraint equations in an asymptotically flat manifold, are the weighted Sobolev spaces H k,δ , where k ∈ N and δ ∈ R . These spaces were introduced by Nirenberg and Walker [21]and Cantor [5], and they are the completion of C ∞ ( R )-functions under the norm(1.7) k u k k,δ = X | α |≤ k Z (cid:0) (1 + | x | ) δ + | α | | ∂ α u | (cid:1) dx. Due to the presence of the equation of state (1.5) and the Makino variable (1.6), wehave to estimate k w γ − k k,δ . So it is perhaps worth discussing the estimate of the Sobolev’snorm of u β in more details for β >
1. For simplicity we discuss this in the ordinary Sobolevspace H k = H k ( R ). The simplest case is when β ∈ N , then k u β k H k ≤ C ( k u k L ∞ ) k u k H k andthere is no restriction on k . When β N , then we obtain the same estimate, provided that k ≤ β . This bound on k was improved by Runst and Sickel [25] to k < β + . Applyingthis to β = γ − , and taking into account the Sobolev embedding k u k L ∞ ≤ C k u k H k for k > , we get a lower and upper bound for k :(1.8) 32 < k < γ − . The only exception is the case when γ − is an integer. Note that for certain values of γ ,inequalities (1.8) possesses no integer solution. Hence, for these values of γ it is impossibleto obtain a solution to the Einstein–Euler system in Sobolev spaces of integer order. Soin order to be able to solve the coupled system for the maximal range of the power γ ,and in addition, to improve the regularity of the solutions, we are considering the Cauchyproblem in the weighted fractional spaces H s,δ , where s is real number (see Definition U. BRAUER AND L. KARP H k,δ to a fractionalorder. In [4] the authors constructed initial data for coupled systems (1.1), (1.2) and (1.4)with the equations of state (1.5). This includes the solution to the constraint equations(2.10), as well as the solution to the compatibility problem between the matter variable( z, j a ) and the fluid variables ( w, u α ), (2.11), in the H s,δ -spaces. Here we will establish thewell-posedness of Einstein–Euler systems in the weighted fractional Sobolev spaces H s,δ .The common way to prove well-posedness is to rewrite the evolution equations as asymmetric hyperbolic system. So our first step is to use the Makino variable (1.6) and toreduce the Euler equations (1.4) to a uniformly first order symmetric hyperbolic system.This result was announced in [3] and here we present a detailed proof of it. Our hyperbolicreduction is based on the fluid decomposition; for alternative reductions see [24].It is well-known that the Einstein equations can be written as a system of quasi-linearwave equations under the harmonic gauge condition [6, 7, 29]. The proofs of existence anduniqueness either use second order techniques [6, 8, 13, 14, 17], or transferring the equationsto a first order symmetric hyperbolic system. Fischer and Marsden used the first ordertechniques and obtained the well-posedness of the reduced vacuum Einstein equations in H s and for s > [11]. This result was improved by Hughes, Kato and Marsden [14], whoobtained ( g αβ , ∂ t g αβ ) ∈ H s +1 × H s for s > . They used second order theory, and tookadvantage of the specific form of the quasi-linear system of wave equations, namely, thatthe coefficients depend only on the semi-metric g αβ , but not on its first order derivatives.Our aim is to prove existence and uniqueness of the reduced Einstein–Euler system (1.1),(1.2) and (1.4) with the equation of state (1.5). In addition, we would like to achieve thesame regularity of the metric as in [14]. But since we have here a coupled system whichone of them is a first order, the second order techniques of Hughes, Kato and Marsden in[14] are no longer available for the present problem.In asymptotically flat spacetimes the initial metric g αβ (0) differs from the Minkowskimetric by a term which is O (1 /r ) at spatial infinity, and this term does not belong to H s .It is therefore more appropriate to consider both the constraint and evolution equationsin the H s,δ spaces rather than the unweighted spaces H s . For the vacuum equations thesecond author obtained well-posedness of the reduced Einstein equations with ( g αβ , ∂ t g αβ ) ∈ H s +1 ,δ × H s,δ +1 , s > and δ > − , see [16]. But unlike Hughes, Kato and Marsden [14],he treated the quasi-linear system of wave equations as a first order symmetric hyperbolicsystem. The first order techniques have the advantage that they enable, in a convenientway, the coupling of the gravitational field equations to other matter models, in particular,to perfect fluids. In the Appendix we explain the main idea of [16] which allows us toobtain the regularity index s > by means of first order hyperbolic systems.A crucial step in the proof of existence and uniqueness of any hyperbolic system is toestablish energy estimates for the linearized system. In order to achieve this we define anappropriated inner–product of the H s,δ spaces, which takes into account the coefficients ofthe linearized system (see Section 5 ). A similar inner–product was used in [16], and herewe rely on these energy estimates. OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 5
Once we have obtained the energy estimates for the linearized system, we use Majda’siterative scheme in order to obtain existence and uniqueness of the quasi–linear symmetrichyperbolic system [18]. This procedure uses the fact that solutions to a linear first ordersymmetric hyperbolic system with C ∞ coefficients and initial data, are also C ∞ . Buthere we encounter a further difficulty, namely, the right hand side of (1.1) contains thefractional power w γ − , see (3.16). So even when w ∈ C ∞ and w ≥ w γ − might not bea C ∞ function. We solve that problem by using the fact that ǫ = w γ − satisfies a certainfirst order linear equation. Gamblin encountered a similar problem for the Euler–Poissonequations [12], but he solved it in a somewhat different way.Our results improve the existence theory of solutions locally in time of self gravitatingrelativistic perfect fluids in several aspects. Rendall studied this problem in [24], but heassumed time symmetry, which means that the extrinsic curvature of the initial manifold iszero, and therefore the Einstein constraint equations are reduced to a single scalar equation.In addition, he dealt only with C ∞ -solutions. In his study of the Newtonian limit of perfectfluids, Oliynyk obtained existence locally in time in the weighted space of integer order H k,δ , for k ≥ γ − is an integer.The paper is organized as follows: In the next section we define the weighted Sobolevspaces of fractional order H s,δ and state the main result. Section 3 has two subsections:the first one deals with the hyperbolic reduction of the Euler equations (1.4); in the secondone we spell out the matrices which describe the coupled equations (1.1), (1.2) and (1.4)as a hyperbolic system.In Section 4 we present tools and properties of the H s,δ -spaces which we need in thecourse of the publication. We also define the corresponding product spaces. The energyestimates for the linearized system are considered in Section 5, there we also define theappropriate inner-product. In Section 6 we treat the iteration procedure. Parts of thesteps are standard and known, but some of them require special attention due to thespecific form of the system (3.24) and the product spaces. In this Section we will use thefact that the coefficients of the first order derivatives depend only on the semi-metric g αβ .Finally, in Section 7 we prove the main result. In the Appendix we give a heuristic ideaexplaining how that fact that the coefficients of the system of wave equations depend onlyon the semi–metric g αβ enable us to obtain the desired regularity by means of symmetrichyperbolic systems. 2. The main results
We obtain the well-posdeness in the weighted Sobolev spaces of fractional order. So wefirst recall their definition.Let { ψ j } ∞ j =0 ⊂ C ∞ ( R ) be a sequence of cutoff function such that, ψ j ( x ) ≥ j ≥
0, supp( ψ j ) ⊂ { x : 2 j − ≤ | x | ≤ j +1 } , ψ j ( x ) = 1 on { x : 2 j − ≤ | x | ≤ j } for j = 1 , , ... , supp( ψ ) ⊂ { x : | x | ≤ } , ψ ( x ) = 1 on { x : | x | ≤ } and | ∂ α ψ j ( x ) | ≤ C α −| α | j , U. BRAUER AND L. KARP where the constant C α does not depend on j .We restrict ourselves to the case p = 2 and denote the Bessel potential spaces by H s with the norm given by k u k H s = Z (1 + | ξ | ) s | ˆ u ( ξ ) | dξ, where ˆ u is the Fourier transform of u . Definition 2.1.
For s, δ ∈ R ,(2.1) (cid:0) k u k H s,δ (cid:1) = ∞ X j =0 ( + δ )2 j k ( ψ j u ) (2 j ) k H s , where f ε ( x ) = f ( εx ) denotes the scaling by a positive number ǫ . The space H s,δ is the setof all tempered distributions having a finite norm given by (2.1).The H s,δ -norm of a distribution u in an open set Ω ⊂ R is given by k u k H s,δ (Ω) = inf f | Ω = u k f k H s,δ ( R ) . Definition 2.2.
Let M be a 3 dimensional smooth connected manifold and let h be ametric on M such that ( M , h ) is complete. We say that ( M , h ) is asymptotically flat of the class H s,δ if h ∈ H s loc ( M ) and there is a compact set S ⊂ M such that:1. There is a finite collection of charts { ( U i , ϕ i ) } Ni =1 which covers M \ S ;2. For each i , ϕ − i ( U i ) = E r i := { x ∈ R : | x | > r i } for some positive r i ;3. The pull-back ( ϕ i ∗ h ) ab is uniformly equivalent to the Euclidean metric δ ab on E r i for each i ;4. For each i , ( ϕ i ∗ h ) ab − δ ab ∈ H s,δ ( E r i ).The H s,δ -norm on the manifold M is defined as follows. Let U ⊂ M be an open set suchthat S ⊂ U and U ⋐ M . Let { χ , χ i } be a partition of unity subordinate to { U , U i } ,then(2.2) k u k H s,δ ( M ) := k χ u k H s ( U ) + N X i =1 k ϕ ∗ i ( χ i u ) k H s,δ ( R ) is the norm of the weighted fractional Sobolev space H s,δ ( M ). For the definition of thenorm k χ u k H s (Ω) on the manifold M see e.g. [1]. Note that the norm (2.2) depends onthe partition of unity, but different partitions of unity result in equivalent norms. In thefollowing we will omit the notation M , that is, we will write k u k H s,δ instead of k u k H s,δ ( M ) .Since the principal symbol of the field equations (1.1) is characteristic in every direction(see e.g. [10]), it is impossible to solve (1.1) in the present form. We study these equationsunder the harmonic gauge condition (2.3) F µ = g βγ Γ µβγ = 0 , OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 7 where g αβ is the inverse matrix of g αβ . Then the field equations (1.1) are equivalent to the reduced Einstein equations (2.4) g µν ∂ µ ∂ ν g αβ = H αβ ( g, ∂g ) − πT αβ + 8 πg µν T µν g αβ , where H αβ ( g, ∂g ) is a quadratic function of the semi–metric g αβ and its first order deriva-tives. Since g µν has a Lorentzian signature, (2.4) is a system of quasi-linear wave equations.Taking into account the equation of state (1.5), the normalization condition (1.2), and theMakino variable (1.6), then the system of wave equations (2.4) become(2.5) g µν ∂ µ ∂ ν g αβ = H αβ ( g, ∂g ) − πw γ − (cid:0) (1 − Kw ) g αβ + 2(1 + Kw ) u α u β (cid:1) . So the unknowns of the system (2.5) coupled with the Euler equations (1.4) are thesemi–metric g αβ , the velocity vector u α and the Makino variable w . Note that even if w isa smooth function, w γ − might not be smooth in certain regions. The initial data consistof the triple ( M , h ab , K ab ), where M is a space-like manifold, h ab is a proper Riemannianmetric on M and K ab is its second fundamental form (extrinsic curvature).The semi–metric g αβ takes the following data on M :(2.6) (cid:26) g | M = − , g a | M = 0 , g ab | M = h ab − ∂ g ab | M = K ab , a, b = 1 , , . The remaining initial data for ∂ g α | M are determined through the harmonic condition F µ = 0. We compute them by inserting the initial data (2.6) in the harmonic gaugecondition (2.3). Since ∂ a g | M = ∂ a g b | M = 0, this results in the following expressions for ∂ g α | M :(2.7) (cid:26) ∂ g | M = 2 h ab ( K ab ) ∂ g c | M = (cid:0) h ab ( ∂ a h bc − ∂ c h ab ) (cid:1) . In addition, the initial data of the velocity vector u α and the Makino variable w are givenon M . We denote the Minikowski metric by η αβ . Theorem 2.3 (Main result) . Let < s < γ − + and − ≤ δ . Assume M is asymp-totically flat of class H s +1 ,δ , K ab ∈ H s,δ +1 , ( u − , u a , w ) (cid:12)(cid:12) M ∈ H s +1 ,δ +1 , w (0) ≥ and u α (0) is a timelike vector. Then there exists a positive T , a unique semi–metric g αβ , aunit timelike vector u α and w satisfying the reduced Einstein equations (2.5) and the Eulerequations (1.4) such that (2.8) ( g αβ ( t ) − η αβ ) ∈ C ([0 , T ] , H s +1 ,δ ) ∩ C ([0 , T ] , H s,δ +1 ) and (2.9) (cid:0) u − , u a , w (cid:1) ∈ C ([0 , T ] , H s +1 ,δ +1 ) ∩ C ([0 , T ] , H s,δ +2 ) . Remark . Note that we have a lower and an upper bound ofthe differentiability index s , however, in case γ − is an integer, then there is no upperbound. U. BRAUER AND L. KARP
A necessary and sufficient condition for the equivalence between the reduced Einsteinequations (2.5) and the field equations (1.1) is that the geometric data ( h, K ab ) satisfy theconstraint equations(2.10) (cid:26) R ( h ) − K ab K ab + ( h ab K ab ) = 16 πz (3) ∇ b K ab − (3) ∇ b ( h bc K bc ) = − πj a . Here R ( h ) = h ab R ab is the scalar curvature with respect to the metric h ab . The righthand-side ( z, j a ) consists of the energy density and the momentum density respectively.Note that the harmonic coordinates F µ satisfy a homogeneous wave equation. That iswhy F µ ≡
0, if F µ | M = 0 and ∂ F µ | M = 0. The first condition follows from (2.7), andthe second holds if the reduced Einstein equations (2.5) and the constraint equations (2.10)are satisfied [7], [27, § § T αβ is divergence free.Thus solving the constraint equations (2.10) ensures that the solution of (2.5) satisfiesthe original system (1.1). However, before we solve the constraints, we need to treat thecompatibility problem between the matter variables ( z, j a ) and the initial data for thevelocity u α and the Makino variable w .This problem can be described as follows: Let ¯ u α denote the projection of the velocityvector u α on the initial manifold M and n α the timelike unit normal vector to M . Theenergy density z is the double projection of T αβ on n α and the momentum density j a isonce the projection of T αβ on n α and once on M . Applying these projections to the perfectfluid (1.2) results in(2.11) ( z = w γ − (1 + (1 + Kw )) h ab ¯ u a ¯ u b j a = w γ − (1 + Kw ) ¯ u a √ h bc ¯ u b ¯ u c . So the compatibility problem consists of solving (2.11) for w , ¯ u a , when z and j a aregiven. This problem combined with a solution to the constraint equations (2.10) wassolved in the H s,δ -spaces in [4, Theorem 2.6]. The conditions for this result are that < s < γ − + , where the metric h ab − δ ab ∈ H s +1 ,δ with δ ∈ ( − , − ), while for themater variables ( z, j a ) ∈ H s,δ and δ is just bounded below by − . Note that for thehyperbolic equations we need one more degree of regularity, so we need to require that( z, j a ) ∈ H s +1 ,δ +1 . But then the Makino variable (1.6) causes that the upper bound for s becomes γ − − . Given this restriction, we have by Theorem 2.5 of [4] and Proposition4.9 below that ( w, u − , u a ) (cid:12)(cid:12) M ∈ H s +1 ,δ +1 .Thus relying on [4], we conclude that there is an initial data set ( h ab , K ab ) and ( w, u α )belonging to the H s,δ -spaces that satisfies both the constraints (2.10) and the compatibilityproblem (2.11). The parameter γ of these initial data, however, belongs to the interval(1 , Corollary 2.5.
Under the assumptions of Theorem 2.3 and in addition under the assump-tion that the initial data ( h ab , K ab ) and ( w, u α ) satisfy the constraint equations (2.10) andcompatibility problem (2.11), there exists a positive T , a semi–metric g , a unit timelike OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 9 vector u α and w satisfying the Einstein (1.1) and the Euler equations (1.4) for t ∈ [0 , T ] .The regularity of g , u α and w are the same as in Theorem 2.3.Remark . Existence, uniqueness and regularityof solutions of the Einstein equations (1.1) hold relative to the harmonic coordinate con-dition (2.3). Geometrical uniqueness requires usually one degree more of differentiability[11]. Planchon and Rodnianski [23](see also [9]) gave an argument for the vacuum case toget rid of this additionally regularity. For the Einstein–Euler system, and other matterfields, however, this problem remains still open.3.
Symmetric Hyperbolic Systems
The main result is proved by transforming the coupled system (2.5) and (1.4) into asymmetric hyperbolic system. We therefore recall its definition.
Definition 3.1 (Symmetric hyperbolic system) . A first order quasi–linear k × k system is symmetric hyperbolic system in a region G ⊂ R k , if it is of the form(3.1) L [ U ] = A α ( U ) ∂ α U + B ( U ) = 0 , where the matrices A α ( U ) are symmetric and for every arbitrary U ∈ G , there exists acovector ξ such that(3.2) ξ α A α ( U )is positive definite. The covectors ξ α for which (3.2) is positive definite, are called timelikewith respect to equation (3.1).If ξ can be chosen to be the vector (1 , , , A ( U ) is a positive definite matrix, and we may write system (3.1) in the form(3.3) A ( U ) ∂ t U = A a ( U ) ∂ a U + B ( U ) . The Euler equations written as a symmetric hyperbolic system.
It is notobvious that the Euler equations written in the conservative form ∇ α T αβ = 0 are symmetrichyperbolic. In fact these equations have to be transformed in order to be expressed in asymmetric hyperbolic form. Rendall presented such a transformation of these equationsin [24], however, its geometrical meaning is not entirely clear and it might be difficult togeneralize it to the non time symmetric case. Hence we will present a different hyperbolicreduction of the Euler equations and discuss it in some details, for we have not seen itanywhere in the literature.The basic idea is to perform the standard fluid decomposition and then to modify theequation by adding, in an appropriate manner, the normalization condition (1.3) which willbe considered as a constraint equation. The fluid decomposition method consists of theprojection of equation ∇ ν T νβ = 0 onto u α which leads to u β ∇ ν T νβ = 0, and the projectionof these equations on rest pace O orthogonal to u α of a fluid which leads to P αβ ∇ ν T νβ = 0, where P αβ = g αβ + u α u β . Inserting this decomposition into (1.2) results in a system of thefollowing form: u ν ∇ ν ǫ + ( ǫ + p ) ∇ ν u ν = 0;(3.4a) ( ǫ + p ) P αβ u ν ∇ ν u β + P να ∇ ν p = 0 . (3.4b)Note that we have beside the evolution equations (3.4a) and (3.4b) the following con-straint equation: g αβ u α u β = −
1. We will show in Subsection 3.1.1 that this constraintequation is conserved under the evolution equation. In order to obtain a symmetric hy-perbolic system we have to modify it in the following way. The normalization condition(1.3) gives that u β u ν ∇ ν u β = 0, so we add ( ǫ + p ) u β u ν ∇ ν u β = 0 to equation (3.4a) and u α u β u ν ∇ ν u β = 0 to (3.4b), which results in u ν ∇ ν ǫ + ( ǫ + p ) P νβ ∇ ν u β = 0(3.5a) Γ αβ u ν ∇ ν u β + σ ( ǫ + p ) P να ∇ ν ǫ = 0 , (3.5b)where σ := q ∂p∂ǫ is the speed of sound andΓ αβ = P αβ + u α u β = g αβ + 2 u α u β is a reflection with respect to the rest subspace O . As mentioned above, we will introducea new matter variable which is given by (1.6). The idea behind is the following: Thesystem (3.5a) and (3.5b) is almost of symmetric hyperbolic form, it is symmetric if wemultiply the system by appropriate factors, for example, (3.5a) by ∂p∂ǫ = σ and (3.5b) by( ǫ + p ). However, doing so we will be faced with a system in which the coefficients willeither tend to zero or to infinity, as ǫ →
0. Hence, it is impossible to represent this systemin a non-degenerate form using these multiplications.The central point is now to introduce a new variable w = M ( ǫ ) which will regularizethe equations even for ǫ = 0. We do this by multiplying equation (3.5a) by κ M ′ = κ ∂M∂ǫ .This results in the following system which we have written in matrix form:(3.6) κ u ν κ ( ǫ + p ) M ′ P νβσ ( ǫ + p ) M ′ P να Γ αβ u ν ∇ ν (cid:18) wu β (cid:19) = (cid:18) (cid:19) . In order to obtain symmetry we have to demand that(3.7) M ′ = σ ( ǫ + p ) κ , where κ ≫ w . If we choose κ = γ − √ Kγ Kǫ γ − , then (3.7) holds and consequently the system (3.6) is transferred to the OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 11 symmetric system(3.8) κ u ν σκP νβ κσP να Γ αβ u ν ∇ ν (cid:18) wu β (cid:19) = (cid:18) (cid:19) . The covariant derivative ∇ ν takes in local coordinates the form ∇ ν = ∂ ν + Γ( g γδ , ∂g αβ ),where Γ denotes the Christoffel symbols. This expresses the fact that system (3.8) iscoupled to system (1.1) for the gravitational field g αβ . In addition, from the definition ofthe Makino variable (1.6), we see that ǫ γ − = w , so κ = γ − √ Kγ Kw and σ = √ γKw . Thusthe fractional power of the equation of state (1.5) does not appear in the coefficients of thesystem (3.8), and these coefficients are C ∞ functions of the scalar w , the four vector u α and the gravitational field g αβ .Now we want to show that A of our system (3.8) is indeed positive definite. In orderto do it we analyze the principal symbol of this system. For each ξ α ∈ T ∗ x V , the principalsymbol is a linear map from R × E x to R × F x , where E x is a fiber in T x V and F x is afiber in the cotangent space T ∗ x V . Since in local coordinates ∇ ν = ∂ ν + Γ( g γδ , ∂g αβ ), theprincipal symbol of system (3.8) is(3.9) ξ ν A ν = κ ( u ν ξ ν ) σκP νβ ξ ν σκP να ξ ν ( u ν ξ ν )Γ αβ . The characteristics are the set of covectors ξ for which ( ξ ν A ν ) is not an isomorphism. Hencethe characteristics are the zeros of Q ( ξ ) := det( ξ ν A ν ). The geometric advantages of thefluid decomposition are the following. The operators in the blocks of the matrix (3.9) arethe projection on the rest hyperplane O , P να , and the reflection with respect to the samehyperplane, Γ αβ . Therefore, the following relations hold:Γ αγ Γ γβ = δ βα , Γ αγ P γ ν = P αν and P βα P αν = P νβ , which yields(3.10) αγ ( ξ ν A ν ) = κ ( u ν ξ ν ) σκP νβ ξ ν σκP αν ξ ν ( u ν ξ ν ) (cid:0) δ αβ (cid:1) . It is now fairly easy to calculate the determinate of the right hand side of (3.10) and wehave det κ ( u ν ξ ν ) σκP νβ ξ ν σκP αν ξ ν ( u ν ξ ν ) (cid:0) δ αβ (cid:1) = κ ( u ν ξ ν ) (cid:0) ( u ν ξ ν ) − σ P αν ξ ν P να ξ ν (cid:1) . Since P αβ is a projection, P αν ξ ν P να ξ ν = g νβ ξ ν P αβ P να ξ ν = g νβ ξ ν P νβ ξ ν = P νβ ξ ν ξ β , and since Γ γβ is a reflection,det (cid:18) αγ (cid:19) = det (cid:0) g αβ Γ γβ (cid:1) = − (det ( g αβ )) − . Consequently,(3.11) Q ( ξ ) := det( ξ ν A ν ) = − κ det( g αβ )( u ν ξ ν ) (cid:8) ( u ν ξ ν ) − σ P αβ ξ α ξ β (cid:9) and therefore the characteristic covectors are given by two simple equations: ξ ν u ν = 0;(3.12) ( ξ ν u ν ) − σ P αβ ξ α ξ β = 0 . (3.13) Remark . The characteristics conormal cone is a union of two hypersurfaces in T ∗ x V .One of these hypersurfaces is given by the condition (3.12) and it is a three dimensionalhyperplane O with the normal u α . The other hypersurface is given by the condition (3.13)and forms a three dimensional cone, the so called, sound cone .Let us now consider the timelike vector u ν and insert the covector − u ν into the principalsymbol (3.9), then − u ν A ν = κ
00 Γ αβ is positive definite. Indeed, Γ αβ is a reflection with respect to a hyperplane having atimelike normal. Hence, − u ν is a timelike covector with respect to the hydrodynamicalequations (3.8). Herewith, we have showed relatively elegant and elementary that therelativistic hydrodynamical equations are symmetric–hyperbolic. We want now to showthat the covector t α = (1 , , ,
0) is also timelike with respect to the system (3.8). Since P αβ u α = 0, the covector − u ν belongs to the sound cone(3.14) ( ξ ν u ν ) − σ P αβ ξ α ξ β > . Inserting t ν = (1 , , ,
0) the right hand side of (3.14) yields(3.15) ( u ) (1 − σ ) − σ g . Since the sound velocity is always less than the light speed, that is σ = ∂p∂ǫ < c = 1, weconclude from (3.15) that t ν also belongs to the sound cone (3.14). Hence, the vector − u ν can be continuously deformed to t ν while condition (3.14) holds along the deformationpath. Consequently, the determinant of (3.11) remains positive under this process andhence t ν A ν = A is also positive definite. Thus we have proved. Theorem 3.3.
Let ǫ be non-negative density function, then the Euler system (1.4) coupledwith the equation of state (1.5) can be written as a symmetric hyperbolic system of the form(3.3), and where A is a positive definite. OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 13
Conservation of the unit length vsctor of the fluid.
Proposition 3.4.
The constraint condition g αβ u α u β = − is conserved along stream lines u α .Proof. Let k ( t ) be a curve such that k ′ ( t ) = u α and set Z ( t ) = ( u ◦ k ) β ( u ◦ k ) β . In order toestablish the conservation of the constraint condition it suffices to establish the followingrelation ddt Z ( t ) = 2 u β ∇ k ′ ( t ) u β = 2 u ν u β ∇ ν u β = 0 . Multiplying the four last rows of the Euler system (3.8) by u α and recalling that P να isthe projection on the rest space O orthogonal to u α , we have0 = u α (cid:0) Γ αβ u ν ∇ ν u β + κσP να ∇ ν w (cid:1) = u α P αβ u ν ∇ ν u β − u ν u β ∇ ν u β + κσu α P να ∇ ν w = − u ν u β ∇ ν u β . Therefore, if g αβ u α u β = − u α . (cid:3) The coupled hyperbolic system.
In this section we will transform the coupledsystem (2.5) and (3.8) into a symmetric hyperbolic system. We will pay attention to thefact that the system will be in a form in which we can apply the energy estimates of [16].That allows us to obtain the same regularity for the gravitational fields as Hughes, Katoand Marsden [14] got for the Einstein vacuum equations. Note that our system is slightlydifferent from the symmetric hyperbolic system obtained by Fisher and Marsden [11], sinceour system contains a constant matrix C a as given by (3.21).We consider a spacetime ( V, g αβ ) of the type R ×M , where M is a Riemannian manifold,and we denote local coordinates by ( t, x a ). Set h αβγ = ∂ γ g αβ , then the reduced Einstein equations (2.5) takes the form(3.16) ∂ t g αβ = h αβ − g ∂ t h αβ = (cid:8) g a ∂ a h αβ + g ab ∂ a h αβb + H αβ ( g, ∂g ) − πw γ − ((1 − Kw ) g αβ + 2(1 + Kw ) u α u β ) o g ab ∂ t h αβa = g ab ∂ a h αβ . In order to apply the energy estimates of [16], we need that the coefficients of ∂ t h αβ willbe independent of t . This is because of the specific form of the inner-product in H s,δ spaceswhich takes into account the matrix A of the system (3.1). In Section 5 we will furtherclarify this issue. Therefore we divide the second row by − g and in order to preserve thesymmetry of the system, we also multiply the third row by ( − g ) − . Thus the system of wave equations (2.5) are equivalent to the system(3.17) ∂ t g αβ = h αβ ∂ t h αβ = ( − g ) − (cid:8) g a ∂ a h αβ + g ab ∂ a h αβb + H αβ ( g, ∂g ) − πw γ − ((1 − Kw ) g αβ + 2(1 + Kw ) u α u β ) o ( − g ) − g ab ∂ t h αβa = ( − g ) − g ab ∂ a h αβ . To shorten and simplify the notation, we introduce the auxiliary variables( v, ∂ t v, ∂ x v ) = ( g αβ − η αβ , ∂ t g αβ , ∂ x g αβ ) , where η αβ denotes the Minkowski metric and ∂ x denotes the set of all spatial derivatives.We also set ( e ) α = (1 , , ,
0) and W = ( w, u α − e α ) stands for the Makino and the fluidvariables. Finally, U = ( v, ∂ t v, ∂ x v, W )represents the unknowns of the coupled system.We write the matrices in a block form, A = ( a ij ), the k × k identity matrix is denotedby e k and m × n is the zero matrix.The coupled system (3.17) and (3.8) can be written in the form of (3.1), where A α are55 ×
55 symmetric matrices which depend only on v and W . We shall describe now thestructure of these matrices:(3.18) A ( v, W ) = e × × × × e × × × × a × × × × a , where a = 1 − g g e g e g e g e g e g e g e g e g e , and a = a ( g αβ , w, u α ) is given by (3.8) when ν = 0. From (3.17) we see that thecoefficients of ∂ a U , a = 1 , ,
3, have the form × × × × a a × × × a a , where a a = a a ( g αβ , w, u α ) is from the system (3.8) of the fluid and(3.19) a a ( g αβ ) = 1 g g a e g a e g a e g a e g a e g a e × g a e . It is essential to demand that a a ( g αβ ) ∈ H s,δ , whenever g αβ − η αβ ∈ H s,δ . Obviously,this does not hold for the matrix in (3.19). Therefore we need to modify these matrices by OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 15 a constant matrix c a = × δ a e δ a e δ a e δ a e δ a e × δ a e , then ( a a − c a ) ( v ) ∈ H s,δ whenever v ∈ H s,δ . So we set(3.20) A a ( v, W ) = × × × × a a − c a × × × a a and a constant matrix(3.21) C a = × × × × c a × × × × . We turn now to the lower order terms. The presence of the fractional power w / ( γ − in(3.17) causes substantial technical difficulties. We set(3.22) f ( v, W ) := − πw γ − g (cid:0) (1 − Kw ) g αβ + 2(1 + Kw ) u α u β (cid:1) , then we can write B ( U ) in the form B ( U ) = B ( U )( v, ∂ t v, ∂ x v ) T + F ( v, W ) , where F ( v, W ) = (0 , f ( v, W ) , , T and(3.23) B ( U ) = × e × × × × b b b b × × × × × × b b b b . The block b j , j = 2 , , ,
5, appear from the quadratic terms in (2.4): H αβ ( g, ∂g ) = C ǫζηκλµαβγδρσ h ǫζη h κλµ g γδ g ρσ , where C ǫζηκλµγδαβρσ are a combination of Kronecker deltas with integer coefficients. Thus b j = ( − g ) − C ǫζηκλµαβγδρσ h ǫζη g γδ g ρσ , µ = j − . The block b j , j = 2 , , ,
5, appear from the multiplication of the reflection Γ αβ and u ν in(3.8) with the Christoffel symbols. So its coefficients consist of multiplications of g αβ , g αβ and u ν .In summary, we can write the coupled systems (3.17) and (3.8) as a symmetric hyperbolicsystem(3.24) A ( v, W ) ∂ t U = (( A a ( v, W ) + C a ) ∂ a U + B ( U ) v∂ t v∂ x v + F ( v, W ) , where A ( U ) is positive definite in the neighborhood of the initial data (2.6), A (0) − e = A a (0) = 0 and C a is a constant symmetric matrix.4. The H s,δ spaces and their properties The definition of the weighted Sobolev spaces of fractional order H s,δ , Definition 2.1, isdue to Trieble [28]. Here we quote the propositions and properties which are needed forthe proof of the main result. For their proofs see [4, 20, 28].We start with some notations. • Let { ψ j } be the sequence of functions in Definition 2.1. For any positive γ we set(4.1) k u k H s,δ,γ = ∞ X j =0 ( + δ )2 j k ( ψ γj u ) j k H s , and we will use the convention k u k H s,δ, = k u k H s,δ . The subscripts 2 j mean a scalingby 2 j , that is, ( ψ γj u ) j ( x ) = ( ψ γj u )(2 j x ). • For a non-negative integer m and β ∈ R , the space C mβ is the set of all functionshaving continuous partial derivatives up to order m and such that the norm(4.2) k u k C mβ = X | α |≤ m sup R (cid:0) (1 + | x | ) β + | α | | ∂ α u ( x ) | (cid:1) is finite. • We will use the notation A . B to denote an inequality A ≤ CB where the positiveconstant C does not depend on the parameters in question.We recall that { ψ j } are cutoff functions, hence ψ βj ∈ C ∞ ( R ) for any positive β . Fur-thermore, for a given α , there are two constants C ( β, α ) and C ( β, α ) such that C ( β, α ) | ∂ α ψ j ( x ) | ≤ | ∂ α ψ βj ( x ) | ≤ C ( β, α ) | ∂ α ψ j ( x ) | and these inequalities are independent of j . Therefore Proposition 4.1 below is a conse-quence of [28, Theorem 1]. Proposition 4.1.
For any positive γ , there are two positive constants c ( γ ) and c ( γ ) suchthat c ( γ ) k u k H s,δ ≤ k u k H s,δ,γ ≤ c ( γ ) k u k H s,δ . Proposition 4.2.
For any nonnegative integer m , positive γ and δ it holds k u k H m,δ,γ . k u k m,δ . k u k H m,δ,γ , where k u k m,δ is defined by (1.7). Proposition 4.3. If s ≤ s and δ ≤ δ , then k u k H s ,δ ≤ k u k H s ,δ . Proposition 4.4. If u ∈ H s,δ , then k ∂ i u k H s − ,δ +1 ≤ k u k H s,δ . OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 17
Proposition 4.5.
Let s , s ≥ s , s + s > s + , s + s ≥ and δ + δ ≥ δ − . If u ∈ H s ,δ and v ∈ H s ,δ , then (4.3) k uv k H s,δ . k u k H s ,δ k v k H s ,δ . Remark . If for a fixed constant c , u − c ∈ H s ,δ and v ∈ H s ,δ , then we can applythe multiplication property (4.3) to ( u − c ) v and obtain k uv k H s,δ . (cid:16) k u − c k H s ,δ + | c | (cid:17) k v k H s ,δ . Proposition 4.7.
Let u ∈ H s,δ ∩ L ∞ , < β , < s < β + and δ ∈ R , then k| u | β k H s,δ ≤ C ( k u k L ∞ ) k u k H s,δ . Proposition 4.8. If s > + m and δ + ≥ β , then (4.4) k u k C mβ . k u k H s,δ , where k u k C mβ is given by (4.2). Proposition 4.9.
Let F : R m → R l be a C N +1 -function such that F (0) = 0 and where N ≥ [ s ] + 1 . Then there is a constant C such that for any u ∈ H s,δ (4.5) k F ( u ) k H s,δ ≤ C k F k C N +1 (cid:0) k u k NL ∞ (cid:1) k u k H s,δ . Proposition 4.10. (a)
The class C ∞ ( R ) is dense in H s,δ . (b) Given u ∈ H s,δ , s ′ > s ≥ and δ ′ ≥ δ . Then for ρ > there is u ρ ∈ C ∞ ( R ) and apositive constant C ( ρ ) such that k u ρ − u k H s,δ ≤ ρ and k u ρ k H s ′ ,δ ′ ≤ C ( ρ ) k u k H s,δ . Product spaces.
The unknown of system (3.24) is a vector valued function U = ( v, ∂ t v, ∂ x v, W ) , where v = g αβ − η αβ stands for the field variables and W = ( w, u α − e α ) stands for thefluid variables. We consider it in the space(4.6) X s,δ := H s,δ × H s,δ +1 × H s,δ +1 × H s +1 ,δ +1 , with the norm (see (4.1)) k U k X s,δ = k v k H s,δ, + k ∂ t v k H s,δ +1 , + k ∂ x v k H s,δ +1 , + k W k H s +1 ,δ +1 , . (4.7) Remark . Note that if U = ( v, ∂ t v, ∂ x v, W ) ∈ X s,δ , then v ∈ H s +1 ,δ . Because U ∈ X s,δ implies that v ∈ H s,δ and ∂ x v ∈ H s,δ +1 , so by the integral representation of the norm H s,δ (see [4, § k v k H s +1 ,δ . (cid:16) k v k H s,δ + k ∂ x v k H s,δ +1 (cid:17) . Energy estimates
In this section we will derive the energy estimates for a linear symmetric hyperbolicsystem, a system which we have obtained by linearising (3.24). So we consider(5.1) A ∂ t U = ( A a + C a ) ∂ a U + B v∂ t v∂ x v + F + D , where U = ( v, ∂ t v, ∂ x v, W ), the matrices A , A a , B and C a have the same structural formas the corresponding matrices in (3.24), C a is a constant matrix, and the vectors F and D have the form (0 , f, ,
0) and (0 , d , d , d ) respectively. Assumptions . All the matrices have the same block structure as (3.18), (3.20) and(3.23) and satisfy: (cid:0) A ( t, · ) − e (cid:1) , A a ( t, · ) ∈ H s +1 ,δ ;(5.2a) ∃ c ≥ c − V T V ≤ V T A V ≤ c V T V, ∀ V ∈ R ;(5.2b) ∂ t A ( t, · ) ∈ L ∞ ;(5.2c) b j ( t, · ) , b j ( t, · ) ∈ H s,δ +1 , j = 2 , , , F ( t, · ) , D ( t, · ) ∈ H s,δ +1 . (5.2e)5.1. X s,δ –energy estimates. We turn now to the definition of an inner–product of thespace X s,δ which takes into account the structure of the matrix A of the system (5.1).Let F ( u ) denote the Fourier transform of a distribution u and setΛ s ( u ) = (1 − ∆) s = F − (cid:16)(cid:0) | ξ | (cid:1) s F (cid:17) ( u ) . The standard inner–product of the Bessel-potential spaces H s is h u , u i s = h Λ s ( u ) , Λ s ( u ) i L . Taking into account the term–wise definition of the norm (2.1), we define the inner–productof H s,δ as follows:(5.3) h u , u i s,δ := ∞ X j =0 δ + ) j D Λ s (cid:0) ψ j u (cid:1) j , Λ s (cid:0) ψ j u (cid:1) j E L , where ( u ) j denotes scaling by 2 j . By Proposition 4.1, h u, u i s,δ = k u k H s,δ, ≃ k u k H s,δ . Toeach component of the space X s,δ := H s,δ × H s,δ +1 × H s,δ +1 × H s +1 ,δ +1 we assign its own inner–product. Since A = ( a ij ), where a ij is the zero matrix for i = j , a ii is the identity for i = 1 ,
2, we assign to the first two components the inner–product(5.3), while for the other terms we insert A termwise. OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 19
Definition 5.2 (Inner–product in X s,δ ) . Let U i = ( v i , ∂ t v i , ∂ x v i , W i ) ∈ X s,δ , i = 1 , A satisfies Assumption 5.1, then we denote the inner–product of X s,δ : h U , U i X s,δ, A := h v , v i s,δ + h ∂ t v , ∂ t v i s,δ +1 + h ∂ x v , ∂ x v i s,δ +1 , a + h W , W i s +1 ,δ +2 , a , (5.4)where the terms are defined in the following way: • An inner–product of H s,δ of the form: h v , v i s,δ , where the inner–product h· , ·i s,δ isdefined by (5.3); • An inner–product of H s,δ +1 of the form: h ∂ t v , ∂ t v i s,δ +1 , where h· , ·i s,δ +1 is definedby (5.3) with δ + 1; • An inner-product on H s,δ +1 of the form: h ∂ x v , ∂ x v i s,δ +1 , a := ∞ X j =0 δ +1+ ) j D Λ s (cid:0) ψ j ∂ x v (cid:1) j , (cid:0) a (cid:1) j Λ s (cid:0) ψ j ∂ x v (cid:1) j E L ;(5.5) • An inner–product of H s +1 ,δ +1 of the form: h W , W i s +1 ,δ +2 , a := ∞ X j =0 δ +1+ ) j D Λ s +1 (cid:0) ψ j W (cid:1) j , (cid:0) a (cid:1) j Λ s +1 (cid:0) ψ j W (cid:1) j E L ;(5.6)We denote by k U k X s,δ, A the norm associated with the inner–product (5.4). Since thematrix A satisfies (5.2b), the following equivalence(5.7) k U k X s,δ . k U k X s,δ, A . k U k X s,δ holds. In order to simplify the notation we set U ( t ) = U ( t, x , x , x ). Lemma 5.3.
Let s > , δ ≥ − and assume the coefficients of (5.1) satisfy Assumptions5.1. If U ( t ) ∈ C ∞ ( R ) is a solution of (5.1), then (5.8) ddt h U ( t ) , U ( t ) i X s,δ, A ≤ Cc (cid:16) h U ( t ) , U ( t ) i X s,δ, A + 1 (cid:17) , where the constant C depends on the corresponding norms of the coefficients, s and δ . The corresponding energy estimates for the vacuum Einstein equations were obtainedin [16]. The same techniques can be applied here with some obvious modifications. Wetherefore give only a short sketch of the proof.
Sketch of the proof.
From the inner–product (5.4) we see that12 ddt h U, U i X s,δ, A = h v, ∂ t v i s,δ + h ∂ t v, ∂ t v i s,δ +1 + h ∂ x v, ∂ x ∂ t v i s,δ +1 , a + h W, ∂ t W i s +1 ,δ +2 , a + ∞ X j =0 δ +1+ ) j D Λ s (cid:0) ψ j ∂ x v (cid:1) j , ∂ t (cid:0) a (cid:1) j Λ s (cid:0) ψ j ∂ x v (cid:1) j E L + ∞ X j =0 δ +2+ ) j D Λ s +1 (cid:0) ψ j W (cid:1) j , ∂ t (cid:0) a (cid:1) j Λ s +1 (cid:0) ψ j W (cid:1) j E L . By the Cauchy Schwarz inequality, we obtain |h v, ∂ t v i s,δ | ≤ k v k H s,δ, k ∂ t v k H s,δ, ≤ (cid:16) k v k H s,δ, + k ∂ t v k H s,δ +1 , (cid:17) , and by Assumption (5.2c), the first infinite sum is less than C (cid:13)(cid:13) ∂ t a (cid:13)(cid:13) L ∞ ∞ X j =0 δ +1+ ) j D Λ s (cid:0) ψ j ∂ x v (cid:1) j , Λ s (cid:0) ψ j ∂ x v (cid:1) j E L = C (cid:13)(cid:13) ∂ t a (cid:13)(cid:13) L ∞ k ∂ x v k H s,δ +1 , . A similar estimate holds for the second infinite sum. The most difficult part is the estimate(5.9) (cid:12)(cid:12)(cid:12) h ∂ t v, ∂ t v i s,δ +1 + h ∂ x v, ∂ x ∂ t v i s,δ +1 , a (cid:12)(cid:12)(cid:12) . (cid:16) k U k X s,δ + 1 (cid:17) , and here it is essential to use the assumption that A α ∈ H s +1 ,δ and s > . We presenthere the main ideas of this estimate and for a detailed proof we refer to [16, § E ∂ t ( j ) = D Λ s (cid:0) ψ j ∂ t v (cid:1) j , Λ s (cid:0) ψ j ( ∂ t v ) (cid:1) j E L and(5.11) E ∂ x ( j ) = D Λ s (cid:0) ψ j ∂ x v (cid:1) j , (cid:0) a (cid:1) j Λ s (cid:0) ψ j ∂ t ( ∂ x v ) (cid:1) j E L . In order to use equation (5.1) we need to commute ( a ) j both with the operator Λ s and ψ j . An essential ingredient is the Kato–Ponce commutator estimate (8.3) (see Appendix).Usually this estimate is used in similar situations with the operator Λ s , we, however, applyit to the pseudodifferential operator P = Λ s ∂ x . This enables us to use estimate (8.3) withthe index s + 1 and to exploit the assumption that A α ∈ H s +1loc .Let Ψ k = (cid:16)P ∞ j =0 ψ j (cid:17) − ψ k , then supp(Ψ k ψ j ) = ∅ for k = j − , . . . , j + 1, and hence wehave that(5.12) E ∂ x ( j ) = j +1 X k = j − D Λ s (cid:0) ψ j ∂ x v (cid:1) j , (cid:0) a (cid:1) j Λ s (cid:0) Ψ k ψ j ∂ t ( ∂ x v ) (cid:1) j E L =: j +1 X k = j − E ∂ x ( j, k ) , OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 21 and similarly(5.13) E ∂ t ( j ) = j +1 X k = j − D Λ s (cid:0) ψ j ∂ t v (cid:1) j , Λ s (cid:0) Ψ k ψ j ∂ t ( ∂ t v ) (cid:1) j E L =: j +1 X k = j − E ∂ t ( j, k ) . Writing(5.14) (cid:0) Ψ k ψ j ∂ t ( ∂ x v ) (cid:1) j = 2 − j ∂ x (cid:0) Ψ k ψ j ∂ t v (cid:1) j − ∂ x (cid:0) Ψ k ψ j (cid:1) j ( ∂ t v ) j , and Λ s (cid:16) ∂ x (cid:0) Ψ k ψ j ∂ t v (cid:1) j (cid:17) = (Λ s ∂ x ) (cid:0) Ψ k ψ j ∂ t v (cid:1) j − (Ψ k ) j (Λ s ∂ x ) (cid:16)(cid:0) ψ j ∂ t v (cid:1) j (cid:17) + (Ψ k ) j (Λ s ∂ x ) (cid:16)(cid:0) ψ j ∂ t v (cid:1) j (cid:17) , (5.15)we obtain that E ∂ x ( j, k ) = 2 − j D Λ s (cid:0) ψ j ∂ x v (cid:1) , (cid:0) Ψ k a (cid:1) j (Λ s ∂ x ) (cid:0) ψ j ∂ t v (cid:1) j E + R ( a, j, k )=: E ∂ x ( a, j, k ) + R ( a, j, k ) . (5.16)We estimate the remainder term R ( a, j, k ) by using the Cauchy–Schwarz inequality, theSobolev embedding theorem and of course the Kato–Ponce commutator estimate. Thisresults in(5.17) | R ( a, j, k ) | . (cid:13)(cid:13)(cid:0) a (cid:1) j (cid:13)(cid:13) L ∞ (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ x v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:16)(cid:13)(cid:13) ( ψ j ∂ t v ) j (cid:13)(cid:13) H s + (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:17) . Next, we commute (Ψ k a ) j with Λ s ∂ x , that is, we write, (cid:0) Ψ k a (cid:1) j (Λ s ∂ x ) (cid:0) ψ j ∂ t v (cid:1) j = (cid:0) Ψ k a (cid:1) j (Λ s ∂ x ) (cid:0) ψ j ∂ t v (cid:1) j − (Λ s ∂ x ) (cid:16)(cid:0) Ψ k a (cid:1) j (cid:0) ψ j ∂ t v (cid:1) j (cid:17) + Λ s (cid:16)(cid:0) ∂ x (cid:0) Ψ k a ψ j (cid:1)(cid:1) j ( ∂ t v ) j (cid:17) + 2 j Λ s (cid:16)(cid:0) Ψ k a ψ j (cid:1) j ( ∂ t ∂ x v ) j (cid:17) , (5.18)then E ∂ x ( a, j, k ) = D Λ s (cid:0) ψ j ∂ x v (cid:1) , Λ s (cid:0) Ψ k a ψ j ∂ t ∂ x v (cid:1) j E L + R ( b, j, k ):= E ∂ x ( b, j, k ) + R ( b, j, k ) . (5.19)Since a ∈ H s +1loc , the Kato–Ponce commutator estimate (8.3) implies that (cid:13)(cid:13)(cid:13)(cid:0) Ψ k a (cid:1) j (Λ s ∂ x ) (cid:0) ψ j ∂ t v (cid:1) j − (Λ s ∂ x ) (cid:16)(cid:0) Ψ k a (cid:1) j (cid:0) ψ j ∂ t v (cid:1) j (cid:17)(cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13) ∇ (cid:0) Ψ k a (cid:1) j (cid:13)(cid:13) L ∞ (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s + (cid:13)(cid:13)(cid:0) Ψ k a (cid:1) j (cid:13)(cid:13) H s +1 (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) L ∞ . (5.20)Taking into account that k (Ψ k a ) j k H s +1 ≃ k ( ψ k a ) k k H s +1 , we obtain in a similarmanner, as in as in the estimate of the previous remainder term, the following | R ( b, j, k ) | . (cid:16)(cid:13)(cid:13)(cid:0) ψ k (cid:0) a − e (cid:1)(cid:1) k (cid:13)(cid:13) H s +1 + 1 (cid:17) (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ x v (cid:1) j (cid:13)(cid:13)(cid:13) H s × (cid:16)(cid:13)(cid:13) ( ψ j ∂ t v ) j (cid:13)(cid:13) H s + (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:17) . (5.21) Adding E ∂ t ( j, k ) to E ∂ x ( b, j, k ) enables us now to use equation (5.1), that is, E ∂ t ( j, k ) + E ∂ x ( b, j, k ) = (cid:28) Λ s (cid:18) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j , Λ s (cid:18) Ψ k ψ j (cid:18) e × × a (cid:19) ∂ t (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (cid:29) L = X a =1 (cid:28) Λ s (cid:18) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j , Λ s (cid:18) Ψ k ψ j ( a a + c a ) ∂ a (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (cid:29) L + (cid:28) Λ s (cid:18) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j , Λ s (cid:0) Ψ k ψ j B (cid:1) j (cid:29) L , where the matrices a a and c a are defined in subsection 3.2 and c a are constant ma-trices. In the last expression B contains the zero and first order derivatives of v . It isstraightforward to estimate this term since it does not contain second order derivatives.In order to estimate the second order terms, we write (cid:18) Ψ k ψ j ( a a + c a ) ∂ a (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j = 2 − j ∂ a (cid:18) Ψ k ( a a + c a ) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j − (cid:18) ∂ a (cid:0) Ψ k ψ j ( a a + c a ) (cid:1) (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (5.22)and then we commute Λ s ∂ a with Ψ k ( a a + c a ) in the first term of the right hand side of(5.22). This results in(5.23) (cid:28) Λ s (cid:18) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j , ∂ a (Ψ k ( a a + c a )) j Λ s (cid:18) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (cid:29) L , plus a remainder term which can be estimate by the Kato–Ponce commutator estimate(8.3) in a similar manner to the previous steps. We use the common method of integrationby parts in order to estimate the inner–product (5.23), and as to the second term of (5.22),we have that (cid:18) ∂ a (cid:0) Ψ k ψ j ( a a + c a ) (cid:1) (cid:18) ∂ t v∂ x v (cid:19)(cid:19) = ( ∂ a (Ψ k ( a a + c a )) ψ j + 2Ψ k ( a a + c a ) ∂ a ψ i ) ψ j (cid:18) ∂ t v∂ x v (cid:19) . So by the Cauchy–Schwraz inequality we obtain that (cid:12)(cid:12)(cid:12) (cid:28) Λ s (cid:18) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j , Λ s (cid:18) ∂ a (Ψ k ( a a )) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (cid:29) L (cid:12)(cid:12)(cid:12) . (cid:16)(cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s + (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ x v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∂ a (Ψ k ( a a )) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (cid:13)(cid:13)(cid:13)(cid:13) H s , (5.24)and since A a ∈ H s +1 ,δ , s > , we have by the multiplicity property of the H s spaces that (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∂ a (Ψ k ( a a )) ψ j (cid:18) ∂ t v∂ x v (cid:19)(cid:19) j (cid:13)(cid:13)(cid:13)(cid:13) H s . k ( ψ k a a ) j k H s +1 (cid:16)(cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s + (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ x v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:17) . (5.25) OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 23
The other term can be estimated by similar arguments. From inequalities (5.17), (5.21),(5.24) and (5.25), we conclude that ∞ X j =0 j +1 X k = j − ( δ +1+ )2 j (cid:12)(cid:12)(cid:12) E ∂ t ( j, k ) + E ∂ x ( j, k ) (cid:12)(cid:12)(cid:12) . ∞ X j =0 j +1 X k = j − ( δ +1+ )2 j k ( ψ k a a ) k k H s +1 (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:13)(cid:13) ( ψ j ∂ x v ) j (cid:13)(cid:13) H s + · · · , (5.26)where · · · represent terms that are sums of similar structure. For example ψ j is replacedby ψ j , a a is replaced by ( a − e ), or the H s +1 –norm is replaced by the L ∞ –norm.Applying the H¨older inequality and using the equivalence of norms (Propositions 4.1),we have that ∞ X j =0 j +1 X k = j − ( δ +1+ )2 j k ( ψ k a a ) k k H s +1 (cid:13)(cid:13)(cid:13)(cid:0) ψ j ∂ t v (cid:1) j (cid:13)(cid:13)(cid:13) H s (cid:13)(cid:13) ( ψ j ∂ x v ) j (cid:13)(cid:13) H s . k a a k H s +1 ,δ k ∂ t v k H s +1 ,δ k ∂ x v k H s +1 ,δ ≤ k a a k H s +1 ,δ (cid:16) k ∂ t v k H s +1 ,δ + k ∂ x v k H s +1 ,δ (cid:17) . (5.27)So recalling the properties of the inner–products (5.3), (5.5), (5.4), (5.10), (5.11), (5.12)and (5.13), we obtain (5.9). The estimate of h W, ∂ t W i s +1 ,δ +2 , a relies on similar ideas tothose of (5.9), but it is simpler, since W ∈ H s +1loc . Having collected the estimates of all theterms, we have 12 ddt h U, U i X s,δ, A . (cid:16) k U k X s,δ + 1 (cid:17) and by the equivalence (5.7) we obtain (5.8). (cid:3) Remark . Note that if the coefficients of ∂ t v in the matrix A were dependent on t , thenwe would have to reiterate the commutation (5.15), but with the operator Λ s ∂ t instead ofΛ s ∂ x . However, Λ s ∂ t is pseudodifferential operator of oder s , and hence we would not getthe desired regularity. This is the reason for dividing equations (3.16) by − g .5.2. L δ –energy estimates. The next section deals with the existence of solutions to thenonlinear symmetric hyperbolic systems by means of an iteration scheme. The X s,δ –energyestimates are used to obtain boundedness of the sequence, while L δ –energy estimates areneeded in order to establish the contraction.Let h u , u i δ = Z R (1 + | x | ) δ u T ( x ) u ( x ) dx, denote a weighted L inner–product, where u T u denote the scalar product between twovectors. The L δ ( R )–space is the closure of all continuous functions under the norm k u k L δ = h u, u i δ , and this norm is equivalent to the norm k u k H ,δ (see [28]). Similar to(4.6), we set Y δ = L δ × L δ +1 × L δ +1 × L δ +1 and k U k Y δ = k v k L δ + k ∂ t v k L δ +1 + k ∂ x v k L δ +1 + k W k L δ +1 . We also define the inner–productof Y δ in accordance with to the system (5.1): h U , U i Y δ , A = h v , v i δ + h ∂ t v , ∂ t v i δ +1 + (cid:10) ∂ x v , a ∂ x v (cid:11) δ +1 + (cid:10) W , a W (cid:11) δ +1 , and the associated norm k U k Y δ , A = h U, U i Y δ , A . By assumption (5.2b), k U k Y δ , A ≃ k U k Y δ . Lemma 5.5.
Assume the coefficients of (5.1) satisfy Assumptions 5.1. If U ( t ) ∈ X ,δ is asolution of (5.1), then (5.28) ddt h U ( t ) , U ( t ) i Y δ , A ≤ Cc (cid:16) h U ( t ) , U ( t ) i Y δ , A + kF k L δ +1 + kDk L δ +1 (cid:17) , where the constant C depends upon the L ∞ –norms of A α , ∂ α A α and B .Proof. We find that12 ddt h U ( t ) , U ( t ) i Y δ , A = h v, ∂ t v i δ + (cid:10) ∂ t v, ∂ t v (cid:11) δ +1 + (cid:10) ∂ x v, a ∂ t ∂ x v (cid:11) δ +1 + (cid:10) W, a ∂ t W (cid:11) δ +1 + (cid:10) ∂ x v, ∂ t ( a ) ∂ x v (cid:11) δ +1 + (cid:10) W, ∂ t ( a ) W (cid:11) δ +1 . By the Cauchy–Schwarz inequality | h v, ∂ t v i δ | ≤ ( k v k L δ + k ∂ t v k L δ ), | h ∂ x v, ∂ t ( a ) ∂ x v i δ +1 | . k ∂ t ( a ) k L ∞ k ∂ x v k L δ +1 and | h W, ∂ t ( a ) W i δ +1 | . k ∂ t ( a ) k L ∞ k W k L δ +1 . Since the system(5.1) is semi-decoupled, we may estimate the expressions with ∂ t v and ∂ x v first, and laterthe term W . Using equation (5.1) and recalling the structure matrices (3.18), (3.20) and(3.23), we have (cid:10) ∂ t v, ∂ t v (cid:11) δ +1 + (cid:10) ∂ x v, a ∂ t ∂ x v (cid:11) δ +1 = X a =1 (cid:28)(cid:18) ∂ t v∂ x v (cid:19) , ( a a + c a ) ∂ a (cid:18) ∂ t v∂ x v (cid:19)(cid:29) δ +1 + h ∂ t v, b ∂ t v + ( b + b + b ) ∂ x v i δ +1 + h ∂ t v, f + d i δ +1 + h ∂ x v, d i δ +1 . Exploring the symmetry of the matrices and using integration by parts, we have that2 (cid:28)(cid:18) ∂ t v∂ x v (cid:19) , ( a a + c a ) ∂ a (cid:18) ∂ t v∂ x v (cid:19)(cid:29) δ +1 = − (cid:28)(cid:18) ∂ t v∂ x v (cid:19) , ∂ a ( a a ) (cid:18) ∂ t v∂ x v (cid:19)(cid:29) δ +1 − δ + 1) (cid:28)(cid:18) ∂ t v∂ x v (cid:19) , x a | x | ( a a + c a ) (cid:18) ∂ t v∂ x v (cid:19)(cid:29) δ +1 . Thus (cid:12)(cid:12)(cid:12) (cid:10) ∂ t v, ∂ t v (cid:11) δ +1 + (cid:10) ∂ x v, a ∂ t ∂ x v (cid:11) δ +1 (cid:12)(cid:12)(cid:12) . X a =1 ( k ∂ a ( a a ) k L ∞ + k a a k L ∞ ) + kBk L ∞ ! (cid:16) k ∂ t v k L δ +1 + k ∂ x v k L δ +1 (cid:17) + k ∂ t v k L δ +1 + k ∂ x v k L δ +1 + k f k L δ +1 + k d k L δ +1 + k d k L δ +1 . OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 25
And for W we have, (cid:10) W, a ∂ t W (cid:11) δ +1 = X a =1 h W, a a ∂ a W i δ +1 + h W, d i δ +1 , so similar arguments give that (cid:12)(cid:12)(cid:12) (cid:10) W, a ∂ t W (cid:11) δ +1 (cid:12)(cid:12)(cid:12) . X a =1 ( k ∂ a ( a a ) k L ∞ + k a a k L ∞ ) + 1 ! k W k L δ +1 + k d k L δ +1 . Finally, using the equivalence of the norms we obtain (5.28). (cid:3) The iteration process
In this section we adopt Majda’s iterative scheme[18] in order to prove the well-posdenessof the coupled hyperbolic system (3.24) in the H s,δ -spaces. A similar approach was carriedout in [16] for the vacuum Einstein equations, but in the presence of a perfect fluid thereare additional difficulties. The zero order term B is not necessarily a C ∞ –function since itcontains the fractional power of the Makino variable w / ( γ − . Hence we could not apply thestandard iteration scheme in order to prove the existence theorem for symmetric hyperbolicsystems. We denote the initial data by U = ( φ, ϕ, ∂ x φ, W ), where W = ( w , ( u ) α − e α )represents the initial data for the Makino variable w and the four velocity vector u α − e α . Theorem 6.1.
Let < s < γ − + , − ≤ δ . Assume U ∈ X s,δ , w ≥ and that thereexits a positive constant µ such that (6.1) 1 µ V T V ≤ V T A ( φ, W ) V ≤ µV T V for all V ∈ R . Then there exists a positive constant T and a unique solution U ( t ) = ( v ( t ) , ∂ t v ( t ) , ∂ x v ( t ) , W ( t )) to the system (3.24) such that U (0 , x ) = U ( x ) , (6.2) U ∈ C ([0 , T ] , X s,δ ) and W ∈ C ([0 , T ] , H s,δ +2 ) . The size of the time interval depends only on the norms of the initial data.
The proof of Theorem 6.1 will be carried out in several steps: Setting up the iterative scheme; Proving that the fractional power ( w k ) / ( γ − is a C ∞ -function; Boundedness of the iteration sequence in the X s,δ -norm; Contraction in a lower norm; Convergence; Uniqueness; Continuity in the X s,δ –norm.A part of the above proofs are standard, but some of them require a special attentiondue to the specific form of the system (3.24) and use of the space X s,δ . Moreover, the factthat the matrices A α = A α ( v, W ) are not dependent on the derivative of the semi-metricplays an essential role here. Step 1.
From condition (6.1) and the embedding into the continuous, Proposition 4.8, wesee that there is a bounded domain G ⊂ R containing the initial value U and a constant c ≥ c V T V ≤ V T A ( v, W ) V ≤ c V T V for all U = ( v, ∂ t v, ∂ x v, W ) ∈ G and V ∈ R . By means of the density properties of H s,δ ,Proposition 4.10, there exist positive constants C and R , and a sequence(6.4) (cid:8) U k (cid:9) ∞ k =0 = (cid:8) ( φ k , ϕ k , ∂ x φ k , w k , ( u α ) k − e α ) (cid:9) ∞ k =0 ⊂ C ∞ ( R ) , such that(6.5) (cid:13)(cid:13) U (cid:13)(cid:13) X s +1 ,δ ≤ C k U k X s,δ , (6.6) (cid:13)(cid:13) U − U (cid:13)(cid:13) X s,δ ≤ R ⇒ U ∈ G, and(6.7) (cid:13)(cid:13) U k − U (cid:13)(cid:13) X s,δ ≤ R − k c . The iterative scheme is defined as follows: let U ( t, x ) = U ( x ) and U k +1 ( t, x ) = ( v k +1 ( t, x ) , ∂ t v k +1 ( t, x ) , ∂ x v k +1 ( t, x ) , W k +1 ( t, x ))be a solution of the linear Cauchy problem(6.8) A ( v k , W k ) ∂ t U k +1 = (cid:0) A a ( v k , W k ) + C a (cid:1) ∂ a U k +1 + B ( U k ) v k +1 ∂ t v k +1 ∂ x v k +1 + F ( v k , W k ) U k +1 (0 , x ) = U k +10 ( x ) , where F ( v k , W k ) = (0 , f ( v k , W k ) , , f ( v k , W k ) is given by (3.22) and W k = ( w k , ( u α ) k − e α ). Step 2.
The iterative method relies on the fact that solutions of linear symmetric hyper-bolic systems with C ∞ coefficients and initial data, are also C ∞ . However, even if w k ≥ w k ∈ C ∞ , it does not guarantee that ( w k ) / ( γ − is a C ∞ -function. Since the function f ( v k , W k ) contains the term ( w k ) / ( γ − , we must assure that it is a C ∞ -function. Proposition 6.2.
Let u ∈ H s,δ be non-negative and β > . Then there is a sequence { u k } ⊂ C ∞ such that u k → u in the H s,δ -norm and ( u k ) β ∈ C ∞ .Proof. Let ε >
0, then by Proposition 4.10 there is u ε ∈ C ∞ with k u − u ε k H s,δ < ε . Takenow a positive number M so that supp( u ε ) ⊂ {| x | ≤ M } , and let χ M be the cut-off functionsatisfying χ M ( x ) = 1 for | x | ≤ M and χ M ( x ) = 0 for | x | ≥ M + 1. For any positive number ̺ , we set u ε,̺ ( x ) = χ M ( x ) ( u ε ( x ) + ̺ ) . OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 27
Then ( u ε,̺ ) β ∈ C ∞ , since ( u ε + ̺ ) > χ M is a cut-off function. Moreover, u ε,̺ − u ε = χ M ̺ , hence u ε,̺ → u ε in the H s,δ -norm as ̺ → (cid:3) Thus we may assume that { w k } ∞ k =0 , the C ∞ approximation of the initial data of theMakino variable w in (6.4), satisfies ( w k ) / ( γ − ∈ C ∞ . We turn now showing that for t ≥ ǫ k ( t, x ) = (cid:0) w k (cid:1) γ − ( t, x )is also a C ∞ -function. Proposition 6.3.
For each integer k ≥ , ǫ k ( t, · ) ∈ C ∞ ( R ) .Proof. We conduct the proof by induction. Obviously the statement holds when k = 0.Now the 51 th row of the system (6.8) is equivalent to (3.5a), so the linearization of it resultsin ( u ) k ∂ t ǫ k +1 + ( u a ) k ∂ a ǫ k +1 + ǫ k (1 + K ( w k ) ) P να (cid:0) g kαβ , ( u β ) k (cid:1) ∂ v ( u α ) k +1 + ǫ k (1 + K ( w k ) ) P να (cid:0) g kαβ , ( u β ) k (cid:1) (cid:0) Γ ανµ (cid:1) k ( u µ ) k = 0 , (6.9)where the P να ( · , · ) is the projection of equation (3.5a) and (cid:0) Γ ανµ (cid:1) k are the Christoffel symbolswith respect to the semi-metric g kαβ . It follows from [4, Theorem 2.6], that u (0 , x ) ≥ u ) k (0 , x ) ≥ u ) k and we conclude that ǫ k +1 satisfies a first order linear equation of the form(6.10) (cid:26) ∂ t ǫ k +1 + b a ( t, x ) ∂ a ǫ k +1 + c ( t, x ) = 0 ǫ k +1 (0 , x ) = ( w k +10 ( x )) γ − . Note that c ( t, x ) contains the term ǫ k , but this term is C ∞ by the induction hypothesis.Hence all the coefficients of (6.10) are C ∞ -functions. We solve (6.10) by means of thecharacteristic method. So let Φ( s, y ) be the solution of the system dtds = 1 , dx a ds = b a ( t, x ) , t (0) = 0 , x (0) = y. Then ǫ k +1 ( t, x ) = Z (Φ − ( t, x )), where Z ( s, y ) is the solution of then initial value problem dZds = − c ( s, x ) , Z (0 , y ) = ( w k +10 ( y )) γ − . Obviously Z ( s, y ) = ( w k +10 ( y )) γ − − Z s c ( τ, x ( τ )) dτ. Since c ( τ, · ) ∈ C ∞ and ( w k +10 ) γ − ∈ C ∞ by Proposition 6.2, Z ( s, · ) also belongs to C ∞ ,and hence also Z (Φ − ( t, · )) = ǫ k +1 ( t, · ) = (cid:0) w k +1 (cid:1) γ − ( t, · ) ∈ C ∞ . (cid:3) Step 3.
We conclude from Proposition 6.2, and the theory of linear symmetric hyperbolicsystems (c.f. [15]), that for each k there is a solution U k ( t, x ) of the linear system (6.8)such that U k ( t, · ) ∈ C ∞ ( R ). Therefore by (6.5), (6.6) and (6.7), for each k we have T k = sup (cid:26) T : sup ≤ t ≤ T (cid:13)(cid:13) U k ( t ) − U (cid:13)(cid:13) X s,δ ≤ R (cid:27) > . Proposition 6.4.
There are positive constants T ∗ and L such that (6.11) sup (cid:26) T : sup ≤ t ≤ T (cid:13)(cid:13) U k ( t ) − U (cid:13)(cid:13) X s,δ ≤ R (cid:27) ≥ T ∗ for all k, and (6.12) sup ≤ t ≤ T ∗ (cid:13)(cid:13) ∂ t W k (cid:13)(cid:13) H s,δ +2 ≤ L for all k. Proof.
We prove it by induction. Set V k +1 = U k +1 − U , then V k satisfies the linear initialvalue problem A ( v k , W k ) ∂ t V k +1 = (cid:0) A a ( v k , W k ) + C a (cid:1) ∂ a V k +1 + B ( U k ) v k +1 − φ ∂ t v k +1 − ϕ ∂ x (cid:0) v k +1 − φ (cid:1) + F ( v k , W k ) + D k V k +1 (0 , x ) = U k +10 ( x ) − U ( x ) , (6.13)where(6.14) D k = (cid:0) A a ( v k , W k ) + C a (cid:1) ∂ a U + B ( U k ) φ ϕ ∂ x φ . In order to apply the energy estimate, Lemma 5.3, to (6.13) we have to check thatAssumptions 5.1 are satisfied and the corresponding norms are independent of k . Notethat all the matrices have the same structure, and from (6.3) we see that condition 5.2bholds. By the induction hypothesis, (cid:13)(cid:13) v k ( t ) − φ (cid:13)(cid:13) H s,δ, + (cid:13)(cid:13) ∂ x v k ( t ) − ∂ x φ (cid:13)(cid:13) H s,δ +1 , ≤ (cid:13)(cid:13) U k ( t ) − U (cid:13)(cid:13) X s,δ ≤ R , therefore by Remark 4.11, (cid:13)(cid:13) v k ( t ) − φ (cid:13)(cid:13) H s +1 ,δ . R . Applying the Moser type estimate,Proposition 4.9, we have (cid:13)(cid:13) A ( v k , W k ) − e (cid:13)(cid:13) H s +1 ,δ . (cid:16)(cid:13)(cid:13) v k (cid:13)(cid:13) H s +1 ,δ + (cid:13)(cid:13) W k (cid:13)(cid:13) H s +1 ,δ (cid:17) . (cid:13)(cid:13) U k − U (cid:13)(cid:13) X s,δ + (cid:13)(cid:13) U (cid:13)(cid:13) X s,δ . (cid:16) R + (cid:13)(cid:13) U (cid:13)(cid:13) X s,δ (cid:17) . In a similar way we get that (cid:13)(cid:13) A a ( v k , W k ) (cid:13)(cid:13) H s +1 ,δ is bounded by a constant depending on R . By Propositions 4.3, 4.5 ,4.9, Remark 4.6 and the structure of the matrix B in (3.23),we obtain that (cid:13)(cid:13) b j ( U k ) (cid:13)(cid:13) H s,δ +1 . k U k k X s,δ . (cid:0) R + k U k X s,δ (cid:1) and a similar estimate holdsfor (cid:13)(cid:13) b j ( U k ) (cid:13)(cid:13) H s,δ +1 , j = 2 , , ,
5. We recall the F ( v k , W k ) = (0 , f ( v k , W k ) , , OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 29 f ( v k , W k ) is given by (3.22). Applying again Propositions 4.5, 4.9 and Remark 4.6, weobtain that (cid:13)(cid:13) f ( v k , W k ) (cid:13)(cid:13) H s,δ +1 . (cid:13)(cid:13)(cid:13) ( w k ) γ − (cid:13)(cid:13)(cid:13) H s,δ +1 (cid:16)(cid:13)(cid:13) v k (cid:13)(cid:13) H s,δ +1 + (cid:13)(cid:13) ( u α ) k − e α (cid:13)(cid:13) H s,δ +1 + 1 (cid:17) . (6.15)Now, for < s < γ − + , we apply the estimate for the fractional power, Proposition 4.7,and obtain that(6.16) (cid:13)(cid:13)(cid:13) ( w k ) γ − (cid:13)(cid:13)(cid:13) H s,δ +1 . (cid:13)(cid:13) w k (cid:13)(cid:13) H s,δ +1 ≤ (cid:13)(cid:13) w k (cid:13)(cid:13) H s +1 ,δ +1 . Since (cid:13)(cid:13) v k (cid:13)(cid:13) H s,δ +1 and (cid:13)(cid:13) W k (cid:13)(cid:13) H s,δ +1 are bounded by a constant independent of k , it followsfrom (6.15) and (6.16) that also (cid:13)(cid:13) f ( v k , W k ) (cid:13)(cid:13) H s,δ +1 is bounded by a constant independentof k .The required estimate for D k defined in (6.14) follows from the multiplicity property (4.3)and the estimates which we have already obtained for (cid:13)(cid:13) A α ( v k , W k ) (cid:13)(cid:13) H s +1 ,δ and (cid:13)(cid:13) B ( U k ) (cid:13)(cid:13) H s,δ +1 .It remains to verify (5.2c); note that by the induction hypothesis (6.11), condition (6.6)and the embedding (4.4), we have (cid:13)(cid:13) ∂ t A ( v k , W k ) (cid:13)(cid:13) L ∞ ≤ sup G (cid:12)(cid:12)(cid:12) ∂ A ∂v ( v, W ) (cid:12)(cid:12)(cid:12) (cid:13)(cid:13) ∂ t v k (cid:13)(cid:13) L ∞ + sup G (cid:12)(cid:12)(cid:12) ∂ A ∂W ( v, W ) (cid:12)(cid:12)(cid:12) (cid:13)(cid:13) ∂ t W k (cid:13)(cid:13) L ∞ . (cid:16)(cid:13)(cid:13) ∂ t v k (cid:13)(cid:13) H s,δ +1 + (cid:13)(cid:13) ∂ t W k (cid:13)(cid:13) H s,δ +2 (cid:17) . Since (cid:13)(cid:13) ∂ t W k (cid:13)(cid:13) H s,δ +2 is bounded by hypothesis (6.12), we see that (cid:13)(cid:13) ∂ t A ( v k , W k ) (cid:13)(cid:13) L ∞ is alsobounded by a constant depending on R but not on k . We now apply Lemma 5.3, andwith the combination of Gronwall’s inequality, condition (6.7) and the equivalence (5.7),we conclude that there is a constant C depending on R such that(6.17) sup ≤ t ≤ T (cid:13)(cid:13) V k +1 ( t ) (cid:13)(cid:13) X s,δ ≤ e Cc T (cid:18) R Cc T (cid:19) . We turn now to show (6.12). It follows from the structure of the matrices A , A a and B (see (3.18), (3.20) and (3.23)) that ∂ t W k +1 = (cid:0) a ( v k , W k ) (cid:1) − " X a =1 a a ( v k , W k ) ∂ a W k +1 + b ( U k ) ∂ t v k +1 + X a =1 b a +2) ( U k ) ∂ a v k +1 . (6.18)Note that k v k ( t ) k H s +1 ,δ , k W k ( t ) k H s +1 ,δ +1 and k U k ( t ) k X s,δ are bounded by a constant inde-pendent of k , while k ∂ t v k +1 ( t ) k H s,δ +1 , k ∂ a v k +1 ( t ) k H s,δ +1 and k ∂ a W k +1 ( t ) k H s,δ +2 are boundedby (6.17). Hence applying the calculus of the H s,δ –spaces to (6.18), we obtain that (cid:13)(cid:13) ∂ t W k +1 (cid:13)(cid:13) H s,δ +2 is also bounded by a constant independent of k . Choosing T ∗ so that e Cc T ∗ (cid:18) R Cc T ∗ (cid:19) < R completes the proof of the Proposition. (cid:3) Step 4.
Here we show contraction in the weighted L –norm. Our method relies on the L δ -energy estimates, Lemma 5.5 Proposition 6.5.
There exists positive constants T ∗∗ ≤ T ∗ and Λ < , and a positivesequence { β k } with P β k < ∞ such that (6.19) sup ≤ t ≤ T ∗∗ (cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y δ , A ≤ Λ sup ≤ t ≤ T ∗∗ (cid:13)(cid:13) U k ( t ) − U k − ( t ) (cid:13)(cid:13) Y δ , A + β k . Proof.
The function ( U k +1 − U k ) satisfies the linear system A ( v k , W k ) ∂ t ( U k +1 − U k ) = A a ( v k , W k ) ∂ a ( U k +1 − U k ) + B ( U k ) v k +1 − v k ∂ t ( v k +1 − v k ) ∂ x ( v k +1 − v k ) + F ( v k , W k ) − F ( v k − , W k − ) + D k , (6.20)where D k = − (cid:0) A ( v k , W k ) − A ( v k − , W k − ) (cid:1) ∂ t U k + (cid:0) A a ( v k , W k ) − A a ( v k − , W k − ) (cid:1) ∂ a U k + (cid:0) B ( U k ) − B ( U k − ) (cid:1) v k ∂ t v k ∂ x v k . (6.21)By (6.11), (6.12) and Proposition 4.8, kA α ( v k , W k ) k L ∞ , k ∂ α A α ( v k , W k ) k L ∞ and (cid:13)(cid:13) B ( U k ) (cid:13)(cid:13) L ∞ are bounded by a constant independent of k , so we may apply Lemma 5.5 and get that ddt (cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y, A ≤ C (cid:16)(cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y, A + (cid:13)(cid:13) F ( v k , W k ) − F ( v k − , W k − ) (cid:13)(cid:13) L δ +1 + (cid:13)(cid:13) D k (cid:13)(cid:13) L δ +1 (cid:17) . (6.22)Thus our main task is to estimate the two last terms of the above inequality by meansof the difference k U k − U k − k Y δ . Recall that F ( v, W ) = (0 , f ( v, w ) , , T , where f is ascalar function given in (3.22), so f ( v k , W k ) − f ( v k − , W k − )= (cid:18) ∂f∂v (cid:0) τ v k + (1 − τ ) v k − , τ W k + (1 − τ ) W k − (cid:1)(cid:19) (cid:0) v k − v k − (cid:1) + (cid:18) ∂f∂W (cid:0) τ v k + (1 − τ ) v k − , τ W k + (1 − τ ) W k − (cid:1)(cid:19) (cid:0) W k − W k − (cid:1) OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 31 for some τ ∈ [0 , v k − v k − ) belong to L δ , while we need to estimate the L δ +1 -norm of the above expressions. However, since the Makino w ∈ H s +1 ,δ +1 ⊂ H s,δ +1 ,we get from (3.22), and Propositions 4.7, 4.5 and 4.9 that ∂f∂v ∈ H s,δ +1 . Therefore, bythe equivalence k u k L δ ≃ k u k H ,δ (Proposition 4.2) and the multiplicity property (4.3), weobtain (cid:13)(cid:13)(cid:13)(cid:13) ∂f∂v (cid:0) v k − v k − (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) H ,δ +1 . (cid:13)(cid:13)(cid:13)(cid:13) ∂f∂v (cid:13)(cid:13)(cid:13)(cid:13) H s,δ +1 (cid:13)(cid:13) v k − v k − (cid:13)(cid:13) H ,δ . (cid:13)(cid:13)(cid:13)(cid:13) ∂f∂v (cid:13)(cid:13)(cid:13)(cid:13) H s,δ +1 (cid:13)(cid:13) v k − v k − (cid:13)(cid:13) L δ . The other term is somewhat easier to treat, since ( W k − W k − ) ∈ L δ +1 and hence (cid:13)(cid:13)(cid:13)(cid:13) ∂f∂W (cid:0) W k − W k − (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L δ +1 ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂f∂W (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (cid:13)(cid:13) W k − W k − (cid:13)(cid:13) L δ +2 . Now (cid:13)(cid:13) ∂f∂W (cid:13)(cid:13) L ∞ is bounded by a constant depending on (cid:13)(cid:13) U k (cid:13)(cid:13) X s,δ and (cid:13)(cid:13) U k − (cid:13)(cid:13) X s,δ , and theseare independent of k . Thus(6.23) (cid:13)(cid:13) F ( v k , W k ) − F ( v k − , W k − ) (cid:13)(cid:13) L δ +1 ≤ C (cid:13)(cid:13) U k − U k − (cid:13)(cid:13) Y,δ , where the constant C is independent of k . We shall now estimate the first term of D k in(6.21). From the structure of A ( v, W ) we see that (cid:0) A ( v k , W k ) − A ( v k − , W k − ) (cid:1) ∂ t U k = (cid:0) a ( v k ) − a ( v k − ) (cid:1) ∂ t ∂ x v k + (cid:0) a ( v k , W k ) − a ( v k − , W k − ) (cid:1) ∂ t W k . Now a ( v k ) − a ( v k − ) = ∂ a ∂v (cid:0) τ v k + (1 − τ ) v k − (cid:1) (cid:0) v k − v k − (cid:1) for some τ ∈ [0 , v k − v k − ) L δ +1 , we cannot use the L ∞ – L estimates. Wetherefore apply the algebra (or multiplication) property, once with s = 1, s = s − s = 0, and once with s = s , s = 1 and s = 1, which results in the following inequality: (cid:13)(cid:13)(cid:0) a ( v k ) − a ( v k − ) (cid:1) ∂ t ∂ x v k − (cid:13)(cid:13) H ,δ +1 . (cid:13)(cid:13)(cid:13)(cid:13) ∂ a ∂v (cid:0) τ v k + (1 − τ ) v k − (cid:1) (cid:0) v k − v k − (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) H ,δ (cid:13)(cid:13) ∂ t ∂ x v k − (cid:13)(cid:13) H s − ,δ +2 . (cid:13)(cid:13)(cid:13)(cid:13) ∂ a ∂v (cid:0) τ v k + (1 − τ ) v k − (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) H s,δ (cid:13)(cid:13) v k − v k − (cid:13)(cid:13) H ,δ (cid:13)(cid:13) ∂ t v k − (cid:13)(cid:13) H s,δ +1 . (6.24)We use now the third Moser inequality (4.5) and k v k − v k − k L ∞ . k v k − v k − k H s +1 ,δ inorder to obtain(6.25) (cid:13)(cid:13)(cid:13)(cid:13) ∂ a ∂v (cid:0) τ v k + (1 − τ ) v k − (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) H s +1 ,δ ≤ C (cid:16)(cid:13)(cid:13) v k (cid:13)(cid:13) H s +1 ,δ , (cid:13)(cid:13) v k − (cid:13)(cid:13) H s +1 ,δ (cid:17) . where C (cid:16)(cid:13)(cid:13) v k (cid:13)(cid:13) H s +1 ,δ , (cid:13)(cid:13) v k − (cid:13)(cid:13) H s +1 ,δ (cid:17) denotes that the constant depends in some way on the terms k v k k H s +1 ,δ and (cid:13)(cid:13) v k − (cid:13)(cid:13) H s +1 ,δ .In a similar way we obtain (cid:13)(cid:13)(cid:0) a ( v k , W k ) − a ( v k − , W k − ) (cid:1) ∂ t W k (cid:13)(cid:13) L δ +1 . (cid:13)(cid:13)(cid:13)(cid:13) ∂ a ∂v (cid:0) τ v k + (1 − τ ) v k − (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) H s,δ +1 (cid:13)(cid:13) v k − v k − (cid:13)(cid:13) H ,δ + (cid:13)(cid:13)(cid:13)(cid:13) ∂ a ∂W (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (cid:13)(cid:13) W k − W k − (cid:13)(cid:13) L δ +1 (cid:19) (cid:13)(cid:13) ∂ t W k (cid:13)(cid:13) H s,δ +2 . (6.26)We recall that (cid:13)(cid:13) ∂ t W k (cid:13)(cid:13) H s,δ +2 is bounded by (6.12) and (cid:13)(cid:13) v k − v k − (cid:13)(cid:13) H ,δ ≃ (cid:13)(cid:13) v k − v k − (cid:13)(cid:13) L δ + (cid:13)(cid:13) ∂ x v k − ∂ x v k − (cid:13)(cid:13) L δ +1 , therefore from (6.24), (6.25) and (6.26) we obtain that(6.27) (cid:13)(cid:13)(cid:0) A ( v k , W k ) − A ( v k − , W k − ) (cid:1) ∂ t U k (cid:13)(cid:13) L δ +1 ≤ C (cid:13)(cid:13) U k − U k − (cid:13)(cid:13) Y δ . In a similar manner we can estimate the difference involving the A a matrices. Theestimate of B ( U k ) − B ( U k − ) is simpler, since its first column of the matrix B is zero andtherefore this expression does not contain the element ( v k − v k − ). Thus we conclude frominequalities (6.22), (6.23) and (6.27) that ddt (cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y, A ≤ C (cid:16)(cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y, A + (cid:13)(cid:13) U k ( t ) − U k − ( t ) (cid:13)(cid:13) Y, A (cid:17) , where C is independent of k . Therefore by Gronwall’s inequality, we obtain that for any T ∗∗ ≤ T ∗ , sup ≤ t ≤ T ∗∗ (cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y, A ≤ e C T ∗∗ (cid:16)(cid:13)(cid:13) U k +10 − U k (cid:13)(cid:13) Y δ , A + T ∗∗ C sup ≤ t ≤ T ∗∗ (cid:13)(cid:13) U k ( t ) − U k − ( t ) (cid:13)(cid:13) Y, A (cid:19) , and hence inequality (6.19) holds if we chose T ∗∗ so that Λ := √ C e C T ∗∗ T ∗∗ <
1, and set β k := √ e C T ∗∗ (cid:13)(cid:13) U k +10 − U k (cid:13)(cid:13) Y δ , A . (cid:3) Step 5.
We discuss here the convergence. It follows from Proposition 6.5 that X k (cid:13)(cid:13) U k +1 ( t ) − U k ( t ) (cid:13)(cid:13) Y δ < ∞ , hence { U k ( t ) } is a Cauchy sequence in L ∞ ([0 , T ∗∗ ] , Y δ ). Applying the Gagliardo-Nirenberg-Moser estimate k u k H s ′ ≤ k u k s ′ s H s k u k − s ′ s L term–wise to the norm (2.1), we get that(6.28) k u k H s ′ ,δ ≤ k u k s ′ s H s,δ k u k − s ′ s L δ for 0 < s ′ < s and δ ∈ R . OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 33
Hence { U k ( t ) } is a Cauchy sequence in L ∞ ([0 , T ∗∗ ] , X s ′ ,δ ) and therefore U k ( t ) → U ( t ) inthe X s ′ ,δ –norm for any 0 < s ′ < s and δ ≥ − . Furthermore, by Remark 4.6, v k ( t ) → v ( t )in H s ′ +1 ,δ -norm. Thus if we choose < s ′ < s , then by the embedding (4.4), v k ( t ) → v ( t ) , W k ( t ) → W ( t ) in C and ∂ t v k ( t ) → ∂ t v ( t ) , ∂ t W k ( t ) → ∂ t W ( t ) in C . Thus, U ( t ) = ( v ( t ) , ∂ t v ( t ) , ∂ x v ( t ) , W ( t )) is a solution of the system (3.24). Proposition 6.6.
For any Φ ∈ X s,δ , (6.29) lim k (cid:10) U k ( t ) , Φ (cid:11) X s,δ = h U ( t ) , Φ i X s,δ uniformly for ≤ t ≤ T ∗∗ , and where h· , ·i X s,δ denote the inner-product (5.4) with A beingthe identity matrix. As a consequence of the weak convergence (6.29), we have that k U ( t ) k X s,δ ≤ lim inf k (cid:13)(cid:13) U k ( t ) (cid:13)(cid:13) X s,δ . Thus the solution U ( t ) belongs to X s,δ . For the proof of Proposition 6.6 see [16, § Step 6.
Here we shall prove uniqueness.
Proposition 6.7.
Suppose U ( t ) , U ( t ) ∈ X s,δ are two solutions of (3.24) with the sameinitial data, then U ( t ) ≡ U ( t ) .Proof. Let V ( t ) = U ( t ) − U ( t ), then it satisfies the same type of a linear system as (6.20),therefore by similar estimates as in Step 4, we obtain that ddt k V ( t ) k Y δ, A . k V ( t ) k Y δ, A , and since V (0) ≡
0, Gronwall’s inequality implies that V ( t ) ≡ (cid:3) Step 7.
Since X s,δ is a Hilbert space it suffices to show that(6.30) lim sup t → + k U ( t ) k X s,δ, A ≤ k U (0) k X s,δ, A in order to establish the continuity in the norm. Here A depends on the initial data φ and W , that is, A = A ( φ, W ). The proof of (6.30) relies on the same arguments as in[18] and therefore we leave it out. This completes the proof of Theorem 6.1. Proof of the main result
The proof of the main result, Theorem 2.3, actually follows from Theorem 6.1, we justhave to check whether the initial data of the gravitational fields and of the fluid satisfythe assumptions of Theorem 6.1. We recall that v ( t ) = g αβ ( t ) − η αβ , so setting φ = v (0),we have by the assumptions of Theorem 2.3 that φ ∈ H s +1 ,δ . The initial data for the timederivative ϕ are given by ∂ t g αβ (0), where ∂ t g ab (0) = − K ab ( a, b = 1 , , ∂ t g α (0) isgiven by expression (2.7). By the assumption of Theorem 2.3, K ab ∈ H s,δ +1 and thereforeby Propositions 4.5 and 4.9, ∂ t g α (0) also belongs to H s,δ +1 . Thus ϕ = ∂ t g αβ (0) satisfies theinitial condition of Theorem 6.1. Note that a (0) = h ab , where h ab is a proper Riemannianmetric. Since w (0) ≥ u α (0) is a unit timelike vector, a (0) is a positive definitematrix by Theorem 3.3. Hence A ( φ, W ) satisfy condition (6.1) and we conclude that U ( t ) = ( g αβ ( t ) − η αβ , ∂ t g αβ ( t ) , ∂ x g αβ ( t ) , W ( t )) ∈ C ([0 , T ] , X s,δ ). Hence (2.8) follows fromRemark 4.11, and (2.9) from (6.2). That completes the proof.8. Appendix
The classical paper of Hughes, Kato and Marsden [14] established the short time exis-tence of the vacuum Einstein equations by solving a second order quasi–linear hyperbolicsystem whose solutions ( g αβ , ∂ t g αβ ) belong to H s +1 × H s for s > .On the other hand, Fisher and Marsden treated the Einstein vacuum equation by meansof the theory of symmetric hyperbolic systems. However, they only obtained the regularityof s > . In [16] we generalized the result of [14] to the H s,δ spaces, treating however, theEinstein equations as a symmetric hyperbolic system. Since the techniques of [16], and inparticular the energy estimates, play an essential role in the present paper, we outline itsmain idea that enables us to obtain the same regularity as in [14].We present a heuristic argument explaining the essential idea. First, if a function v satisfies a wave equation, then the vector V = ( v, ∂ t v, ∂ x v ) satisfies a symmetric hyperbolicsystem. The general condition for existence and uniqueness in the H s ( R ) spaces is s > .Hence, we have by this method that ∂ t v, ∂ x v ∈ H s for s > .However, in our case we improve this regularity to ( v, ∂ t v, ∂ x v ) ∈ H s +1 × H s × H s for s > . This is because we do not consider a general quasi–linear symmetric hyperbolicsystem where the matrices A a ( V ) depend on V , but a system in which the matrices A a ( v )only depend on v but not on its derivatives.In order to see how this fact allows us to improve the regularity of the solution we willderive energy estimates for the linearized symmetric hyperbolic system. For the sake ofclarity we consider a simple hyperbolic system ∂ t V = A a ( v ) ∂ a V, then its linearized form is(8.1) ∂ t V = ˜ A a ∂ a V. Note that in each iteration we solve the linear system (8.1) with e A a = e A a ( v k ), and since V k = ( v k , ∂ t v k , ∂ x v k ) ∈ H s , v k ∈ H s +1 and hence e A a = e A a ( v k ) ∈ H s +1 by Moser type OCAL EXISTENCE OF RELATIVISTIC PERFECT FLUIDS 35 estimates. The crucial step is to derive the energy estimate(8.2) ddt (cid:18) k V k H s (cid:19) ≤ C k V k H s for s > and whenever V satisfies the linear system (8.1). We recall that k V k H s = k Λ s V k L , where Λ s is the pseudodifferential operator (1 − ∆) s .One of the basic tools for obtaining (8.2) are the commutator’s estimates. Here weshall use the following Pseudodifferential operators version of the Kato–Ponce estimate[26, § P be a differential operator in the class OP S s , , then(8.3) k P ( f g ) − f P ( g ) k L ≤ C {k∇ f k L ∞ k g k H s − + k f k H s k g k L ∞ } , for any f ∈ H s ∩ C and g ∈ H s − ∩ L ∞ .The standard way to obtain (8.2) is to differentiate k V k H s with respect to time, to insertthe differential equation (8.1) and then apply a suitable commutator which leads to12 ddt k V k H s = h Λ s ( V ) , Λ s ( ∂ t V ) i L = D Λ s ( V ) , Λ s (cid:16) e A a ∂ a V (cid:17)E L = D Λ s ( V ) , e A a (Λ s ( ∂ a V )) E L + D Λ s ( V ) , h Λ s (cid:16) e A a ∂ a V (cid:17) − e A a (Λ s ( ∂ a V )) iE L , and then the first term is taken care of by integration by parts and the second one is byapplying the above Kato–Ponce estimate to the operator Λ s . But this procedure resultsin a term of the form k ∂ a V k L ∞ which contains k ∂ a ∂ x v k L ∞ . In order to estimate it by k ∂ a ∂ x v k H s − . k ∂ x v k H s we need to require that s − > , and hence we do not get thedesired result.We circumvent this difficulty by writing e A a ∂ a V = ∂ a (cid:16) e A a V (cid:17) − (cid:16) ∂ a e A a (cid:17) V, and making the commutationΛ s (cid:16) ∂ a (cid:16) e A a V (cid:17)(cid:17) = h (Λ s ∂ a ) (cid:16) e A a V (cid:17) − e A a (Λ s ∂ a ) ( V ) i + e A a (Λ s ∂ a ) ( V ) . which we insert into the first row of equation (8). Then we have to estimate three terms: I = D Λ s ( V ) , h (Λ s ∂ a ) (cid:16) e A a V (cid:17) − e A a (Λ s ∂ a ) ( V ) iE L ,II = D Λ s ( V ) , e A a (Λ s ∂ a ) ( V ) E L and III = D Λ s ( V ) , Λ s (cid:16) ( ∂ a e A a ) V (cid:17)E L . For the first term we apply the Kato-Ponce commutator (8.3). However, this time wedo it for the operator (Λ s ∂ a ) which has order s + 1, and hence | I | ≤ k V k H s (cid:13)(cid:13)(cid:13) (Λ s ∂ a ) (cid:16) e A a V (cid:17) − e A a (Λ s ∂ a ) ( V ) (cid:13)(cid:13)(cid:13) L . k V k H s n k∇ e A a k L ∞ k V k H s + k e A a k H s +1 k V k L ∞ o . So by Sobolev embedding theorem, we see that | I | . k e A a k H s +1 k V k H s . Likewise, since H s is an algebra for s > , | III | . k V k H s k ( ∂ a e A a ) V k H s . k V k H s k ( ∂ a e A a ) k H s . k e A a k H s +1 k V k H s . Since Λ s ∂ a = ∂ a Λ s and e A a is symmetric, we obtain a similar estimate for II by usingintegration by parts. Hence we conclude that the energy estimate (8.2) holds. Note thatin the estimate of all three terms above we have used the fact that e A a ∈ H s +1 .For the general case where A = I , one has to define an appropriated inner-productwhich takes into account the matrix A . Details for the vacuum equations in the weightedspaces H s,δ and a positive definite A can be found in [16, §
4] and only slight modificationsare needed in order to extend the energy estimates of [16] to the coupled system (3.24).
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