The global nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FRW geometry
aa r X i v : . [ g r- q c ] D ec The Global Nonlinear Stability of Self-Gravitating IrrotationalChaplygin Fluids in a FRW Geometry
Philippe G. LeFLOCH ∗ and Changhua WEI † Abstract
We analyze the global nonlinear stability of FRW (Friedmann-Robertson-Walker) spacetimesin presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called(generalized) Chaplygin equation of state p = − A ρ α relating the pressure to the mass-energydensity, in which A > α ∈ (0 ,
1] are constant. We express the Einstein equations in wavegauge as a systems of coupled nonlinear wave equations and by performing a suitable conformaltransformation, we are able to analyze the global behavior of solutions in future timelike directions.We establish that the (3+1)-spacetime metric and the mass density and velocity vector describingthe evolution of the fluid remain globally close to a reference FRW solution, under small initialdata perturbations. Our analysis provides also the precise asymptotic behavior of the perturbedsolutions in the future directions.
Contents ∗ Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Universit´e Pierre et Marie Curie(Paris 6), 4 Place Jussieu, 75252 Paris, France. Email: contact@philippelefloch.org † Corresponding author: School of Mathematical Sciences, Fudan University, Shanghai, 200433, China. Email:[email protected].
Key words and phrases : FRW cosmology; generalized Chaplygin gas; conformal transformation; wave coordinates. Introduction
Recent cosmological observations predict that our Universe is currently enjoying a phase of acceleratedexpansion, which could be described for instance by introducing the notion of dark energy. An analysisbased on the Big Bang model reveals that our Universe is spatially flat and consists of 70 percentsof dark energy (with negative pressure), while the remaining 30 percents consist of dust matter (i.e.cold dark matter plus baryons), as well as negligible radiation. It has been predicted that the darkenergy may be responsible for the present acceleration of our Universe. These predictions from physicsrely on several possible theories of dark energy. One possibility is to include a cosmological constantin the Einstein equations, while another quite interesting approach models the matter content as aChaplygin gas or, more generally a generalized Chaplygin gas (GCG) or a modified Chaplygin gas(MCG). In the past decade, these models have been studied extensively by elementary methods ofanalysis and via numerical simulations. For a more detailed description of these models, we refer to[1–3, 6, 8–10, 16, 22] and the references therein.In this paper, we consider the nonlinear future stability of self-gravitating fluids governed by the(generalized) Chaplygin equation of state and, therefore, analyze the global existence problem for theEinstein-Euler system.
Definition 1.1. A generalized Chaplygin gas, by definition, is a perfect fluid governed by theequation of state p = − A ρ α , (1.1)relating the pressure p = p ( ρ ) to the mass-energy density ρ ≥ A is a positiveconstant and α ∈ (0 , Chaplygin gas when α = 1.The Einstein-Euler system for a generalized Chaplygin gas read as follows: e G µν = e T µν , e ∇ µ e T µν = 0 , (1.2)where e G µν = e Ric µν − e R e g µν is Einstein’s curvature tensor of an unknown metric e g = e g µν dx µ dx ν ,while e Ric µν and e R are the Ricci and scalar curvature of e g , respectively, and e ∇ µ denotes the covariantderivative of e g . Here, e T µν denotes the stress energy tensor e T µν = ( ρ + p ) e u µ e u ν + p e g µν (1.3)where ρ denotes the energy density, p = p ( ρ ) denotes the pressure and is given by (1.1), e u =( e u , · · · , e u ) denotes the unit, future-directed, timelike 4-velocity, e g µν is the inverse of e g µν . As isstandard, we use Einstein’s summation convention, i.e. we sum over repeated lower and upper in-dices.We are interested in irrotational full fluids, namely, under the equation of state (1.1), fluid suchthat there exists a potential function Ψ allowing us to express the stress energy tensor in form e T µν = ( ρ + p ) e u µ e u ν + p e g µν = A α +1 I − αα +1 " e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 − ( I − e h α +12 α ) αα +1 e g µν , (1.4)where I is a constant depending only on the initial data of the state variables and e h = − e g µν e ∂ µ Ψ e ∂ ν Ψ ≥
0. The derivation of (1.4) from (1.1) will be presented in Section 2.2, below.It is well-known that there exists a particular family of solutions to (1.2), that is, the Friedmann-Robertson-Walker cosmological spacetimes, and our main result in the present work can be simplydescribed as follows. 2 heorem 1.2 (The nonlinear future stability of Friedmann-Robertson-Walker spacetimes. Prelimi-nary version) . The FRW solutions to the Einstein-Euler system for a generalized Chaplygin gas arenonlinearly stable toward the future, when the initial data set prescribed on an initial hypersurface isa small perturbation of this FRW solution.A more technical version of the above theorem will be provided below after we will introducesome necessary notations. At this junction, we want to recall some eariler work on the Einstein-Eulersystem, especially for the expanding spacetimes of interest in the present work.
One natural problem one may ask is whether or not our universe dominated by above models is stable.Due to the importance of this problem in mathematics and physics, it attracts a lot of attention ingeneral relativity and great improvements have been made on the Einstein-Euler equations withpositive cosmological constant in recent years. We will describe these improvements clearly below.While we still know nothing on the future nonlinear stability of our universe dominated by Chaplygingas (or GCG, MCG) in 1+3 spacetime dimensions, that is one motivation for us to study this problem.The Einstein-Euler equations with positive cosmological constant read e G µν = e T µν − Λ e g µν , e ∇ µ e T µν = 0 , (1.5)where Λ > ?? ) and stress energy tensor (1.3) is focused on the fluid with equation of state p = C s ρ, (1.6)where C s ≥ C s = 0, it is called the dust universe, and when C s = q , it is used to describe the radiation universe.There are mainly two approaches to deal with the Einstein-Euler system (1.5) mathematically:working with the given spacetime metric, or alternatively working with a conformally equivalentmetric. Both approaches play an important role in the mathematical theory of general relativity. Thewell-known family of Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) solutions to (1.5) represents ahomogeneous, fluid filled universe that is undergoing accelerated expansion. Rodnianski and Speck[19] established the future global stability of a class of FLRW solutions under the assumption ofzero vorticity and 0 < C s < q . The so-called wave gauge approach in [19] was first used byRingstr¨om in [18] who treated scalar fields without positive cosmological constant: e T µν = e ∂ µ Ψ e ∂ ν Ψ − [ e g µν e ∂ µ Ψ e ∂ ν Ψ + V (Ψ)] e g µν . The presence of V (Ψ) plays an analogue role as Λ, under the assumptionthat V (0) > , V ′ (0) = 0 , V ′′ (0) >
0. The main observation in these papers is two-fold: on one hand,the Einstein-nonlinear scalar field system can be formulated as a system of nonlinear wave equationsprovided one introduces generalized wave coordinates, inspired by the standard wave coordinatesused earlier (for instance in Lindblad and Rodnianski [13] for the vacuum case, revisited recently inLeFloch and Ma [11]); on the other hand, the problem under consideration describes the expansionof the universe, and the expansion provides one dispersive terms, which leads to exponential decayfor solutions. Later, Speck (and collaborator) [7, 20] proved that above nonlinear stability resultremains true even for a fluid with non-vanishing vorticity when 0 ≤ C s < q . When C s = q ,L¨ubbe and Valiente Kroon [14] have shown the desired stability property by relying on Friedrich’sconformal method [4, 5] —an approach entirely different from the method in [7, 19, 20]. More recently,a very efficient method was proposed by Oliynyk [15], which combine the conformal method with wavecoordinates in order to handle the case 0 < C s ≤ q with non-vanishing vorticity. One advantageof the latter method is that, under a conformal transformation, the whole Einstein-Euler system3an be turned into a symmetric hyperbolic system (with singular terms) and solutions defined onfinite interval of time. The singular terms enjoy good positivity properties and, by a standard energyestimate, one can then get the global nonlinear stability of a family of FLRW solutions and establishedthe asymptotic behavior of perturbed solutions in the far future. Finally, we recall that, in the regime C s > q , Rendall [17] has found some evidence for instability. We thus study the Einstein-Euler system (1.2) with the stress energy tensor (1.4). In order to describeour ideas and main results clearly, we must fix our notations at first. The spacetime that we considerare of the form (0 , × T . For the coordinates, we use x i (i=1,2,3) to denote the spacial coordinatesand use x = τ to denote the time coordinate. We always use the Greek indices to denote thespacetime coordinates that run form 0 to 3 and Latin indices to denote the spacial coordinates thatrun from 1 to 3. In this system of coordinates, when we say the fluid velocity is future directed, wemean that e u < . (1.7)As Oliynyk in [15], we do not consider the original metric e g directly, but instead consider the confor-mally transformed metric g µν = e − e g µν , or g µν = e e g µν , (1.8)where Φ = − ln( τ ) . (1.9)Under the conformal transformation (1.8) and (1.9), the equations (1.2) that we consider in this paperis the following Cauchy problem G µν = T µν := e e T µν + 2( ∇ µ ∇ ν Φ − ∇ µ Φ ∇ ν Φ) − (2 (cid:3) g Φ + |∇ Φ | g ) g µν , ∇ µ e T µν = − e T µν ∇ µ Φ + g κλ e T κλ g µν ∇ µ Φ ,g µν | τ =1 = g µν ( x ) , ∂ τ g µν | τ =1 = g µν ( x ) , ∂ µ Ψ | τ =1 = m µ ( x ) . (1.10) Remark 1.3.
Above initial data set ( g µν ( x ) , g µν ( x ) , m µ ( x )) can not be chosen arbitrarily. They mustsatisfy the Gauss-Codazzi equations, which are equivalent to ( G µ − T µ ) | τ =1 = 0. Furthermore, theywill also satisfy the wave coordinates condition Z µ | τ =1 = 0, the precise definition of Z µ can be foundin Section 3.1.We know that there exist a family of FRW solutions ( e η, Ψ( τ )) to the original Einstein-Eulerequations of GCG (1.2). e η takes the following form e η = 1 τ − w ( τ ) dτ + X i =1 ( dx i ) ! , where τ = 1 a ( t ) , w = 1 a ( t ) da ( t ) dt := ˙ a ( t ) a ( t ) , for some scale factor a ( t ) with t ∈ [0 , + ∞ ). Under conformal transformation, the conformal metric g can be seen as small perturbations to the conformal background metric η which is given by η = − w dτ + X i =1 ( dx i ) . Define the densitized three-metric g ij = det (ˇ g lm ) g ij , where ˇ g lm = ( g lm ) − , and introduce the variable q = g − η + η g ij )) . With above notations, our main result can be described as follows.4 heorem 1.4 (The nonlinear future stability of Friedmann-Robertson-Walker spacetimes. Statementin wave coordinates) . Suppose k ≥ g µν ( x ) ∈ H k +1 ( T ), g µν ( x ) , m µ ( x ) ∈ H k ( T ) for all x ∈ T .Then there exists a small parameter ǫ >
0, such that if the initial data sets satisfy the constraintequations of Remark 1.3 and k g µν − η µν (1) k H k +1 + k g µν − ∂ τ η µν (1) k H k + k m µ ( x ) − ∂ µ Ψ(1) k H k < ǫ, then there exists a unique classical solution g µν , Ψ ∈ C ((0 , × T ) to the conformal Einstein-Eulersystem of GCG (1.10) and it has the following regularity g µν ∈ C ((0 , , H k +1 ( T )) ∩ C ([0 , , H k ( T )) ∩ C ((0 , , H k ( T )) ∩ C ([0 , , H k − ( T ))and ∂ µ Ψ ∈ C ((0 , , H k ( T )) ∩ C ([0 , , H k − ( T )) . The solution also satisfies k g µν ( τ ) − η µν ( τ ) k H k +1 + k g µνκ ( τ ) − ∂ κ η µν ( τ ) k H k + k ∂ µ Ψ( τ ) − ∂ µ Ψ( τ ) k H k < Cǫ, for some positive constants C . Moreover, there exists γ µ ∈ H k − ( T ), such that the solution for all τ ∈ [0 ,
1] satisfies k ∂ τ g µ ( τ ) − τ − ( g µ ( τ ) − η µ ( τ )) + γ µ k H k − ≤ Cǫτ, k ∂ τ g µ ( τ ) − τ − ( g µ ( τ ) − η µ ( τ )) k H k − + k ∂ i g µ ( τ ) k H k − ≤ − Cǫτ ln τ, k q ( τ ) − q (0) k H k + k ∂ τ q ( τ ) k H k − ≤ Cǫτ, k g ij ( τ ) − g ij (0) k H k + k ∂ τ g ij ( τ ) k H k − ≤ Cτ, k ∂ µ Ψ( τ ) − ∂ µ Ψ(0) k H k − ≤ Cǫτ, k ∂ τ Ψ(0) − ∂ τ Ψ(0) k H k − ≤ Cǫ, k ∂ i Ψ(0) k H k − ≤ Cǫ, with, furthermore, p ρ (0) = α. Remark 1.5.
1. The above results show that in the future of the Einstein-Euler equations of GCG,the speed of sound is √ α , especially for Chaplygin gas, it means that the speed of sound is equal tothe speed of light. This phenomenon is very interesting and quite different from the problem of a fluidmoving in a Universe with positive cosmological constant Λ.2. When α = 1, the Euler system is linearly degenerate, while when 0 < α <
1, this system isgenuinely nonlinear and shocks generally form in finite time, no matter how small the perturbationsare in the flat Minkowski spacetime. Our result shows that the spacetime expansion stabilize the fluidand prevent the formation of shocks.The main idea of the proof is turning system (1.10) into a symmetric hyperbolic system underappropriate wave coordinates Z µ = 0. Here we would like to see some differences between our paperand Oliynyk’s and others’ mentioned above. Firstly, we consider the nonlinear future stability ofthe nontrivial FRW solutions, thus, we need to choose different coordinates, which can be seen asa generalization of Oliynyk’s work; second, for irrotational fluids, we have to choose an appropriateconformal factor in order to solve the difficulty brought by the degeneracy of the enthalpy pe h .An outline of this paper is as follows. In Section 2.1, we give some preliminaries on the notationsand norms used in this paper. Sections 2.2 and 2.3 are aimed at giving the detailed formulation of theproblem and studying the properties of the FRW background solutions. Section 3 is the main part ofthe whole paper, we present our choice of coordinates in Section 3.1. In Sections 3.2, 3.3 and 3.4, weturn the whole system into a symmetric hyperbolic system and analyze the structure of this system.Sections 3.5 and 3.6 contain the proof of the main results.5 Formulation of the problem
Greek indices µ range from 0 to 3, while Latin indices i range from 1 to 3. Repeated lower and upperindices means summation with their corresponding metric. For the metrics in this paper, we use e g and e η to denote the original metric and original background metric respectively; we also use g and η to denote the conformal metric and the conformal background metric. e Γ, e Γ, Γ and Γ denote theChristoffel symbols with respect to e g , e η , g and η , respectively, similar conventions are used for thecurvature tensors e R , e R , R , R and the norms e h , e h , h and h , whose square root denotes the physicalquantity “enthalpy”.For convenience, we use A ∼ B to denote the equivalence relationship between A and B , whichmeans that there exists a positive constant C >
1, such that AC ≤ B ≤ CA . e ∂ µ = e ∂ x µ and ∂ µ = ∂ x µ are the partial derivatives of the original spacetime and the conformal spacetime. Similar definitionsare used for the covariant derivatives e ∇ and ∇ .For a function u ( t, x ), we define the following standard sobolev norms k u ( t, x ) k L ( T n ) := (cid:18)Z T n | u ( t, x ) | dx (cid:19) , k u ( t, x ) k H k ( T n ) := k X I =0 k D I u ( t, x ) k L ( T n ) , and k u ( t, x ) k L ∞ ( T ) := ess sup x ∈ T | u ( t, x ) | . In this section, we focus on the derivation of the irrotational energy momentum tensor e T µν . Forisentropic fluids, the pressure is given by p = n ∂ρ∂n − ρ, (2.1)where n denotes the number of particles per unit volume. We also have e ∇ µ ( n e u µ ) = 0 . In this paper, we consider the generalized Chaplygin gas, whose equation of state is given by p = − A ρ α , (2.2)where A is a positive constant and 0 < α ≤ ρ α +1 − A = J (1) n α +1 , where J (1) = ρ α +1 (1) − A n α +1 (1) dependsonly on the initial data of the state variables ( ρ, n ) at τ = 1. Define the enthalpy in the originalspacetime by pe h = ρ + pn = J α +1 (1) (cid:18) − A ρ α +1 (cid:19) αα +1 , then ρ = " A J α (1) J α (1) − e h α +12 α α +1 . J α (1) = I , then ρ = A α +1 I α +1 ( I − e h α +12 α ) α +1 , (2.3)and p = − A ρ α = − A α +1 I − αα +1 ( I − e h α +12 α ) αα +1 . (2.4)By above two equalities (2.3) and (2.4), we can easily get p + ρ = A α +1 I − αα +1 e h α +12 α ( I − e h α +12 α ) α +1 , and ρ − p = A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 . Assume that the fluid is irrotational, then there locally exists a potential function Ψ( τ, x ) such that e u µ = − e ∂ µ Ψ pe h . Based on the normalization condition e g µν e u µ e u ν = − , and e u < , in our spacetime, we have e h = − e g µν e ∂ µ Ψ e ∂ ν Ψ ≥ , e ∂ τ Ψ ≥ . Thus, we can define our stress energy tensor as follows e T µν = ( ρ + p ) e u µ e u ν + p e g µν = A α +1 I − αα +1 " e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 − ( I − e h α +12 α ) αα +1 e g µν . (2.5)At the same time, we have e T = e g µν e T µν = − ρ − p + 4 p = A α +1 I − αα +1 − I + 3 e h α +12 α ( I − e h α +12 α ) α +1 . (2.6) Remark 2.1.
1. It is easy to choose appropriate initial data for the state variables ( ρ, n ) such that I − e h α +12 α > In this section, we study some properties of the background FRW solutions. At first, we are readyto give the detailed information of our spacetime. The metric endowed by the original spacetime(0 , × T is e g = e g µν dx µ dx ν , (2.7)which can be seen as the perturbation of the following metric e η = 1 τ − w ( τ ) dτ + X i =1 ( dx i ) ! , (2.8)7here τ = a ( t ) and w ( t ) = ˙ a ( t ) a ( t ) , for some scale factor a ( t ) with t ∈ [0 , + ∞ ). Obviously dτ = − a ( t ) ˙ a ( t ) = − τ w ( t ) dt. Note in passing that the metric (2.8) is equivalent to ds = − dt + a ( t ) X i =1 ( dx i ) , which is the original model studied in physics.As discussed in Section 1.3, we do not work with the spacetime metric (2.7) directly but insteaduse the conformally transformed metric g µν = e − e g µν , (2.9)where Φ = − ln( τ ). We will consider above g (2.9) as a small perturbation of the following metric η = − w ( τ ) dτ + X i =1 ( dx i ) , (2.10)which is the conformal transformation of (2.8). Under above conformal transformations, the Einsteinequations are equivalent to ( G µν = T µν := e e T µν + 2( ∇ µ ∇ ν Φ − ∇ µ Φ ∇ ν Φ) − (2 (cid:3) g Φ + |∇ Φ | g ) g µν , ∇ µ e T µν = − e T µν ∇ µ Φ + g κλ e T κλ g µν ∇ µ Φ . (2.11)Expanding the first equation of (2.11), we have by inserting e T µν defined by (2.5) − R µν = − ∇ µ ∇ ν Φ + 4 ∇ µ Φ ∇ ν Φ − (cid:3) g Φ + 2 |∇ Φ | g (2.12)+ 12 A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 e ] g µν − e A α +1 I − αα +1 e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 , or equivalently − R µν = − ∇ µ ∇ ν Φ + 4 ∇ µ Φ ∇ ν Φ − (cid:3) g Φ + 2 |∇ Φ | g (2.13)+ 12 A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 e ] g µν − A α +1 I − αα +1 e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 . Now we consider the background FRW solution to (2.11). Assume that there exists Ψ( τ, x ) = Ψ( τ )satisfies (2.11) with background metric (2.10). Then it is easy to see that e h := − e η e ∂ τ Ψ e ∂ τ Ψ = 1 τ w ( e ∂ τ Ψ) . e T = A α +1 I − αα +1 e h α +12 α τ w ( I − e h α +12 α ) α +1 + ( I − e h α +12 α ) αα +1 τ w , e T i = 0 , and e T ij = A α +1 I − αα +1 (cid:20) − ( I − e h α +12 α ) αα +1 τ (cid:21) δ ij . λµν = 12 η ( ∂ τ η ) δ λ δ µ δ ν = − ∂ τ ww δ λ δ µ δ ν , then R µν = ∂ α Γ αµν − ∂ µ Γ ααν + Γ ααλ Γ λµν − Γ αµλ λ ααν = 0 . With above preparations, we have the following important lemma, which are the represent formulasof the background FRW solutions.
Lemma 2.2.
There exists a family of solutions ( w ( τ ) , Ψ( τ )) to (2.11)-(2.13), and the solutions satisfythe following e h = ( IK ) αα +1 τ α (1 + Kτ α ) ) αα +1 , e ∂ τ Ψ = ( IK ) αα +1 τ α +1 w (1 + Kτ α ) ) αα +1 , w = A α +1 I α +1 ( I − e h α +12 α ) α +1 ,∂ t w = − A α +1 I − αα +1 e h α +12 α I − e h α +12 α ) α +1 , (2.14)where K is a constant depending only on the initial data. Proof.
At first, we consider the 0-th component of the fluid equations, we have by neglecting A α +1 I − αα +1 ∇ µ e T µ = ∂ τ e T + ∂ i e T i + 2Γ e T = ∂ τ τ w I ( I − e h α +12 α ) α +1 − ∂ τ ww τ w I ( I − e h α +12 α ) α +1 = 6 τ e h α +12 α τ w ( I − e h α +12 α ) α +1 + ( I − e h α +12 α ) αα +1 τ w − w τ τ e h α +1 α ( I − e h α +1 α ) α +1 + 4 τ ( I − e h α +12 α ) αα +1 . Solving above ODE, we easily get
Iτ ∂ τ I − e h α +12 α ) α +1 = 3 e h α +12 α ( I − e h α +12 α ) α +1 , (2.15)(2.15) is equivalent to Iτ ∂ τ e h = 6 α e h ( I − e h α +12 α ) . By setting ξ = e h α +12 α we have Idξξ ( I − ξ ) = 3( α + 1) dττ . (2.16)9olving (2.16), we have ξ = IKτ α +1) Kτ α +1) , where K = ξ (1) I − ξ (1) and thus, e h = ( IK ) αα +1 τ α (1 + Kτ α +1) ) αα +1 . (2.17)Obviously, we have e ∂ τ Ψ = ( τ w e h ) = ( IK ) αα +1 τ α +1 w (1 + Kτ α ) ) αα +1 . (2.18)The i -th components of the fluid equations hold obviously.Now we consider the Einstein equations.At first, from R ij = 0, we have − w τ + w∂ τ wτ − w τ + A α +1 I − αα +1 τ I − e h α +12 α ( I − e h α +12 α ) α +1 = 0 . (2.19)Combining (2.19) with R = 0, we have −
4( 1 τ − ∂ τ wτ w ) + 4 τ − A α +1 I − αα +1 e h − α α e ∂ τ Ψ e ∂ τ Ψ( I − e h α +12 α ) α +1 = 0 . (2.20)From (2.19), (2.20) and ∂ τ w = − ∂ t wτw , we easily get w − A α +1 I α +1 ( I − e h α +12 α ) α +1 = 0 , − ∂ t w = A α +1 I − αα +1 e h α +12 α I − e h α +12 α ) α +1 . (2.21)The lemma holds by combing (2.17), (2.18) and (2.21). Corollary 2.1.
By the above lemma and via a Taylor expansion, we see that when τ → , w − A α +1 ∼ τ α +1) , − ∂ t w ∼ τ α +1) ,∂ τ ( ˙ w ) ∼ τ α +2 ,∂ τ w ∼ τ α +1 . (2.22)The asymptotic behavior above plays very important roles in the analysis of the source terms inSection 3.4. Proof.
The first and second asymptotic behavior can be obtained directly from (2.14). The thirdasymptotic behavior can be derived by differentiating ˙ w directly and use (2.17). For the last, we have ∂ τ w = ∂ τ ( − ˙ wτ w ) = − ∂ τ ˙ wτ w + ˙ ww ( τ w ) − ( ˙ w ) τ w ∼ τ α +1 . emark 2.3. Since we have confined τ ∈ (0 , a (0) = 1. Thus,we have to prove that when t → ∞ , a ( t ) → + ∞ . From (2.14), there must exist two positive constants A and A such that A ≤ w ( t ) = ˙ a ( t ) a ( t ) ≤ A , then by comparison theorem of ODE, e A t ≤ a ( t ) ≤ e A t . It is obvious that a ( t ) = + ∞ , when t → + ∞ . Our first task is to introdude appropriate coordinates. For the background metric e η , we have e Γ = − τ + ˙ wτ w , and e Γ ii = − e η ∂ τ e η ii = − w τ . Then, we obtain e Γ = e η e Γ + e η ii e Γ ii = − τ w − τ ˙ w, and e e Γ = − w τ − ˙ wτ . (3.1)Direct calculations show that e Γ i = 0 , ( i = 1 , , . (3.2)For the conformal metric η , we have Γ µ = − ˙ wτ δ µ . (3.3)On the other hand, under conformal transformation (2.9)Γ µ = g αβ Γ µαβ = 2 ∇ µ Φ + e e Γ µ . (3.4)Define the wave coordinates as Z µ = Γ µ + Y µ = Γ µ + 2 τ (cid:18) g µ + ( w + ˙ w δ µ (cid:19) := Γ µ + 2 τ (cid:18) g µ + Λ( τ )3 δ µ (cid:19) . (3.5)In (3.5), we have denoted w + ˙ w by Λ( τ )3 for convenience. Remark 3.1.
1. By results in Zengino˘glu [23], we see that if initially Z µ = 0, then Z µ ≡ Y µ = − ∇ µ Φ − e e Γ µ . From (3.1)-(3.4) we see that for thebackground metric η , Z µ ≡
0. This fact is very important for the disappear of the linear parts of theconformal Einstein and conformal fluid equations (1.10).3. When w = Λ3 with Λ a positive constant, this is exactly the case considered by Oliynyk [15].11 .2 The reduced conformal Einstein equations With the wave coordinates Z µ defined by (3.5), we can consider the following equivalently reducedconformal Einstein equations by assuming Z µ | τ =1 = 0 − R µν + 2 ∇ ( µ Z ν ) + A µνκ Z κ = − ∇ µ ∇ ν Φ + 4 ∇ µ Φ ∇ ν Φ − " (cid:3) g Φ + 2 |∇ Φ | g + 12 A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 e g µν − e A α +1 I − αα +1 e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 . (3.6)With A µνκ = − Γ ( µ δ ν ) κ + Y ( µ δ ν ) κ and ∇ ( µ Z ν ) = 12 ( ∇ µ Z ν + ∇ ν Z µ ) , we have 2 ∇ ( µ Z ν ) = 2 ∇ ( µ Γ ν ) + ∇ µ Y ν + ∇ ν Y µ = 2 ∇ ( µ Γ ν ) + ∂ τ (cid:18) τ )3 τ (cid:19) ( g i δ ν + g ν δ µ ) − τ )3 τ ∂ τ g µν − ∇ µ ∇ ν Φand A µνk Z κ = ( − Γ ( µ δ ν ) κ + Y ( µ δ ν ) κ )(Γ κ + Y κ )= − Γ ( µ Γ ν ) + 4 ∇ µ Φ ∇ ν Φ − τ )3 τ ( ∇ µ Φ δ ν + ∇ ν Φ δ µ ) + 4Λ ( τ )9 τ δ µ δ ν . The Einstein equations (2.12) become − R µν + 2 ∇ ( µ Γ ν ) − Γ µ Γ ν = 2Λ( τ )3 τ ∂ τ g µν − ∂ τ Λ( τ )3 τ ( g µ δ ν + g ν δ µ ) − τ )3 τ (cid:18) g + Λ( τ )3 (cid:19) δ µ δ ν − τ )3 τ g i δ ( µ δ ν ) i − τ g µν (cid:18) g + Λ( τ )3 (cid:19) + 2 τ Λ( τ ) − A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 ! g µν − e A α +1 I − αα +1 e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 . Expanding the left hand of above and inserting Λ( τ )3 = w + ˙ w , we get − g κλ ∂ κ ∂ λ g µν = 2 w τ ∂ τ g µν − w τ ( g + w ) δ µ δ ν − w τ g i δ ( µ δ ν ) i − τ g µν ( g + w ) − wτ ( g + w + ˙ w δ µ δ ν − ˙ wτ ∂ τ g µν − τ w ˙ wδ µ δ ν − wτ g i δ ( µ δ ν ) i − ˙ wτ g µν − ∂ τ Λ( τ )3 τ ( g µ δ ν + g ν δ µ )+ 2 τ w + 3 ˙ w − A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 ! g µν − e A α +1 I − αα +1 e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 + Q µν ( g, ∂g )12= 2 w τ ∂ τ g µν − w τ ( g + w ) δ µ δ ν − w τ g i δ ( µ δ ν ) i − τ g µν ( g + w ) + M µν or equivalently − g κλ ∂ κ ∂ λ ( g µν − η µν ) = 2 w τ ∂ τ ( g µν − η µν ) − w τ ( g + w ) δ µ δ ν − w τ g i δ ( µ δ ν ) i − τ g µν ( g + w ) + ˆ M µν . (3.7)We haveˆ M µν = M µν + g κλ ∂ κ ∂ λ η µν + 2 w τ ∂ τ η µν = ( g κλ − η κλ ) ∂ κ ∂ λ η µν − wτ ( g + w ) δ µ δ ν − ˙ wτ ∂ τ ( g µν − η µν ) − wτ g i δ ( µ δ ν ) i − ˙ wτ ( g µν − η µν ) − ∂ τ Λ( τ )3 τ (cid:0) ( g µ − η µ ) δ ν + ( g ν − η ν ) δ µ (cid:1) + 2 τ A α +1 I − αα +1 I − e h α +12 α ( I − e h α +12 α ) α +1 − I − e h α +12 α ( I − e h α +12 α ) α +1 g µν − e A α +1 I − αα +1 e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 − e h − α α e ∂ µ Ψ e ∂ ν Ψ( I − e h α +12 α ) α +1 + Q µν ( g, ∂g ) − Q µν ( η, ∂η ) (3.8) Remark 3.2.
1. In above, Q µν ( g, ∂g ) are quadratic in ∂g = ( ∂ κ g µν ) and analytical in g = ( g µν ).Thus, in ˆ M µν , each term contains g − η or e ∂ Ψ − e ∂ Ψ, which will be proved later.2. Equation (2.12) is equivalent to (3.7), provided that Z µ = 0. On the other hand, the backgroundsolution ( η, Ψ( τ )) defined in Section 2.3 satisfies (3.7) obviously, since Z µ ≡ η , whichis also the reason for the disappear of the linear parts of (3.8)The main part of this paper is turning the Einstein-Euler system of GCG into a symmetric hyper-bolic system, thus we make this process clear in the following.Recall the densitized three metric defined in Section 1.3 g ij = det (ˇ g lm ) g ij , (3.9)where ˇ g lm = ( g lm ) − , and the variable q = g + w − w g pq )) . (3.10)It is easy to check that ∂ µ g ij = (det(ˇ g pq )) L ijlm ∂ µ g lm , (3.11)where L ijlm = δ il δ jm −
13 ˇ g lm g ij . Obviously, L ijlm is trace-free, i.e., L ijlm g lm = 0 . u ν = g ν − η ν τ , (3.12) u ν = ∂ τ ( g ν − η ν ) − g ν − η ν )2 τ , (3.13) u νi = ∂ i ( g ν − η ν ) , (3.14) u ij = g ij − δ ij , (3.15) u ijµ = ∂ µ g ij , (3.16) u = q , (3.17) u µ = ∂ µ q . (3.18)At first, we consider the equation satisfied by g µ − η µ .Utilizing (3.12)-(3.14) and inserting g µ − η µ = 2 τ u µ , ∂ i ( g µ − η µ ) = u µi and ∂ τ ( g µ − η µ ) = u µ + 3 u µ into (3.7), we have − g ∂ τ ( u µ + 3 u µ ) − g i ∂ i ( u µ + 3 u µ ) − g ij ∂ j u µi = 2 w τ ( u µ + 3 u µ ) − w τ (2 τ u ) δ µ − w τ (2 τ u i ) δ ( µ δ i − τ g µ u + ˆ M µ . Based on above, we get − g ∂ τ u µ − g i ∂ i u µ − g ij ∂ j u µi = 3 g ∂ τ (cid:18) g µ − η µ τ (cid:19) + 6 g i ∂ i (cid:18) g µ − η µ τ (cid:19) + 2 τ (2 τ u − g )( u µ + 3 u µ ) − τ (2 τ u − g ) u δ µ − τ (2 τ u − g ) u i δ ( µ δ i − τ g µ u + ˆ M µ = 3 g u µ + 3 u µ τ − g u µ τ + 6 u i u µi + 4 u u µ + 12 u u µ − τ g u µ − τ g u µ − u u δ µ + 8 τ g u δ µ − u u i δ ( µ δ i + 8 g τ u i δ ( µ δ i − τ g µ u + ˆ M µ = τ [ − g ( u + u )] + 6 u i u + 4 u u − u u + ˆ M , µ = 0 τ [ − g ( u k + u k )] + 6 u i u k + 4 u u k − u u k + ˆ M k , µ = k = 1 τ (cid:20) − g u µ + u µ ) (cid:21) + 6 u i u µ + 4 u u µ − u u µ + ˆ M µ . (3.19)Now we consider g ij − δ ij , direct calculations give − g κλ ∂ κ ∂ λ g ij = − g κλ ∂ κ [( det (ˇ g pq )) L ijlm ∂ λ g lm ]= ( det (ˇ g pq )) L ijlm ( − g κλ ∂ κ ∂ λ g lm ) − g κλ ∂ κ [( det (ˇ g pq )) L ijlm ] ∂ λ g lm = ( det (ˇ g pq )) L ijlm (cid:18) w τ ∂τ ( g lm − η lm ) − τ g lm ( g + w ) + ˆ M lm (cid:19) − g κλ ∂ κ [( det (ˇ g pq )) L ijlm ] ∂ λ g lm = 2 w τ ∂ τ g ij + ( det (ˇ g pq )) L ijlm ˆ M lm − g κλ ∂ κ [( det (ˇ g pq )) L ijlm ] ∂ λ g lm := 2 w τ ∂ τ g ij + ´ M ij . (3.20)14e have ´ M ij = ( det (ˇ g pq )) L ijlm ˆ M lm − g κλ ∂ κ [( det (ˇ g pq )) L ijlm ] ∂ λ g lm . Then by (3.15)-(3.16), we have − g ∂ τ u ij − g i ∂ i u ij − g pq ∂ p u ijq = 2 τ (2 τ u − g ) u ij + ´ M ij = − τ g u ij + 4 u u ij + ´ M ij . (3.21)At last, we consider q , we have by (3.10) ∂ λ q = ∂ λ ( g + w ) − w g pq ∂ λ g pq − w∂ τ wδ λ g pq )) . (3.22)Then by (3.22) ∂ κ ∂ λ q = ∂ κ ∂ λ ( g + w ) − w g pq ∂ κ ∂ λ g pq − w ∂ κ g pq ∂ λ g pq − w∂ τ wδ κ g pq ∂ λ g pq − ∂ τ w ) δ λ δ κ ln(det( g pq )) − w∂ τ w δ λ δ κ ln(det( g pq )) − w∂ τ w δ λ g pq ∂ κ g pq := ∂ κ ∂ λ ( g + w ) − w g pq ∂ κ ∂ λ g pq + R q , (3.23)where R q = − w ∂ κ g pq ∂ λ g pq − w∂ τ wδ κ g pq ∂ λ g pq − ∂ τ w ) δ λ δ κ ln(det( g pq )) − w∂ τ w δ λ δ κ ln(det( g pq )) − w∂ τ w δ λ g pq ∂ κ g pq . Thus, we have by (3.17)-(3.18) − g κλ ∂ κ ∂ λ q = − g κλ ∂ κ ∂ λ ( g + w ) + w g pq g κλ ∂ κ ∂ λ g pq − g κλ R q = 2 w τ ∂ τ ( g + w ) − w τ ( g + w ) − τ g ( g + w ) + ˆ M − w g pq ( 2 w τ ∂ τ g pq − τ g pq ( g + w ) + ˆ M pq ) − g κλ R q = 2 w τ ∂ τ q − (cid:18) g + w τ (cid:19) + ˆ M − w g pq ˆ M pq + 4 w ∂ τ w τ ln(det g pq ) − g κλ R q := 2 τ (2 τ u − g ) ∂ τ q − u ) + ˆ R q = − τ g ∂ τ q + 4 u ∂ τ q − u ) + ˆ R q . (3.24)We have ˆ R q = ˆ M − w g pq ˆ M pq + 4 w ∂ τ w τ ln(det( g pq )) − g κλ R q . − g ∂ τ u − g i ∂ i u − g ij ∂ i u j = − τ g u + 4 u u − u ) + ˆ R q . (3.25)From (3.19), (3.21) and (3.25), we easily transform the Einstein equations into the following symmetrichyperbolic system A κ ∂ κ u µ u µj u µ = 1 τ AP u µ u µj u µ + F µ , (3.26) A κ ∂ κ u lm u lmj u lm = 1 τ ( − g )Π u lm u lmj u lm + F lm , (3.27)and A κ ∂ κ u u j u = 1 τ ( − g )Π u u j u + F q , (3.28)where A = − g g ij
00 0 − g , A k = − g k − g jk − g ik , P = δ jk , A = − g g jk
00 0 − g , Π = , F µ = u i u µ + 4 u u µ − u u µ + ˆ M µ , and F ij = u u ij + ´ M ij g u lm , F q = u u − u ) + ˆ R q g u lm . In this section, we transform the conformal fluid equation into a symmetric hyperbolic system. Atfirst, we choose an appropriate conformal factor for the potential function Ψ( τ, x ).Define e ∂ µ Ψ = e − λ Φ ∂ µ Ψ = τ λ ∂ µ Ψ , (3.29)where λ will be determined later. Then e h = − e g µν e ∂ µ Ψ e ∂ ν Ψ = − τ λ − g µν ∂ µ Ψ ∂ ν Ψ := τ λ − h. (3.30)16n terms of h and ∂ Ψ, from (3.29), (3.30), we have by neglecting some unnecessary constants A α +1 I − αα +1 e T µν = [ τ λ − h ] − α α τ λ ∂ µ Ψ ∂ ν Ψ[ I − ( τ λ − h ) α +12 α ] α +1 − [ I − ( τ λ − h ) α +12 α ] αα +1 τ g µν (3.31)and so e T = 3 p − ρ = − I − τ λ − h ] α +12 α [ I − ( τ λ − h ) α +12 α ] α +1 . (3.32)Then we have by (1.10) and (3.31)-(3.32) ∇ µ e T µν = ∇ µ [ τ λ − h ] − α α τ λ ∂ µ Ψ ∂ ν Ψ[ I − ( τ λ − h ) α +12 α ] α +1 − [ I − ( τ λ − h ) α +12 α ] αα +1 τ g µν ! = ∇ µ ( τ λ − h ) − α α τ λ ∂ µ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 ! ∂ ν Ψ + ( λ − τ λ − h ) − α α τ λ − hg ν [ I − ( τ λ − h ) α +12 α ] α +1 − [ I − ( τ λ − h ) α +12 α ] αα +1 τ g ν = 6 τ [ τ λ − h ] − α α τ λ ∂ τ Ψ ∂ ν Ψ[ I − ( τ λ − h ) α +12 α ] α +1 − [ I − ( τ λ − h ) α +12 α ] αα +1 τ g ν ! + τ g ν I − τ λ − h ] α +12 α [ I − ( τ λ − h ) α +12 α ] α +1 . (3.33)From (3.33), we get easily ∇ µ ( τ λ − h ) − α α τ λ ∂ µ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 ! ∂ ν Ψ= 6 τ [ τ λ − h ] − α α τ λ ∂ τ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 ! ∂ ν Ψ + τ g ν (2 − λ )( τ λ − h ) α +12 α [ I − ( τ λ − h ) α +12 α ] α +1 . (3.34)Contracting (3.34) with ∂ ν Ψ and using the fact h = − ∂ ν Ψ ∂ ν Ψ ≥
0, we have ∇ µ ( τ λ − h ) − α α τ λ ∂ µ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 ! = 6 τ [ τ λ − h ] − α α τ λ ∂ τ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 ! − (2 − λ ) τ ( τ λ − h ) − α α τ λ ∂ τ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 = 4 + λτ ( τ λ − h ) − α α τ λ ∂ τ Ψ[ I − ( τ λ − h ) α +12 α ] α +1 (3.35)Expanding (3.35) directly gives[ I − ( τ λ − h ) α α ] (cid:18) (cid:3) g Ψ − − αα ∂ µ Ψ ∂ ν Ψ h ∇ µ ∇ ν Ψ (cid:19) − α ( τ λ − h ) − α α τ λ − ∂ µ Ψ ∂ ν Ψ ∇ µ ∇ ν Ψ + 1 α ( λ − τ λ − h ) − α α τ λ − h = 4 − λ − λ − − α α τ ∂ τ Ψ[ I − ( τ λ − h ) α α ] . (3.36)We choose λ such that 4 − λ − λ −
1) 1 − α α = 0 , and thus λ = 3 α + 1 . emark 3.3. If λ = 3 α + 1, then e ∂ Ψ has the same asymptotic behavior as the background solutionΨ( τ ) when τ →
0. This property plays very important roles in the non-degeneracy of above waveequation.Define a function Θ( τ, x ) such that e ∂ µ Ψ − e ∂ µ Ψ = τ α +1 ∂ µ Θ . (3.37)Then by (3.29) e ∂ τ Ψ = τ α +1 ∂ τ Ψ = τ α +1 ∂ τ Θ + ( IK ) αα +1 w (1 + Kτ α +1) ) αα +1 ! := τ α +1 ( ∂ τ Θ + f ( τ )) . (3.38)We have e ∂ i Ψ = τ α +1 ∂ i Θ . (3.39)Based on (3.37)-(3.39), we have h = − g ( ∂ τ Ψ) − g i ∂ τ Ψ ∂ i Ψ − g ij ∂ i Ψ ∂ j Ψ= − g f ( τ ) − g µ f ( τ ) ∂ µ Θ − g ( ∂ τ Θ) − g i ∂ i Θ ∂ τ Θ − g ij ∂ i Θ ∂ j Θ . (3.40) Remark 3.4.
It is obvious that ( e η, e ∂ τ Ψ) defined in Section 2.3 is a solution to (3.36), which meansthat ( η, τ α +1 f ( τ )) is the solution to (3.36).Define B = ( I − ( τ λ − h ) α α )( g − − αα ∂ τ Ψ ∂ τ Ψ h ) − α ( τ λ − h ) − α α τ λ − ∂ τ Ψ ∂ τ Ψ ,B i = ( I − ( τ λ − h ) α α )( g i − − αα ∂ τ Ψ ∂ i Ψ h ) − α ( τ λ − h ) − α α τ λ − ∂ τ Ψ ∂ i Ψ ,B jk = ( I − ( τ λ − h ) α α )( g jk − − αα ∂ j Ψ ∂ k Ψ h ) − α ( τ λ − h ) − α α τ λ − ∂ j Ψ ∂ k Ψand similarly B = ( I − ( τ λ − h ) α α )( η − − αα ∂ τ Ψ ∂ τ Ψ h ) − α ( τ λ − h ) − α α τ λ − ∂ τ Ψ ∂ τ Ψ ,B i = ( I − ( τ λ − h ) α α )( η i − − αα ∂ τ Ψ ∂ i Ψ h ) − α ( τ λ − h ) − α α τ λ − ∂ τ Ψ ∂ i Ψ ,B jk = ( I − ( τ λ − h ) α α )( η jk − − αα ∂ j Ψ ∂ k Ψ h ) − α ( τ λ − h ) − α α τ λ − ∂ j Ψ ∂ k Ψ , where h = − η ∂ τ Ψ ∂ τ Ψ.Define p µ = ∂ µ Θ = g ν ∂ ν Θ . (3.41)Expanding (3.36) in terms of Θ, we have B ∂ τ ( ∂ τ Θ) + B ∂ τ ( g f ( τ )) + 2 B i ∂ i ∂ τ Θ + 2 B i ∂ i ( g f ( τ ))+ B ij ∂ i ∂ j Θ + B ij ∂ i ( g j f ( τ )) + T = 0 (3.42)and T = (cid:20) − Γ κ ( ∂ κ Θ + g κ f ( τ )) + 1 − αα ∂ µ Ψ ∂ ν Ψ h Γ κµν ( ∂ κ Θ + g κ f ( τ )) (cid:21) [ I − ( τ λ − h ) α α ]18 1 α ( τ λ − h ) − α α [ τ λ − ∂ µ Ψ ∂ ν ΨΓ κµν ( ∂ κ Θ + g κ f ( τ )) + ( λ − τ λ − h ] . From (3.42), we get B ∂ τ p + 2 B i ∂ i p + B ij ∂ i p j = ˆ T − ˆ T , (3.43)where ˆ T = − T − B ∂ τ ( g f ( τ )) − B i ∂ i ( g f ( τ )) − B ij ∂ i ( g j f ( τ ))= (cid:20) Γ κ ( ∂ κ Θ + g κ f ( τ )) − − αα ∂ µ Ψ ∂ ν Ψ h Γ κµν ( ∂ κ Θ + g κ f ( τ )) (cid:21) [ I − ( τ λ − h ) α α ] − α ( τ λ − h ) − α α [ τ λ − ∂ µ Ψ ∂ ν ΨΓ κµν ( ∂ κ Θ + g κ f ( τ )) + ( λ − τ λ − h ] − B ∂ τ ( g f ( τ )) − B i ∂ i ( g f ( τ )) − B ij ∂ i ( g j f ( τ )) . (3.44)When we use the background metric η instead of g and Ψ instead of Ψ to calculate ˆ T , we get ˆ T .Clearly ˆ T = 0 since ( η, Ψ) is the solution to the fluid equation. From (3.36), (3.43) and (3.44), we caneasily rewrite the fluid equation into the following symmetric hyperbolic system B κ ∂ κ p p p p = ˆ T − ˆ T , (3.45)where B = B − B − B − B − B − B − B − B − B − B , B i = B i B i B i B i B i B i B i . In order to analyze the structure of the symmetric hyperbolic system, in this section, we mainly focuson the source terms of above two subsections, especially the terms ˆ M µ , ˆ M ij , ´ M ij , ˆ R q and ˆ T − ˆ T .We need the following basic lemmas. At first we have the following algebraic relationship between g − and g . Lemma 3.5.
Assume g − = ( g µν ) is a symmetric (1 + 3) × (1 + 3) Lorentz metric with g < g ij ) positive definite, then g = 1 g − d , (3.46) g i = g ij g j d − g , (3.47)where d = g ij g i g j . Proof.
The proof can be found in Lemmas 1 and 2 of [18].The following two lemmas will be repeatedly used in this section.
Lemma 3.6.
Suppose that a i ( i = 1 , · · · , n ) and a i ( i = 1 , · · · , n ) are smooth functions, then we have n Y i =1 a i − n Y i =1 a i = n X j =1 F j ( a i , a i − a i ) , (3.48)19ere F j ( a i , a i − a i ) = j Y k =1 n Y l = j +1 ( a i k − a i k ) a i l , where i k ∈ (1 , · · · , n ). Proof.
We can get this result by induction. At first, for i = 1, it holds obviously. Assume that (3.48)holds for i = n −
1, namely n − Y i =1 a i − n − Y i =1 a i = n − X j =1 F j ( a i , a i − a i ) , then for i = n , we have n Y i =1 a i − n Y i =1 a i = a n n − Y i =1 a i − a n n − Y i =1 a i = ( a n − a n )( n − Y i =1 a i − n − Y i =1 a i ) + a n ( n − Y i =1 a i − n − Y i =1 a i ) + ( a n − a n ) n − Y i =1 a i = ( a n − a n )( n − X j =1 F j ( a i , a i − a i )) + a n ( n − X j =1 F j ( a i , a i − a i )) + ( a n − a n ) n − Y i =1 a i = n X j =1 F j ( a i , a i − a i ) . The proof of the following result is straighforward
Lemma 3.7.
Let f ( x ) be analytical in the neighborhood of a point x , and assume that f ′ ( x ) = 0,then there exists a small parameter δ , such that when x ∈ [ x − δ, x + δ ], we have f ( x ) − f ( x ) ∼ f ′ ( x )( x − x ) . (3.49)Based on Lemmas 3.5-3.7, we have the following important estimates in terms of the unknowns. Lemma 3.8.
In terms of the unknowns ( u µ , u µν , u ij , u ijµ , q , q µ , p µ ), we have g ij − η ij ∼ u ij + η ij w (2 τ u − q ) ,∂ λ ( g ij − η ij ) ∼ u ijλ + 2 ˙ wη ij δ λ τ w (2 τ u − q )+ η ij w (2 u δ λ + 2 τ u i δ iλ + ( u + u ) δ λ − q λ ) ,g ij − η ij ∼ X k =1 F k ( η ij , g ij − η ij ) + ( η ij ) ∗ τ u − q ) w ,g − η ∼ − τ u + P k =1 F k ( η, η − , ( g ij − η ij ) , ( g i − η i ))( η ) ,g i − η i ∼ P k =1 F k ( η, η − , ( g ij − η ij ) , ( g i − η i ) , ( g − η ))( η ) ,∂ τ Ψ − ∂ τ Ψ = 2 τ u ν p ν + η ν p ν , i Ψ − ∂ i Ψ ∼ ( g iν − η iν ) p ν + η iν p ν ,h − h = X k =1 F k ( η, ∂ µ Ψ , ( g µν − η µν ) , ( ∂ µ Ψ − ∂ µ Ψ)) , provided that k ( u µ , u µν , u ij , u ijµ , q , q µ , p µ ) k L ∞ ( T ) is sufficiently small. In above, ( η ij ) ∗ denotes thecofactor of η ij of the positive definite matrix ( η ij ). Proof.
At first, from the definition of g ij and q , i.e. (3.9)-(3.10), we have det ( g ij ) = e τ u − q ) w , (3.50)and g ij = e τ u − q w g ij . (3.51)Then we have by (3.51) g ij − η ij = e τ u − q w g ij − η ij = e τ u − q w ( g ij − η ij ) + η ij ( e τ u − q w − ∼ u ij + η ij τ u − q w . And we have ∂ λ ( g ij − η ij ) = ∂ λ ( e τ u − q w g ij − η ij ) ∼ u ijλ + 2 ˙ wη ij δ λ τ w (2 τ u − q ) + η ij w (2 u δ λ + 2 τ ∂ λ u − q λ ) . Since ∂ τ u = u + u τ and ∂ i u = u i we obtain ∂ λ ( g ij − η ij ) ∼ u ijλ + 2 ˙ wη ij δ λ τ w (2 τ u − q )+ η ij w (2 u δ λ + 2 τ u i δ iλ + ( u + u ) δ λ − q λ ) . According to (3.50) and (3.51), we have g ij = ( g ij ) ∗ det ( g ij ) = ( g ij ) ∗ e τ u − q ) w . (3.52)From (3.52), we easily see g ij − η ij = ( g ij ) ∗ e τ u − q ) w − ( η ij ) ∗ = ( g ij ) ∗ − ( η ij ) ∗ + ( η ij ) ∗ (1 − e τ u − q ) w ) e τ u − q ) w ∼ X k =1 F k ( η, ( g ij − η ij )) + ( η ij ) ∗ (cid:18) τ u − q ) w (cid:19) . In the second equality, we have used Lemma 3.5, since ( g ij ) ∗ = ( g − ) × ( g − ).21efore we estimating g µ − η µ , we have to estimate d , obviously, d = 0, thus d − d = g ij g i g j − η ij η i η j = X k =1 F k ( η, η − , ( g ij − η ij ) , ( g i − η i )) . (3.53)Utilizing (3.53) and Lemma 3.5, we have g − η = 1 g − d − η − d = η − g − d + d ( g − d )( η − d ) ∼ − τ u + P k =1 F k ( η, η − , ( g ij − η ij ) , ( g i − η i ))( g − η + η − d )( η − d ) ∼ − τ u + P k =1 F k ( η, η − , ( g ij − η ij ) , ( g i − η i ))( η ) and g i − η i = g ij g j d − g − η ij η j d − η = g η ij η j − η g ij g j + g ij g i d − η ij η i d ( d − g )( d − η ) ∼ P k =1 F k ( η, η − , ( g − η ) , ( g ij − η ij ) , ( g j − η j ))( η ) . For the fluid variables, we have the following ∂ τ Ψ − ∂ τ Ψ = ∂ τ Θ = ( g ν − η ν ) ∂ ν Θ + η ν ∂ ν Θ = 2 τ u p ν + η ν p ν .∂ i Ψ − ∂ i Ψ = ( g iν − η iν ) ∂ ν Θ + η iν ∂ ν Θ = ( g iν − η iν ) p ν + η iν p ν , and h − h = − g µν ∂ µ Ψ ∂ ν Ψ + η µν ∂ µ Ψ ∂ ν Ψ = X k =1 F k ( η, ∂ µ Ψ , ( g µν − η µν ) , ( ∂ µ Ψ − ∂ µ Ψ)) . Remark 3.9.
Above lemma shows that when k ( u µ , u µν , u ij , u ijµ , q , q µ , p µ ) k L ∞ ( T ) is sufficientlysmall, the differences between the unknowns and the background solutions are small and dependonly on the unknowns and τ . Thus, for convenience of analysis, we denote the differences by H ( τ, u ) := { f ( τ, u ) | f ( τ,
0) = 0 } with u = ( u µ , u µν , u ij , u ijµ , q , q µ , p µ ) and f ( τ, u ) denote smooth functions, which are regular with τ ∈ [0 , I − ( e h ) α +12 α ( I − ( e h ) α +12 α ) α +1 − I − ( e h ) α +12 α ( I − ( e h ) α +12 α ) α +1 (3.54)22nd ( e h ) − α α e ∂ µ Ψ e ∂ µ Ψ( I − ( e h ) α +12 α ) α +1 − ( e h ) − α α e ∂ µ Ψ e ∂ µ Ψ( I − ( e h ) α +12 α ) α +1 . (3.55)Based on Lemmas 3.5-3.8, we estimate (3.54)-(3.55) as follows. Lemma 3.10.
Assume that k u k L ∞ ( T ) is sufficiently small, we have I − ( e h ) α +12 α ( I − ( e h ) α +12 α ) α +1 − I − ( e h ) α +12 α ( I − ( e h ) α +12 α ) α +1 ∼ τ α +1) H ( τ, u )and ( e h ) − α α e ∂ µ Ψ e ∂ µ Ψ( I − ( e h ) α +12 α ) α +1 − ( e h ) − α α e ∂ µ Ψ e ∂ µ Ψ( I − ( e h ) α +12 α ) α +1 ∼ ( τ α +5 + τ α +8 ) H ( τ, u ) . Proof.
From (3.29), (3.30) and λ = 3 α + 1, we see that e h = τ α h, e h = τ α h and e ∂ µ Ψ = τ α +1 ∂ µ Ψ , e ∂ µ Ψ = τ α +1 ∂ µ Ψ . Then direct calculations give I − ( e h ) α +12 α ( I − ( e h ) α +12 α ) α +1 − I − ( e h ) α +12 α ( I − ( e h ) α +12 α ) α +1 = − τ α +1) [ h α +12 α − h α +12 α ]( I − ( τ α h ) α +12 α ) α +1 + ( I − τ α +1) h α +12 α ) × ( I − τ α +1) h α +12 α ) α +1 − [ I − τ α +1) h α +12 α + τ α +1) ( h α +12 α − h α +12 α )] α +1 [( I − τ α +1) h α +12 α )( I − τ α +1) h α +12 α )] α +1 ∼ − τ α +1) [ h α +12 α − h α +12 α ]( I − ( τ α h ) α +12 α ) α +1 + ( I − τ α +1) h α +12 α ) τ α +1) α +1 ( h α +12 α − h α +12 α )[( I − τ α +1) h α +12 α )( I − τ α +1) h α +12 α )] α +1 ∼ τ α +1) H ( τ, u ) . In the fifth line, we have used Lemma 3.7 with f ( x ) = x α +1 and in the last line we have used f ( x ) = x α +12 α and Lemma 3.8.For (3.55), we have( e h ) − α α e ∂ µ Ψ e ∂ ν Ψ( I − ( e h ) α +12 α ) α +1 − ( e h ) − α α e ∂ µ Ψ e ∂ ν Ψ( I − ( e h ) α +12 α ) α +1 = τ α +5 [ h − α α ∂ µ Ψ ∂ ν Ψ − h − α α ∂ µ Ψ ∂ ν Ψ]( I − τ α +1) h α +12 α ) α +1 + τ α +5 h − α α ∂ µ Ψ ∂ ν Ψ[( I − τ α +1) h α +12 α ) α +1 − ( I − τ α +1) h α +12 α ) α +1 ][( I − τ α +1) h α +12 α )( I − τ α +1) h α +12 α )] α +1 ∼ τ α +5 [ h − α α ∂ µ Ψ ∂ ν Ψ − h − α α ∂ µ Ψ ∂ ν Ψ]( I − τ α +1) h α +12 α ) α +1 τ α +8 h − α α ∂ µ Ψ ∂ ν Ψ[ h α α − h α α ]( α + 1)[( I − τ α +1) h α +12 α )( I − τ α +1) h α +12 α )] α +1 ∼ ( τ α +5 + τ α +8 ) H ( τ, u ) . In the fifth line, we have used Lemma 3.7 with f ( x ) = x α +1 and in the last line we have used Lemma3.6 and 3.8.Since Q µν ( g, ∂g ) is quadratic in ( ∂g µν ) and analytical in ( g µν ), by Lemmas 3.6 and 3.8, we easilyget the following statement. Lemma 3.11.
We have Q µν ( g, ∂g ) − Q µν ( η, ∂η ) ∼ H ( τ, u ) . The other terms in ˆ M µν , ´ M ij and ˆ R q can be easily expressed by the unknowns u , we neglect thedetailed analysis for those terms but give the lemma to conclude the analysis of these source terms. Lemma 3.12.
Assume that k u k L ∞ ( T ) is sufficiently small, then we have the following equivalentrelationship.ˆ M µ ∼ τ u ∂ τ w δ µ − wτ u δ µ − ˙ wτ ( u µ + 3 u µ ) − wτ u i δ ( µ δ i − wτ u µ − ∂ τ ( w + ˙ w u µ δ ν + u ν δ µ )+(2 τ α +1 − τ α +4 ) A α +1 I − αα +1 H ( τ, u ) + H ( τ, u ) ∼ H ( τ, u ) , ˆ M ij ∼ − ˙ wτ ∂ τ ( g ij − η ij ) − ˙ wτ ( g ij − η ij )+(2 τ α +1 − τ α +4 ) A α +1 I − αα +1 H ( τ, u ) + H ( τ, u ) ∼ H ( τ, u ) , ´ M ij = ( det (ˇ g ab )) L ijlm ˆ M lm − g κλ ∂ κ (( det (ˇ g ab )) L ijlm ) ∂ λ ( g lm − η lm ) ∼ H ( τ, u ) , ˆ R q ∼ ˆ M − w g pq ˆ M pq + 4 w∂ τ wτ (2 τ u − q )+ g κλ ( − w ∂ κ g pq ∂ λ ( g pq − η pq ) − w∂ τ wδ κ g pq ∂ λ ( g pq − η pq ) − w∂ τ w δ λ g pq ∂ κ ( g pq − η pq )) + g κλ (cid:18) ∂ τ w ) + 2 w∂ τ ww δ λ δ κ (2 τ u − q ) (cid:19) ∼ H ( τ, u ) . Remark 3.13.
The regularity of above source terms with respect to τ can be derived directly fromCorollary 2.1.Now what remains is to estimate the source term of the fluid equation ˆ T − ˆ T . Lemma 3.14.
Suppose that k u k L ∞ ( T ) is sufficiently small, we haveˆ T − ˆ T ∼ H ( τ, u ) . Proof.
As before, this can be derived by checking the following termsΓ κ ( g κ f ( τ )) − Γ κ ( η κ f ( τ )) , α +2 ( h α α − h α α ) ,B ∂ τ ( g f ( τ )) − B ∂ τ ( η f ( τ )) ,B i ∂ i ( g f ( τ )) − B i ∂ i ( η f ( τ )) ,B ij ∂ i ( g j f ( τ )) − B ij ∂ i ( η j f ( τ )) ,∂ µ Ψ ∂ ν Ψ h Γ κµν ( g κ f ( τ )) − ∂ µ Ψ ∂ ν Ψ h Γ κµν ( η κ f ( τ )) ,τ α +1) [ h − α α ∂ µ Ψ ∂ ν ΨΓ κµν ( g κ f ( τ )) − h − α α ∂ µ Ψ ∂ ν ΨΓ κµν ( η κ f ( τ )] . Above seven terms are easy to analyze based on Lemmas 3.6-3.8. We need to analyze the regularityof the following term with respect to τ∂ τ ( f ( τ )) = ∂ τ ( IK ) αα +1 w ( I + Kτ α +1) ) αα +1 ! = − ( IK ) αα +1 ˙ wτ w ( I + Kτ α +1) ) αα +1 − ( IK ) αα +1 αKwτ α +2 ( I + Kτ α +1) ) α +1 α +1 . According to Corollary 2.1, this term is regular when τ → Proposition 3.15.
Under the wave coordinates Z µ = 0, the whole system (1.10) is equivalent to thefollowing symmetric hyperbolic system B µ ∂ µ u = 1 τ BPu + H ( τ, u ) [1 , × T , u = u in 1 × T , (3.56)where B µ , B and P are defined by (3.26)-(3.28), (3.45) and satisfy the constraints of the generalsymmetric hyperbolic system discussed in Section 3.5. Consider the following symmetric hyperbolic system. B µ ∂ µ u = 1 t BP u + H in [ T , T ] × T n , (3.57) u = u in T × T n , (3.58)where(i) T < T ≤ P is a constant, symmetric projection operator, i.e., P = P , P T = P ,(iii) u = u ( t, x ) and H ( t, u ) are R N -valued maps, H ∈ C ([ T , , C ∞ ( R N )) and satisfies H ( t,
0) =0, (iv) B µ = B µ ( t, u ) and B = B ( t, u ) are M N × N -valued maps, and B µ , B ∈ C ([ T , , C ∞ ( R N ))and they satisfy ( B µ ) T = B µ , [ P , B ] = PB − BP = 0 , (v) there exists constants κ, γ , γ such that1 γ I ≤ B ≤ κ B ≤ γ I for all ( t, u ) ∈ [ T , × R N ,(vi) for all ( t, u ) ∈ [ T , × R N , we have P ⊥ B P = P B P ⊥ = 0 , P ⊥ = I − P is the orthogonal projection operator.We will be able to conclude our argument with tthe help of the following result, whose proof relieson standard energy estimates; see [15]. Proposition 3.16.
Suppose that k ≥ n + 1, u ∈ H k ( T n ) and assumptions (i)-(vi) are fulfilled.Then there exists a T ∗ ∈ ( T , u ∈ C ([ T , T ∗ ] × T n ) that satisfies u ∈ C ([ T , T ∗ ] , H k ) ∩ C ([ T , T ∗ ] , H k − ) and the energy estimate k u ( t ) k H k − Z tT τ k P u k H k ≤ Ce C ( t − T ) ( k u ( T ) k H k )for all T ≤ t < T ∗ , where C = C ( k u k L ∞ ([ T ,T ∗ ) ,H k ) , γ , γ , κ ) , and can be uniquely continued to alarger time interval [ T , T ∗ ) for all T ∗ ∈ ( T ∗ ,
0] provided k u k L ∞ ([ T ,T ∗ ) ,W , ∞ ) < ∞ .Moreover, there exists a δ > k u k H k ≤ δ , then the solution exists on the time interval[ T ,
0) and can be uniquely extended to [ T ,
0] as an element of C ([ T , , H k − ) satisfying k P u ( τ ) k H k − ≤ Cδ − t if κ > ,t ln( tT ) if κ = 1 , ( − t ) k if κ < k P ⊥ u ( τ ) − P ⊥ u (0) k H k − ≤ Cδ (cid:26) − t if κ ≥ B , P ] = 0 − t + ( − t ) κ if κ < T ≤ t ≤ Based on the analysis of subsections 3.1-3.5, we have transformed the Einstein-Euler equations of GCGinto a symmetric hyperbolic system (3.56). After a simple coordinate transformation τ → − τ , thederived system is just the one considered in Section 3.5, thus we can use the main result of the Section3.5 to get Theorems 1.2 and 1.4 directly under the assumption that the matrix − B of subsection3.3 is positive definite. Thus, we need to prove that − B is positive definite based on the a prioriestimates of Proposition 3.16.Obviously, from the smallness of the unknowns of Proposition 3.16, we have the following equivalentrelationships h = − g µν ∂ µ Ψ ∂ ν Ψ ∼ − g f ( τ ) ∼ h, (3.59) ∂ τ Ψ = g µ ∂ µ Ψ ∼ ∂ τ Ψ ∼ f ( τ ) , (3.60) ∂ i Ψ = g iµ ∂ µ Ψ ∼ Cǫ. (3.61)Thus, we have − B ∼ − ( I − ( τ λ − h ) α α )(1 + 1 − αα ) η + 1 α ( τ λ − h ) − α α τ λ − ( f ( τ )) > ,B jk ∼ ( I − ( τ λ − h ) α α ) (cid:18) δ jk − − αα C ǫ − g f ( τ ) (cid:19) − α ( τ λ − h ) − α α τ λ − C ǫ ∼ ( I − ( τ λ − h ) αα ) δ jk . Clearly, from (3.59)-(3.61) and Remarks 2.1, 2.3, − B is a positive definite matrix. Hence, all theassumptions of Section 3.5 are satisfied. Then we have by Proposition 3.16 k u ( τ ) k H k ≤ Cǫ.
26y Lemma 3.8, we can equivalently get k g µν ( τ ) − η µν ( τ ) k H k +1 + k ∂ κ g µν ( τ ) − ∂ κ η µν ( τ ) k H k + k ∂ µ Ψ( τ ) − ∂ µ Ψ( τ ) k H k ≤ Cǫ. (3.62)Moreover, based on (3.56), we have the following asymptotic behavior: k P ( u µ ( τ ) , u µi ( τ ) , u µ ( τ )) T k H k − ≤ − Cǫτ ln ( τ ) , (3.63) k Π( u ij ( τ ) , u ijk ( τ ) , u ij ( τ )) T , Π( u ( τ ) , u k ( τ ) , u ( τ )) T k H k − ≤ Cǫτ. (3.64)And k ( I − P )( u µ , u µi , u µ ) T ( τ ) − ( I − P )( u µ (0) , u µi (0) , u µ (0)) T k H k − ≤ Cǫτ, (3.65) k ( I − Π)( u ij , u ijk , u ij ) T ( τ ) − ( I − Π)( u ij (0) , u ijk (0) , u ij (0)) T k H k − ≤ Cǫτ, (3.66) k ( I − Π)( u , u k , u ) T ( τ ) − ( I − Π)( u (0) , u k (0) , u (0)) T k H k − ≤ Cǫτ, (3.67) k p µ − p µ (0) k H k − ≤ Cǫτ. (3.68)Where in (3.63) and (3.64), we have used κ = 1 and κ = 2 respectively. From (3.62)-(3.68), we have k ∂ τ ( g µ ( τ ) − η µ ( τ )) − τ ( g µ ( τ ) − η µ ( τ )) k H k − + k ∂ i g µ ( τ ) k H k − ≤ − Cǫτ ln ( τ ) , k ∂ τ q ( τ ) k H k − + k ∂ τ g ij ( τ ) k H k − ≤ Cǫτ, k ∂ τ ( g µ ( τ ) − η µ ( τ )) − τ ( g µ ( τ ) − η µ ( τ )) + γ µ k H k − ≤ Cǫτ, k ∂ i q ( τ ) − ∂ i q (0) k H k − + k q ( τ ) − q (0) k H k − + k ∂ i g ij ( τ ) − ∂ i g ij (0) k H k − + k g ij ( τ ) − g ij (0) k H k − ≤ Cǫτ, k ∂ µ Ψ( τ ) − ∂ µ Ψ(0) k H k − ≤ Cǫτ, k ∂ µ Ψ(0) − ∂ µ Ψ(0) k H k − ≤ Cǫ, k ∂ i Ψ(0) k H k − ≤ Cǫ, where we have set γ µ = lim τ → − [ ∂ τ ( g µ ( τ ) − η µ ( τ )) − τ ( g µ ( τ ) − η µ ( τ ))] = lim τ → ( u µ ( τ ) − u µ ( τ )) . On the other hand, we have ρ ∼ A α +1 I α +1 (cid:16) I − τ α +1) ( − η f ( τ )) α +12 α (cid:17) α +1 . Thus, when τ →
0, we have ρ → A α +1 and p ρ = αA ρ α +1 → α. The proof of Theorem 1.4 is completed.
Acknowledgement
This paper was written in the Fall 2015 at the Institut Henri Poincar´e (IHP, Paris) during the TrimesterProgram “Mathematical General Relativity”, which was organized by L. Andersson, S. Klainerman,and P.G. LeFloch. The authors would like to thank Chao Liu for helpful discussions on the subject ofthis paper. The first author (PGLF) gratefully acknowledges support from the Agence Nationale de laRecherche through grant ANR SIMI-1-003-01. The second author (CW) is grateful to the Fondationdes Sciences Math´ematiques de Paris (FSMP) for financial support during his stay at IHP.27 eferences [1] H. B. Benaoum,
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