Ann. Henri Poincar´e 19 (2018), 2157–2243 c 2018 Springer International Publishing AG,  part of Springer Nature 1424-0637/18/072157-87 published online June 4, 2018 https://doi.org/10.1007/s00023-018-0686-2

Annales Henri Poincar´ e

Newtonian Limits of Isolated Cosmological Systems on Long Time Scales Chao Liu and Todd A. Oliynyk Abstract. We establish the existence of 1-parameter families of -dependent solutions to the Einstein–Euler equations with a positive cosmological constant Λ > 0 and a linear equation of state p = 2 Kρ, 0 < K ≤ 1/3, for the parameter values 0 <  < 0 . These solutions exist globally to the future, converge as   0 to solutions of the cosmological Poisson– Euler equations of Newtonian gravity, and are inhomogeneous nonlinear perturbations of FLRW fluid solutions.

1. Introduction Gravitating relativistic perfect fluids are governed by the Einstein–Euler equations. The dimensionless version of these equations with a cosmological constant Λ is given by ˜ μν + Λ˜ g μν = T˜μν , G ˜ μ T˜μν = 0, ∇

(1.1) (1.2)

˜ μν is the Einstein tensor of the metric where G xμ d¯ xν , g˜ = g˜μν d¯ and T˜μν = (¯ ρ + p¯)˜ v μ v˜ν + p¯g˜μν is the perfect fluid stress–energy tensor. Here, ρ¯ and p¯ denote the fluid’s proper energy density and pressure, respectively, while v˜ν is the fluid four-velocity, which we assume is normalized by v˜μ v˜μ = −1.

(1.3)

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In this article, we assume a positive cosmological constant Λ > 0 and restrict our attention to barotropic fluids with a linear equation of state of the form 1 0
(1.4) (1.5) (1.6)

where ρH (1) is the initial density (freely specifiable) and a(t) satisfies   3 Λ ρH (t)  − ta (t) = a(t) + , a(1) = 1. (1.7) Λ 3 3 Throughout this article, we will refer to the global coordinates (¯ xμ ) on M as relativistic coordinates. In order to discuss the Newtonian limit and the 1

n Here, Tn  = [0, ] / ∼ where ∼ is equivalence relation that follows from the identification of the sides of the box [0, ]n . When  = 1, we will simply write Tn .

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sense in which solutions converge as   0, we need to introduce the spatially rescaled coordinates (xμ ) defined by t=x ¯0 = x0

and x ¯i = xi ,

 > 0,

(1.8)

which we refer to as Newtonian coordinates. We note that the Newtonian coordinates define a global coordinate system on the -independent manifold M := M1 = (0, 1] × T3 . Remark 1.1. Due to our choice of time coordinate t on (0, 1], the future lies in the direction of decreasing t and timelike infinity is located at t = 0. Remark 1.2. The nonstandard form of the FLRW solution and the -dependence in the manifold M is a consequence of our starting point for the Newtonian limit, which differs from the standard formulation in that the time interval has been compactified from [0, ∞) to (0, 1] and the light cones of the metric (1.4) do not flatten as   0. For comparison, we observe that the standard formulation can be obtained by first switching to Newtonian coordinates, which removes the -dependence from the spacetime manifold, followed by the introduction of a new time coordinate according to √Λ (1.9) t = e− 3 τ , which undoes the compactification of the time interval. These new coordinates define a global coordinate system on the -independent manifold [0, ∞) × T3 on which the FLRW metric can be expressed as

where a ˆ(τ ) = a(e−

√Λ 3

ˆ = −dτ dτ + 2 a ˆ(τ )δij dxi dxj h τ

). Dividing through by 2 yields the metric

ˆ  = − 1 dτ dτ + a h ˆ(τ )δij dxi dxj , 2 which is now in the standard form for taking the Newtonian limit. In particular, we observe that the light cones of this metric flatten out as   0. Remark 1.3. Throughout this article, we take the homogeneous initial density ρH (1) to be independent of . All of the results established in this article remain true if ρH (1) is allowed to depend on  in a C 1 manner, that is the map [0, 0 )   −→ ρH (1) ∈ R>0 is C 1 for some 0 > 0. Remark 1.4. As we show in Sect. 2.1, FLRW solutions {a, ρH } depend regularly on  and have well-defined Newtonian limits. Letting ρH = lim ρH ˚ a = lim a and ˚ 0

0

(1.10)

denote the Newtonian limit of a and ρH , respectively, it then follows from (1.6) and (1.7) that {˚ a, ˚ ρH } satisfy ˚ ρH =

˚ ρH (1) ˚ a(t)3

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  ρH (t) 3 Λ ˚ + , ˚ a(1) = 1, a(t) −t˚ a (t) = ˚ Λ 3 3 which define the Newtonian limit of the FLRW equations.

and



In the articles [47,48], the second author established the existence of 1parameter families of solutions2 {˜ gμν , ρ¯ , v˜μ }, 0 <  < 0 , to (1.1)–(1.2), which include the above family of FLRW solutions, on spacetime regions of the form (T1 , 1] × T3 ⊂ M , for some T1 ∈ (0, 1], that converge, in a suitable sense, as   0 to solutions of the cosmological Poisson–Euler equations of Newtonian gravity. Although this result rigorously established the existence of a wide class of solutions to the Einstein–Euler equations that admit a (cosmological) Newtonian limit, the local-in-time nature of the result left open the question of what happens on long time scales. In particular, the question of the existence of 1-parameter families of solutions that converge globally to the future as   0 was not addressed. In light of the importance of Newtonian gravity in cosmological simulations [9,15,62,63], this question needs to be resolved in order to understand on what time scales Newtonian cosmological simulations can be trusted to approximate relativistic cosmologies. In this article, we resolve this question under a small initial data condition. Informally, we establish the existence of 1-parameter families of -dependent solutions to (1.1)–(1.2) that: (i) are defined for  ∈ (0, 0 ) for some fixed constant 0 > 0, (ii) exist globally on M , (iii) converge, in a suitable sense, as   0 to solutions of the cosmological Poisson–Euler equations of Newtonian gravity, and (iv) are small, nonlinear perturbations of the FLRW fluid solutions (1.4)–(1.7). The precise statement of our results can be found in Theorem 1.7. Before proceeding with the statement of Theorem 1.7, we first fix our notation and conventions, and define a number of new variables and equations. 1.1. Notation 1.1.1. Index of Notation. An index containing frequently used definitions and nonstandard notation can be found in “Appendix D.” 1.1.2. Indices and Coordinates. Unless stated otherwise, our indexing convention will be as follows: we use lower case Latin letters, e.g., i, j, k, for spatial indices that run from 1 to n, and lower case Greek letters, e.g., α, β, γ, for spacetime indices that run from 0 to n. When considering the Einstein–Euler equations, we will focus on the physical case where n = 3, while all of the PDE results established in this article hold in arbitrary dimensions. For scalar functions f (t, x ¯i ) of the relativistic coordinates, we let f (t, xi ) := f (t, xi ) 2

(1.11)

To convert the 1-parameter solutions to the Einstein–Euler equations from [47, 48] to solutions of (1.1)–(1.2), the metric, four-velocity, time coordinate and spatial coordinates must each be rescaled by an appropriate powers of , after which the rescaled time coordinate must be transformed according to the formula (1.9).

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denote the representation of f in Newtonian coordinates. 1.1.3. Derivatives. Partial derivatives with respect to the Newtonian coordinates (xμ ) = (t, xi ) and the relativistic coordinates (¯ xμ ) = (t, x ¯i ) will be μ μ ¯ ¯ , respectively, and we use Du = (∂j u) denoted by ∂μ = ∂/∂x and ∂μ = ∂/∂ x and ∂u = (∂μ u) to denote the spatial and spacetime gradients, respectively, ¯ = (∂¯μ u) to denote with respect to the Newtonian coordinates, and similarly ∂u the spacetime gradient with respect to the relativistic coordinates. We also use Greek letters to denote multi-indices, e.g., α = (α1 , α2 , . . . , αn ) ∈ Zn≥0 , and employ the standard notation Dα = ∂1α1 ∂2α2 · · · ∂nαn for spatial partial derivatives. It will be clear from context whether a Greek letter stands for a spacetime coordinate index or a multi-index. Given a vector-valued map f (u), where u is a vector, we use Df and Du f interchangeably to denote the derivative with respect to the vector u, and use the standard notation  d  Df (u) · δu := f (u + tδu) dt t=0 for the action of the linear operator Df on the vector δu. For vector-valued maps f (u, v) of two (or more) variables, we use the notation D1 f and Du f interchangeably for the partial derivative with respect to the first variable, i.e.,  d  f (u + tδu, v), Du f (u, v) · δu := dt  t=0

and a similar notation for the partial derivative with respect to the other variable. 1.1.4. Function Spaces. Given a finite-dimensional vector space V , we let H s (Tn , V ), s ∈ Z≥0 , denote the space of maps from Tn to V with s derivatives in L2 (Tn ). When the vector space V is clear from context, we write H s (Tn ) instead of H s (T, V ). Letting  u, v = (u(x), v(x)) dn x, Tn

where (·, ·) is a fixed inner product on V , denote the standard L2 inner product, the H s norm is defined by  Dα u, Dα u . u2H s = 0≤|α|≤s

For any fixed basis H s (Tn , V ) by ¯ s (Tn , V ) = H

{eI }N I=1

of V , we follow [47] and define a subspace of

   N   u(x) = uI (x)eI ∈ H s (Tn , V )1, uI  = 0 for I = 1, 2, . . . , N . I=1

Specializing to n = 3, we define, for fixed 0 > 0 and r > 0, the spaces ¯ s (T3 )) × H ¯ s (T3 , R3 ), Xs0 ,r (T3 ) = (−0 , 0 ) × Br (H s+1 (T3 , S3 )) × H s (T3 , S3 ) × Br (H

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where SN denotes the space of symmetric N × N matrices, and here and throughout this article, we use, for any Banach space Y , Br (Y ) = { y ∈ Y | yY < r } to denote the open ball of radius r. To handle the smoothness of coefficients that appear in various equations, we introduce the spaces E p ((0, 0 ) × (T1 , T2 ) × U, V ),

p ∈ Z≥0 ,

which are defined to be the set of V -valued maps f (, t, ξ) that are smooth on the open set (0, 0 ) × (T1 , T2 ) × U , where U ⊂ Tn × RN is open, and for which there exist constants Ck, > 0, (k, ) ∈ {0, 1, . . . , p} × Z≥0 , such that |∂t k Dξ f (, t, ξ)| ≤ Ck, ,

∀ (, t, ξ) ∈ (0, 0 ) × (T1 , T2 ) × U.

If V = R or V is clear from context, we will drop the V and simply write E p ((0, 0 )×(T1 , T2 )×U ). Moreover, we will use the notation E p ((T1 , T2 )×U, V ) to denote the subspace of -independent maps. Given f ∈ E p ((0, 0 )×(T1 , T2 )× U, V ), we note, by uniform continuity, that the limit f0 (t, ξ) := lim0 f (, t, ξ) exists and defines an element of E p ((T1 , T2 ) × U, V ). 1.1.5. Constants. We employ that standard notation ab for inequalities of the form a ≤ Cb in situations where the precise value or dependence on other quantities of the constant C is not required. On the other hand, when the dependence of the constant on other inequalities needs to be specified, for example if the constant depends on the norms uL∞ and vL∞ , we use the notation C = C(uL∞ , vL∞ ). Constants of this type will always be nonnegative, non-decreasing, continuous functions of their arguments, and in general, C will be used to denote constants that may change from line to line. However, when we want to isolate a particular constant for use later on, we will label the constant with a subscript, e.g., C1 , C2 , C3 , etc. 1.1.6. Remainder Terms. In order to simplify the handling of remainder terms whose exact form is not important, we will use, unless otherwise stated, upper case calligraphic letters, e.g., S(, t, x, ξ), T (, t, x, ξ) and  U(, t, x, ξ), to denote 0 (0, 0 ) × (0, 2) × Tn × vector-valued maps that are elements of the space E  N , and upper case letters in typewriter font, e.g., S(, t, x, ξ), T(, t, x, ξ) BR R and U(, t, x, ξ), to denote vector-valued maps that are elements of the space  E 1 (0, 0 ) × (0, 2) × Tn × BR RN .

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1.2. Conformal Einstein–Euler Equations The method we use to establish the existence of -dependent families of solutions to the Einstein–Euler equations that exist globally to the future is based on the one developed in [51]. Following [51], we introduce the conformal metric g¯μν = e2Ψ g˜μν

(1.12)

v¯μ = eΨ v˜μ .

(1.13)

and the conformal four velocity

Under this change in variables, the Einstein equation (1.1) transforms as μν ¯ ν Ψ− ∇ ¯ μν = T¯μν := e4Ψ T˜μν −e2Ψ Λ¯ ¯ μ∇ ¯ μ Ψ∇ ¯ ν Ψ)−(2Ψ+| ¯ 2 )¯ ¯ G g μν +2(∇ ∇Ψ| g ¯ g , (1.14) ¯ μ , |∇Ψ| ¯ μ Ψ∇ ¯ μ∇ ¯ 2g¯ = g¯μν ∇ ¯ ν Ψ, and here and in the following, unless ¯ =∇ where  otherwise specified, we raise and lower all coordinate tensor indices using the conformal metric. Contracting the free indices of (1.14) gives

¯ = 4Λ − T, ¯ R where T¯ = g¯μν T¯μν , which we can use to write (1.14) as ¯ ν Ψ + 4∇ ¯ μν = −4∇ ¯ μ∇ ¯ μ Ψ∇ ¯ νΨ −2R



1 − 2 K 2 2Ψ ¯ ¯ ρ¯ + Λ e − 2 Ψ + 2|∇Ψ| + g¯μν 2 − 2e2Ψ (1 + 2 K)¯ ρv¯μ v¯ν .

(1.15)

We will refer to these equations as the conformal Einstein equations. ¯ γμν denote the Christoffel symbols of the metrics g˜μν ˜ γμν and Γ Letting Γ ˜ γμν − Γ ¯ γμν is readily calculated to be and g¯μν , respectively, the difference Γ  ¯ ν Ψ + g¯να ∇ ¯ μ Ψ − g¯μν ∇ ¯ αΨ . ˜ γμν − Γ ¯ γμν = g¯γα g¯μα ∇ Γ Using this, we can express the Euler equations (1.2) as ¯ μ T˜μν = −6T˜μν ∇μ Ψ + g¯αβ T˜αβ g¯μν ∇ ¯ μ Ψ, ∇

(1.16)

which we refer to as the conformal Euler equations. Remark 1.5. Due to our choice of time orientation, the conformal fluid fourvelocity v¯μ , which we assume is future oriented, satisfies v¯0 < 0. We also note that v¯μ is normalized by v¯μ v¯μ = −1, which is a direct consequence of (1.3), (1.12) and (1.13).

(1.17)

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1.3. Conformal Factor Following [51], we choose Ψ = − ln t

(1.18)

for the conformal factor, and for later use, we introduce the background metric ¯ = − 3 dtdt + E 2 (t)δij d¯ h xi d¯ xj , Λ

(1.19)

E(t) = a(t)t,

(1.20)

where

which is conformally related to the FLRW metric (1.4). Using (1.7), we observe that E(t) satisfies ∂t E(t) =

1 E(t)Ω(t), t

(1.21)

where Ω(t) is defined by

  3 Λ ρH (t) + . Ω(t) = 1 − Λ 3 3

(1.22)

A short calculation then shows that the non-vanishing Christoffel symbols of the background metric (1.19) are given by 0 γ¯ij =

Λ 2 E Ωδij 3t

i and γ¯j0 =

1 i Ωδ , t j

(1.23)

from which we compute ¯ μν γ¯ σ = γ¯ σ := h μν

Λ σ Ωδ . t 0

(1.24)

1.4. Wave Gauge For the hyperbolic reduction of the conformal Einstein equations, we use the wave gauge from [51] that is defined by Z¯ μ = 0,

(1.25)

¯ μ + Y¯ μ Z¯ μ = X

(1.26)

where with ¯ μ := Γ ¯ μ − γ¯ μ = −∂¯ν g¯μν + 1 g¯μσ g¯αβ ∂¯σ g¯αβ − Λ Ωδ μ X 2 t 0



¯ μσν ¯ μ = g¯σν Γ Γ



(1.27)

and ¯ μν )∇ ¯ μ Ψ + 2Λ δ μ = −2(¯ ¯ ν Ψ. Y¯ μ := −2∇ g μν − h 3t 0

(1.28)

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1.5. Variables To obtain variables that are simultaneously suitable for establishing global existence and taking Newtonian limits, we switch to Newtonian coordinates (xμ ) = (t, xi ) and employ the following rescaled version of the variables introduced in [51]: 1 g¯0μ − η¯0μ ,   2t  g 0μ − η¯0μ ) 1 ¯ 0μ 3(¯ = ∂0 g¯ − ,  2t

u0μ =

(1.29)

u0μ 0

(1.30)

u0μ i = uij (t, x) = uij μ = u= uμ = zi = ζ=

1 ¯ 0μ ∂i g¯ ,  1  ij ¯ − η¯ij , g  1 ¯ ij ¯ , ∂μ g  1 ¯q,  1¯ ∂μ ¯q,  1 v¯i ,   2 1 ln t−3(1+ K) ρ¯ , 2 1+ K

(1.31) (1.32) (1.33) (1.34) (1.35) (1.36) (1.37)

and δζ = ζ − ζH ,

(1.38)

where ¯ij = α−1 g¯ij , g

1

1

α := (det gˇij )− 3 = (det g¯kl ) 3 , gˇij = (¯ g ij )−1 , 2Λ Λ ¯q = g¯00 − η¯00 − ln α − ln E, 3 3 Λ η¯μν = − δ0μ δ0ν + δiμ δjν δ ij , 3

(1.39) (1.40) (1.41)

and

 2 1 ln t−3(1+ K) ρH (t) . 2 1+ K As we show below in Sect. 2.1, ζH is given by the explicit formula   2 C0 − t3(1+ K) 2 ζH (t) = ζH (1) − ln , 1 + 2 K C0 − 1 ζH (t) =

where the constants C0 and ζH (1) are defined by  √ Λ + ρH (1) + Λ √ >1 C0 =  Λ + ρH (1) − Λ

(1.42)

(1.43)

(1.44)

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1 ln ρH (1), 1 + 2 K

(1.45)

respectively. Letting ˚ ζH = lim ζH

(1.46)

0

denote the Newtonian limit of ζH , it is clear from the formula (1.43) that  C0 − t3 ˚ ζH (t) = ln ρH (1) − 2 ln . (1.47) C0 − 1 For later use, we also define 1 i v¯ . (1.48)  Remark 1.6. It is important to emphasize that the above variables are defined on the -independent manifold M = (0, 1] × T3 . Effectively, we are treating components of the geometric quantities with respect to the relativistic coordinates as scalars defined on M and we are pulling them back as scalars to M by transforming to Newtonian coordinates. This process is necessary to obtain variables that have a well-defined Newtonian limit. We stress that for any fixed  > 0, the gravitational and matter fields {¯ g μν , v¯μ , ρ¯} on M are completely 0μ ij determined by the fields {u , u , u, zi , ζ} on M . zi =

1.6. Conformal Poisson–Euler Equations The   0 limit of the conformal Einstein–Euler equations on M are the conformal cosmological Poisson–Euler equations, which are defined by  Ω) 3  j 3(1 − ˚ ∂j ˚ ˚ ρ, (1.49) ρ˚ z = ρ+ ∂t ˚ Λ t   Λ Λ 1 j 1 3 j˚  j δ ji  ∂ := ˚ ρ∂t˚ ˚ ρ˚ z − ˚ ρ∂ Φ z j + K∂ j ˚ ρ+˚ ρ˚ z i ∂i˚ zj = ∂ , ˚2 i 3 3 t 2Λ E (1.50) ˚2 ΛE Δ˚ Φ= Π˚ ρ 3 t2 where Π is the projection operator defined by

(Δ := δ ij ∂i ∂j ),

Πu = u − 1, u , 2

3

for u ∈ L (T ),

 ˚ = E(t)

and ˚ Ω(t) =

C0 − t3 C0 − 1

(1.51)

(1.52)

23

2t3 , t3 − C0

(1.53)

(1.54)

with C0 given by (1.44). It will be important for our analysis to introduce the modified density variable ˚ ζ defined by ˚ ρ), ζ = ln(t−3˚

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which is the nonrelativistic version of the variable ζ introduced above, see (1.37). A short calculation then shows that the conformal cosmological Poisson– Euler equations can be expressed in terms of this modified density as follows:  3˚ Ω 3 j ˚ ˚ ˚ z ∂j ζ + ∂ j ˚ (1.55) ∂t ζ + zj = − , Λ t   Λ Λ 1 j 1 3 j˚ (1.56) ∂t˚ ˚ z − ∂ Φ, zj + ˚ z i ∂i˚ z j + K∂ j ˚ ζ= 3 3 t 2Λ Λ ˚2 ˚ Δ˚ Φ = tE Πeζ . (1.57) 3 1.7. Main Theorem We are in the position to state the main theorem of the article. The proof is given in Sect. 7. Theorem 1.7. Suppose s ∈ Z≥3 , 0 < K ≤ 13 , Λ > 0, ρH (1) > 0, r > 0 and the ˘ij ˘ij ∈ Br (H s+1 (T3 , S3 )), u ˘ij free initial data {˘ uij , u ˘0 , ν˘i } is chosen so that u 0 ,ρ 0 ∈ ¯ s (T3 )), ν˘i ∈ H ¯ s (T3 , R3 ). Then for r > 0 chosen small H s (T3 , S3 ), ρ˘0 ∈ Br (H  ˘μν ∈ C ω Xs0 ,r (T3 ), H s+1 enough, there exists a constant 0 > 0 and maps u   μν (T3 , S4 ) , u ˘ ∈ C ω Xs0 ,r (T3 ), H s+1 (T3 ) , u ˘0 ∈ C ω Xs0 ,r (T3 ), H s (T3 , S4 ) ,   u ˘0 ∈ C ω Xs0 ,r (T3 ), H s (T3 ) , z˘ = (˘ zi ) ∈ C ω Xs0 ,r (T3 ), H s (T3 , R3 , and δ ζ˘ ∈  C ω Xs0 ,r (T3 ), H s (T3 ) , such that3 Λ ˘kl , u ˘kl uμ0 |t=1 := u ˘μ0 (, u ˘0 , ν˘k ) =  Δ−1 ρ˘0 δ0μ + O(2 ), 0 ,ρ 6 1 pq ij ij kl kl k 2 ij ij ˘ ,u ˘0 , ρ˘0 , ν˘ ) =  u ˘ − u ˘ δpq δ ˘ (, u u |t=1 := u + O(3 ), 3 2Λ ij ˘kl , u ˘kl ˘ δij + O(3 ), u|t=1 := u ˘(, u ˘0 , ν˘k ) = 2 u 0 ,ρ 9 ν˘j δij ˘kl , u ˘kl ˘0 , ν˘k ) = + O(), zi |t=1 := z˘i (, u 0 ,ρ ρH (1) + ρ˘0  ρ˘0 kl kl k ˘ ˘ ,u ˘0 , ρ˘0 , ν˘ ) = ln 1 + δζ|t=1 := δ ζ(, u + O(2 ), ρH (1) ˘ij , u ˘ij ˘0 (, u ˘0 , ν˘i ) = O() u0 |t=1 := u 0 ,ρ and ˘kl , u ˘kl ˘μν ˘0 , ν˘k ) = O() uμν 0 ,ρ 0 |t=1 := u 0 (, u determine via the formulas (1.29), (1.30), (1.32), (1.34), (1.36), (1.37), and (1.38) a solution of the gravitational and gauge constraint equations, see (6.3)– (6.4) and Remark 6.1. Furthermore, there exists a σ > 0, such that if uij ρ0 H s + ˘ ν i H s ≤ σ, ˘ uij H s+1 + ˘ 0 H s + ˘ then there exist maps 0 s 3 1 s−1 (T3 , S4 )), uμν  ∈ C ((0, 1], H (T , S4 )) ∩ C ((0, 1], H 3

See Lemma 6.7 for details.

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0 s 3 1 s−1 uμν (T3 , S4 )), γ, ∈ C ((0, 1], H (T , S4 )) ∩ C ((0, 1], H

u ∈ C 0 ((0, 1], H s (T3 )) ∩ C 1 ((0, 1], H s−1 ((T3 )), uγ, ∈ C 0 ((0, 1], H s (T3 )) ∩ C 1 ((0, 1], H s−1 ((T3 )), δζ ∈ C 0 ((0, 1], H s (T3 )) ∩ C 1 ((0, 1], H s−1 (T3 )), zi ∈ C 0 ((0, 1], H s (T3 ), R3 )) ∩ C 1 ((0, 1], H s−1 (T3 , R3 )), for  ∈ (0, 0 ), and ˚ Φ ∈ C 0 ((0, 1], H s+2 (T3 )) ∩ C 1 ((0, 1], H s+1 (T3 )), δ˚ ζ ∈ C 0 ((0, 1], H s (T3 )) ∩ C 1 ((0, 1], H s−1 (T3 )), ˚ zi ∈ C 0 ((0, 1], H s (T3 , R3 )) ∩ C 1 ((0, 1], H s−1 (T3 , R3 )), such that  (i) {uμν  (t, x), u (t, x), δζ (t, x), zi (t, x)} determines, via (1.12), (1.13), (1.17), (1.29), (1.32), (1.34), (1.36), and (1.37)–(1.41), a 1-parameter family of solutions to the Einstein–Euler equations (1.1)–(1.2) in the wave gauge (1.25) on M , ˚ −2 δ ij˚ ˚ z i (t, x) := E(t) zj (t, x)}, with ˚ ζH and E (ii) {˚ Φ(t, x), ˚ ζ(t, x) := δ˚ ζ +˚ ζH , ˚ given by (1.47) and (1.53), respectively, solves the conformal cosmological Poisson–Euler equations (1.55)–(1.57) on M with initial data ˚ ζ|t=1 = z i |t=1 = ν˘i /(ρH (1) + ρ˘0 ), ln(ρH (1) + ρ˘0 ) and ˚ (iii) the uniform bounds

δ˚ ζL∞ ((0,1],H s ) + ˚ ΦL∞ ((0,1],H s+2 ) + ˚ zj L∞ ((0,1],H s ) + δζ L∞ ((0,1],H s ) + ˚ zj L∞ ((0,1],H s )  1 and μν uμν  L∞ ((0,1],H s ) + uγ, L∞ ((0,1],H s ) + u L∞ ((0,1],H s )

+ uγ, L∞ ((0,1],H s )  1, hold for  ∈ (0, 0 ), (iv) and the uniform error estimates δζ − δ˚ ζL∞ ((0,1],H s−1 ) + zj − ˚ zj L∞ ((0,1]×H s−1 )  , ΦL∞ ((0,1],H s−1 ) uμν L∞ ((0,1],H s−1 ) + uμν − δ μ δ0ν ∂k ˚

,0 +uμν  L∞ ((0,1],H s−1 )

k,

0

 and uγ, L∞ ((0,1],H s−1 ) + u L∞ ((0,1],H s−1 )   hold for  ∈ (0, 0 ).

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1.8. Future Directions Although the 1-parameter families of -dependent solutions to the Einstein– Euler equations from Theorem 1.7 do provide a positive answer to the question of the existence of non-homogeneous relativistic cosmological solutions that are globally approximated to the future by solutions of Newtonian gravity, it does not resolve the question for initial data that is relevant to our Universe. This is because these solutions have a characteristic size ∼  and should be interpreted as cosmological versions of isolated systems [23,49,50]. This defect was remedied on short time scales in [50]. There the local-in-time existence of 1-parameter families of -dependent solutions to the Einstein–Euler equations that converge to solutions of the cosmological Poisson–Euler equations on cosmological spatial scales was established. In work that is currently in preparation [38], we combine the techniques developed in [50] with a generalization of the ones developed in this article to extend the local-in-time existence result from [50] to a global-in-time result. This resolves the existence question of non-homogeneous relativistic cosmological solutions that are globally approximated to the future on cosmological scales by solutions of Newtonian gravity, at least for initial data that is a small perturbation of FLRW initial data. However, this is far from the end of the story because there are relativistic effects that are important for precision cosmology that are not captured by the Newtonian solutions. To understand these relativistic effects, higher-order post-Newtonian (PN) expansions are required starting with the 1/2-PN expansion, which is, by definition, the  order correction to the Newtonian gravity. In particular, it can be shown [52] that the 1-parameter families of solutions must admit a 1/2-PN expansion in order to view them on large scales as a linear perturbation of FLRW solutions. The importance of this result is that it shows it is possible to have rigorous solutions that fit within the standard cosmological paradigm of linear perturbations of FLRW metrics on large scales while, at the same time, are fully nonlinear on small scales of order . Thus the natural next step is to extend the results of [38] to include the existence of 1-parameter families of -dependent solutions to the Einstein–Euler equations that admit 1/2-PN expansions globally to the future on cosmological scales. This is work that is currently in progress. 1.9. Prior and Related Work The future nonlinear stability of the FLRW fluid solutions for a linear equation of state p = Kρ was first established under the condition 0 < K < 1/3 and the assumption of zero fluid vorticity by Rodnianski and Speck [58] using a generalization of a wave-based method developed by Ringstr¨ om [56]. Subsequently, it has been shown [19,24,40,61] that this future nonlinear stability result remains true for fluids with nonzero vorticity and also for the equation of state parameter values K = 0 and K = 1/3, which correspond to dust and pure radiation, respectively. It is worth noting that the stability results established in [19,40] for K = 1/3 and K = 0, respectively, rely on Friedrich’s conformal method [17,18], which is completely different from the techniques used in [24,58,61] for the parameter values 0 ≤ K < 1/3.

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In the Newtonian setting, the global existence to the future of solutions to the cosmological Poisson–Euler equations was established in [4] under a small initial data assumption and for a class of polytropic equations of state. A new method was introduced in [51] to prove the future nonlinear stability of the FLRW fluid solutions that was based on a wave formulation of a conformal version of the Einstein–Euler equations. The global existence results in this article are established using this approach. We also note that this method was recently used to establish the nonlinear stability of the FLRW fluid solutions that satisfy the generalized Chaplygin equation of state [37]. 1.10. Overview In Sect. 2, we employ the variables (1.29)–(1.38) and the wave gauge (1.25) to write the conformal Einstein–Euler system, given by (1.15) and (1.16), as a non-local symmetric hyperbolic system, see (2.104), that is jointly singular in  and t. In Sect. 3, we state and prove a local-in-time existence and uniqueness result along with a continuation principle for solutions of the reduced conformal Einstein–Euler equations and discuss how solutions to the reduced conformal Einstein–Euler equations determine solutions to the singular system (2.104). Similarly, in Sect. 4, we state and prove a local-in-time existence and uniqueness result and continuation principle for solutions of the conformal cosmological Poisson–Euler equations (1.55)–(1.57). We establish in Sect. 5 uniform a priori estimates for solutions to a class of symmetric hyperbolic equations that are jointly singular in  and t, and include both the formulation (2.104) of the conformal Einstein–Euler equations and the   0 limit of these equations. We also establish error estimates, that is, a priori estimates for the difference between solutions of the singular hyperbolic equation and the corresponding   0 limit equation. In Sect. 6, we construct -dependent 1-parameter families of initial data for the reduced conformal Einstein–Euler equations that satisfy the constraint equations on the initial hypersurface t = 1 and limit as   0 to solutions of the conformal cosmological Poisson–Euler equations. Using the results from Sects. 2 to 6, we complete the proof of Theorem 1.7 in Sect. 7.

2. A Singular Hyperbolic Formulation of the Conformal Einstein–Euler Equations In this section, we employ the variables (1.29)–(1.38) and the wave gauge (1.25) to cast the conformal Einstein–Euler system, given by (1.15) and (1.16), into a form that is suitable for analyzing the limit   0 globally to the future. More specifically, we show that the Einstein–Euler system can be written as a symmetric hyperbolic system that is jointly singular in  and t, and for which the singular terms have a specific structure. Crucially, the -dependent singular terms are of a form that has been well studied beginning with the pioneering

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work of Browning, Klainerman, Kreiss and Majda [5,29,30,33,59,60], while the t-dependent singular terms are of the type analyzed in [51]. 2.1. Analysis of the FLRW Solutions As a first step in the derivation, we find explicit formulas for the functions Ω(t), ρH (t) and E(t) that will be needed to show that the transformed conformal Einstein–Euler systems is of the form analyzed in Sect. 5. We begin by differentiating (1.22) and observe, with the help of (1.6), (1.20) and (1.21), that it satisfies the differential equation 3 3 − t∂t (1 − Ω) + (1 + 2 K)(1 − Ω)2 = (1 + 2 K). (2.1) 2 2 Integrating gives Ω(t) =

2t3(1+ t

3(1+2 K)

2

K)

− C0

,

(2.2)

where C0 is as defined above by (1.44). Then by (1.22), we find that 2

4C0 Λt3(1+ K) ρH (t) = , (C0 − t3(1+2 K) )2

(2.3)

which, in turn, shows that ζH (t), as defined by (1.42), is given by the formula (1.43). It is clear from the above formulas that Ω, ρ and ζH , as functions of (t, ), are in C 2 ([0, 1] × [0, 0 ]) ∩ W 3,∞ ([0, 1] × [− 0 , 0 ]) for any fixed 0 > 0. In particular, we can represent t−1 Ω and ∂t Ω as 2 2 1 Ω = E −1 ∂t E = t2+3 K Q1 (t) and ∂t Ω = t2+3 K Q2 (t), t respectively, where we are employing the notation from Sect. 1.1.6 for the remainder terms Q1 and Q2 . Using (2.2), we can integrate (1.21) to obtain  22    2 2 t 2s2+3 K C0 − t3(1+ K) 3(1+ K) E(t) = exp ds = ≥ 1 (2.4) 3(1+2 K) − C C0 − 1 0 1 s for t ∈ [0, 1]. From this formula, it is clear that E ∈ C 2 ([0, 1] × [− 0 , 0 ]) ∩ ˚ and W 3,∞ ([0, 1] × [− 0 , 0 ]), and that the Newtonian limit of E, denoted E defined by ˚ = lim E(t), E(t) 0

is given by the formula (1.53). Similarly, we denote the Newtonian limit of Ω by ˚ Ω(t) = lim Ω(t), 0

which we see from (2.2) is given by the formula (1.54). For later use, we observe that E, Ω, ρH and ζH satisfy 1 1 − E −1 ∂t2 E + E −1 ∂t E = (1 + 32 K)ρH , t 2Λt2

(2.5)

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5 3 E −1 ∂t2 E + 2E −2 (∂t E)2 − E −1 ∂t E = (1 − 2 K)ρH t 2Λt2

(2.6)

and 2 3 i i γi0 = −¯ γ0i = t2+3 K Q3 (t) (2.7) ∂t ζH = − Ω = −3E −1 ∂t E = −¯ t as can be verified by a straightforward calculation using the formulas (1.43) and (2.2)–(2.4). By (1.47) and (1.54), it is easy to verify

6t2 3 Ω= . ζH = − ˚ ∂t ˚ t C0 − t3

(2.8) 2

We also record the following useful expansions of t1+3 K , E(, t) and Ω(, t):  2 6K  1+3λ2 K t1+3 K = t + 2 X (, t) where X (, t) = 2 λt ln tdλ (2.9)  0 and ˚ + E(, t) E(, t) = E(t)

and

Ω(, t) = ˚ Ω(t) + A(, t)

(2.10)

for (, t) ∈ (0, 0 ) × (0, 1], where X , E and A are all remainder terms as defined in Sect. 1.1.6. 2.2. The Reduced Conformal Einstein Equations The next step in transforming the conformal Einstein–Euler system is to replace the conformal Einstein equations (1.15) with the gauge reduced version given by ¯σ ¯ (μ Z¯ ν) + A¯μν ¯μ ¯ν ¯μ ¯ν ¯ μν + 2∇ 2R σ Z = −4∇ ∇ Ψ + 4∇ Ψ∇ Ψ



1 − 2 K 2 2Ψ ¯ ¯ ρ¯ + Λ e − 2 Ψ + 2|∇Ψ| + ρv¯μ v¯ν , g¯μν − 2e2Ψ (1 + 2 K)¯ 2 (2.11) where ¯ (μ ν) ¯ (μ δσν) . A¯μν σ := −X δσ + Y We will refer to these equations as the reduced conformal Einstein equations. Proposition 2.1. If the wave gauge (1.25) is satisfied, Ψ is chosen as (1.18) and γ¯ ν is given by (1.24), then the following relations hold:  Λ 1 (μ ν) 0(μ ν) Λ ¯ ∇ γ¯ = g¯ δ0 ∂t Ω − Ω − Ω∂t g¯μν , t t 2t 1 1 1 00 0 0 ¯ 2 = 1 g¯00 , ¯ = g¯ − Y¯ + γ¯ , |∇Ψ| Ψ t2 t t t2 2 ¯ μ Ψ∇ ¯ ν Ψ + 8Λ δ (μ g¯ν)0 + 4Λ δ μ δ ν Y¯ μ Y¯ ν = 4∇ 3t2 0 9t2 0 0 and ¯ (μ Y¯ ν) = −2∇ ¯ ν Ψ − 2Λ g¯0(μ δ ν) − Λ ∂¯t g¯μν . ¯ μ∇ ∇ 0 3t2 3t

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Proof. The proof follows from the formulas (1.18), (1.24) and (1.26)–(1.28) via straightforward computation.  Remark 2.2. For the purposes of proving a priori estimates, we can always assume that the wave gauge (1.25) holds since this gauge condition is known to propagate for solutions of the reduced Einstein–Euler equations assuming that the gravitational constraint equations and the gauge constraint Z¯ μ = 0 are satisfied on the initial hypersurface. The implication for our strategy of obtaining global solutions to the future by extending local-in-time solutions via a continuation principle through the use of a priori estimates is that we can assume that the wave gauge Z¯ μ = 0 is satisfied, which, in particular, means that we can freely use the relations4 from Proposition 2.1 in the following. A short computation using the relations from Proposition 2.1 then shows that the reduced conformal Einstein Equations (2.11) can be written as ¯ μν + 2∇ ¯ ν) − X ¯ ν + 2Λ Ω¯ ¯ (μ X ¯ μX − 2R g μν t  2Λ 4Λ Λ 4Λ (μ ν) ∂t g¯μν − 2 g¯00 + = δ0μ δ0ν − 2 g¯0k δ0 δk 3t 3t 3 3t  2 Λ ρ¯ ρ¯ − 2 g¯μν g¯00 + − (1 − 2 K) 2 g¯μν − 2(1 + 2 K) 2 v¯μ v¯ν . t 3 t t

(2.12)

Recalling the formula (e.g., see [20,57]) ¯ μν = 1 g¯λσ ∂¯λ ∂¯σ g¯μν + ∇ ¯ ν) + 1 (Qμν − X ¯ ν ), ¯ (μ Γ ¯ μX R 2 2 where ¯ η g¯ηδ g¯λγ g¯ρν Γ ¯δ Qμν = g¯λσ ∂¯λ (¯ g αμ g¯ρν )∂¯σ g¯αρ + 2¯ g αμ Γ ργ λα η λ δγ αμ ρν μ μ ¯ν ¯ ¯ ¯ + 4Γδη g¯ g¯λ(α Γ g¯ g¯ + (Γ − γ¯ )(Γ − γ¯ ν ), ρ)γ

we can express (2.12) as 2Λ 4Λ ¯ (μ γ¯ ν) − Qμν + 2Λ Ω¯ − g¯λσ ∂¯λ ∂¯σ g¯μν − 2∇ g μν = ∂t g¯μν − 2 t2 3t 3t  4Λ 2 Λ ρ¯ (μ ν) − 2 g¯0k δ0 δk − 2 g¯μν g¯00 + − (1 − 2 K) 2 g¯μν 3t t 3 t ρ¯ − 2(1 + 2 K) 2 v¯μ v¯ν . t

(2.13)

 Λ δ0μ δ0ν g¯00 + 3

(2.14)

¯ μν , ρH , v¯μ }, see (1.5), (1.6), (1.18) By construction, the quadruple {Ψ, h H and (1.19), is the conformal representation of a FLRW solution, and as such, it satisfies the conformal Einstein equations (1.14) under the replacement ¯ μν , ρH , eΨ v˜μ }. Since X ¯ μν , ¯ μ and Y¯ μ vanish when g¯μν −→ h {¯ gμν , ρ¯, v¯} → {h H 4

¯μ = 0 In fact, the only relation from Proposition 2.1 that relies on the gauge condition Z 1 00 1 ¯0 1 0 ¯ being satisfied is Ψ = t2 g¯ − t Y + t γ ¯ .

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it is clear that the conformal Einstein equations (1.14) and the reduced conformal Einstein equations (2.14) coincide under the replacement {¯ gμν , ρ¯, v¯} → ¯ μν satisfies ¯ μν , ρH , eΨ v˜μ }, and thus it follows that h {h H ¯ μν ¯ μν − 2∇ ¯ 00 ∂¯2 h ¯ (μ γ¯ ν) − Qμν + 2Λ Ωh −h 0 H H t2 2Λ ¯ μν ρH ¯ μν ρH Λ ∂t h − (1 − 2 K) 2 h = − 2(1 + 2 K) 2 δ0μ δ0ν , 3t t t 3 ¯ μν , ¯ H is the Levi-Civita connection of h where ∇  2Λ 1 (μ ν) 0(μ ν) Λ ¯ ¯ μν ¯ ∇H γ¯ = h δ0 Ω∂t h ∂t Ω − Ω − t t t

(2.15)

and ¯ λσ ¯ ¯ αμ h ¯ αμ γ¯ η h ¯ ρν )∂¯σ h ¯ αρ + 2h ¯ ¯ λγ ¯ ρν ¯ δ Qμν ργ H = h ∂λ (h λα ηδ h h γ ¯ δγ h ¯ λ(α γ¯ η h ¯ αμ h ¯ ρν . + 4¯ γλ h δη

ρ)γ

(2.16)

¯ μν , it is not difficult Using the formulas (1.23) for the Christoffel symbols of h to verify via a routine calculation that independent components of Eq. (2.15) agree up to scaling by a constant with Eqs. (2.5)–(2.6). Setting ν = 0 and subtracting (2.15) from (2.14), we obtain the equation ¯ μ0 ) − 2(∇ ¯ (μ γ¯ 0) − ∇ ¯ (μ γ¯ 0) ) − (Qμ0 − Qμ0 ) + − g¯λσ ∂¯λ ∂¯σ (¯ g μ0 − h H H  2Λ ¯ μ0 ) − 4Λ g¯00 + Λ δ μ δ00 − 4Λ g¯0k δ (μ δ 0) − g μ0 − h = ∂t (¯ 0 0 k 3t 3t2 3 3t2   1 ¯ μ0 ) ρ − ρH )¯ g μ0 + ρH (¯ g μ0 − h − (1 − 2 K) 2 (¯ t

 1 Λ v μ v¯0 + ρH v¯μ v¯0 − δ0μ − 2(1 + 2 K) 2 (¯ ρ − ρH )¯ t 3

2Λ ¯ μ0 ) Ω(¯ g μ0 − h t2  2 μ0 00 Λ g ¯ + g ¯ t2 3

(2.17)

¯ μ0 . This equation is close to the form that we are for the difference g¯μ0 − h seeking. The final step needed to complete the transformation is to introduce a non-local modification which effectively subtracts out the contribution due to the Newtonian potential. For the spatial components, a more complicated transformation is required to bring those equations into the desired form. The first step is to contract the μ = i, ν = j components of (2.14) with gˇij , where we recall that (ˇ gkl ) = (¯ g kl )−1 . A straightforward calculation, using the identity gˇkl ∂¯μ g¯kl = −1 ¯ −3α∂μ α (recall α = det(¯ g kl )) and (2.17) with μ = 0, shows that ¯q, defined previously by (1.40), satisfies the equation  2 2 1 1 ¯ 0 Λ Ω + 2Λ ΩΓ ¯ j) gˇij ¯ k δ i + 2Λ Ω¯ ¯ − 2Λ¯ ∂t Ω − Ω − 2¯ g λ0 Γ g 0(i Γ − g¯λσ ∂¯λ ∂¯σ q g 00 i0 k λ0 00 t t t 9t 9t  2 2Λ 2Λ 2Λ 00  −1 2 Λ 4Λ ¯+ E ∂t E − E −2 (∂t E)2 − Q + 2 Ω g¯00 − g¯ = ∂¯0 q E −1 ∂t E − 3 t 3 3t 9t    2 Λ 2 ρ¯ Λ ρ¯ Λ − 2 g¯00 + − (1 − 2 K) 2 g¯00 − − 2(1 + 2 K) 2 v¯0 v¯0 − gˇij v¯i v¯j , t 3 t 3 t 9

(2.18)

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where Λ λσ ¯ Λ g¯ ∂λ gˇij ∂¯σ g¯ij − gˇij Qij . 9 9 Ψ μ ¯ Under the replacement {¯ gμν , ρ¯, v¯} → {hμν , ρH , e v˜H }, Eq. (2.18) becomes Q = Q00 +

 1 1 2Λ2 ¯ 0(i j) ˇ 2Λ2 0 Λ k i ¯ λ0 γ¯λ0 δk + ∂t Ω − Ω − 2h Ω+ Ω¯ γi0 Ωh γ¯00 hij t t t 9t 9t 2Λ ¯ 00  −1 2 − h E ∂t E − E −2 (∂t E)2 3   2 2Λ ¯ 00 − Λ = 4Λ E −1 ∂t E − (1 − 2 K) ρH h ¯ 00 − Λ − QH + 2 Ω h t 3 9t t2 3 Λ ρ H − 2(1 + 2 K) 2 , (2.19) t 3 ¯ 00 − 2Λh

where QH = Q00 H +

Λ ¯ λσ ¯ ˇ ¯ ¯ ij Λ ˇ h ∂λ hij ∂σ h − hij Qij H 9 9

ˇ ij := (h ¯ kl )−1 = E 2 δij , and h

which, for the reasons discussed above, is satisfied by the conformal FRLW so¯ μν , ρH , v¯μ }. Taking the difference between (2.18) and (2.19) yields lution {Ψ, h H the following equation for ¯q:  1 λσ ¯ ¯ 00 00 1 ¯ ¯ λ0 γ¯ 0 ) Λ Ω ¯ 0λ0 − h g λ0 Γ − g¯ ∂λ ∂σ ¯q − 2Λ(¯ g −h ) ∂t Ω − Ω − 2(¯ λ0 t t t 2Λ2 ¯ k k + Ω(Γi0 − γ¯i0 )δki 9t 2Λ2 2Λ 00 ¯ 00  −1 2 ¯ 0(i γ¯ j) h ˇ ¯ j) gˇij − h Ω(¯ g 0(i Γ (¯ g − h ) E ∂t E − E −2 (∂t E)2 + 00 00 ij ) − 9t 3 2Λ  00 ¯ 00 + 2 Ω g¯ − h t   2 2Λ ¯ Λ 1 Λ 2 − (1 − 2 K) 2 (¯ ρ − ρH ) g¯00 − − (Q − QH ) = ∂0 ¯q − 2 g¯00 + 3t t 3 t 3  1 Λ − (1 − 2 K) 2 ρH g¯00 + t 3 

1 Λ − 2(1 + 2 K) 2 (¯ ρ − ρH ) v¯0 v¯0 − gˇij v¯i v¯j t 9  Λ Λ + ρH v¯0 v¯0 − − gˇij v¯i v¯j . (2.20) 3 9 Next, denote 1 i j ˇkl g¯ij , Lij kl = δk δl − g 3 and apply

1 ij α Llm

to (2.14) with μ = l, ν = m. A calculation using the identities

¯ ¯lm = ∂¯σ g ¯ij α−1 Lij lm ∂σ g

and

Lij ¯lm = 0, lm g

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¯ij is defined by (1.39), then shows that g ¯ij satisfies the where we recall that g equation 2 ¯ (l γ¯ m) − Q ˜ ij ∇ − g¯λσ ∂¯λ ∂¯σ (¯ gij − δ ij ) − Lij α lm 2Λ 2(1 + 2 K) ρ¯ ij l m ∂t (¯ = gij − δ ij ) − L v¯ v¯ , (2.21) 3t α t2 lm where  1 ij ¯ lm 1 ij lm ˜ ij = −¯ Q Llm ∂σ g¯ + Llm Q . g λσ ∂¯λ α α ¯ μν , ρH , eΨ v˜μ }, Eq. (2.21) becomes Making the replacement {¯ gμν , ρ¯, v¯} → {h H

2 ij ¯ (l γ¯ m) − Q ˜ ij = 0, ∇ L − H αH lm,H H where ¯ λσ ∂¯λ ˜ ij = −h Q H



(2.22)

1 ij ¯ lm + 1 Lij Qlm , Llm,H ∂¯σ h αH αH lm,H H 1

ˇ ij )− 3 = E −2 αH = (det h and 1 i j ij Lij kl,H = δk δl − δkl δ . 3 Subtracting (2.21) by (2.22) gives  1 ij ¯ (l m) 1 ij ¯ (l γ¯ m) − (Q ˜ ij ) ˜ ij − Q − g¯λσ ∂¯λ ∂¯σ (¯ Llm ∇ γ¯ − gij − δ ij ) − 2 Llm,H ∇ H H α αH 2Λ 2(1 + 2 K) ρ¯ ij l m ∂t (¯ = gij − δ ij ) − L v¯ v¯ . (2.23) 3t α t2 lm 2.3. -Expansions and Remainder Terms The next step in the transformation of the reduced conformal Einstein–Euler equations requires us to understand the lowest-order -expansion for a number of quantities. We compute and collect together these expansions in this section. Throughout this section, we work in Newtonian coordinates, and we frequently employ the notation (1.11) for evaluation in Newtonian coordinates, and the notation from Sect. 1.1.6 for remainder terms. First, we observe, using (1.29), (1.34) and (1.40), that we can write α as  3  00 2tu − u . α = E −2 exp  (2.24) Λ Using this, we can write the conformal metric as g¯ij = E −2 δ ij + Θij , where Θij = Θij (, t, u, uμν ) :=

1 α − E −2 (δ ij + uij ) + E −2 uij , 

(2.25)

(2.26)

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and Θij satisfies Θij (, t, 0, 0) = 0 and the E 1 -regularity properties of a remainder term, see Sect. 1.1.6. By the definition of u0μ , see (1.29), we have that g¯0μ = η¯0μ + 2tu0μ

(2.27)

and, see (1.30) and (1.31), for the derivatives 0μ ∂¯0 g¯0μ = (u0μ 0 + 3u ),

and

∂¯i g¯0μ = 2t∂i u0μ = u0μ i .

(2.28)

Additionally, by (2.24), we see, with the help of (1.21), (1.29)–(1.31) and (1.34)–(1.35), that 1 3 23 α(u00 ∂t α = −2α Ω + α (3u00 + u00 0 − u0 ) and ∂i α =  i − ui ). t Λ Λ Then differentiating (2.25), we find, using the above expression and (1.32)– (1.33), that

2 0 3 ij ij ij ij 00 0 00 ij ij ¯ ¯ ¯ ∂σ g¯ = ∂σ h −  δσ ΩΘ + α uσ + (3u δσ + uσ − uσ )(δ + u ) . t Λ (2.29) Since gˇij is, by definition, the inverse of g¯ij , it follows from (2.25) and Lemma B.2 that we can express gˇij as gˇij = E 2 δij + Sij (, t, u, uμν ),

(2.30)

where Sij (, t, 0, 0) = 0. From (2.25), (2.27) and Lemma B.2, we then see that ¯ μν + Ξμν (, t, uσγ , u), g¯μν = h

(2.31)

where Ξμν satisfies Ξμν (, t, 0, 0) = 0 and the E 1 -regularity properties of a remainder term. Due to the identity ∂¯λ g¯μν = −¯ gμσ ∂¯λ g¯σγ g¯γν

(2.32)

we can easily derive from (2.29) and (2.31) that ¯ μν + Sμνσ (, t, uαβ , u, uαβ , uγ ), ∂¯σ g¯μν = ∂¯σ h γ

(2.33)

where Sμνσ (, t, 0, 0, 0, 0) = 0, which in turn, implies that σ ¯ σμν − γ¯μν Γ = Iσμν (, t, uαβ , u, uαβ γ , uγ ),

(2.34)

where Iσμν (, t, 0, 0, 0, 0) = 0. Later, we will also need the explicit form of the ¯ i . To compute this, we first observe next order term in the -expansion for Γ 00 that the expansions 3 0i 2 αβ , u, uαβ ∂¯0 g¯k0 =  δki E 2 [u0i 0 + (3 + 4Ω)u ] +  Sk00 (, t, u γ , uγ ) Λ and  2 3 2 αβ u00 , u, uαβ ∂¯k g¯00 = − k +  S00k (, t, u γ , uγ ), Λ

(2.35)

(2.36)

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where Sk00 (, t, 0, 0, 0, 0) = S00k (, t, 0, 0, 0, 0) = 0, follow from (2.25), (2.27), (2.29), (2.32) and a straightforward calculation. Using (2.35) and (2.36), it is then not difficult to verify that  2 ¯ i =  3 [u0i + (3 + 4Ω)u0i ] +  1 3 Γ E −2 δ ik u00 00 k Λ 0 2 Λ + 2 Ii00 (, t, uαβ , u, uαβ (2.37) γ , uγ ),

¯ i − γ¯ i = Ξkj E −2 Ω δ kj −  1 E 2 δkj − 2 ΩΘij Γ i0 i0 t 2 t  3 ij −2 00 00 ij +E u0 + (3u + u0 − u0 )δ + 2 Iii0 (, t, uαβ , u, uαβ γ , uγ ), Λ (2.38) where Ii00 (, t, 0, 0, 0, 0) = 0 and Iii0 (, t, 0, 0, 0, 0) = 0. Continuing on, we observe from (1.37) that we can express the proper energy density in terms of ζ by ρ := ρ¯ = t3(1+

2

K) (1+2 K)ζ

e

,

(2.39)

and correspondingly, by (1.42), ρH = t3(1+

2

K) (1+2 K)ζH

e

(2.40)

for the FLRW proper energy density. From (1.38), (2.39) and (2.40), it is then clear that we can express the difference between the proper energy densities ρ and ρH in terms of δζ by   2 2 2 (2.41) δρ := ρ − ρH = t3(1+ K) e(1+ K)ζH e(1+ K)δζ − 1 . Due to the normalization v¯μ v¯μ = −1, only three components of v¯μ are independent. Solving v¯μ v¯μ = −1 for v¯0 in terms of the components v¯i , we obtain  −¯ g 0i v¯i + (¯ g ij v¯i v¯j + 1) g 0i v¯i )2 − g¯00 (¯ v¯0 = , g¯00 which, in turn, using definitions (1.29), (1.32), (1.34), (1.36), we can write as v¯0 = − 

1 −¯ g 00

+ 2 V2 (, t, u, uμν , zj ),

(2.42)

where V2 (, t, u, uμν , 0) = 0. From this and the definition v¯0 = g¯0μ v¯μ , we get  v¯0 = −¯ g 00 + 2 W2 (, t, u, uμν , zj ), (2.43) where W2 (, t, u, uμν , 0) = 0. We also observe that v¯k = (2tu0k v¯0 + g¯ik zi ) and z k = 2tu0k v¯0 + g¯ik zi

(2.44)

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follow immediately from definitions (1.36) and (1.48). For later use, note that g μν , zj ) by z k can also be written in terms (¯    i0  g ¯ 1 1 z i = g¯ij zj + 00 −¯ −¯ g 00 1 − 00 2 (¯ g 0j zj )2 + 2 g¯jk zj zk . g 0j zj + g¯  g¯ (2.45) ˜ ij . Using the above expansions, we are able to expand Qμν , Q and Q ˜ ij admit the following expansions: Proposition 2.3. Qμν , Q and Q μν Qμν − Qμν H = Q ,

Q − QH = Q,

and

˜ ij − Q ˜ ij = Q ˜ ij , Q H

where Ω μνγ Ω μν R (t)u00 R (t, u) + S μν (, t, uαβ , u, uαβ γ + σ , uσ ), t t Ω Ω Q = E −2 Rγ (t)u00 R(t, u) + S(, t, uαβ , u, uαβ γ + σ , uσ ), t t Ω ˜ ij ˜ ij = E −2 Ω R ˜ ijγ (t)u00 Q R (t, u) + S˜ij (, t, uαβ , u, uαβ γ + σ , uσ ), t t ij μν μνγ ˜ ij ˜ ijγ } with5 u = (uαβ , u, u0i , Rγ , R σ , uσ , uσ ), {R , R, R } linear in u, {R satisfy ˜ ijk (t)|  t2 , |∂t Rμνk (t)| + |∂t Rk (t)| + |∂t R Qμν = E −2

and the terms {S μν , S, S˜ij } vanish for (, t, uαβ , u, uαβ σ , uσ ) = (, t, 0, 0, 0, 0). ¯g ), where Proof. First, we observe that we can write Qμν as Qμν = Qμν (¯ g , ∂¯ μν μν μν ¯g ) is quadratic in ∂¯ ¯g = (∂¯γ g¯ ) and analytic in g¯ = (¯ g , ∂¯ g ) on the region Q (¯ μν μν ¯ ¯¯ μν ¯ ¯ + T ) to det(¯ g ) < 0. Since QH = Q (h, ∂ h), we can expand Q (h + S, ∂¯h get ¯ + S, ∂¯h ¯ + T ) − Qμν Qμν (h H  ¯ ∂¯h, ¯ S, T ¯ ∂¯h) ¯ · S + DQμν (h, ¯ ∂¯h) ¯ · T + 2 G μν , h, (2.46) = DQμν ( h, 1 2 where G μν is analytic in all variables and vanishes for (S, T ) = (0, 0), and μν DQμν 1 and DQ2 are linear in their second variable. By (2.25), (2.27), (2.28) and (2.29), we can choose S = (S μν (, t, u, uαβ ))

and T = (Tγμν (, t, u, uαβ , uσ , uαβ σ ))

for appropriate remainder terms S μν and T μν , so that ¯ + S and ∂¯ ¯g = ∂¯h ¯ + T . g¯ = h Using the fact that  ¯ = − Λ δ μ δ ν + 1 δ μ δ ν δ ij h 3 0 0 E2 i j 5

¯ = E −2 Ω h, and ∂¯h t

Here, in line with our conventions, see Sect. 1.1.6, the quantities written with calligraphic letters, e.g., S and R, denote remainder terms, and consequently also satisfy the properties of remainder terms.

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 where h = −2δγ0 δiμ δjν ), we can, using the linearity on the second variable of the derivatives DQμν  ,  = 1, 2, write (2.46) as Ω ¯ h) · S + E −2 Ω DQμν (h, ¯ h) · T DQμν (h, 2 t  1 t ¯ ∂¯h, ¯ S, T . + 2 G μν , h,

−2 Qμν − Qμν H = E

(2.47)

ij

Expanding Θ , recall (2.26), as  1 3 ij 00 ij ij (2tu − u)δ + u Θ = 2 + Aij (, t, u, uμν ), E Λ where Aij (, t, 0, 0) = 0, we see from (2.25), (2.27), (2.28) and (2.29) that 1 ¯ μν = 2tδ μ δ ν u00 + 4tδ (μ δ ν) u0j S μν = g¯μν − h 0 0 0 j   1 3 (2tu00 − u)δ ij + uij + B μν (, t, u, uαβ ), (2.48) + δiμ δjν 2 E Λ where B μν (, t, 0, 0) = 0, and 1 ¯ μν = δ μ δ ν δ 0 (u00 + 3u00 ) Tγμν = ∂¯γ g¯μν − ∂¯γ h 0 0 0 γ  (μ ν) 0 0j (μ ν) μ ν k 00 + δ0 δ0 δγ uk + 2δ0 δj δγ (u0 + 3u0j ) + 2δ0 δj δγk u0j k

2 μ ν 0 1 3 (2tu00 − u)δ ij + uij − Ωδi δj δγ 2 t E Λ

1 3 ij 00 0 00 ij + 2 uσ + (3u δσ + uσ − uσ )δ + Cγμν (, t, u, uαβ , uσ , uαβ σ ), E Λ (2.49) where Cγμν (, t, 0, 0, 0, 0) = 0. The stated expansion for Qμμ is then an immediate consequence of (2.47), (2.48), (2.49) and the boundedness and regularity ˜ ij properties of E and Ω, see Sect. 2.1 for details. The expansions for Q and Q can be established in a similar fashion.  Finally, we collect the last -expansions that will be needed in the following proposition. The proof follows from the same arguments used to prove Proposition 2.3 above. Proposition 2.4. The following expansions hold: ¯ μ0 ) = E μ0 , ¯ (μ γ¯ 0) − ∇ ¯ (μ γ¯ 0) ) − 2Λ Ω(¯ 2(∇ g μ0 − h H t2  2Λ2 ¯ k 0 Λ k ¯ 00 ) Λ ∂t Ω − 1 Ω + 2(¯ ¯ λ0 γ¯λ0 ¯ 0λ0 − h 2(¯ g 00 − h ) Ω− )δki g λ0 Γ Ω(Γi0 − γ¯i0 t t t 9t  2Λ2 2Λ 00 ¯ 00  −1 2 ˇ ¯ 0(i γ¯ j) h ¯ j) gˇij − h − Ω(¯ g 0(i Γ (¯ g − h ) E ∂t E − E −2 (∂t E)2 00 00 ij ) + 9t 3 2Λ  00 ¯ 00  − 2 Ω g¯ − h = E t

and

  1 ¯ (l γ¯ m) − |h| 13 Lij ∇ ¯ (l γ¯ m) = E˜ij , ∇ 2 |g| 3 Lij lm lm,H H

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where Ω μ0γ Ω μ0 F (t)u00 F (t, u) + S μ0 (, t, uαβ , u, uαβ γ + σ , uσ ) t t Ω Ω E = E −2 F γ (t)u00 F(t, u) + S(, t, uαβ , u, uαβ γ + σ , uσ ), t t Ω Ω ij E˜ = E −2 F˜ ijγ (t)uγ + F˜ ij (t, u) + S˜ij (, t, uαβ , u, uαβ σ , uσ ), t t

E μ0 = E −2

ij μ0γ , F γ , F˜ ijγ } satisfy with u = (uαβ , u, u0i σ , uσ , uσ ), {F

|∂t F μνγ (t)| + |∂t F γ (t)| + |∂t F˜ ijγ (t)|  t2 , {F μ0 , F, F˜ ij } are linear in u, and the remainder terms {S μ0 , S, S˜ij } vanish for (, t, uαβ , u, uαβ σ , uσ )=(, t, 0, 0, 0, 0).

2.4. Newtonian Potential Subtraction Switching to Newtonian coordinates, a straightforward calculation, with the help of Propositions 2.3 and 2.4, shows that the reduced conformal Einstein equations given by (2.17), (2.20) and (2.23) can be written in first-order form using the variables (1.29)–(1.38) and (1.48) as follows: ⎛ 0μ ⎞ ⎛ 0μ ⎞ ⎛ 0μ ⎞ ⎛ 0μ ⎞ u0 u0 u0 u0 1 ˜ k ⎜ 0μ ⎟ 1˜ ⎜ 0μ ⎟ ˆ 0μ ⎟ 0μ ⎟ ˜ 0 ∂0 ⎜ ˜ k ∂k ⎜ B ∂k ⎝ul ⎠ = BP ⎝uk ⎠ + B ⎝ul ⎠ + C 2 ⎝ul ⎠ + S1 ,  t u0μ u0μ u0μ u0μ ⎛ ij ⎞ ⎛ ij ⎞ ⎛ ij ⎞ ⎛ ij ⎞ u0 u0 u0 u0 2 00 2E g¯ 1 ˜ k ⎜ ij ⎟ ⎟ ⎜ ij ⎟ ˆ ij ⎟ ˘ ˜ 0 ∂0 ⎜ ˜ k ∂k ⎜ B C ∂ + B + = − P u u u l ⎠ + S2 , ⎝uij ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 2 k k l l  t ij ij ij u u u uij ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u0 u0 u0 u0 2E 2 g¯00 ⎟ ˘ 2 ⎝ ul ⎠ + Sˆ3 , ˜ 0 ∂0 ⎜ ˜ k ∂k ⎝ u l ⎠ = − ˜ k ∂k ⎝ u l ⎠ + 1 C B P ⎝uk ⎠ + B  t u u u u

where

⎛

− g¯00 0 2 ˜ =E ⎝ 0 B 0 ⎛

0 C˜ k = ⎝− δ kl 0

0 g¯kl 0 −δ 0 0

kl

⎛1 2

P2 = ⎝ 0 1 2

⎞ 0 0 ⎠, − g¯00 ⎞

0 0⎠ , 0 0 δil 0

1 2

⎛

− 4tu0k k 2 ˜ = E ⎝ − Θkl B 0 ⎛

− g¯ ˜ = E2 ⎝ 0 B 0 ⎞

0⎠ , 1 2

00

⎛

1 ˘ 2 = ⎝0 P 0

0

3 ki ¯ 2g

0

0 0 0

⎞ 0 0⎠ , 0

− Θkl 0 0

(2.50)

(2.51)

(2.52)

⎞ 0 0⎠ , 0 ⎞

(2.53)

0 0 ⎠, − g¯00 (2.54) (2.55)

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⎛ 0⎞ Sˆ1 Sˆ1 = E 2 ⎝ 0 ⎠ , 0

(2.56)

0μ 00 0μ 2 2+32 K Sˆ10 = Q0μ + E μ0 + 4u00 u0μ + 6u0k u0μ 0 − 4u u k −2u (1 −  K)t 2 e(1+ K)(ζH +δζ) + fˆ0μ , 2 2 fˆ0μ = −2(1 + 2 K)t1+3 K e(1+ K)(ζH +δζ)         Λ Λ 1 μ μ v¯0 − v¯0 + δ0 + v¯0 z k δk ×  3 3

Sˆ2

Sˆ3

2 2 2 1Λ 1 μ − KΛt1+3 K e(1+ K)ζH (e(1+ K)δζ − 1)δ0μ − δ δρ,  3 t2 0 ⎛ ⎞ ˜ ij + E˜ij + 4u00 uij + fˆij Q 0 ⎜ ⎟ = E2 ⎝ 0 ⎠, − g¯00 uij 0 ⎛ ⎞ 00 Q + E + 4u u0 − 8(u00 )2 + fˆ ⎠, = E2 ⎝ 0 00 − g¯ u0

1+32 K (1+2 K)ζ k l e z z, fˆij = −2(1 + 2 K)α−1 Lij kl t

(2.57)

(2.58)

(2.59) (2.60)

and

4Λ 1+32 K (1+2 K)ζH  (1+2 K)δζ t e e −1 fˆ = −K 3 2 2 Λ + 2(1 + 2 K) gˇij t1+3 K e(1+ K)ζ z i z j 9 2 2 − 2(1 − 2 K)u00 t2+3 K e(1+ K)(ζH +δζ)       Λ 1 Λ 2 1+32 K (1+2 K)ζ 0 0 − 2(1 +  K)t e v + v¯ − . 3  3

(2.61)

At this point, it is important to stress that Eqs. (2.50)–(2.52) are completely equivalent to the reduced conformal Einstein equations for  > 0. Moreover, these equations are almost of the form that we need in order to apply the results of Sect. 5. Since the term

1 

v¯0 −

Λ 3

is easily seen to be regular

in  from the expansion (2.43) , the only -singular term left is − 1 Λ3 t12 E 2 δρδ0μ , which can be found in the quantity fˆ0μ . Following the method introduced in [45] and then adapted to the cosmological setting in [47], we can remove the singular part of this term while preserving the required structure via the introduction of the shifted variable 0 μ wk0μ = u0μ k − δ0 δ0 ∂k Φ,

where Φ is the potential defined by solving the Poisson equation Λ 1 2 1+12 K Λ Φ := E Πρ = E 2 teζH Πeδζ (Δ = δ ij ∂i ∂j ), 2 3t 3

(2.62)

(2.63)

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which, as we shall show, reduces to the (cosmological) Newtonian gravitational field equations in the limit   0. In this sense, we can view (2.62) as the subtraction of the gradient of the Newtonian potential from the gravitational field component u00 k . Using (2.63), we can decompose − 1 Λ3 t12 E 2 δρδ0μ as 2 1 Λ 1Λ 1 2 E δρδ0μ = − δ0μ ΔΦ − δ0μ E 2 t1+3 K φ + E 2 S μ (, t, δζ), (2.64)  3 t2  3 where    2 2 1 1 1 (2.65) φ := 1, 3(1+2 K) δρ = (1 − Π)e(1+ K)ζH e(1+ K)δζ − 1   t and " 2 2 2 Λ 1 ! ζH δζ S μ (, t, δζ) = Π te (e − 1) − t1+3 K e(1+ K)ζH (e(1+ K)δζ − 1) δ0μ , 2 3 which obviously satisfies S μ (, t, 0) = 0. Although it is not obvious at the moment, φ is regular in , and consequently, with this knowledge, it is clear from (2.64) that − 1 δ0μ ΔΦ is the only -singular term in − 1 Λ3 t12 E 2 δρ. A straightforward computation using (2.63) and (2.64) along with the expansions from Propositions 2.3 and 2.4 then shows that replacing u0μ k in (2.50) with the shifted variable (2.62) removes the -singular term − 1 δ0μ ΔΦ and yields the equation ⎛ 0μ ⎞ ⎛ 0μ ⎞ ⎛ 0μ ⎞ ⎛ 0μ ⎞ u0 u0 u0 u0 ⎟ ˜ k ⎜ 0μ ⎟ 1 ˜ k ⎜ 0μ ⎟ 1 ˜ ⎜ 0μ ⎟ ˜ ˜ 0 ∂0 ⎜ B ∂k ⎝wl ⎠ + C ∂k ⎝wl ⎠ = BP2 ⎝wl ⎠ + G1 + S˜1 , ⎝wk0μ ⎠ + B  t u0μ u0μ u0μ u0μ (2.66)

−

where

⎛ 0⎞ ˜ G 1 ˜1 = E2 ⎝ 0 ⎠ , G 0

(2.67)

" ! ˜ 0 = −E −2 Ω D0μ0 (t)u00 + D0μk (t)w00 − Ω D0μ (t, u) G 1 0 k t t 0μ 00 u , + 4u00 u0μ 0 − 4u ⎞ ⎛ Ω ⎛ μ ⎞ − t D0μk ∂k Φ + Θkl δ0μ ∂k ∂l Φ S (, t, uαβ , u, uαβ σ , uσ ) ⎟ ⎜ ⎠, 0 S˜1 = ⎝ 3 1 δ0μ g¯kl ∂l Φ − g¯kl δ0μ ∂0 ∂l Φ ⎠ +  ⎝ 2 t 0 0

(2.68) D0μν (t) = −R0μν (t) − F 0μν (t),

D0μ (t, u) = −R0μ (t, u) − F 0μ (t, u) (2.69)

and 2

2

f 0μ = −2(1 + 2 K)t1+3 K e(1+ K)(ζH +δζ)        Λ Λ 1 μ 0 0 0 k μ v¯ − v¯ + δ0 + v¯ z δk ×  3 3 − ΛKt1+3

2

K (1+2 K)ζH

e

(e(1+

2

K)δζ

− 1)δ0μ

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2

− 2u0μ (1 − 2 K)t2+3 K e(1+ K)(ζH +δζ) 2 Λ − t1+3 K φδ0μ + S μ (, t, δζ). 3

(2.70)

With the help of the expansions from Propositions 2.3 and 2.4, we further decompose Sˆ2 and Sˆ3 into a sum of local and non-local terms given by ˜ 2 + S˜2 Sˆ2 = G

˜ 3 + S˜3 , and Sˆ3 = G

(2.71)

where ⎛

! ij0 " ˜ (t)u00 + D ˜ ijk (t)w00 − D − E −2 Ω 0 k t 2⎜ ˜ G2 = E ⎝ 0 − g¯00 uij 0

Ω ˜ ij D (t, u) t

ˆij + 4u00 uij 0 +f

⎞ ⎟ ⎠,

(2.72)

⎛ ij ⎞ S (, t, uαβ , u, uαβ − σ , uσ ) ⎠, ⎠ + ⎝ 0 (2.73) S˜2 = ⎝ 0 0 0 ! 0 " Ω ⎛ ⎞ k 00 D (t)u00 − t D(t, u) + 4u00 u0 − 8(u00 )2 + fˆ − E −2 Ω 0 + D (t)wk t ˜3 = E2 ⎝ ⎠, G 0 − g¯00 u0 ⎛

⎞

Ω ˜ ijk D ∂k Φ t

⎛ − ˜ S3 = ⎝

Ω k D ∂k Φ t

0 0

⎛ ⎞ S(, t, uαβ , u, uαβ σ , uσ ) ⎠ + ⎝ ⎠, 0 0 ⎞

˜ ijν (t) − F˜ ijν (t), ˜ ijν (t) = −R D μ

μ

μ

D (t) = −R (t) − F (t)

˜ ij (t, u) = −R ˜ ij (t, u) − F˜ ij (t, u), D

and

D(t, u) = −R(t, u) − F (t, u).

(2.74) (2.75) (2.76) (2.77)

Not only is the system of Eqs. (2.51) (2.52) and (2.66) completely equivalent to the reduced conformal Einstein equations for any  > 0, but it is now of the form required to apply the results from Sect. 5. This completes our transformation of the reduced conformal Einstein equations.

2.5. The Conformal Euler Equations With the transformation of the reduced conformal Einstein equations complete, we now turn to the problem of transforming the conformal Euler equations. We begin observing that it follows from the computations from [50, §2.2] that conformal Euler equations (1.16) can be written in Newtonian coordinates as    ˆ 2 ζi + S, ¯ 0 ∂0 ζi + B ¯ k ∂k ζi = 1 B¯P ¯ B (2.78) z z z t

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Newtonian Limits on Long Time Scales

where

⎛ ¯0 = ⎝ B ⎛ ¯k = B

L0



S¯ =

⎠, ⎞

Lk i v ¯0

k

k

K −1 Mij 1 vv¯¯0

1



⎠=

1 k v ¯0 z 1 k v ¯0 δj



0 g ¯

− K −1 (1 − 32 K) v¯0ik v ¯0 0 , δjk ¯ iμν v¯ν − Lμ Γ

0 

ˆ2 = P

K −1 Mij

 v¯0j

¯ 1v 0 ⎝  v¯k Lj v ¯0

¯ = B

⎞

L0

 v¯0i

1

0 0 

i

− K −1 (1 − 32 K) v¯10 g¯0j − K

Lμi = δiμ −

2185

1 k v ¯0 δi −1 1 K v¯0 Mij z k

 ,

,

1 v ¯0 −1

 ¯ i v¯ν 10 Mij v¯μ 1 Γ μν v ¯

,

v¯i μ δ v¯0 0

and Mij = g¯ij −

v¯j g¯00 v¯i g¯0j − g¯0i + v¯i v¯j . v¯0 v¯0 (¯ v0 )2

In order to bring (2.78) into the required form, we perform a change in variables from z i to zj , which are related via a map of the form z i = z i (zj , g¯μν ), see (2.45). Denoting the Jacobian of the transformation by J im :=

∂z i , ∂zm

we observe that ∂σ z i = J im ∂σ zm + δσ0

i ∂z i ¯ μν j ∂z ¯ μν g ¯ + δ ∂ ∂j g¯ . 0 σ ∂¯ g μν ∂¯ g μν

Multiplying (2.78) by the block matrix diag (1, J jl ) and changing variables from (ζ, z i ) to (δζ, zj ), where we recall from (1.38) that δζ = ζ − ζH , we can write the conformal Euler equations (2.78) as    1 ˆ δζ δζ δζ ˆ + B k ∂k = BP + S, (2.79) B 0 ∂0 2 zm zm zm t where

⎛ 1

L0

 v¯0i J im

⎞

⎠, B 0 = ⎝ L0  v¯0j J jl K −1 Mij J jl J im  1  1 km k v ¯0 z v ¯0 J k B = 1 kl , K −1 v¯10 Mij J jl J im z k v ¯0 J

(2.80)

(2.81)

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 B= and

⎛ Sˆ = ⎝

1 0

−K

−1

Ann. Henri Poincar´e

0 (1 − 32 K) v¯01v¯0 J ml

i ¯ i − Lμ Γ ¯ i ¯j 10 + (¯ ¯i ) γi0 −Γ − L0i Γ 00 i0 i μj v v ¯

(2.82)

⎞

⎠ 0 ¯ i v¯ν 10 +  L0j J jl γ¯ i − K −1 J jl Mij v¯μ 1 Γ μν i0 v ¯ v ¯ ⎛ ⎞ L0i ∂z i ¯ μν δik ∂z i ¯ μν  v¯0 ∂ g¯μν ∂0 g¯ +  v¯0 ∂ g¯μν ∂k g¯ ⎠. −⎝ i i ¯0 g¯μν + K −1 M ¯k g¯μν ¯ ij zk0 J jl ∂zμν K −1 Mij J jl ∂∂z ∂ ∂ μν g ¯ v ¯ ∂g ¯

By direct calculation, we see from (2.45) and the expansions (2.25) and (2.27) that J ik = E −2 δ ik + Θik + 2 Sik (, t, u, uμν , zj ), (2.83) where Sik (, t, 0, 0, 0) = 0. Similarly, it is not difficult to see from (2.45) and the expansions (2.25) and (2.27)–(2.29) that δσ0

i ∂z i ¯ μν j ∂z ¯ μν g ¯ + δ ∂ ∂j g¯ 0 σ ∂¯ g μν ∂¯ g μν   Ω 3 0i (u0 + 3u0i ) + S i (, t, u, uαβ , uγ , uαβ = −2δσ0 E −2 zj δ ij + γ , zj ) t Λ (2.84)

and δ k ∂z i 6  i0 μν ∂¯k g¯μν = − u0i δ k + 2 S(, t, u, uαβ , uγ , uαβ γ , zj ), v¯ ∂¯ g Λ k i

(2.85)

where S i (, t, 0, 0, 0, 0, 0) = 0 and S(, t, 0, 0, 0, 0, 0) = 0. We further note that ¯ jμν v¯ν 10 found in Sˆ above is not singular in . This the term − K −1 J jl Mij v¯μ 1 Γ v ¯ can be seen from the expansions (2.25), (2.27), (2.31) and (2.37), which can be used to calculate 1 ¯j μ ν ¯ j v0 v0 ¯ j v 0 z i + Γ ¯ j zizk + 1 Γ Γ v v = 2Γ 00 0i ik  μν   " 1 3 −2 ik 00 Λ 2Ω −2 ij ! 0i E zj δ + u0 + (3 + 4Ω)u0i + E δ uk = 3 t 2Λ + S j (, t, u, uαβ , uγ , uαβ (2.86) γ , zj ), where S j (, t, 0, 0, 0, 0, 0) = 0. Using the expansions (2.83), (2.5) and (2.86) in conjunction with (2.25), (2.27), (2.31), (2.34), (2.42), (2.43) and (2.44), we can expand the matrices {B 0 , B k , B} and source term S defined above as follows: B0 =

 1 0 

Bk =

3 Λ

0 K −1 E −2 δ lm 

k

z E −2 δ kl





+

0 0

−2 km

0 K −1 Θlm

E δ K −1 E −2 δ lm z k





+ 2 S0 (, t, u, uαβ , uγ , uαβ γ , zj ),

(2.87)

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3 tu00 z k Λ 3 Θ + Λ tu00 E −2 δ kl 2 Sk (, t, u, uαβ , uγ , uαβ γ , zj ),

+ +  1 B= 0

3 Λ

km 00 −2 km 3 Θ  lm+ Λ3tu 00E −2δ lm k Θ + Λ tu E δ z K

kl

0 K −1 (1 − 32 K)E −2 δ lm

2187

−1



 +

0 0

0 K −1 Θlm

(2.88)



+ 2 S(, t, u, uαβ , uγ , uαβ γ , zj )

(2.89)

and 

0

  3 −1





1  3 3 −2 lk 00 0l −u0l + 2 Λ 2 E δ uk −K 0 + (−3 + 4Ω)u Λ  Sˆ0 + 2 S2 (, t, u, uαβ , uγ , uαβ + γ , zj ), S1 (, t, u, uαβ , uγ , uαβ γ , zj )

Ω 1 2 Sˆ0 = −Ξkj E −2 δ kj + E 2 δkj − ΩΘkj t 2 t

 kj 3 6 −2 00 00 u0 + (3u + u0 − u0 )δ kj + u0i +E δk (2.90) Λ Λ k i Sˆ =

where the remainder terms S0 , Sk , S, S1 and S2 all vanish for (u, uαβ , uγ , uαβ γ , zj ) ˆ into a local and non-local part by writing = (0, 0, 0, 0, 0). We further decompose S Sˆ = G + S, (2.91) where



G= and

−K 

S=

0

  3 −1

−K

Λ

0l −u0l + 0 + (−3 + 4Ω)u

0  3 −1 1 3 2 2

Λ

1 2

 3 32 Λ

  , 00

E −2 δ lk wk

(2.92)

 E −2 δ lk ∂k Φ

+ S(, t, u, uαβ , uγ , uαβ γ , zj ).

(2.93)

2.6. The Complete Evolution System To complete the transformation of reduced conformal Einstein–Euler equations, we need to treat φ, defined by (2.65), as an independent variable and derive an evolution equation for it. To do so, we see from (2.79) that we can write the time derivative of δζ as

 

1 ˆ δζ δζ ˆ ∂t δζ = e0 (B 0 )−1 −B k ∂k + BP + S 2 zm zm t  Λ k z ∂k δζ + E −2 δ km ∂k zm + S(, t, u, uαβ , uγ , uαβ =− γ , zj ), (2.94) 3 where e0 = (1, 0, 0, 0) and S vanishes for (u, uαβ , uγ , uαβ γ , zj ) = (0, 0, 0, 0, 0). Noting that (2.7) is equivalent to 1 3 3 ∂t ρH = ρH − ΩρH , 1 + 2 K t t

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it follows directly from the definition of δζ, ρ and δρ, see (1.38), (2.39) and (2.41), that ∂t (δζ) =

1 δρ 1 3 ∂t (δρ) + (Ω − 1) 1 + 2 K ρ t ρ

(2.95)

and

1 1 ∂k ρ. (2.96) 2 1+ K ρ Using (2.95), (2.96) and (2.87)–(2.90), we can write (2.94) as  1 3 3 ∂t (δρ) + (Ω − 1)δρ + ∂k (ρz k ) 1 + 2 K t Λ  32  2 2 3 = − t4+3 K ∂k u00 e(1+ K)ζ z k Λ 2 3(1+2 K) ˇ + t S + 2 t3(1+ K) S(, t, u, uαβ , uγ , uαβ γ , zj , ∂k zj , δζ, ∂k δζ), (2.97) ∂k (δζ) =

where

 1 2 3 2 kj kj −2 00 00 kj ˇ S = E δkj − ΩΘ + E u0 + (3u + u0 − u0 )δ 2 t Λ 2 2 6 × e(1+ K)(ζH +δζ) + u0i δ k e(1+ K)(ζH +δζ) Λ k i 2 Ω − Ξkj E −2 δ kj e(1+ K)(ζH +δζ) t and the remainder term S satisfies S(, t, 0, 0, 0, 0, 0, 0, δζ, 0) = 0. Taking the 2 L2 inner product of (2.97) with 1 and then multiplying by 1/(t3(1+ K) ), we obtain the desired evolution equation for φ given by ´ + S, ´ ∂t φ = G

(2.98)

2 # $ ´ = (1 + 2 K) 1, Sˇ − 3(1 +  K)Ω φ G t

(2.99)

where

and

# $ S´ = (1 + 2 K) 1, S(, t, u, uαβ , uγ , uαβ γ , zj , ∂k zj , δζ, ∂k δζ) .

(2.100)

Next, we incorporate the shifted variable (2.62) into our set of gravitational variables by defining the vector quantity 0μ ij ij 0μ ij T U1 = (u0μ 0 , wk , u , u0 , uk , u , u0 , uk , u) ,

(2.101)

and then combine this with the fluid variables and φ by defining U = (U1 , U2 , φ)T ,

(2.102)

U2 = (δζ, zi )T .

(2.103)

where

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Gathering (2.51), (2.52), (2.66), (2.79) and (2.98) together, we arrive at the following complete evolution equation for U: 1 1 B0 ∂t U + Bi ∂i U + Ci ∂i U = BPU + H + F,  t

(2.104)

where ⎛

⎛ ˜ B ⎜0 ⎜ B=⎜ ⎜0 ⎝0 0

˜0 B 0 ⎜0 ˜ B0 ⎜ B0 = ⎜ 0 ⎜0 ⎝0 0 0 0 ⎛ i ˜ B 0 ⎜0 ˜i B ⎜ Bi = ⎜ 0 ⎜0 ⎝0 0 0 0 ⎛ i C˜ 0 ⎜0 ˜ Ci ⎜ i ⎜ C =⎜0 0 ⎝0 0 0 0 0 − 2E 2 g¯00 I 0 0 0

0 0 − 2E 2 g¯00 I 0 0

0 0 ˜0 B 0 0

0 0 0 B0 0

0 0 ˜i B 0 0

0 0 0 Bi 0

0 0 C˜ i 0 0 0 0 0 B 0

0 0 0 0 0 ⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 1 ⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 0 ⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 0

(2.105) ⎛

P2 ⎜0 ⎜ P=⎜ ⎜0 ⎝0 0

0 ˘2 P 0 0 0

⎞ 0 0 0 0 0 0⎟ ⎟ ˘ 2 0 0⎟ , P ⎟ ˆ 0⎠ 0 P 2

0 0 0 (2.106)

T

˜1, G ˜2, G ˜ 3 , G, G) ´ H = (G

and

T

´ . F = (S˜1 , S˜2 , S˜3 , S, S)

(2.107)

The importance of Eq. (2.104) is threefold. First, solutions of the reduced conformal Einstein–Euler equations determine solutions of (2.104) as we shall show in the following section. Second, Eq. (2.104) is of the required form so that the a priori estimates from Sect. 5 apply to its solutions. Finally, estimates for solutions of (2.104) that are determined from solutions of the reduced conformal Einstein–Euler equations imply estimates for solutions of the reduced conformal Einstein–Euler equations. In this way, we are able to use the evolution Eq. (2.104) in conjunction with the a priori estimates from Sect. 5 to establish, for appropriate small data, the global existence of 1-parameter families of -dependent solutions to the conformal Einstein–Euler equations that exist globally to the future and converge in the limit   0 to solutions of the cosmological conformal Poisson–Euler equations of Newtonian gravity.

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3. Reduced Conformal Einstein–Euler Equations: Local Existence and Continuation In this section, we consider the local-in-time existence and uniqueness of solutions to the reduced Einstein–Euler equations and discuss how these solutions determine solutions of (2.104). Furthermore, we establish a continuation principle for the Einstein–Euler equations which is based on bounding the H s norm of U for s ∈ Z≥3 . Proposition 3.1. Suppose s ∈ Z≥3 , 0 > 0,  ∈ (0, 0 ), T0 ∈ (0, 1], (¯ g0μν ) ∈ μν μ s+1 3 s 3 s 3 4 (T , S4 ), and (¯ g1 ) ∈ H (T , S4 ), (¯ v0 ) ∈ H (T , R ) and ρ¯0 ∈ H s (T3 ), H μ μ ν g0μν ) < 0 and ρ¯0 > 0 on where v¯0 is normalized by g¯0μν v¯0 v¯0 = −1, and det(¯ 3 T . Then there exists a T1 ∈ (0, T0 ] and a unique classical solution (¯ g μν , v¯μ , ρ¯) ∈

2 %

C  ((T1 , T0 ], H s+1− (T3 )) ×

=0

×

1 %

C  ((T1 , T0 ], H s− (T3 ))

=0

1 %

C  ((T1 , T0 ], H s− (T3 )),

=0

of the reduced conformal Einstein–Euler equations, given by (1.16) and (2.12), on the spacetime region (T1 , T0 ] × T3 that satisfies g μν , g¯μν , v¯μ , ρ¯0 ). (¯ g μν , ∂¯0 g¯μν , v¯μ , ρ¯)|t=T = (¯ 0

0

1

0

Moreover,

&1  ¯ s+2− (T3 )) that solves (i) there exists a unique Φ ∈ =0 C ((T1 , T0 ], H Eq. (2.63), (ii) the vector U, see (2.102), is well-defined, lies in the space 1 %

U∈

C  ((T1 , T0 ], H s− (T3 , V)),

=0

where V = R4 × R12 × R4 × S3 × (S3 )3 × S3 × R × R3 × R × R × R3 × R, and solves (2.104) on the spacetime region (T1 , T0 ] × T3 , and (iii) there exists a constant σ > 0, independent of  ∈ (0, 0 ) and T1 ∈ (0, T0 ), such that if U satisfies UL∞ ((T1 ,T0 ],H s (T3 )) < σ, μν

then the solution (¯ g , v¯μ , ρ¯) can be uniquely continued as a classical solution with the same regularity to the larger spacetime region (T1∗ , T0 ]×T3 for some T1∗ ∈ (0, T1 ). Proof. We begin by noting that the reduced conformal Einstein–Euler equations are well defined as long as the conformal metric g¯μν remains non-degenerate and the conformal fluid four-velocity remains future directed, that is, det(¯ g μν ) < 0 and v¯0 < 0. (3.1)

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Since it is well known that the reduced Einstein–Euler equations can be written as a symmetric hyperbolic system6 provided that ρ remains strictly positive, we obtain from standard local existence and continuation results for symmetric hyperbolic systems, e.g., Theorems 2.1 and 2.2 of [41], the existence of a unique local-in-time classical solution 2 1 % % μν μ  s+1− 3 C ((T1 , T0 ], H (T )) × C  ((T1 , T0 ], H s− (T3 )) (¯ g , v¯ , ρ¯) ∈ =0

×

=0

1 %

C  ((T1 , T0 ], H s− (T3 ))

(3.2)

=0

of the reduced conformal Einstein–Euler equations, for some time T1 ∈ (0, T0 ), that satisfies g0μν , g¯1μν , v¯0μ , ρ¯0 ) (¯ g μν , ∂¯0 g¯μν , v¯μ , ρ¯)|t=T0 = (¯ for given initial data (¯ g μν , ∂¯0 g¯μν , v¯μ , ρ¯0 ) ∈ H s+1 (T3 , S4 ) × H s (T3 , S4 ) × H s (T3 , R4 ) × H s (T3 ) 







satisfying (3.1) and ρ¯0 > 0 on the initial hypersurface t = T0 . Moreover, if the solution satisfies det(¯ g μν (¯ xγ )) ≤ c1 < 0,

v¯0 (¯ xγ ) ≤ c2 < 0

(3.3)

and ρ¯(¯ xγ ) ≥ c3 > 0 for all (¯ xγ ) ∈ (T1 , T0 ] × T3 , for some constants ci , i = 1, 2, 3, and ¯g μν L∞ ((T ,T ],W 1,∞ (T3 )) ¯ g μν L∞ ((T ,T ],W 1,∞ (T3 )) + ∂¯ 1

0



1

0



+ ¯ v μ L∞ ((T1 ,T0 ],W 1,∞ (T3 )) + ¯ ρL∞ ((T1 ,T0 ],W 1,∞ (T3 )) < ∞, then there exists a time T1∗ ∈ (0, T1 ) such that the solution uniquely extends to the spacetime region (T1∗ , T0 ] × T3 with the same regularity as given by (3.2). Next, we set u = (uμν , uμν γ , u, uγ , δζ, zi ), , u, u , δζ and z are computed from the solution (3.2) via where uμν , uμν γ i γ the definitions from Sect. 1.5. From the definitions (1.37) and (1.38), the formulas (1.43)–(1.45), the expansions (2.25)–(2.27) and (2.43), and Sobolev’s inequality, see Theorem A.1, that there exists a constant σ > 0, independent of T1 ∈ (0, T0 ) and  ∈ (0, 0 ), such that (uμν , u, δζ, zi )L∞ ((T1 ,T0 ],H s (T3 )) < σ

(3.4)

−3(1+2 K)

implies that the inequalities (3.3) and t ρ¯ ≥ c3 > 0 hold for some constants ci , i = 1, 2, 3. Moreover, for σ small enough, we see from the Moser 6 This follows from writing the wave Eq. (2.14) in first-order form and using one of the various methods for expressing the relativistic conformal Euler equations as a symmetric hyperbolic system. One particular way of writing the conformal Euler equations in symmetric hyperbolic form is given in Sect. 2.5 which is a variation of the method introduced by Rendall [54]. For other elegant approaches, see [3, 16, 65].

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inequality from Lemma A.3 and the expansions (2.25)–(2.29) and (2.43)–(2.44) that ¯g μν L∞ ((T ,T ],W 1,∞ (T3 )) ¯ g μν L∞ ((T1 ,T0 ],W 1,∞ (T3 )) + ∂¯ 1 0 

+ ¯ v μ L∞ ((T1 ,T0 ],W 1,∞ (T3 )) + ¯ ρL∞ ((T1 ,T0 ],W 1,∞ (T3 ))  ≤ C(σ) uL∞ ((T1 ,T0 ],H s (T3 )) + 1 .

Thus by the continuation principle, there exists a σ > 0 such that if (3.4) holds and uL∞ ((T1 ,T0 ],H s (T3 )) < ∞, (3.5) then the solution (3.2) can be uniquely continued as a classical solution with the same regularity to the larger spacetime region (T1∗ , T0 ] × T3 for some T1∗ ∈ (0, T1 ). ¯ s+2 (T3 ) −→ H ¯ s (T3 ) is an isomorphism, we can solve (2.63) Since Δ : H to get 1 % 1 Λ ¯ s+2− (T3 )). Φ = E 2 eζH Δ−1 Πeδζ ∈ C  ((T1 , T0 ], H (3.6) t 3 =0

As ζH and E are uniformly bounded on (0, 1], see (1.43) and (2.4), it then follows via the Moser inequality from Lemma A.3 that the derivative ∂k Φ satisfies the bound  t−1 ∂k Φ(t)H s+1 (T3 )) ≤ C δζ(t)H s (T3 ) δζ(t)H s (T3 ) uniformly for (t, ) ∈ (T1 , T0 ] × (0, 0 ), where C is independent of initial data and the times {T1 , T2 }. But, this implies via the definition of U, see (2.102), that  uL∞ ((T1 ,T0 ],H s (T3 )) ≤ C UL∞ ((T1 ,T0 ],H s (T3 )) UL∞ ((T1 ,T0 ],H s (T3 )) . Since (uμν , u, δζ, zi )L∞ ((T1 ,T0 ],H s (T3 )) ≤ UL∞ ((T1 ,T0 ],H s (T3 )) , we find that UL∞ ((T1 ,T0 ],H s (T3 )) < σ

(3.7)

implies that the inequalities (3.4) and (3.5) both hold. In particular, this shows that if (3.7) holds for σ > 0 small enough, then the solution (3.2) can be uniquely continued as a classical solution with the same regularity to the larger spacetime region (T1∗ , T0 ] × T3 for some T1∗ ∈ (0, T1 ). 

4. Conformal Cosmological Poisson–Euler Equations: Local Existence and Continuation In this section, we consider the local-in-time existence and uniqueness of solutions to the conformal cosmological Poisson–Euler equations, and we establish ζ, ˚ z j ). a continuation principle that is based on bounding the H s norm of (˚

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Proposition 4.1. Suppose s ∈ Z≥3 , ˚ ζ0 ∈ H s (T3 ) and (˚ z0i ) ∈ H s (T3 , R3 ). Then there exists a T1 ∈ (0, T0 ] and a unique classical solution Φ) ∈ (˚ ζ, ˚ zi, ˚

1 %

C  ((T1 , T0 ], H s− (T3 )) ×

=0

×

1 %

1 %

C  ((T1 , T0 ], H s− (T3 , R3 ))

=0

C  ((T1 , T0 ], H s+2− (T3 )),

=0

of the conformal cosmological Poisson–Euler equations, given by (1.55)–(1.57), on the spacetime region (T1 , T0 ] × T3 that satisfies ζ0 , ˚ z0i ) (˚ ζ, ˚ z i )|t=T0 = (˚ on the initial hypersurface t = T0 . Furthermore, if (˚ ζ, ˚ z i )L∞ ((T ,T ],H s ) < ∞, 1

0

i

Φ) can be uniquely continued as a classical solution then the solution (˚ ζ, ˚ z ,˚ with the same regularity to the larger spacetime region (T1∗ , T0 ] × T3 for some T1∗ ∈ (0, T1 ). ¯ s+2 −→ H ¯ s is an isomorphism, we can solve Proof. Using the fact that Δ : H the Poisson equation (1.51) by setting Λ ˚2 −1 ˚ ˚ Φ = tE Δ Πeζ . (4.1) 3 We can use this to write (1.55)–(1.57) as  3˚ Ω 3 j ˚ ˚ z ∂j ζ + ∂ j ˚ (4.2) zj = − , ζ+ ∂t ˚ Λ t   Λ Λ 1 j 1 ˚2 j −1 ˚ j i j j˚ ∂t˚ ˚ z − tE ∂ Δ Πeζ . z +˚ z ∂i˚ z + K∂ ζ = (4.3) 3 3 t 2 It is then easy to see that this system  can be cast in symmetric hyperbolic ˚2 K −1 3 . Even though the resulting system form by multiplying (4.3) by E Λ

is non-local due to the last term in (4.3), all of the standard local existence and uniqueness results and continuation principles that are valid for local symmetric hyperbolic systems, e.g., Theorems 2.1 and 2.2 of [41], continue to apply. Therefore it follows that there exists a unique local-in-time classical solution 1 1 % % (˚ ζ, ˚ zi) ∈ C  ((T1 , T0 ], H s− (T3 )) × C  ((T1 , T0 ], H s− (T3 , R3 )) (4.4) =0

=0

of (4.2)–(4.3) for some time T1 ∈ (0, T0 ) that satisfies (˚ ζ, ˚ z i )t=T0 = (˚ ζ0 , ˚ z0i ) z0i ) ∈ H s (T3 ) × H s (T3 , R3 ). Moreover, if the solution for given initial data (˚ ζ0 , ˚ satisfies ˚ ζL∞ ((T1 ,T0 ],W 1,∞ ) + ˚ z i L∞ ((T1 ,T0 ],W 1,∞ ) < ∞,

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then there exists a time T1∗ ∈ (0, T1 ) such that the solution (4.4) uniquely extends to the spacetime region (T1∗ , T0 ] × T3 with the same regularity. By Sobolev’s inequality, see Theorem A.1, this is clearly implied by the stronger condition (˚ ζ, ˚ z i )L∞ ((T1 ,T0 ],H s ) < ∞. Finally from (4.1), (4.4) and the Moser inequality from Lemma A.3, it is clear that 1 % ˚ Φ∈ C  ((T1 , T0 ], H s+2− (T3 )). =0

 Corollary 4.2. If the initial modified density ζ0 ∈ H s (T3 ) from Proposition 4.1 is chosen so that  ˚ ρH (T0 ) + ρ˘0 ˚ ζ0 = ln , T03 3 ¯ s (T3 ), and ˚ ρH (T0 ) + ρ˘0 > 0 in T3 , then the where ˚ ρH = 4C0 Λt3 2 , ρ˘0 ∈ H (C0 −t )

solution (˚ ζ, ˚ zi, ˚ Φ) to the conformal cosmological Poisson–Euler equations from Proposition 4.1 satisfies Π˚ ρ = δ˚ ρ := ˚ ρ−˚ ρH

in (T1 , T0 ] × T3 .

˚

Proof. Since ˚ ρ = t3 eζ satisfies (1.49), we see after applying 1, · to this equations that 1, ˚ ρ satisfies 3(1 − ˚ Ω(t)) d 1, ˚ ρ(t) = 1, ˚ ρ(t) , dt t while from the choice of initial data, we have

T1 < t ≤ T 0 ,

1, ˚ ρ(T0 ) = ˚ ρH (T0 ). 3

4C0 Λt By a direct computation, we observe with the help of (1.54) that ˚ ρH = (C 3 2 0 −t ) satisfies the differential equation d 3(1 − ˚ Ω(t)) ˚ ρH (t) = ˚ ρH (t) 0 < t ≤ T0 , (4.5) dt t and hence, that (4.6) 1, ˚ ρ(t) = ˚ ρH (t), T1 < t ≤ T0 , by the uniqueness of solutions to the initial value problem for ordinary differential equations. The proof now follows since (1.52)

(4.6)

Π˚ ρ = ˚ ρ − 1, ˚ ρ = ˚ ρ−˚ ρH (t)

in (T1 , T0 ] × T3 . 

Remark 4.3. Letting δ˚ ζ =˚ ζ −˚ ζH , where, see (1.10), (1.47) and (2.3), ˚ ρH ), ζH = ln(t−3˚

(4.7) (4.8)

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it is clear that the initial condition

 ˚ ρH (T0 ) + ρ˘0 ˚ ζ|t=T0 = ln , T03 from Corollary 4.2 is equivalent to the initial condition  ρ˘0 δ˚ ζ|t=T0 = ln 1 + ˚ ρH (T0 ) for δ˚ ζ.

5. Singular Symmetric Hyperbolic Systems In this section, we establish uniform a priori estimates for solutions to a class of symmetric hyperbolic systems that are jointly singular in  and t, and include both the formulation of the reduced conformal Einstein–Euler equations given by (2.104) and the   0 limit of these equations. We also establish error estimates, that is, a priori estimates for the difference between solutions of the -dependent singular symmetric hyperbolic systems and their corresponding   0 limit equations. The -dependent singular terms that appear in the symmetric hyperbolic systems we consider are of a type that have been well studied, see [5,29,30, 33,59,60], while the t-dependent singular terms are of the type analyzed in [51]. The uniform a priori estimates established here follow from combining the energy estimates from [5,29,30,33,59,60] with those from [51]. Remark 5.1. In this section, we switch to the standard time orientation, where the future is located in the direction of increasing time, while keeping the singularity located at t = 0. We do this in order to make the derivation of the energy estimates in this section as similar as possible to those for non-singular symmetric hyperbolic systems, which we expect will make it easier for readers familiar with such estimates to follow the arguments below. To get back to the time orientation used to formulate the conformal Einstein–Euler equations, see Remark 1.1, we need only apply the trivial time transformation t → −t. 5.1. Uniform Estimates We will establish uniform a priori estimates for the following class of equations: 1 1 (5.1) A0 ∂0 U + Ai ∂i U + C i ∂i U = APU + H in [T0 , T1 ) × Tn ,  t where U = (w, u)T ,  0 A1 (, t, x, w) 0 A0 = , 0 A02 (, t, x, w)  i A1 (, t, x, w) 0 , Ai = 0 Ai2 (, t, x, w)  i  C1 0 P1 0 , P = , Ci = 0 C2i 0 P2

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0 A= , A2 (, t, x, w)   F1 (, t, x) H1 (, t, x, w) , + H= H2 (, t, x, w, u) + R2 F2 (, t, x) 1 R2 = M2 (, t, x, w, u)P3 U, t and the following assumptions hold for fixed constants 0 , R > 0, T0 < T1 < 0 and s ∈ Z>n/2+1 : A1 (, t, x, w) 0

Assumptions 5.2. 1. The Cai , i = 1, . . . , n and a = 1, 2, are constant, symmetric Na × Na matrices. 2. The Pa , a = 1, 2, are constant, symmetric Na × Na projection matrices, i.e., P2a = Pa . We use P⊥ a = 1 − Pa to denote the complementary projection matrix. 3. The source terms Ha (, t, x, w),  a = 1, 2, Fa (, t, x), a = 1, 2, and M 2 (, t, x, w, u) satisfy H1 ∈ E 0 (0, 0 ) × (2T0 , 0) × Tn × BR (RN1 ), RN1 , H2 ∈ E 0 (0, 0 )×(2T0 , 0)×Tn ×BR (RN 1 )×BR (RN2 )×BR ((RN1 )n ), RN2 , Fa ∈ C 0 (0, 0 )×[T0 , T1 ), H s (T n , RNa ) , M2 ∈ E 0 (0, 0 )×(2T0 , 0)×Tn × BR (RN1 ) × BR (RN2 ), MN2 ×N2 , and H1 (, t, x, 0) = 0,

H2 (, t, x, 0, 0) = 0

and

M2 (, t, x, 0, 0) = 0

n

for all (, t, x) ∈ (0, 0 ) × (2T0 , 0) × T . i 4. The matrix  valued maps Aa (,nt, x, w), iN= 0, . . . , n and a = 1, 2, satisfy i 0 Aa ∈ E (0, 0 ) × (2T0 , 0) × T × BR (R a ), SNa . 5. The matrix valued maps A0a (, t, x, w), a = 1, 2, and Aa (, t, x, w), a = 1, 2, can be decomposed as ˚0a (t) + A˜0a (, t, x, w), (5.2) A0a (, t, x, w) = A ˜ a (, t, x, w), Aa (t) + A (5.3) Aa (, t, x, w) = ˚   0 1 1 0 ˚ ˜ ˚ where Aa ∈ E (2T0 , 0), SNa , Aa ∈ E (2T0 , 0), MNa ×Na , Aa ∈ E 1   N1 ˜ a ∈ E 0 (0, 0 ) × (2T0 , 0) × (0, 0 ) × (2T0 , 0) × Tn × , A B (R ), S R N a Tn × BR (RN1 ), MNa ×Na , and7 ˜ a (, t, x, 0) = Dx A˜0 (, t, x, 0) = 0 Dx A a

(5.4)

for all (, t, x) ∈ (0, 0 ) × (2T0 , 0) ∈ Tn . 6. For a = 1, 2, the matrix Aa commutes with Pa , i.e., [Pa , Aa (, t, x, w)] = 0 n

(5.5) N1

for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × T × B(R ). 7. P3 is a symmetric (N1 + N2 ) × (N1 + N2 ) projection matrix that satisfies PP3 = P3 P = P3 , i

P3 A 7

(, t, x, w)P⊥ 3

=

P3 C i P⊥ 3

=

(5.6) P3 A(, t, x, w)P⊥ 3

=0

˜ a |w=0 and A ˜0 |w=0 depend only on (, t). Or in other words, the matrices A a

(5.7)

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and [P3 , A0 (, t, x, w)] = 0 n

(5.8) N1

for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × T × BR (R ), where defines the complementary projection matrix. 8. There exists constants κ, γ1 , γ2 > 0, such that 1 1 1 ≤ A0a (, t, x, w) ≤ Aa (, t, x, w) ≤ γ2 1 γ1 κ

P⊥ 3

= 1 − P3

(5.9)

for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × Tn × B(RN1 ) and a = 1, 2. 9. For a = 1, 2, the matrix A0a satisfies 0 ⊥ 0 ⊥ ⊥ P⊥ a Aa (, t, x, P1 w)Pa = Pa Aa (, t, x, P1 w)Pa = 0 n

(5.10)

N1

for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × T × B(R ). 0 0 −1 A1 P1 w]P⊥ 10. For a = 1, 2, the matrix P⊥ a [Dw Aa · (A1 ) a can be decomposed as !  0 −1 " 0 P⊥ A1 (, t, x, w)P1 w P⊥ a Dw Aa (, t, x, w) · A1 (, t, x, w) a = tSa (, t, x, w) + Ta (, t, x, w, P1 w) (5.11)  for some Sa ∈ E 0 (0, 0 ) × (2T0 , 0) × Tn × BR (RN1 ), MNa ×Na , a = 1, 2, and Ta ∈ E 0 (0, 0 )×(2T0 , 0)×Tn ×BR (RN1 )×RN1 , MNa ×Na , a = 1, 2, where the Ta (, t, x, w, ξ) are quadratic in ξ. Before proceeding with the analysis, we take a moment to make a few observations about the structure of the singular system (5.1). First, if A = 0, then the singular term 1t APU disappears from (5.1) and it becomes a regular symmetric hyperbolic system. Uniform -independent a priori estimates that are valid for t ∈ [T1 , 0) would then follow, under a suitable small initial data assumption, as a direct consequence of the energy estimates from [5,29,30,33,59,60]. When A = 0, the positivity assumption (5.9) guarantees that the singular term 1t APU acts like a friction term. This allows us to generalize the energy estimates from [5,29,30,33,59,60] in such a way as to obtain, under a suitable small initial data assumption, uniform -independent a priori estimates that are valid on the time interval [T1 , 0); see (5.39), (5.40) and (5.41) for the key differential inequalities used to derive these a priori estimates. Remark 5.3. The equation for w decouples from the system (5.1) and is given by 1 1 A01 ∂0 w + Ai1 ∂i w + C1i ∂i w = A1 P1 w + H1 + F1 in [T0 , T1 ) × Tn .  t (5.12) Remark 5.4. 1. By Taylor expanding A0a (, t, x, P⊥ 1 w+P1 w) in the variable P1 w, it follows 1 ˆ0 ˘0 from (5.10) that there exist matrix  N valued maps Aa , Aa ∈ E (0, 0 ) × n 1 , MNa ×Na , a = 1, 2, such that (2T0 , 0) × T × BR R 0 ⊥ ˆ0 P⊥ a Aa (, t, x, w)Pa = Pa [Aa (, t, x, w) · P1 w]Pa

(5.13)

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and ⊥ ˘0 Pa A0a (, t, x, w)P⊥ a = Pa [Aa (, t, x, w) · P1 w]Pa

2.

(5.14)

for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × Tn × B(RN1 ). It is not difficult to see that the assumptions (5.9) and (5.10) imply that  0  −1 −1 ⊥ ⊥ Pa = Pa A0a (, t, x, P⊥ Pa = 0 P⊥ a Aa (, t, x, P1 w) 1 w) for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × Tn × B(RN1 ). By Taylor ex−1 in the variable P1 w, it follows that panding (A0a (, t, x, P⊥ 1 w + P1 w))  ˆa0 , B ˘a0 ∈ E 1 (0, 0 ) × (2T0 , 0) × Tn × there exist matrix valued maps B  BR RN1 , MNa ×Na , a = 1, 2, such that  0 −1 ˆ0 Pa = P⊥ (5.15) P⊥ a Aa (, t, x, w) a [Ba (, t, x, w) · P1 w]Pa and

 −1 ⊥ ˘ 0 (, t, x, w) · P1 w]P⊥ Pa A0a (, t, x, w) Pa = Pa [B a a

(5.16)

for all (, t, x, w) ∈ (0, 0 ) × (2T0 , 0) × Tn × B(RN1 ). To facilitate the statement and proof of our a priori estimates for solutions of the system (5.1), we introduce the following energy norms: Definition 5.5. Suppose w ∈ L∞ ([T0 , T1 ) × Tn , RN1 ), k ∈ Z≥0 , and {Pa , A0a }, a = 1, 2, are as defined above. Then for maps fa , a = 1, 2, and U from the torus Tn into RNa and RN1 × RN2 , respectively, the energy norms, denoted |||fa |||a,H s and |||U |||H s , of fa and U are defined by   2 Dα fa , A0a , t, ·, w(t, ·) Dα fa |||fa |||a,H k := 0≤|α|≤k

and 2

|||U |||H k :=



 Dα U, A0 , t, ·, w(t, ·) Dα U ,

0≤|α|≤k

respectively. In addition to the energy norms, we also define, for T0 < T ≤ T1 , the spacetime norm of maps fa , a = 1, 2, from [T0 , T ) × Tn to RNa by    12 T 1 2 Pa fa (t)H k dt . fa MP∞a ,k ([T0 ,T )×Tn ) := fa L∞ ([T0 ,T ),H k ) + − T0 t Remark 5.6. For w ∈ L∞ ([T0 , T1 )×Tn , RN1 ) satisfying wL∞ ([T0 ,T1 )×Tn ) < R, we observe, by (5.9), that the standard Sobolev norm  · H k and the energy norms ||| · |||a,H k , a = 1, 2, are equivalent since they satisfy 1 √ √  · H k ≤ ||| · |||a,H k ≤ γ2  · H k . γ1 With the preliminaries out of the way, we are now ready to state and prove a priori estimates for solutions of the system (5.1) that are uniform in .

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Theorem 5.7. Suppose R > 0, s ∈ Z≥n/2+1 , T0 < T1 < 0, 0 > 0,  ∈ (0, 0 ), Assumptions 5.2 hold, the map U = (w, u) ∈

1 %

C  ([T0 , T1 ), H s− (Tn , RN1 ))

=0

×

1 %

C  ([T0 , T1 ), H s−1− (Tn , RN2 ))

=0

defines a solution of the system (5.1), and for t ∈ [T0 , T1 ), the source terms Fa , a = 1, 2, satisfy the estimates F1 (, t)H s ≤ C(wL∞ ([T0 ,t),H s ) )w(t)H s

(5.17)

and F2 (, t)H s−1  ≤ C wL∞ ([T0 ,t),H s ) , uL∞ ([T0 ,t),H s−1 ) (w(t)H s + u(t)H s−1 ), 

(5.18)

where the constants C(wL∞ ([T0 ,t),H s ) ) and C wL∞ ([T0 ,t),H s ) , uL∞ ([T0 ,t),H s−1 ) ) are independent of  ∈ (0, 0 ) and T1 ∈ (T0 , 0]. Then there exists a σ > 0 independent of  ∈ (0, 0 ) and T1 ∈ (T0 , 0), such that if initially w(T0 )H s ≤ σ

and

u(T0 )H s−1 ≤ σ,

then

R 2 and there exists a constant C > 0, independent of  ∈ (0, 0 ) and T1 ∈ (T0 , 0), such that  t 1 P3 U H s−1 dτ ≤ Cσ wMP∞,s ([T0 ,t)×Tn ) + uMP∞,s−1 ([T0 ,t)×Tn ) − 1 2 τ T0 wL∞ ([T0 ,T1 )×Tn ) ≤

for T0 ≤ t < T1 . Proof. Letting CSob denote the constant from the Sobolev inequality, we have that w(T0 )L∞ ≤ CSob w(T0 )H s ≤ CSob σ. We then choose σ to satisfy  ˆ R σ ≤ min 1, , (5.19) 4 ˆ= where R

R 2CSob ,

so that w(T0 )L∞ ≤

R . 8

Next, we define K1 (t) = wL∞ ([T0 ,t),H s )

and K2 (t) = uL∞ ([T0 ,t),H s−1 ) ,

ˆ and observe that K1 (T0 ) + K2 (T0 ) ≤ R/2, and hence, by continuity, either ˆ K1 (t) + K2 (t) < R for all t ∈ [T0 , T1 ), or else there exists a first time T∗ ∈

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ˆ Letting T∗ = T1 if the first case holds, (T0 , T1 ) such that K1 (T∗ )+K2 (T∗ ) = R. we then have that ˆ K1 (t) + K2 (t) < R,

0 ≤ t < T∗ ,

(5.20)

where T∗ = T1 or else T∗ is the first time in (T0 , T1 ) for which K1 (T∗ ) + ˆ K2 (T∗ ) = R. Before proceeding the proof, we first establish a number of preliminary estimates, which we collect together in the following Lemma. Lemma 5.8. There exists constants C(K1 (t)) and C(K1 (t), K2 (t)), both independent of  ∈ (0, 0 ) and T∗ ∈ (T0 , T1 ], such that the following estimates hold for T0 ≤ t < T∗ < 0: −

2  1 Dα w, A01 [(A01 )−1 A1 , Dα ]P1 w ≤ − C(K1 )wH s P1 w2H s , (5.21) t t |α|≤s

2 t

−



Dα u, A02 [(A02 )−1 A2 , Dα ]P2 u

|α|≤s−1

1 ≤ − C(K1 )(uH s−1 + wH s )(P2 u2H s−1 + P2 w2H s ), t Dα w, A01 [Dα , (A01 )−1 Ai1 ]∂i w ≤ C(K1 )w2H s , −

(5.22) (5.23)

|α|≤s



−

Dα u, A02 [Dα , (A02 )−1 Ai2 ]∂i u ≤ C(K1 )u2H s−1 ,

(5.24)

|α|≤s−1



−

Dα w, [A˜01 , Dα ](A01 )−1 C1i ∂i w ≤ C(K1 )w2H s ,

(5.25)

|α|≤s



−  |α|≤s

Dα u, [A˜02 , Dα ](A02 )−1 C2i ∂i u ≤ C(K1 )u2H s−1 ,

(5.26)

|α|≤s−1

1 D w, (∂t A01 )Dα w ≤ C(K1 )w2H s − C(K1 )wH s P1 w2H s , t α



(5.27)

1 Dα u, (∂t A02 )Dα u ≤ C(K1 )u2H s−1 − C(K1 , K2 )(uH s−1 t

|α|≤s−1

+ wH s )(P2 u2H s−1 + P1 w2H s )

(5.28)

and 

1 Dα P3 U, (∂t A0 )Dα P3 U ≤ − C(K1 )P1 wH s P3 U 2H s−1 t

|α|≤s−1

+ C(K1 )P3 U 2H s−1 .

(5.29) 

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T Proof. Using the properties P21 = P1 , P1 + P⊥ 1 = 1, P1 = P1 , and DP1 = 0 of the projection matrix P1 repeatedly, we compute

−

2  Dα w, A01 [(A01 )−1 A1 , Dα ]P1 w t |α|≤s

=−

2  Dα P1 w, A01 [(A01 )−1 A1 , Dα ]P1 w t |α|≤s

2  ⊥ 0 0 −1 − Dα P⊥ A1 , Dα ]P1 w 1 w, P1 A1 [(A1 ) t |α|≤s

=−

2  Dα P1 w, A01 [(A01 )−1 A1 , Dα ]P1 w t |α|≤s

−

2  ⊥ 0 0 −1 Dα P⊥ P1 A1 , Dα ]P1 w 1 w, P1 A1 [(A1 ) t

(by (5.5))

|α|≤s

=−

2  Dα P1 w, A01 [(A01 )−1 A1 , Dα ]P1 w t |α|≤s

−

2  ⊥ 0 ⊥ ⊥ 0 −1 Dα P⊥ P1 A1 , Dα ]P1 w 1 w, P1 A1 P1 [P1 (A1 ) t |α|≤s

2  ⊥ 0 0 −1 − Dα P⊥ P1 A1 , Dα ]P1 w . 1 w, P1 A1 P1 [P1 (A1 ) t |α|≤s

From this expression, we obtain, with the help the Cauchy–Schwarz inequality, the calculus inequalities from “Appendix A,” the expansions (5.2)–(5.3), the relations (5.4), (5.13), and (5.15), and the inequality (5.20), the estimate −

1  Dα w, A01 [(A01 )−1 A1 , Dα ]P1 w t |α|≤s

 1  − A01 H s P1 wH s D (A01 )−1 A1 ]H s−1 t  ⊥ 0 −1 + A01 H s P⊥ P1 A1 H s−1 1 wH s D P1 (A1 )

 ⊥ 0 ⊥ 0 −1 + P1 A1 P1 H s P1 wH s D P1 (A1 ) P1 A1 H s−1 P1 wH s−1 1 ≤ −C(K1 ) wH s P1 w2H s t

for T0 ≤ t < T∗ , where the constant C(K1 ) is independent of  ∈ (0, 0 ) and T∗ ∈ (T0 , T1 ]. This establishes the estimate (5.21). By a similar calculation, we find that

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Dα u, A02 [(A02 )−1 A2 , Dα ]P2 u

|α|≤s−1

=−

2 t

2 − t 2 − t



Dα P2 u, A02 [(A02 )−1 A2 , Dα ]P2 u

|α|≤s−1



⊥ 0 ⊥ ⊥ 0 −1 Dα P⊥ P2 A2 , Dα ]P2 u 2 u, P2 A2 P2 [P2 (A2 )

|α|≤s−1



⊥ 0 0 −1 Dα P⊥ P2 A2 , Dα ]P2 u 2 u, P2 A2 P2 [P2 (A2 )

|α|≤s−1

1 1 ≤ − C(K1 )wH s P2 u2H s−1 − C(K1 )uH s−1 P1 wH s P2 uH s−1 t t 1 − C(K1 )uH s−1 P1 wH s P2 uH s−1 t 1 ≤ − C(K1 )(uH s−1 + wH s )(P2 u2H s−1 + P2 w2H s ), t which establishes the estimate (5.22). Next, using the calculus inequalities from “Appendix A,” we observe that  Dα u, −A02 [Dα , (A02 )−1 Ai2 ]∂i u 0≤|α|≤s−1

 A02 L∞ u2H s−1 D((A02 )−1 Ai2 )H s−1 ≤ C(K1 )u2H s−1 , which establishes the estimate (5.24). Since the estimates (5.23), (5.25) and (5.26) can be obtained in a similar fashion, we omit the details. Finally, we consider the estimates (5.27)–(5.28). We begin establishing these estimates by writing (5.12) as 1 ∂0 w =  (A01 )−1 A1 P1 w − (A01 )−1 Ai1 ∂i w − (A01 )−1 C1i ∂i w t + (A01 )−1 H1 + (A01 )−1 F1 . Using this and the expansion (5.2), we can express the time derivatives ∂t A0a , a = 1, 2, as ∂t A0a = Dw A0a · ∂t w + Dt A0a = −Dw A0a · (A01 )−1 Ai1 ∂i w − [Dw A˜0a · (A01 )−1 C1i ∂i w] + [Dw A0a · (A01 )−1 H1 ] + Dt A0a + [Dw A0a · (A01 )−1 F1 ] 1 + [Dw A0a · (A01 )−1 A1 P1 w]. t

(5.30)

Using (5.30) with a = 2, we see, with the help of the calculus inequalities from “Appendix A,” the Cauchy–Schwarz inequality, the estimate (5.17), and the expansion (5.11) for a = 2, that

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Dα u, (∂t A02 )Dα u

|α|≤s−1

≤

 ! 0 ⊥ α α ⊥ 0 α Dα u, P⊥ 2 (∂t A2 )P2 D u + D u, P2 (∂t A2 )P2 D u

|α|≤s−1

" α α 0 α + Dα u, P2 (∂t A02 )P⊥ 2 D u + D u, P2 (∂t A2 )P2 D u 2 ≤ C(K1 )u2H s−1 − uH s−1 (A01 )−1 A1 L∞ Dw A02 L∞ P2 uH s−1 t 1 × P1 wH s−1 − P1 wH s (A0 )−1 AL∞ Dw A02 L∞ P2 u2H s−1 t 1 2 − uH s−1 C(K1 )P1 w2H s−1 t 1 ≤ C(K1 )u2H s−1 − C(K1 , K2 )(uH s−1 +wH s )(P2 u2H s−1 +P1 w2H s ). t This establishes the estimate (5.28). Since the estimate (5.27) can be established using similar arguments, we omit the details. The last estimate (5.29) can also be established using similar arguments with the help of the identity  P3 P = PP3 = P3 . We again omit the details. Applying A0 Dα (A0 )−1 to both sides of (5.1), we find that 1 A0 ∂0 Dα U + Ai ∂i Dα U + C i ∂i Dα U  = −A0 [Dα , (A0 )−1 Ai ]∂i U − [A˜0 , Dα ](A0 )−1 C i ∂i U 1 1 + ADα PU + A0 [Dα , (A0 )−1 A]PU + A0 Dα [(A0 )−1 H], t t where in deriving this we have used

(5.31)

1 0 α 0 −1 i (5.2) 1 ˚0 + A˜0 , Dα ](A0 )−1 C i ∂i U [A , D ](A ) C ∂i U = [A   =[A˜0 , Dα ](A0 )−1 C i ∂i U and

  A0 [Dα , (A0 )−1 ]C i ∂i U = A0 Dα (A0 )−1 C i ∂i U − Dα C i ∂i U  = A0 Dα (A0 )−1 C i ∂i U − Dα (A0 (A0 )−1 C i ∂i U = [A0 , Dα ](A0 )−1 C i ∂i U. 1

1

Writing A0a , a = 1, 2, as A0a = (A0a ) 2 (A0a ) 2 , which we can do since A0a is a real symmetric and positive-definite, we see from (5.9) that 1

1

(A0a )− 2 Aa (A0a )− 2 ≥ κ1.

(5.32)

Since, by (5.5), 1 1 1 1 2 2 α D f, Aa Dα Pa f = Dα Pa f, (A0 ) 2 [(A0a )− 2 Aa (A0a )− 2 ](A0a ) 2 Dα Pa f , t t a = 1, 2,

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it follows immediately from (5.32) that 

2 t

Dα u, A2 Dα P2 u ≤

0≤|α|≤s−1

and 2 t



Dα w, A1 Dα P1 w ≤

0≤|α|≤s

2κ 2 |||P2 u|||2,H s−1 t

2κ 2 |||P1 w|||1,H s . t

(5.33)

Then, differentiating Dα w, A01 Dα w with respect to t, we see, the identities Dα w, C1i ∂i Dα w = 0 and 2 Dα w, Ai1 ∂i Dα w = − Dα w, (∂i Ai1 )Dα w , the block decomposition of (5.31), which we can use to determine Dα ∂t w, the estimates (5.17) and (5.33) together with those from Lemma 5.8 and the calculus inequalities from “Appendix A,” that   2 Dα w, (∂t A01 )Dα w + 2 Dα w, A01 Dα ∂t w ∂t |||w|||1,H s = 0≤|α|≤s

0≤|α|≤s

1 ≤ C(K1 )w2H s − C(K1 )wH s P1 w2H s + t



Dα w, (∂i Ai1 )Dα w

0≤|α|≤s

=0

−

2 

−2

() *  '  Dα w, A01 [Dα , (A01 )−1 Ai1 ]∂i w Dα w, C1i ∂i Dα w − 2 0≤|α|≤s



0≤|α|≤s

2 + t +2

0≤|α|≤s

2 Dα w, [A˜01 , Dα ](A01 )−1 C1i ∂i w + t





Dα w, A1 Dα P1 w

0≤|α|≤s

Dα w, A01 [(A01 )−1 A1 , Dα ]P1 w

0≤|α|≤s



Dα w, A01 Dα [(A01 )−1 (H1 + F1 )]

0≤|α|≤s 2

≤ C(K1 )|||w|||1,H s +

" 1! 2 2κ − C1 (K1 )wH s |||P1 w|||1,H s t

(5.34)

for t ∈ [T0 , T∗ ). By similar calculation, we obtain from differentiating Dα u, A02 Dα u with respect to t the estimate   2 Dα u, (∂t A02 )Dα u + 2 Dα u, A02 Dα ∂t u ∂t |||u|||2,H s−1 = 0≤|α|≤s−1

0≤|α|≤s−1

1 ≤ C(K1 )u2H s−1 − C(K1 , K2 )(uH s−1 +wH s )(P2 u2H s−1 +P1 w2H s ) t 

D

α

u, (∂i Ai2 )Dα u

0≤|α|≤s−1

−2



0≤|α|≤s−1

D

α

2 − 

 0≤|α|≤s−1

u, A02 [Dα , (A02 )−1 Ai2 ]∂i u

=0

' () * Dα u, C2i ∂i Dα u

Vol. 19 (2018)

−2

Newtonian Limits on Long Time Scales



Dα u, [A˜02 , Dα ](A02 )−1 C2i ∂i u

0≤|α|≤s−1

2 + t +2

2205



Dα u, A2 Dα P2 u −

0≤|α|≤s





0≤|α|≤s−1

2 t



Dα u, A02 [(A02 )−1 A2 , Dα ]P2 u

0≤|α|≤s−1

 1 Dα u, A02 Dα [(A02 )−1 H2 + M2 P3 U + F2 ] t 2

2

≤ C(K1 , K2 )(|||u|||2,H s−1 + |||w|||1,H s ) 1 2 − C2 (K1 , K2 )(uH s−1 + wH s )|||P1 w|||1,H s 2t " 1! 2 2κ − C2 (K1 , K2 )(uH s−1 + wH s ) |||P2 u|||2,H s−1 + t 1 2 2 − C(K1 ) (|||u|||2,H s−1 + |||w|||1,H s )|||P3 U |||H s−1 (5.35) t for t ∈ [T0 , T∗ ). Applying the operator A0 Dα P3 (A0 )−1 to (5.1), we see, with the help of (5.6)–(5.8), that 1 A0 ∂0 Dα P3 U + P3 Ai P3 ∂i Dα P3 U + P3 C i P3 ∂i Dα P3 U  = −A0 [Dα , (A0 )−1 P3 Ai P3 ]∂i P3 U − [A˜0 , Dα ](A0 )−1 P3 C i P3 ∂i P3 U 1 1 + P3 AP3 Dα P3 U + A0 [Dα , (A0 )−1 P3 AP3 ]P3 U + A0 Dα [(A0 )−1 P3 H]. t t (5.36) Then, by similar arguments used to derive (5.34) and (5.35), we obtain from (5.36) the estimate ∂t |||P3 U |||2H s−1  = Dα P3 U, (∂t A0 )Dα P3 U  + 2 0≤|α|≤s−1



Dα P3 U, P3 A0 P3 Dα ∂t P3 U 

0≤|α|≤s−1

1 ≤ − C(K1 )P1 wH s P3 U 2H s−1 + C(K1 )P3 U 2H s−1 t 

+

2 Dα P3 U, (∂i Ai )Dα P3 U  − 

0≤|α|≤s−1

− 2



α

0

α

0 −1

D P3 U, A [D , (A )

0≤|α|≤s−1

2 + t + 2



Dα P3 U, ADα P3 U  +

0≤|α|≤s



0≤|α|≤s−1

#

α

0

α

2 t

=0

' () * Dα P3 U, C i ∂i Dα P3 U 

0≤|α|≤s−1

i

A ]∂i P3 U + [A˜0 , Dα ](A0 )−1 C i ∂i P3 U  

Dα P3 U, A0 [(A0 )−1 A, Dα ]P3 U 

0≤|α|≤s−1

0 −1

D P3 U, A D [(A )



P3 H]

$

 ≤ C(K1 )P3 U 2H s−1 + C(K1 )P3 U H s−1 H1 H s−1 + H2 H s−1 + F1 H s−1  1   2κ − C2 (K1 , K2 ) wH s + uH s−1 |||P3 U |||2H s−1 + F2 H s−1 + t

2206

C. Liu, T. A. Oliynyk

Ann. Henri Poincar´e

 ≤ C(K1 )|||P3 U |||2H s−1 + C(K1 , K2 ) |||w|||1,H s + |||u|||2,H s−1 ) |||P3 U |||H s−1   1 2κ − C2 (K1 , K2 ) wH s + uH s−1 |||P3 U |||2H s−1 . + t

Dividing the above estimate by |||P3 U |||H s−1 gives

 ∂t |||P3 U |||H s−1 ≤ C(K1 )|||P3 U |||H s−1 + C(K1 , K2 ) |||w|||1,H s + |||u|||2,H s−1 )  1 C2 (K1 , K2 )  wH s + uH s−1 |||P3 U |||H s−1 . + κ− t 2 (5.37)

Next, we choose σ > 0 small enough so that   ˆ + 2C2 (R, ˆ R) ˆ σ<κ C1 (R) 2 in addition to (5.19). Then since  2κ − C1 (K1 (T0 ))w(T0 )H s + C2 (K1 (T0 ), K2 (T0 ))   × w(T0 )H s + u(T0 )H s−1 > κ, we see by continuity that either    2κ − C1 (K1 (t))w(t)H s + C2 (K1 (t), K2 (t)) w(t)H s + u(t)H s−1 > κ, 0 ≤ t < T∗ , or else there exists a first time T ∗ ∈ (0, T∗ ) such that  2κ − C1 (K1 (T ∗ ))w(T ∗ )H s + C2 (K1 (T ∗ ), K2 (T ∗ ))(w(T ∗ )H s  + u(T ∗ )H s−1 = κ. Thus if we let T ∗ = T∗ if the first case holds, then we have that    2κ − C1 (K1 (t))w(t)H s +C2 (K1 (t), K2 (t)) w(t)H s +u(t)H s−1 > κ, 0 ≤ t < T ∗ ≤ T∗ .

(5.38)

Taken together, the estimates (5.20), (5.34), (5.35), (5.37) and (5.38) imply that κ 2 2 2 ˆ ∂t |||w|||1,H s ≤ C(R)|||w||| (5.39) 1,H s + |||P1 w|||1,H s , t  2 ˆ |||u|||2 s−1 + |||w|||2 s ∂t |||u|||2,H s−1 ≤ C(R) 2,H 1,H  1 2 ˆ |||u||| s−1 + |||w|||2 s |||P3 U ||| s−1 − C3 (R) H 2,H 1,H t κ κ 2 2 + |||P1 w|||1,H s + |||P2 u|||2,H s−1 (5.40) 2t t and  ˆ |||P3 U ||| s−1 + |||w||| s + |||u||| s−1 + κ |||P3 U ||| s−1 ∂t |||P3 U |||H s−1 ≤C(R) H 1,H 2,H H 2t (5.41) for 0 ≤ t < T ∗ ≤ T∗ .

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Newtonian Limits on Long Time Scales

2207

Next, we set 2

2

2

2

X = |||w|||1,H s + |||u|||2,H s−1 , Y = |||P1 w|||1,H s + |||P2 u|||2,H s−1 , and Z = |||P3 U |||H s−1 . ˆ ˆ ˆ Since C3 (R)X(T 0 )/σ ≤ C(R)σ, we can choose σ small enough so that C3 (R) ˆ X(T0 )/σ < κ/4. Then by continuity, either C3 (R)X(t)/σ ≤ κ/4 for t ∈ ˆ [T0 , T ∗ ), or else there exists a first time T ∈ (T0 , T ∗ ) such that C3 (R)X(T )/σ = κ/4. Thus if we set T = T ∗ if the first case holds, then we have that ˆ C3 (R)

X(t) < κ/4, σ

T0 ≤ t < T ≤ T ∗ ≤ T ∗ .

(5.42)

Adding the inequalities (5.39) and (5.40) and dividing the results by σ, we obtain, with the help of (5.42), the inequality  X ˆ X − κ Z + κ Y , T0 ≤ t < T ≤ T ∗ ≤ T ∗ , ∂t (5.43) ≤ C(R) σ σ 4t 2t σ while the inequality  ˆ Z + σ + X + κ Z, ∂t Z ≤ C(R) σ 2t

T0 ≤ t < T ∗ ≤ T∗

(5.44)

follows from (5.41) and Young’s inequality. Adding (5.43) and (5.44), we find that    X κ t 1 Y ∂t +Z − + Z dτ + σ σ 4 T0 τ σ   t  1 Y ˆ X +Z − κ ≤ C(R) + Z dτ + σ (5.45) σ 4 T0 τ σ ˆ 2 and Z(T0 )  σ, it follows for T0 ≤ t < T ≤ T ∗ ≤ T∗ . Since X(T0 ) ≤ C(R)σ directly from (5.45) and Gr¨ onwall’s inequality that   X κ t 1 Y ˆ ˆ +Z − +Z dτ +σ ≤ eC(R)(t−T0 ) C(R)σ, T 0 ≤ t < T ≤ T ∗ ≤ T∗ , σ 4 T0 τ σ from which it follows that  wMP∞,s ([T0 ,t)×Tn ) + uMP∞,s−1 ([T0 ,t)×Tn ) − 1

T0 ≤ t < T ≤ T ∗ ≤ T ∗ ,

2

t

T0

1 ˆ P3 U H s−1 dτ ≤ C(R)σ, τ (5.46)

ˆ is independent of  and the times T , where we stress that the constant C(R) ∗ T , T∗ , and T1 . Choosing σ small enough, it is then clear from the estimate (5.46) and the definition of the times T , T ∗ , and T1 that T = T ∗ = T∗ = T1 , which completes the proof. 

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C. Liu, T. A. Oliynyk

Ann. Henri Poincar´e

5.2. Error Estimates In this section, we consider solutions of the singular initial value problem 1 A01 (, t, x, w)∂0 w + Ai1 (, t, x, w)∂i w + C1i ∂i w  1 (5.47) = A1 (, t, x, w)P1 w + H1 + F1 in [T0 , T1 ) × Tn , t 0 0 n w(x)|t=T0 = w ˚ (x) + s (, x) in {T0 } × T , (5.48) where the matrices A01 , Ai1 , i = 1, . . . , n, and A1 and the source terms H1 and F1 satisfy the conditions from Assumption 5.2. Our aim is to use the uniform a priori estimates from Theorem 5.7 to establish uniform a priori estimates for solutions of (5.47)–(5.48) and to establish an error estimate between solutions of (5.47)–(5.48) and solutions of the limit equation, which is defined by 1˚ ˚i ∂i w ˚1 + F ˚1 in [T0 , T1 ) × Tn , (5.49) ˚0 ∂0 w ˚+ A ˚ − C1i ∂i v + H A 1 1 ˚ = A1 P1 w t C1i ∂i w ˚= 0 in [T0 , T1 ) × Tn , (5.50) ˚0 (x) w ˚(x)|t=T0 = w

in

{T0 } × Tn .

(5.51)

˚0 and ˚ In this system, A A1 are defined by (5.2) and (5.3) with a = 1, respec1 i ˚ ˚ tively, A1 and H1 are defined by the limits ˚i1 (t, x, w A ˚) = lim Ai1 (, t, x, w ˚) 0

˚1 (t, x, w and H ˚) = lim H1 (, t, x, w ˚), (5.52) 0

respectively, and the following assumptions hold for fixed constants R > 0, T0 < T1 < 0 and s ∈ Z>n/2+1 : Assumptions 5.9.

 ˚1 and v satisfy F ˚1 ∈ C 0 [T0 , T1 ), H s (Tn , RN1 ) and 1. The source terms8 F  &1 v ∈ =0 C  [T0 , T1 ), H s+1− (Tn , RN1 ) . ˚i , i = 1, . . . , n and the source term H ˚1 satisfy9 tA ˚i 2. The matrices A 1  N   N1 ∈ 1 n 1 n 1 ˚1 ∈ E (2T0 , 0) × T × BR R 1 , E (2T , SN1 , tH 0 , 0) × T × BR R RN1 , and  ˚1 (t, x, 0 = 0. Dt tH

We are now ready to state and establish uniform a priori estimates for solutions of the singular initial value problem (5.47)–(5.48) and the associated limit equation defined by (5.49)–(5.51). , T0 < T1 ≤ 0, 0 > 0, w ˚0 ∈ Theorem 5.10. Suppose R > 0, s ∈ Z>n/2+1 H s (Tn , RM ), s0 ∈ L∞ (0, 0 ), H s (Tn , RN1 ) , Assumptions 5.2 and 5.9 hold, the maps 8

˚1 should be thought of as the   0 limit of F1 . This is made precise The source term F by the hypothesis (5.57) of Theorem 5.10. 9 From the assumptions, see Assumption 5.2.(5.2)–(5.2), on Ai and H , it follows directly 1   1  ˚i ∈ E 0 (2T0 , 0) × Tn × BR RN1 , SN and H ˚1 ∈ E 0 (2T0 , 0) × Tn × from the (5.52) that A 1 1  N N BR R 1 , R 1 .

Vol. 19 (2018)

(w, w ˚) ∈

1 %

Newtonian Limits on Long Time Scales

2209

1   %   C  [T0 , T1 ), H s− Tn , RN1 × C  [T0 , T1 ), H s− Tn , RN1

=0

=0

define a solution to the initial value problems (5.47)–(5.48) and (5.49)–(5.51), and for t ∈ [T0 , T1 ), the following estimate holds:  1 w(t)H s , wL∞ ([T0 ,t),H s ) ˚ v(t)H s+1 − P1 v(t)H s+1 + ∂t v(t)H s ≤ C ˚ t (5.53)  ˚1 (t)H s−1 ≤ C ˚ ˚1 (t)H s + t∂t F w(t)H s , F wL∞ ([T0 ,t),H s ) ˚ (5.54)  F1 (, t)H s ≤ C wL∞ ([T0 ,t),H s ) w(t)H s , (5.55)  ˚i (t, ·, w ˚(t)) − A ˚(t))H s−1 ≤ C ˚ w(t)L∞ ([T0 ,t),H s ) (5.56) Ai1 (, t, ·, w 1 and ˚1 (t, ·, w ˚1 (t)H s−1 H1 (, t, ·, w ˚(t)) − H ˚(t))H s−1 + F1 (, t) − F  ≤ C wL∞ ([T0 ,t),H s ) , ˚ wL∞ ([T0 ,t),H s ) (w(t)H s + z(t)H s−1 + ˚ w(t)H s ),

(5.57)

where

1 ˚ − v) z = (w − w    ∞ ([T ,t),H s ) , ∞ ([T ,t),H s ) C ˚ w  and and the constants C w L L 0 0  C wL∞ ([T0 ,Tt ),H s ) , ˚ wL∞ ([T0 ,t),H s ) are independent of  ∈ (0, 0 ) and the time T1 ∈ (T0 , 0). Then there exists a small constant σ > 0, independent of  ∈ (0, 0 ) and T1 ∈ (T0 , 0), such that if initially ˚ w0 H s + s0 H s ≤ σ

and

C1i ∂i w ˚0 = 0,

(5.58)

then

R (5.59) 2 and there exists a constant C > 0, independent of  ∈ (0, 0 ) and T1 ∈ (T0 , 0), such that max{wL∞ ([T0 ,T1 )×Tn ) , wL∞ ([T0 ,T1 )×Tn ) } ≤

∞ wMP∞,s ([T0 ,T1 )×Tn ) + ˚ wMP∞,s ([T0 ,T1 )×Tn ) + t∂t w ˚M1,s−1 ([T0 ,T1 )×Tn ) 1 1  t  t 1 P1 w + ∂t w ˚H s−1 dτ − ˚H s−1 dτ ≤ Cσ, (5.60) T0 T0 τ (5.61) w − w ˚L∞ ([T0 ,t),H s−1 ) ≤ Cσ

and

 −

for T0 ≤ t < T1 .

t

T0

1 P1 (w − w ˚)2H s−1 dτ ≤ 2 Cσ 2 τ

(5.62)

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C. Liu, T. A. Oliynyk

Ann. Henri Poincar´e

Proof. First, we observe, by (5.2) and (5.10), that A01 satisfies ˚0 ˚0 ⊥ P⊥ 1 A1 P1 = P1 A1 P1 .

(5.63)

Using this, we find, after applying P1 to the limit Eq. (5.49), that b = P1 w ˚

(5.64)

˚01 P1 ∂t b + P1 A ˚i1 P1 ∂i b = 1 P1 ˚ ˚1 + P1 F¯2 , P1 A A1 P1 b + P1 H t

(5.65)

satisfies the equation

where ˚i1 P⊥ ˚1 − P1 C1i ∂i v. F¯2 = −P1 A ˚ + P1 F 1 ∂i w Clearly, F¯2 satisfies  w(t)H s F¯2 (t)H s−1 ≤ C ˚ wL∞ (T0 ,t),H s ˚

(5.66)

for 0 ≤ t < T1 by (5.53), (5.54) and the calculus inequalities from “Appendix A,” while b(T0 )H s−1 ≤ ˚ w0 H s ≤ σ, (5.67) ˚1 (t, x, w ˚) satisfies by the assumption (5.58) on the initial data, and P1 H ˚1 (t, x, 0) = 0 P1 H

(5.68)

by Assumption 5.2.(3). Next, we set y = t∂t w ˚. In order to derive an evolution equation for y, we apply t∂t to (5.49) and use the identity ˚] = Dt (tf ) − f + [Dw ˚], t∂t f = tDt f + [Dw ˚f · t∂t w ˚f · t∂t w

f = f (t, x, w ˚(t, x)),

to obtain

 ˚0 ∂t y + A ˚i ∂i y = 1 P1 ˚ ˚0 y − 1 ˚ ˜2 + H ˜ 2 + F˜2 , A A1 P1 + A A1 b + R 1 1 1 t t where ˚ ˜ 2 = Dt (tH ˚ ˚0 ˚1 ) − H ˚1 + [Dw H ˚H1 · y] + (Dt A1 )b − (Dt A )y

(5.69)

1

and ˚i ˚1 + tC1i ∂i ∂t v. ˚i1 )∂i w ˚i1 ∂i w F˜2 = −[Dw ˚ − Dt (tA ˚+ A ˚ + t∂t F ˚A1 · y]∂i w Note that in deriving the above equation, we have used the identity ˚ A1 P1 = P1 ˚ A1 = P1 ˚ A1 P1 ,

(5.70)

which follows directly from (5.3) and (5.5). We further note by (5.53), (5.54) ˜2 = H ˜ 2 (t, x, w and Assumptions 5.2.(4) and 5.9.(2), it is clear that F˜2 and H ˚, b, y) satisfy  wH s (yH s−1 + ˚ wH s ) (5.71) F˜2 (t)H s−1 ≤ C ˚ for T0 ≤ t < T1 and ˜ 2 (t, x, 0, 0, 0) = 0, H (5.72)

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Newtonian Limits on Long Time Scales

2211

respectively. Using (5.49) and (5.58), we see that  ˚i ∂i w ˚0 )−1 ˚ ˚0 )−1 A ˚0 −1 C i ∂i v ˚ − t(A A1 P1 w y|t=T0 = (A 1 1 1 ˚ − t(A1 ) 1  ˚1 + t(A ˚1  ˚01 )−1 H ˚01 )−1 F + t(A , t=T0

which in turn, implies, via (5.58), (5.53)–(5.54), and the calculus inequalities from “Appendix A,” that yH s−1 (T0 ) ≤ C(σ)σ. A short computation using (5.47), (5.49) and (5.50) shows that 1 1 ˆ 2 + Fˆ2 , A01 ∂t z + Ai1 ∂i z + C1i ∂i z = A1 P1 z + R (5.73)  t where 1 ˚1 ) + 1 (F1 − F ˚1 ) − 1 (Ai1 − A ˚i1 )∂i w Fˆ2 = (H1 − H ˚ − Ai1 ∂i v − A01 ∂t v    1 + P1 A1 P1 v t and ˜ 1 b, ˆ 2 = − 1 A˜0 y + 1 A R t 1 t ˜ 1 are defined by the expansions (5.2)–(5.3). Next, and we recall that A˜01 and A we estimate 1 ˚1 (t, ·, w H1 (, t, ·, w(t)) − H ˚(t))H s−1  1 ≤ H1 (, t, ·, w(t)) − H1 (, t, ·, w ˚(t))H s−1  1 ˚1 (t, ·, w + H1 (, t, ·, w ˚(t)) − H ˚(t))H s−1   ≤ C wL∞ ([T0 ,t),H s ) , ˚ wL∞ ([T0 ,t),H s ) w(t)H s ), × (w(t)H s + z(t)H s−1 + ˚

(5.74)

for T0 ≤ t < T1 , where in deriving the second inequality, we used (5.57), Taylor’s Theorem (in the last variable), and the calculus inequalities. By similar arguments and (5.56), we also get that 1 ˚i1 (t, ·, w (Ai1 (, t, ·, w(t)) − A ˚(t))H s−1   ≤ C wL∞ ([T0 ,t),H s ) , ˚ wL∞ ([T0 ,t),H s ) (w(t)H s +z(t)H s−1 +˚ w(t)H s ), (5.75) again for T0 ≤ t < T1 . Taken together, the estimates (5.53), (5.57), (5.74) and (5.75) along with the calculus inequalities imply that Fˆ2 (, t)H s−1 ≤ C(wL∞ ([T0 ,t),H s ) , ˚ wL∞ ([T0 ,t),H s ) )(w(t)H s +z(t)H s−1 +˚ w(t)H s ) (5.76)

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Ann. Henri Poincar´e

for T0 ≤ t < T1 . Furthermore, we see from (5.53) and (5.58) that we can estimate z at t = T0 by zH s−1 (T0 ) ≤ C(σ)σ. 

(5.77)

We can combine two Eqs. (5.47) and (5.49) together into the equation    i   1 C1i 0 A01 0 0 A1 w w w ∂i + + ˚0 ∂t w ˚i ∂i w 0 0 ˚ ˚ w ˚ 0 A 0 A  1 1      1 A1 0 F1 H1 P1 w 0 = , + ˚ + ˚ ˚ w ˚ 0 P1 H1 F1 − C1i ∂i v A1 t 0 (5.78)

and collect three Eqs. (5.65), (5.69) and (5.73) together into the equation ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ b b b b 1 1 A02 ∂t ⎝y ⎠ + Ai2 ∂i ⎝y ⎠ + C2i ∂i ⎝y ⎠ = A2 P2 ⎝y ⎠ + H2 + R2 + F2 ,  t z z z z (5.79) where

⎞ ⎞ ⎛ ˚0 P1 ˚i P1 0 0 0 0 P1 A P1 A 1 1 i ˚0 ˚i A02 := ⎝ 0 0 ⎠ , A2 := ⎝ 0 0 ⎠ , (5.80) A A 1 1 0 0 0 A1 0 0 Ai1 ⎞ ⎞ ⎛ ⎛ 0 0 0 0 P1 0 1 0 ⎠, 0 ⎠ , P2 := ⎝ 0 C2i := ⎝0 0 0 0 P1 0 0 C1i ⎞ ⎛ 0 0 A1 P1 P1 ˚ ˚0 (5.81) A2 = ⎝−P1 ˚ 0 ⎠, A1 P1 P1 ˚ A1 P1 + A 1 0 0 A1 ⎛ ⎛ ⎞ ⎛ ¯⎞ ⎞ ˚ P1 F2 0 P1 H1 ˜ 2 ⎠ , R2 := ⎝ 0 ⎠ and F2 := ⎝ F˜2 ⎠ . H2 := ⎝ H (5.82) ˆ2 R Fˆ2 0 ⎛

We remark that due to the projection operator P1 that appears in the definition (5.64) of b and in the top row of (5.80), the vector (b, y, z)T takes values in the vector space P1 RN1 × RN1 × RN1 and (5.80) defines a symmetric hyperbolic system, i.e., A02 and Ai2 define symmetric linear operators on P1 RN1 ×RN1 ×RN1 and A02 is non-degenerate. Setting ⎛ ⎞ 0 0 0 0 0 ⎜0 0 0 0 0⎟ ⎜ ⎟ ⎜ P3 := ⎜0 0 P1 0 0⎟ ⎟, ⎝0 0 0 1 0⎠ 0 0 0 0 0 it is then not difficult to verify from the estimates (5.53), (5.55), (5.66), (5.71) and (5.76), the initial bounds (5.58), (5.67) and (5.77), the relations (5.63), ˚0 , (5.68), (5.70) and (5.72), and the assumptions on the coefficients {A01 , Ai1 , A 1

Vol. 19 (2018)

Newtonian Limits on Long Time Scales

2213

˚i , A1 , ˚ A A1 , H, F }, see Assumptions 5.2 and 5.9, that the system consisting of 1 (5.78) and (5.79) and the solution U = (w, w ˚, b, y, z)T satisfy the hypotheses of Theorem 5.7, and thus, for σ > 0 chosen small enough, there exists a constant C > 0 independent of  ∈ (0, 0 ) and T1 ∈ (T0 , 0) such that (w, w ˚)L∞ ([T0 ,T1 )×Tn ) ≤

R 2

(5.83)

and (w, w ˚)MP∞,s ([T0 ,t)×Tn ) + (b, y, z)MP∞,s−1 ([T0 ,t)×Tn ) 1 2  t 1 P3 U H s−1 dτ ≤ Cσ − T0 τ

(5.84)

for T0 ≤ t < T1 . This completes the proof since the estimates (5.59)–(5.62) follow immediately from (5.83) and (5.84). 

6. Initial Data As is well known, the initial data for the reduced conformal Einstein–Euler equations cannot be chosen freely on the initial hypersurface ΣT0 = {T0 } × T3 ⊂ M = (0, T0 ] × T3

(T0 > 0).

Indeed, a number of constraints, which we can separate into gravitational, gauge and velocity normalization, must be satisfied on ΣT0 . There are a number of distinct methods available to solve these constraint equations. Here, we will follow the method used in [47,48], which is an adaptation of the method introduced by Lottermoser [39]. The goal of this section is to construct 1-parameter families of -dependent solutions to the constraint equations that behave appropriately in the limit   0. In order to use the method from [47,48] to solve the constraint equaˆμν ˆμν and u tions, we need to introduce new gravitational variables u σ defined via the formulas ¯ μν + 2 u ˆ μν + 2 E 3 u ˆμν := ∂¯σ u ˆμν ) = h ˆμν ˆμν , gˆμν := θ¯ g μν = E 3 (h and u σ

(6.1) respectively, where   |¯ g| Λ ˆ μν = E 3 h ¯ μν and θ= = |¯ g |, |¯ g | = − det g¯μν , h 3 |¯ η| 3 (6.2) |¯ η | = − det η¯μν = . Λ Notation In the following, we will use upper case script letters, e.g., Q(ξ), R(ξ), S (ξ), to denote analytic maps of the variable ξ whose exact form is not important. The domain of analyticity of these maps will be clear from context. Generally, we will use S to denote maps that may change line to line, while other letters will be used to denote maps that need to be distinguished for

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later use. We also introduce the following derivative notation to facilitate the statements, 1 ∂ˆμ = δμi ∂i + δμ0 ∂0 .  The total set of constraints that we need to solve on ΣT0 are: ¯ 0μ − T¯0μ )|t=T = 0 (Gravitational Constraints), (6.3) (G 0   3 3 2 2Λ θ − E μ θ − E Λ μ  ˆμν ) − E 3 u ˆμ0 − Ωδ ∂ˆν (E 3 u δ0 + =0 t 3t 2 2 t 0 t=T0 (Gauge constraint)

(6.4)

and (¯ v μ v¯μ + 1)|t=T0 = 0

(Velocity Normalization).

(6.5)

Remark 6.1. It is not difficult to verify that the constraint (6.4) is equivalent to the wave gauge condition Z¯ μ = 0 on the initial hypersurface ΣT0 . Indeed, ˆ μν ) = −E 3 Λ Ωδ μ and it is enough to notice that ∂¯ν (h 0 t  1 Λ ¯ μ = −∂ˆν g¯μν − g¯μν  ∂ˆν |¯ g | − Ωδ0μ X t |¯ g| 1 Λ 1 Λ (−θ∂ˆν g¯μν − g¯μν ∂ˆν θ) − Ωδ0μ = − ∂ˆν gˆμν − Ωδ0μ . θ t θ t 6.1. Reduced Conformal Einstein Equations =

Before proceeding, we state in the following lemma a result that will be used repeatedly in this section. The proof follows from the definition of θ, see (6.2), and a direct calculation. We omit the details. Lemma 6.2. μν



3 ¯ μν + 2 u ˆμν ) det (h Λ  1 2 3 3 00 3 2 ij ˆμν ), (6.6) ˆ δij + 4 S (, t, E, u ˆ +E u =E +  E − u 2 Λ

ˆ )=E θ(, u

6

−

where S (, t, E, 0) = 0. Using this lemma, we can express the gauge constraint (6.4) as follows:  ⎧ 1 2 3 00 Λ 3 3 00 3 00 3 0k 2 ij ⎪ ⎪ ˆ ) = − ∂k (E u ˆ )+ E u ˆ + E − u ˆ δij ˆ +E u ⎪ ∂t (E u ⎪  t 3t Λ ⎪ ⎪  ⎨ 1 Λ 3 00 ˆμν ) (6.7) ˆij δij + 2 S (, t, E, Ω/t, u ˆ + E2u − E3 Ω − u ⎪ 2 t Λ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 ⎩ ∂ (E 3 u ˆj0 ) = − ∂k (E 3 u ˆjk ) + E 3 u ˆj0 t  t where S (, t, E, Ω/t, 0) = 0. The importance of the relations (6.7) is that they ˆμ0 on the initial hypersurface allow us to determine the time derivatives ∂0 u μν ˆ and their spatial derivatives on ΣT0 . ΣT0 from the metric variables u

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3 ˆμν , ∂l u ˆij ˆ0k + 2 A (, t, E, Ω/t, u ˆμν , u E 3 ∂k u 0 ) 2Λ

(6.8)

Lemma 6.3. ∂t (θ − E 3 ) = and ∂i θ = −

3 2 3 00 1 2 5 ˆμν , ∂l u ˆij ˆ +  E δkl ∂i u ˆkl + 4 Si (, t, E, Ω/t, u ˆμν , u  E ∂i u 0 ), 2Λ 2 (6.9)

ˆij ˆμν , u where the A and Si are linear in (∂l u 0 ) and vanish for (, t, E, Ω/t, 0, 0, 0) = 0. Proof. The proof of this Lemma follows from straightforward calculations; we only prove (6.8). Noticing 1 1 gμν = gˆμν ∂ˆ g μν (ˆ gμν = θ−1 g¯μν ), (6.10) θ−1 ∂θ = g¯μν ∂¯ 2 2 it is not difficult to verify that 3 Ω Ω Ω 1 ˆ0k + 3E 3 − 3E 3 + 2 A ∂t (θ − E 3 ) = θˆ gμν ∂0 gˆμν − 3E 3 = E 3 ∂k u 2 t 2Λ t t follows from (6.7).  We proceed by differentiating (6.7) with respect to time t to obtain, with the help of Lemma 6.3, the following:  ⎧ Λ θ−E 3 2 1 14 2 3 00 ⎪ 2 3 00 3 ik 3 k0 ⎪ ˆ ) = 2 ∂k ∂i (E u ˆ )− ˆ )+ 2 E u ˆ + ∂ ∂ −Ω (E (E u u k ⎪ t ⎪  t t 3 t2 2 ⎪ ⎪ ⎪  ⎪ 3 3 ⎪ θ−E θ−E Λ Λ 2 ⎪ ⎪ ∂t Ω + − Ω ∂t − ⎪ ⎪ 2 ⎪  t t 3 2 ⎪ ⎪  ⎪ ⎪ 1 13 1 ⎪ ⎨ ˆik ) − ˆk0 = 2 ∂k ∂i (E 3 u 1 + Ω E 3 ∂k u  t 2 ⎪ ⎪ 1 ⎪ ⎪ ˆμν , ∂l u ˆij ˆμk , u ⎪ + 2 S (, t, E, Ω/t, ∂t Ω, u 0 ) ⎪ ⎪ t ⎪ ⎪ ⎪ 1 2 21 ⎪ ⎪ ˆj0 ) = − ∂k ∂0 (E 3 u ˆjk ) + 2 E 3 u ˆj0 − ˆjk ) ⎪ ∂k (E 3 u ∂t2 (E 3 u ⎪ ⎪  t t ⎪ ⎪ ⎪ ⎪ 1 2 3 j0 2 + 3Ω 1 ⎩ ˆjk ˆ − ˆjk ) = − E 3 ∂k u ∂k (E 3 u 0 + 2E u  t t  (6.11) where S (, t, E, Ω/t, ∂t Ω, 0, 0, 0) = 0. Next, we consider the following reduced version of the conformal Einstein equations (1.14), which we write using the metric variable gˆμν defined by (6.1): 1 λσ ˆ ˆ μν ¯ (μ ¯ ν) 1 ¯g , θ) ¯ λΓ ¯λ + 1 Q ˆ μν (ˆ gˆ ∂λ ∂σ gˆ + ∇ Γ − gˆμν ∇ g , ∂ˆ 2 2θ 2θ θ2 1 ¯ μ ¯ ν ¯ (μ ¯ ν) 1 ¯ λ Z¯ λ − 1 A¯μν Z¯ λ X − ∇ Z + gˆμν ∇ − X 2 2θ 2 λ 1 ¯ νΨ − ∇ ¯ μ∇ ¯ μ Ψ∇ ¯ ν Ψ) − (2Ψ ¯ 2g¯ ) 1 gˆμν , ¯ + |∇Ψ| = e4Φ T˜μν − e2Φ Λˆ g μν + 2(∇ θ θ (6.12)

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where ˆg , θ) = 1 θ2 Qμν − 1 gˆμν gˆαβ (θ2 Qαβ −θ2 X ¯ αX ¯ β )− 1 gˆλσ gˆαβ ∂ˆσ gˆμν ∂ˆλ gˆαβ ˆ μν (ˆ Q g , ∂ˆ 2 4 2 1 λσ μν 1 λσ μν ˆ γρ ˆ αβ ˆ + gˆ gˆ gˆγρ gˆαβ ∂λ gˆ ∂σ gˆ + gˆ gˆ ∂λ gˆαβ ∂ˆσ gˆαβ 8 4 with Qμν as defined previously by (2.13). By (1.27), (2.13), (6.1), (6.10) and the identity 1 ¯ λμν = −ˆ Γ gσ(μ ∂ˆν) gˆλσ + gˆλσ gˆαμ gˆβν ∂ˆσ gˆαβ 2  1 λ ˆ 2ˆ gαβ δ(μ + ∂ν) gˆαβ − gˆλσ gˆμν gˆαβ ∂ˆσ gˆαβ , 4

(6.13)

ˆg μν and θ. From this and the it is obvious that θ2 Qμν is analytic in gˆμν , ∂ˆ ˆg μν and θ. Moreover, ˆ μν is analytic in gˆμν , ∂ˆ formula (6.13), it is clear that Q using (6.7) and (6.13), it can be verified by a straightforward calculation that ˆ μν satisfies Q ¯g , θ) − Q ˆ ∂¯h, ˆ E3) ˆ μν (ˆ ˆ μν (h, Q g , ∂ˆ H ˆαβ , ∂k u ˆij ) ˆαβ + 2 Qˆμν (, t, E, Ω/t, x, u ˆαβ , u = T μνk (t)∂k u αβ

0

μνk ˆαβ , for coefficients Tαβ that depend only on t and where Qˆμν (, t, E, Ω/t, x, u 0, 0) = 0. Using the easy to verify identities

¯ λλ0 = 1 g¯λσ ∂ˆ0 g¯λσ = 1 gˆλσ ∂ˆ0 gˆλσ = 1 ∂ˆ0 θ, Γ 2 2 θ 1 1 ¯ λ γ¯ λ = ¯ λλ0 Ω ¯λ , ¯ λ Y¯ λ = −2Ψ ¯ − 2Λ + 2Λ Γ ∇ and ∇ ∂t Ω − Ω + Γ t t 3t2 3t λ0 we can write the reduced conformal Einstein equations (6.12) as  1 λσ ˆ ˆ μν ¯ (μ ν) 1 μν 1 1 1 ˆ μν 1 μν 1 0 λ ¯ γ¯ gˆ ∂λ ∂σ gˆ + ∇ γ¯ − gˆ + gˆ ∂t Ω − Ω+ Γλ0 Ω + 2 Q 2θ2 2θ t t θ θ t



Λ 1 ˆ μν 2Λ Λ (μ ν) =− g¯00 + ∂0 gˆ + 2 δ0μ δ0ν + g¯0k δk δ0 3t θ 3t 3 1 11 + 2 (1 + 2 K)ρ¯ v μ v¯ν + 2 2 Kρˆ g μν . (6.14) t θt This equation is satisfied for the FLRW solutions (1.4)–(1.7), i.e., we ˆ μν , ρH , eΨ v˜μ }. Dividing the resulting FLRW can substitute {ˆ g μν , ρ¯, v¯μ } → {h H equation through by θ2 , we get  1 ˆ λσ ˆ ˆ ˆ μν E 6 ¯ (μ ν) E 3 ˆ μν 1 1 λ h ∂λ ∂σ h + 2 ∇H γ¯ − 2 h ∂t Ω − Ω + γ¯λ0 Ω 2θ2 θ 2θ t t 1 ˆ μν E 3 ˆ μν 1 0 γ¯ + 2 QH + 2 h θ θ t

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Newtonian Limits on Long Time Scales

Λ E 3 ˆ ˆ μν E 6 2Λ =− ∂0 h + 2 2 3t θ2 θ 3t E3 1 2 ˆ μν . + 2 2  KρH h θ t

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 Λ E6 Λ 1 00 ¯ (1 + 2 K)ρH δ0μ δ0ν h + δ0μ δ0ν + 2 3 θ 3 t2 (6.15)

Subtracting (6.15) from (6.14) yields  6 ˆ μν ) + (ˆ ˆ λσ )∂ˆλ ∂ˆσ h ˆ μν + 2θ2 ∇ ¯ (μ γ¯ ν) − E ∇ ¯ (μ γ¯ ν) gˆλσ ∂ˆλ ∂ˆσ (ˆ g μν − h g λσ − h θ2 H   ˆ μν ) 1 ∂t Ω − 1 Ω + (E 3 − θ)h ˆ μν 1 ∂t Ω − 1 Ω − θ(ˆ g μν − h t t t t 1 ˆ μν − gˆμν ) γ¯ λ Ω + θ(h t λ0 1 λ ˆ μν 1 γ¯ λ Ω + θˆ ¯ λ )Ω + 2(Q ˆ μν − Q ˆ μν ) γλ0 − Γ g μν (¯ + (E 3 − θ)h λ0 λ0 H t t  3 1 E ˆ μν + 2θ γ¯ 0 gˆμν − h t θ  3 2Λ ˆ μν ) − 2Λ θ 1 − E ˆ μν g μν − h = − θ∂ˆ0 (ˆ ∂ˆ0 h 3t 3t θ

 

4Λ Λ E 6 ¯ 00 Λ (μ ν) + 2 θ2 + g¯00 + − 2 h δ0μ δ0ν + g¯0k δk δ0 3t 3 θ 3  6 1 E Λ v μ v¯ν − 2 ρH δ0μ δ0ν + 2θ2 2 (1 + 2 K) ρ¯ t θ 3  3 E 1 2 μν μν ˆ ρH h g − + 2θ 2  K ρˆ . (6.16) t θ 6.2. Transformation Formulas Before proceeding, we collect in the following lemma a set of formulas that can be used to transform from the gravitational variables used in this section to those introduced previously in Sect. 1.5 for the formulation of the evolution equations. Lemma 6.4. The evolution variables u0μ , uij and u can be expressed in terms ˆμν by the following expressions: of the gravitational variables u u0μ =

 2t



1 00 μ Λ ˆ0k δkμ + E 2 u ˆij δij δ0μ ˆ δ0 + u u 2 6



ˆαβ ), + 3 S μ (, t, E, Ω/t, u (6.17)

2Λ ˆαβ ), ˆij δij + 3 S (, t, E, Ω/t, u u =  E2u 9  1 kl ˆαβ ), ˆij − u ˆ δkl δ ij + 3 S ij (, t, E, Ω/t, u uij = E 2 u 3

(6.18) (6.19)

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where all of the remainder terms vanish for (, t, E, Ω/t, 0) = 0. Moreover, the 0-component of the conformal fluid four-velocity v¯μ can be written as  Λ 0 ˆαβ , zj ). + 2 S (, t, E, Ω/t, u v¯ = (6.20) 3 Proof. First, we observe that (6.17) follows directly from (2.27) and Lemma 6.2. Next, using (6.1), it is not hard to show that ˆij δij ) + 4 S det (¯ g kl ) = (θE −3 )−3 (E −6 + 2 E −4 u  9 00 1 ˆαβ ), ˆij δij + 4 S (, t, E, Ω/t, u ˆ − E2u = E −6 + 2 E −6 u 2 Λ from which it follows that  1 2 9 00 2 2 ij ˆαβ ) ˆ δij + 4 S (, t, E, Ω/t, u ˆ −E u αE = 1 +  u 6 Λ

(6.21)

by (1.39). Then by (1.34), (1.40) and (6.21), we have u = 2tu00 − while

1Λ 2Λ ˆαβ ), ˆij δij + 3 S (, t, E, Ω/t, u ln[1 + (αE 2 − 1)] =  E 2 u 3 9  1 ˆ ij (αθ)−1 gˆij − E −1 h   1 kl 2 ij ij ˆαβ ) ˆ − u ˆ δkl δ = E u + 3 S ij (, t, E, Ω/t, u 3

uij =

follows from (2.25), (6.1), (6.21) and 1 ij ˆμν ). u δij + 4 S (, t, E, Ω/t, u (αθ)−1 = E −1 − 2 Eˆ 3 Finally, (6.20) follows from (2.43), (2.27) and (6.17)–(6.19)



6.3. Solving the Constraint Equations We now need to write the constraint equations in a form that is suitable to used the methods from [47,48]. We begin by defining the rescaled variables ˆij |t=T0 = ˘ u uij ,

ˆij ˘ij u 0 |t=T0 = u 0 ,

ˆ0μ |t=T0 = u ˘0μ u

and

ˆ0μ ˘0μ u 0 |t=T0 = u 0 ,

and noting that ˆij |t=T0 = ∂k u ˘ij . ∂k u

(6.22)

We then observe that the following terms from (6.16) can be represented as  1 2 3 λσ ˆ ˆ ˆ μ0 2 ¯ (μ 0) 6 ¯ (μ 0) 2 3 μ0 1 ˆ ∂λ ∂σ h + 2(θ ∇ γ¯ − E ∇H γ¯ ) −  θE u ˆ  E u ∂t Ω − Ω t t  2 0 0μ 1 1 ˆ μν ˆ μ0 ) + (E 3 − θ)h + γ¯ (θˆ g − E3h ∂t Ω − Ω t t t 1 1 2Λ ˆ μν − gˆμν ) γ¯ λ Ω + (E 3 − θ)h ˆ μν γ¯ λ Ω + ˆ 0μ + θ(h (θ − E 3 )∂t h t λ0 t λ0 3t ˆαβ , ∂k u ˆij ˆαβ , u = 2 S μ (, t, E, Ω/t, ∂t Ω, x, u (6.23) 0 ).

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We also note that 1 E −3 3 λ ¯ λλ0 − γ¯λ0 Γ = ∂t (θ − E 3 ) + ∂t E 3 − Ω θ θE −3 t 3 ˆμν , ∂k u ˆij ˆ0k + 2 S (, t, E, Ω/t, u ˆμν , u =  ∂k u (6.24) 0 ), 2Λ ˆμ0 Using (6.7) and (6.11) to replace the first and second time derivatives of u ij μν ˆ and the time derivatives u ˆ0 in (6.16) with ν = 0, by spatial derivatives of u we obtain, with the help of (6.23)–(6.24), the following elliptic equations on ˆμ0 : ΣT0 for u  2Λ 2 Λ 00k αβ ˘ ) − E 2 (T0 )∂k ∂i u ˘ik Δ˘ u00 − E (T0 )δρ +  ∂k (Tαβ u 2 3T0 3   Λ Λ+1 ˘k0 + Ω E 2 (T0 )∂k u + 3t 2t ˘μν , ∂k u ˘ij ˘μν , u ˘0μ , δρ, z j ) = 0, + 2 R 0 (, u (6.25) 0 , ∂i ∂j u    Λ 2 2 Λ 2 0ik αβ ˘ik ˆ ) E (T0 )∂k u E (T0 )ρz i + ∂k (Tαβ Δ˘ ui0 +  u 0 − 3 3 T02 ˘μν , ∂k u ˘ij ˘μν , u ˘0μ , δρ, z j ) = 0, + 2 R i (, u 0 , ∂i ∂j u 0μk Tαβ

(6.26)

0μk Tαβ (T0 )

where the coefficients = are constant on ΣT0 and the remainder terms satisfy R μ (, 0, 0, 0, 0, 0, 0) = 0. Remark 6.5. From the above calculations, it is not difficult to see that the elliptic equations (6.25)–(6.26) are equivalent to the gravitational constraint equations (6.3) provided that the gauge constraint (6.4) is also satisfied. Recalling that (6.7) is equivalent to the gauge constraints, it is clear that we can solve the gauge constraints by using (6.7) to determine the time derivatives ˘μν and their spatial derivatives. ˘μ0 from the metric variables u ∂t u Decomposing δρ = ρ − ρH and ρz i on ΣT0 as δρ|t=T0 = ρ˘0 + φ˘ and (ρz i )|t=T0 = ψ˘i + ν˘i , where 1 φ˘ := 1, δρ |t=T0 ,  ψ˘i := 1, ρz i |t=T0 and ν˘i = Π(ρz i )|t=T0 , ρ˘0 := Πδρ|t=T0 ,

˘ and it is clear that z i |t=T0 and δρ|t=T0 depend analytically on (˘ ν i , ψ˘i , ρ˘0 , φ), in particular, ν˘i + ψ˘i z i |t=T0 = . (6.27) ρH (T0 ) + ρ˘0 + φ˘ From this and the fact that the spatial derivatives ∂i : H s (Tn ) → H s−1 (Tn ) define bounded linear maps, we can, by Lemmas C.1 and C.2 from “Appendix C,” view the remainder terms R μ from (6.25)–(6.26) as defining analytic maps

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(−0 , 0 ) × Br (H s+1 (T3 )) × H s (T3 ) × Br (H s (T3 )) × Br (R) × R3 × H s (T3 ) ˘ ψ˘i , ν˘i ) ˘μν , u ˘ik  (, u ˘0 , φ, 0 ,ρ ˘ ψ˘i , ν˘i ) ∈ H s−1 (T3 ) ˘μν , u ˘ik −→ R μ (, u ˘0 , φ, 0 ,ρ

(6.28)

for r > 0 chosen small enough. Using this observation, we can proceed with the existence proof for solutions to the constraint equations. ˘ij ∈ Br (H s+1 (T3 , S3 )), Theorem 6.6. Suppose s ∈ Z>n/2+1 and r > 0, u s 3 ¯ s (T3 )), ν˘i ∈ H ¯ s (T3 , R3 ). Then for r > 0 cho˘ij ˘0 ∈ Br (H u 0 ∈ H (T , S3 ), ρ sen small enough so that the map (6.28) is well defined and analytic, there exists an 0 > 0, and analytic maps φ˘ ∈ C ω (Xs0 ,r , R), ψ˘l ∈ C ω (Xs0 ,r , R3 ), ω s s 3 4 ¯ s+1 (T3 , R4 )) and u ˘0μ ˘0μ ∈ C ω (Xs0 ,r , H u 0 ∈ C (X0 ,r , H (T , R )) that satisfy ψ˘l (, 0, 0, 0, 0) = 0,

˘ 0, 0, 0, 0) = 0, φ(, ˘0μ (, 0, 0, 0, 0) = 0 u

and

˘0μ u 0 (, 0, 0, 0, 0) = 0

such that ˘ ρ|t=T0 = ρH (T0 ) + ρ˘0 + φ, z i |t=T0 = uμν |t=T0 ∂0 uμν |t=T0

ν˘i + ψ˘i

ρH (T0 ) + ρ˘0 + φ˘  00 ˘ ˘0j u u = , ˘i0 ˘ uij u  00 ˘0 ˘0j u u 0 = , ˘i0 ˘ij u u 0 0

,

˘μ0 where the u 0 are determined by (6.7), solve the constraints (6.3), (6.4) and ˘ ψ˘i , u ˘00 , u ˘0i } satisfy the estimate (6.5). Moreover, the fields {φ, ˘ + |ψ˘i | + ˘ u0μ H s+1 + ˘ u0μ uik H s+1 + ˘ uik ρ0 H s + ˘ ν i H s |φ| 0 H s + ˘ 0 H s  ˘ uniformly for  ∈ (−0 , 0 ) and can be expanded as ˘jk , u ˘jk ˘jk , u ˘jk φ˘ = S (, u ˘0 , ν˘i ), ψ˘i = S i (, u ˘0 , ν˘i ), 0 ,ρ 0 ,ρ 2Λ 2 ˘00 = ˘jk , u ˘jk u E (T0 )Δ−1 ρ˘0 + S (, u ˘0 , ν˘i ) 0 ,ρ 3T02

(6.29)

and ˘jk , u ˘jk ˘0i = S i (, u ˘0 , ν˘i ), u 0 ,ρ

(6.30)

˘jk , where the maps S and S i that are analytic on Xs0 ,r and vanish for (, u jk ˘ = (, 0, 0, 0, 0). ˘0 , ρ˘0 , φ) u Proof. Acting on (6.25) and (6.26) with 1, · and Π , we obtain, with the help of (6.20) and (6.28), the equations  0 / ˘ ψ˘j , ν˘j ˘μν , u ˘jk = 0, (6.31) , ρ ˘ , φ, φ˘ −  1, R 0 , u 0 0

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  2Λ 2 ˘ ψ˘j , ν˘j = 0, ˘μν , u ˘jk Δ˘ u00 − E (T0 )˘ ρ0 + ΠR 0 , u ˘0 , φ, 0 ,ρ 2 3T / 0  0 ˘ ψ˘j , ν˘j ˘jk ˘μν , u ψ˘i +  1, R i , u ˘0 , φ, =0 0 ,ρ and

  ˘ ψ˘j , ν˘j = 0, ˘jk ˘μν , u , ρ ˘ , φ, Δ˘ u0i + ΠR i , u 0 0

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(6.32) (6.33)

(6.34)

which are clearly equivalent to (6.25)–(6.26). Next, we let   ˘ ψ˘i , u ˘ik , ρ˘0 , ν˘i ) and β := φ, ˘0μ , ι := (˘ uik , u 0

and write (6.31)–(6.32) more compactly as F (, ι, β) := L(ι, β) + M (, ι, β) = 0, where

(6.35)

⎞ φ˘ ⎟ ⎜ ψ˘i L(ι, β) = ⎝ ⎠. μ 2Λ 0μ 2 Δ˘ u − 3T 2 E (T0 )δ0 ρ˘0 ⎛

0

¯ s+1 (T3 ) to Recalling that the Laplacian Δ defines an isomorphism from H ¯ s−1 (T3 ), we observe that H ⎞⎞ ⎛ ⎛ 0 ⎠⎠ 0 (0, ι, β) = ⎝0, ι, ⎝ μ −1 2Λ 2 E (T )δ Δ ρ ˘ 2 0 0 0 3T 0

solves (6.35). Since Dβ F (0, ι, β) · δβ = L(0, δβ), we can solve (6.35) via an analytic version of the Implicit Function Theorem [11, Theorem 15.3], at least for small , if we can show that ⎛ ⎞ δ φ˘ ˜ L(δβ) = ⎝ δ ψ˘i ⎠ Δδ˘ u0μ

¯ s+1 (T3 , R4 ) to R × R3 × H ¯ s−1 (T3 , R4 ). defines an isomorphism from R × R3 × H s+1 3 s−1 3 ¯ ¯ (T ) → H (T ) is an isomorphism. Thus, But this is clear since Δ : H for r > 0 chosen small enough and any R > 0, there exists an 0 > 0 and a unique analytic map ¯ s+1 (T3 , R4 ) P : X s → R × R3 × H 0 ,r

that satisfies F (ι, P (, ι), ) = 0  s+1 3  s 3  s 3 ¯ (T ) × H H H forall (, ι) ∈ (− ,  )×B (T , S ) ×B (T , S ) ×B 0 0 R 3 R 3 r s 3 3 ¯ Br H (T , R ) and ⎛ ⎞ 0 ⎠ + O(). 0 P (, ι) = ⎝ (6.36) μ −1 2Λ 2 E (T )δ Δ ρ ˘ 0 0 0 3T 2 0

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Finally, the estimate ˘ + |ψ˘i | + ˘ u0μ H s+1 + u0μ uik H s+1 + ˘ uik ρ0 H s + ˘ ν i H s |φ| 0 H s + ˘ 0 H s  ˘ follows from analyticity of P , (6.36) and (6.7).



6.4. Bounding U|t=T 0 For the evolution problem, we need to bound U|t=T0 , see (2.102), by the free ˘ik ˘0 , ν˘i } uniformly in . The required bound is the content initial data {˘ uik , u 0 ,ρ of the following lemma. Lemma 6.7. Suppose that the hypotheses of Theorem 6.6 hold, and that φ˘ ∈ ¯ s+1 (T3 , R4 )) and u ˘0μ ∈ C ω (Xs0 ,r , H ˘0μ C ω (Xs0 ,r , R), ψ˘l ∈ C ω (Xs0 ,r , R3 ), u 0 ∈ ω s s 3 4 C (X0 ,r , H (T , R )) are the analytic maps from that theorem. Then on the initial hypersurface ΣT0 , the gravitational and matter fields 0μ 0μ {uμν , uij γ , wi , u0 , u, uγ , zj , δζ}

can be expanded as follows: Λ ˘kl , u ˘kl ˘0 , ν˘l ), u0μ |t=T0 =  3 E 2 (T0 )Δ−1 ρ˘0 δ0μ + 2 S μ (, u 0 ,ρ 6T0 2Λ ˘kl , u ˘kl u|t=T0 = 2 E 2 (T0 )˘ uij δij + 3 S (, u ˘0 , ν˘l ), 0 ,ρ 9  1 kl ˘ij − u ˘kl , u ˘kl ˘ δkl δ ij + 3 S ij (, u ˘0 , ν˘l ), uij |t=T0 = 2 E 2 (T0 ) u 0 ,ρ 3 ν˘i δij ˘kl , u ˘kl + Sj (, u ˘0 , ν˘l ), zj |t=T0 = E 2 (T0 ) 0 ,ρ ρH (T0 ) + ρ˘0   ρ˘0 + φ˘ 1 ln 1 + δζ|t=T0 = 1 + 2 K ρH (T0 )  ρ˘0 ˘kl , u ˘kl ˘0 , ν˘l ), + 2 S (, u = ln 1 + 0 ,ρ ρH (T0 ) ˘kl , u ˘kl wi0μ |t=T0 = Siμ (, u ˘0 , ν˘l ), 0 ,ρ μ ˘kl , u ˘kl u0μ ˘0 , ν˘l ), 0 ,ρ 0 |t=T0 = S (, u

˘kl , u ˘kl uγ |t=T0 = Sγ (, u ˘0 , ν˘l ) 0 ,ρ and ij ˘kl , u ˘kl ˘0 , ν˘l ), uij γ |t=T0 = Sγ (, u 0 ,ρ

for maps S that are analytic on Xs0 ,r . Moreover, the estimates uμν |t=T0 H s+1 +u|t=T0 H s+1 +wi0μ |t=T0 H s +u0μ 0 |t=T0 H s +uμ |t=T0 H s + uij uij H s+1 + ˘ uij ρ0 H s + ˘ ν i H s ) μ |t=T0 H s + |φ(T0 )|  (˘ 0 H s + ˘ and uij H s+1 + ˘ uij ρ0 H s + ˘ ν i H s zj |t=T0 H s + δζ|t=T0 H s  ˘ 0 H s + ˘ hold uniformly for  ∈ (− 0 , 0 ).

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Proof. First, we observe by (2.28), (2.62), (2.63), (2.64), (6.1), (6.22) and Lemma 6.2 that wi0μ |t=T0 Λ 1 ˘kl , u ˘kl ˘00 δ0μ − δ0μ 2 E 2 (T0 )∂i Δ−1 ρ˘0 + ∂i u ˘0k δkμ + 2 Siμ (, u = ∂i u ˘0 , ν˘l ), 0 ,ρ 2 3T0 (6.37) where Siμ (, 0, 0, 0, 0) = 0, which in turn, implies by (6.30) that ˘kl , u ˘kl wi0μ |t=T0 = Siμ (, u ˘0 , ν˘l ), 0 ,ρ

(6.38)

where again Siμ (, 0, 0, 0, 0) = 0. Furthermore, by (2.28), (6.1), (6.17), Lemma 6.2 and Theorem 6.6, we see that 1 11 1 ˘ik , u ˘ik ∂0 gˆ0μ − gˆ0μ 2 ∂0 θ − 3u0μ = S μ (, u ˘0 , ν˘i ), (6.39) u0μ 0 ,ρ 0 |t=T0 = θ  θ where S μ (, 0, 0, 0, 0) = 0. Next, we see from (1.21), (1.39), (6.1), (6.8), (6.9) and Theorem 6.6, that we can express ∂μ α as α−3 ∂t α3 = 3α−1 ∂t α = gˇkl ∂t g¯kl = −

6Ω(T0 ) ˘ik , u ˘ik + 2 S (, u ˘0 , ν˘i ) (6.40) 0 ,ρ T0

and α−3 ∂j α3 = 3α−1 ∂j α = 2 = 2

9 ˘ik , u ˘ik ˘00 + 3 S (, u ∂j u ˘0 , ν˘i ) 0 ,ρ 2Λ

3 2 ˘ik , u ˘ik E (T0 )Δ−1 ∂j ρ˘0 + 3 S (, u ˘0 , ν˘i ), 0 ,ρ T02

(6.41)

˘ik , u ˘ik where the error terms S vanish for (, u ˘0 , ν˘i ) = (, 0, 0, 0, 0). Using 0 ,ρ (6.17) and (6.39), we then find with the help of (1.35) and (6.40) that u0 |t=T0 = 3u00 + u00 0 −

1 Λ −3 1 2Λ Ω(T0 ) ˘ik , u ˘ik α ∂t α 3 − = S0 (, u ˘0 , ν˘i ), 0 ,ρ 9  3 T0 (6.42)

while we note that Λ 2 1 Λ ˘ij , u ˘ij uk |t=T0 = wk00 + E (T0 )∂k Δ−1 ρ˘0 − 2 α−3 ∂k α3 = Sk (, u ˘0 , ν˘i ) 0 ,ρ 2 3T0  9 (6.43) follows from (2.63), (6.41) and (6.37). Again the error terms Sμ vanish for ˘ik , u ˘ik (, u ˘0 , ν˘i ) = (, 0, 0, 0, 0). Starting from (1.33) and (1.39), we see, with 0 ,ρ the help of (6.42), Lemmas 6.2 and 6.3 along with Theorem 6.6, that 1 ˘kl , u ˘kl ∂0 (α−1 θ−1 gˆij ) = S ij (, u ˘0 , ν˘l ), (6.44) 0 ,ρ  where S ij (, 0, 0, 0, 0) = 0. By a similar calculation, we find with the help of (6.38) and(6.43) that uij 0 |t=T0 =

uij k |t=T0 =

1 ˘kl , u ˘kl ∂k (α−1 θ−1 gˆij ) = S ij (, u ˘0 , ν˘l ), 0 ,ρ 

(6.45)

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where S ij (, 0, 0, 0, 0) = 0. Noting that φ(T0 ) =

1 3(1+2 K) T0

˘ik ˘ik , u ˘0 , ν˘i ) φ˘ = S (, u 0 ,ρ

by Theorem 6.6, the estimate uμν |t=T0 H s+1 + u|t=T0 H s+1 + wi0μ |t=T0 H s +u0μ 0 |t=T0 H s +uμ |t=T0 H s + uij uij H s+1 + ˘ uij ρ0 H s + ˘ ν i H s ), μ |t=T0 H s + |φ(T0 )|  (˘ 0 H s + ˘ which holds uniformly for  ∈ (−0 , 0 ), follows directly from (6.38), (6.39), (6.42), (6.43), (6.44), (6.45), Lemma 6.4 and Theorem 6.6. Next, we observe from zj = 1 g¯j0 v¯0 + g¯ij z i , (2.43), (6.1), (6.17)–(6.19), (6.27) and Theorem 6.6 that we can write zj |t=T0 as zj |t=T0 = E 2 (T0 )

ν˘i δij ˘ik , u ˘ik + S (, u ˘0 , ν˘i ), 0 ,ρ ρH (T0 ) + ρ˘0

where S (, 0, 0, 0, 0) = 0. In addition, we note that   ρ˘0 + φ˘ 1 ln 1 + δζ|t=T0 = 1 + 2 K ρH (T0 )  ρ˘0 ˘ik , u ˘ik = ln 1 + ˘0 , ν˘i ) + 2 S (, u 0 ,ρ ρH (T0 )

(6.46)

(6.47)

follows from (2.40), (2.41) and Theorem 6.6, where S (, 0, 0, 0, 0) = 0. Together, (6.46) and (6.47) imply that the estimate uij H s+1 + ˘ uij ρ0 H s + ˘ ν i H s zj |t=T0 H s + δζ|t=T0 H s  ˘ 0 H s + ˘ holds uniformly for  ∈ (−0 , 0 ).



7. Proof of Theorem 1.7 7.1. Transforming the Conformal Einstein–Euler Equations The first step of the proof is to observe that the non-local formulation of the conformal Einstein–Euler equations given by (2.104) can be transformed into the form (5.47) analyzed in Sect. 5 by making the simple change in time coordinate t → tˆ := −t

(7.1)

and the substitutions w(tˆ, x) = U(−tˆ, x),

A01 (, −tˆ, w) = B0 (, −tˆ, U),

Ai1 (, tˆ, w) = −Bi (, −tˆ, U), C1i and

i

= −C ,

P1 = P,

A1 (, tˆ, w) = B(, −tˆ, U),

(7.2)

H1 (, tˆ, w) = −H(, −tˆ, U)

F1 (, tˆ, x) = −F(, −tˆ, x, U, ∂k Φ, ∂t ∂k Φ, ∂k ∂l Φ).

(7.3)

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With these choices, it is clear that the evolution Eq. (2.104) on the spacetime region t ∈ (T1 , 1], 0 < T1 < 1, are equivalent to 1 1 for (tˆ, x) ∈ [−1, −T1 ) × T3 , A01 ∂tˆw+Ai1 ∂i w+ C1i ∂i w = A1 P1 w+H1 +F1  tˆ which is of the form studied in Sect. 5.2, see (5.47). Furthermore, it is not difficult to verify (see [51, §3] for details) that matrices {Aμ1 , C1i , A1 , P1 } and the source term H1 satisfy Assumptions 5.2.(5.2)–(5.2) from Sect. 5.1 for some positive constants κ, γ1 , γ2 > 0. To see that Assumption 5.2.(5.2) is also satisfied is more involved. First, we note that this assumption is equivalent to verifying P⊥ [DU B0 · (B0 )−1 BPU]P⊥ admits an expansion of the type (5.11). To see why this is the case, we recall that B0 and P are block matrices, see (2.105)–(2.106), from which it is clear using (2.53)–(2.55) that we can expand P⊥ [DU B0 · (B0 )−1 BPU]P⊥ as ⎛ ⊥ ˜ 0 · W]P⊥ P2 [DU B 2 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0

0 ˘⊥ ˘⊥ [DU B ˜ 0 · W]P P 2 2 0 0 0

0 0 ˘⊥ ˘⊥ [DU B ˜ 0 · W]P P 2 2 0 0

0 0 0 ˆ⊥ ˆ⊥ [DU B 0 · W]P P 2 2 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠

0

(7.4) where W

:= (B0 )−1 BPU ⎛ ˜ 0 −1 ˜ (B ) BP2 0 ˘2 ˜ 0 )−1 P ⎜ 0 −2E 2 g¯00 (B ⎜ =⎜ 0 0 ⎜ ⎝ 0 0 0 0 ⎛ Y 0 0 0 ⎜0 −21 0 0 ⎜ 0 0 −21 0 = P⎜ ⎜ ⎝0 ˆ2 0 0 (B 0 )−1 BP 0 0 0 0

with

⎛ 1 Y = ⎝0 0

0 0 ˘2 ˜ 0 )−1 P −2E 2 g¯00 (B ⎞

0 0

0 0 0 ˆ2 (B 0 )−1 BP 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟U 0⎠ 0

0 0⎟ ⎟ 0⎟ ⎟U 0⎠ 0

0 3 i 2 δj 0

⎞ 0 0⎠ . 1

˜ 0 can be Next, by (1.32), (1.39), (2.24), (2.27), and (2.53), we observe that B expressed as ⎛Λ ⎞ 00 0  0 3 − 2tu ˜0 = E2 ⎝ ⎠. 0 (δ ij + uij )E −2 exp  Λ3 (2tu00 − u) 0 B Λ 00 0 0 3 − 2tu Noting from definition (2.101) of U1 that uij and u are components of the vector P⊥ 1 U1 , where ˘2, P ˘ 2 ), P1 = diag (P2 , P

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˜ 0 , as a map, depends only on the variables (, tU1 , P⊥ U1 ). To it is clear that B 1 ˆ 0 (, tU1 , P⊥ U1 ) := B ˜ 0 (, t, U). Letting make this explicit, we define the map B 1 P denote linear maps that projects out the components U1 from U, i.e., U1 = PU, ˜ 0 with respect to U in the direction W to get we can then differentiate B  ˜ 0 · W = DU B ˆ 0 (, tU1 , P⊥ ˆ0 DU B 1 U1 ) · W = D2 B DU (tU1 ) ˆ 0 DU (P⊥ + D3 B 1 U1 ) · W  ˆ 0 DU (PU) + D3 B ˆ 0 DU (P⊥ = tD2 B 1 PU) · W ˆ 0 PW + D3 B ˆ 0 (P⊥ P)W = tD2 B ˆ 0 PW, = tD2 B 1 where in the above calculations, we employed the identities ⎛ 0 0 0 ⎞ Y ⎛ P2 0 0 0 0 ⎜ 0 −21 0 0 ⎜ ˘2 0 −21 0 0 0 0⎠ ⎜ P PW = ⎝ 0 ⎜0 ˆ2 ˘2 0 0 ⎝ 0 0 0 (B 0 )−1 BP 0 0 P 0 0 0 0 ⎛ ⎞ Y 0 0 0 ⎠ P1 U1 U = ⎝ 0 −21 0 0 −21

(7.5) ⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

and P⊥ 1 PW = 0. By (2.80), it is not difficult to see that ˆ 2 )⊥ [DU B 0 W](P ˆ 2 )⊥ = (P



DU 1 · W 0 = 0, 0 0

(7.6)

which in turn, implies via (7.4), (7.5) and (7.6) that P⊥ [DU B0 · (B0 )−1 BPU]P⊥  ⊥ ˘⊥ ˘⊥ ˘⊥ ˘⊥ ˆ0 ˆ0 ˆ0 = t diag P⊥ 2 D2 B PWP2 , P2 D2 B PWP2 , P2 D2 B PWP2 , 0, 0 . From this it is then clear that P⊥ [DU B0 · (B0 )−1 BPU]P⊥ satisfies Assumption 5.2.(5.2). 7.2. Limit Equations Setting ˚ T, ˚ = (˚ ˚k0μ , ˚ u0μ , ˚ uij uij uij , ˚ u0 , ˚ uk , ˚ u, δ ˚ ζ, ˚ zi , φ) U u0μ 0 ,w 0 ,˚ k ,˚

(7.7)

the limit equation, see Sect. 5.2, associated to (2.104) on the spacetime region (T2 , 1] × T3 , 0 < T2 < 1, is given by ˚ 0 ∂t U ˚ +B ˚ i ∂i U ˚ + Ci ∂i V = 1 BP ˚ U ˚ +H ˚ +˚ B F t ˚=0 Ci ∂i U

in (T2 , 1] × T3 ,

(7.8)

in (T2 , 1] × T3 ,

(7.9)

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where ˚ ˚ := lim Bμ (, t, U), ˚ Bμ (t, U) 0

˚ ˚ := lim B(, t, U), ˚ B(t, U) 0

˚ U) ˚ := lim H(, t, U), ˚ H(t,

(7.10)

0

and ˚ F :=

 ˚ ˚ Ω Ω ˜ ijr ˚ 3 ˚−2 kl ˚ ˚−2 kl μ − D0μj ∂j ˚ Φ, δ0μ E Φ, 0, − D δ ∂l Φ − E δ δ 0 ∂0 ∂l ˚ ∂r Φ, t 2t t T  3 ˚ Ω 1 3 2 ˚−2 lk ˚ 0, 0, − Dj ∂j ˚ Φ, 0, 0, 0, −K −1 E δ ∂k Φ, 0 . t 2 Λ (7.11)

˜ ijr are as defined by (2.69) and (2.76), In ˚ F, the coefficients D0μj and D ˚ ˚ and ˚ Φ is the Newtonian potential, see (1.57), and E Ω are defined by (1.53) and (1.54), respectively. We then observe that under the change in time coordinate (7.1) and the substitutions ˚0 (tˆ, w) = ˚ ˚ tˆ, x), A ˚ w ˚(tˆ, x) = U(− B0 (−tˆ, U), 1 ˚ ˚i (tˆ, w) = −˚ ˚ ˚ Bi (−tˆ, U), A1 (tˆ, w) = ˚ B(−tˆ, U), C1i = −Ci , A 1 ˚ tˆ, U) ˚1 (tˆ, w) = −H(− ˚ v(tˆ, x) = V(−tˆ, x), P1 = P, H ˚1 (tˆ, x) = −˚ F(−tˆ, x), and F

(7.12) (7.13)

the limit equation (7.8)–(7.9) transforms into ˚0 ∂ˆw ˚i ˚ = 1 ˚ ˚1 + F ˚1 A ˚ − C1i ∂i v + H A1 P1 w 1 t ˚ + A1 ∂i w tˆ C1i ∂i w ˚= 0

in

[−1, −T2 ) × T3 ,

in

[−1, −T2 ) × T3 ,

which is of the form analyzed in Sect. 5.2, see (5.49)–(5.50) and (5.52). It is ˚i and the source term H ˚1 satisfy also not difficult to verify the matrices A 1 Assumptions 5.9.(2) from Sect. 5.2. 7.3. Local Existence and Continuation For fixed  ∈ (0, 0 ), we know from Proposition 3.1 that for T1 ∈ (0, 1) chosen close enough to 1 there exists a unique solution U∈

1 %

 C  (T1 , 1], H s− (T3 , V)

=0

to (2.104) satisfying the initial condition  0μ ij ij 0μ U|t=1 = u0μ 0 |t=1 , wk |t=1 , u |t=1 , u0 |t=1 , uk |t=1 , uij |t=1 , u0 |t=1 , uk |t=1 , u|t=1 , δζ|t=1 , zi |t=1 , φ|t=1

T

,

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0μ where the initial data, u0μ 0 |t=1 , wk |t=1 , . . ., is determined from Lemma 6.7. Moreover, this solution can be continued beyond T1 provided that

sup U(t)H s < ∞.

t∈(T1 ,1]

Next, by Proposition 4.1, there exists, for some T2 ∈ (0, 1], a unique solution Φ) ∈ (˚ ζ, ˚ zi, ˚

1 %



C ((T1 , T0 ], H

s−

=0

×

1 %

3

(T )) ×

1 %

C  ((T1 , T0 ], H s− (T3 , R3 ))

=0

C  ((T1 , T0 ], H s+2− (T3 )),

(7.14)

=0

to the conformal cosmological Poisson–Euler equations, given by (1.55)–(1.57), satisfying the initial condition   ν˘i δij ˚ (ζ, ˚ zi )|t=1 = ln ρH (1) + ρ˘0 , . (7.15) ρH (1) + ρ˘0 Setting

  V = V00μ , Vk0μ , V 0μ , 0, Vkij , 0, 0, Vk , 0, 0, 0, 0 ,

where

˚ 1 ˚2 μ ˚ 3 μ ˚ ˚2 μ ˚ ˚2 ∂t Φ , δ 0 Φ + E δ 0 ∂t Φ = − E δ0 Φ + δ0μ tE 2t 2t t  ˚ Ω ˚2 Λ 1 Δ−1 ∂k (˚ Φ + 2E = D0μj Δ−1 ∂k ∂j ˚ ρ˚ z j )δjμ , t 3 t2 Φ Λ ˚4 ˚ −1 1 ˚2 ˚ ΩΔ δ˚ + δ0μ 3 E = δ0μ E ρ, 2 t 3t ˚ Ω ˜ ijr −1 Φ, = D Δ ∂k ∂r ˚ t

(7.16)

˚2 V00μ = −E

(7.17)

Vk0μ

(7.18)

V 0μ Vkij

(7.19) (7.20)

and ˚ Ω j −1 Φ, (7.21) D Δ ∂k ∂j ˚ t it follows from Corollary 4.2 and (7.14) that V is well defined and lies in the space 1 %  V∈ C  (T2 , 1], H s− (T3 , V) . Vk =

=0

Defining ˚ = (0, 0, 0, 0, 0, 0, 0, 0, 0, δ˚ U ζ, ˚ zi , 0),

(7.22)

where we recall, see (4.7), (4.8) and Theorem 1.7.(ii), that δ˚ ζ =˚ ζ −˚ ζH

˚−2 δ ij˚ and ˚ zi = E zj ,

(7.23)

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˚ lies in the space we see from Remark 4.3, (7.7) and (7.14)–(7.15) that U ˚∈ U

1 %

 C  (T2 , 1], H s− (T3 , V)

=0

and satisfies ˚ t=1 = U|



 0, 0, 0, 0, 0, 0, 0, 0, 0, ln 1 +

T ρ˘0 ν˘i δij ,0 . , ρH (1) ρH (1) + ρ˘0

˚ determines a It can be verified by a direct calculation that the pair (V, U) solution of the limit equation (7.8)–(7.9). Moreover, by Proposition 4.1, it is clear that this solution can be continued past T2 provided that ˚ sup U(t) H s < ∞.

t∈(T2 ,1]

7.4. Global Existence and Error Estimates For the last step of the proof, we will use the a priori estimates from Theo˚ to the reduced conformal rem 5.10 to show that the solutions U and (V, U) Einstein–Euler equations and the corresponding limit equation, respectively, can be continued all the way to t = 0, i.e., T1 = T2 = 0, with uniform bounds and error estimates. In order to apply Theorem 5.10, we need to verify that ˚ We begin by the estimates (5.53)–(5.57) hold for the solutions U and (V, U). observing the equation  Ω) 3  j 3(1 − ˚ ∂j ˚ δ˚ ρ (7.24) ∂t δ˚ ρ˚ z + ρ=− Λ t holds in (T2 , 1] × T3 by (1.49), (4.5) and the equivalence of the two formulations (1.49)–(1.51) and (1.55)–(1.57) of the conformal Poisson–Euler equations. From this equation, (1.54) and the calculus inequalities from “Appendix A,” we obtain the estimate 1  1 1 ρ 1 1 1∂t δ˚ 1 3 t 1H s−1  ζ(t)H s + ˚ ≤ C δ˚ ζL∞ ((t,1],H s ) , ˚ zi L∞ ((t,1],H s ) (δ˚ zi (t)H s ), T2 < t ≤ 1

(7.25)

Recalling that we can write the Newtonian potential as Λ 1 ˚2 −1 Λ ˚2 ˚ ˚ ˚ E Δ δ˚ Φ= ρ = tE eζH Δ−1 (eδζ − 1) 2 3t 3

in (T2 , 1] × T3

(7.26)

by (1.47), (4.1) and Corollary 4.2 we see, using the calculus inequalities from ¯ k−1 (T3 ), ¯ k+1 (T3 ) → H “Appendix A” and invertibility of the Laplacian Δ : H 1˚ k ∈ Z≥1 , that we can estimate t Φ by 1 1 11 1  ˚ ˚ 1 ˚ 1 Φ(t) (7.27) 1t 1 s+1 ≤ C δ ζL∞ ((t,1],H s−1 ) δ ζ(t)H s−1 , T2 < t ≤ 1. H

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Dividing (7.26) by t and then differentiating with respect to t, we find using (7.24) that    ˚  k Φ Λ 1 ˚2 1 Λ ˚2 ˚ −1 ∂t E ∂k Δ−1 ˚ ρ˚ z = 0, (7.28) ρ+ + 4 E ΩΔ δ˚ 3 t 3t 3t which, by (7.26), is also equivalent to

  k Λ 1 ˚2 1 Λ 2 −1 ˚ (1 − ˚ ∂t ˚ Φ= E E ∂k Δ−1 ˚ ρ˚ z . (7.29) Ω)Δ δ˚ ρ− 3 2 3t 3t From (7.28) and (7.29), we then obtain, with the help of the calculus inequalities and the invertibility of the Laplacian, the estimate 1  1 1 ˚ Φ 1 1 1 ΦH s+1 + 1t∂t ∂t ˚ 1 1 t 1 s+1 H

≤ C(δ˚ ζL∞ ((t,1],H s ) , ˚ zi L∞ ((t,1]),H s ) )(δ˚ ζ(t)H s + ˚ zi (t)H s ), T2 < t ≤ 1.

(7.30)

Continuing on, we differentiate (7.29) with respect to t to get    k Λ 1 ˚ 5˚ Λ 1 ˚2 −1 2˚ 2 −1 ˚ Δ δ˚ Ω E Δ ∂k ∂t ˚ ρ˚ z Ω−4 E ρ− ∂t Φ = 4 2 3t 2 3t   k Λ 1 ˚2 E (1 − ˚ ρ˚ z , (7.31) + Ω)Δ−1 ∂k ˚ 3 3t where in deriving this we have used the fact that ˚ Ω satisfies (2.1) with  = 0 and that Δ−1 δ˚ ρ is well defined by Corollary 4.2. Adding the conformal cosmological Poisson–Euler equations (1.49)–(1.50) together, we obtain the following equation for ˚ ρ˚ zj :    3  j Ω j 1 3 2 j˚ 3 3  i j 4 − 3˚ K∂ j ˚ ∂i ˚ ˚ ρ˚ z − ρ˚ z + ρ˚ z˚ ∂t ˚ ρ+ z = ˚ ρ∂ Φ. Λ Λ t 2 Λ Substituting this into (7.31) yields the estimate ∂t2 ˚ ΦH s ≤ C(δ˚ ζL∞ ((t,1],H s ) , ˚ zi L∞ ((t,1]),H s ) )(δ˚ ζ(t)H s + ˚ zi (t)H s ), T2 < t ≤ 1,

(7.32)

¯ k−1 (T3 ), k ∈ ¯ k+1 (T3 ) → H by (7.27), the invertibility of the Laplacian Δ : H Z≥1 , and the calculus inequalities from “Appendix A.” Next, from the definition of P, see (2.106), and (7.28), we compute  T 1 0μ 1 0μ 0μ 1 0μ 1 0μ PV = (V + V ), Vi , (V0 + V ), 0, 0, 0, 0, 0, 0, 0, 0, 0 , t 2t 0 t 2t (7.33) where the components are given by   ˚ Φ 1 0μ 1 1 Λ ˚4 ˚ −1 1 μ ˚2 0μ ΩΔ δ˚ (V0 + V ) = δ0 E ∂t + δ0μ 3 E ρ 2t 2 t 2t 3 t

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 k Λ 1 ˚4 E ∂k Δ−1 ˚ ρ˚ z , (7.34) 3 2 3t  ˚ Ω 1 0μ Λ 1 −1 Φ + 2E 2 Vk = 2 D0μj Δ−1 ∂k ∂j ˚ Δ ∂k (˚ ρ˚ z j )δjμ . (7.35) t t 3 t3 Routine calculations also show that the components of ∂t V are given by  ˚ ˚ Ω ˚2 μ ˚ Φ 3 0μ μ ˚2 δ 2˚ ˚2 δ μ ∂t2 ˚ Φ − 2E (7.36) δ0 Φ, ∂t V 0 = E Ω− ∂t +E 0 0 2 t t     2 ˚ ˚ ˚˚ 1 μ Φ Ω Λ δ˚ ρ μ ˚2 Ω Φ μ 0μ 2 4 ˚ ∂t ˚ 4 ∂t V = δ 0 E Ω Δ−1 3 + δ0 E + δ0 E + ∂t ˚ t t 2 t 3 t t  δ˚ ρ Λ ˚4 ˚ −1 ΩΔ ∂t + δ0μ E , (7.37) 3 3 t   ˚ ˚ Ω Ω ˜ ijr −1 ˜ ijr )Δ−1 ∂k ∂r ˚ Φ + (∂t D Φ D Δ ∂k ∂r ˚ ∂t Vkij = ∂t t t   ˚ Ω ˜ ijr −1 + D Φ , (7.38) Δ ∂k ∂r ∂t ˚ t    ˚ ˚ ˚ Ω Ω Ω ∂t V k = ∂ t Φ + (∂t Dj )Δ−1 ∂k ∂j ˚ Φ + Dj Δ−1 ∂k ∂j ∂t ˚ Φ Dj Δ−1 ∂k ∂j ˚ t t t (7.39) =

and ∂t Vk0μ

1 −δ0μ

  ˚ ˚ Ω Ω Φ + (∂t D0μj )Δ−1 ∂k ∂j ˚ Φ D0μj Δ−1 ∂k ∂j ˚ = ∂t t t   ˚ Ω + D0μj Δ−1 ∂k ∂j ∂t ˚ Φ t 

Λ 1 μ 2Ω −1 2 −1 2 j j −1 j + 2E Δ ∂k (˚ δ ρ˚ z ) − Δ ∂k (˚ ρ˚ z ) + Δ ∂k ∂t (˚ ρ˚ z ) . 3 t2 j t t (7.40)

˜ ijk and Dj are remainder terms as defined Recalling the coefficients D0μν , D in Sect. 1.1.6, it is then clear that the estimate V(t)H s+1 + t−1 PV(t)H s+1 + ∂t V(t)H s ≤ C(δ˚ ζL∞ ((t,1],H s ) , ˚ zi L∞ ((t,1),H s ) )(δ˚ ζ(t)H s + ˚ zi (t)H s ),

(7.41)

which holds for T2 < t ≤ 1, follows from the formulas (1.53), (1.54), (7.16)– (7.21) and (7.33)–(7.40), the estimates (7.25), (7.27), (7.30), (7.32), the calculus inequalities and the invertibility of the Laplacian. By similar reasoning, it is also not difficult to verify that ˚ F, defined by (7.11), satisfies the estimate ˚ F(t)H s +t∂t ˚ F(t)H s−1 ˚ ≤ C(δ ζL∞ ((t,1],H s ) , ˚ zi L∞ ((t,1],H s ) )(δ˚ ζ(t)H s +˚ zi (t)H s ), T2 < t ≤ 1. (7.42)

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From the definition of F, see (2.107), along with (2.68), (2.73), (2.75), (2.93) and (2.100), the definitions (2.62) and (2.101)–(2.103), and the calculus inequalities, we see that F can be estimated as  F(t)H s ≤ C UL∞ ((t,1],H s ) , ∂k ΦL∞ ((t,1],H s ) (U(t)H s + ∂l ∂k Φ(t)H s + ∂t ∂k Φ(t)H s ),

T1 < t < 1.

(7.43)

¯ k−1 (T3 ), ¯ k+1 (T3 ) → H Appealing again to the invertibility of the map Δ : H k ∈ Z≥1 , it follows from (2.63) and the calculus inequalities that we can estimate the spatial derivatives of Φ as follows:  ∂k Φ(t)H s + ∂l ∂k Φ(t)H s ≤ C UL∞ ((t,1],H s ) U(t)H s

(7.44)

for T1 < t < 1. Using (2.7) and (3.6), we see that ∂t ∂k Φ satisfies  Λ 2 ζH Λ 2 ζH −1 δζ −1 δζ ∂t ∂k Φ = E e (1 − Ω)∂k Δ Πe + E te ∂k Δ e ∂t δζ . 3 3 Replacing ∂t δζ in the above equation with the right-hand side of (2.94), we see, with the help of the calculus properties and the invertibility of the Laplacian that ∂t ∂k Φ can be estimated by ∂t ∂k ΦH s ≤ C(δζL∞ ((t,1],H s ) )(∂t δζH s−1 + δζH s−1 ) ≤ C(UL∞ ((t,1],H s ) )UH s

(7.45)

for T1 < t < 1. Combining the estimates (7.43)–(7.45) gives  F(t)H s ≤ C UL∞ ((t,1],H s ) U(t)H s ,

T1 < t < 1.

(7.46)

˚1 , v}, as deTogether, (7.41), (7.42) and (7.46) show that source terms {F1 , F fined by (7.3) and (7.13), satisfy the estimates (5.53)–(5.55) from Theorem 5.10 for times −1 ≤ tˆ < −T3 , where T3 = max{T1 , T2 }. This leaves us to verify the Lipschitz estimates (5.56)–(5.57). We begin by noticing, with the help of (2.53), (2.88) and (7.22), that

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Newtonian Limits on Long Time Scales

˚ = 0, ˜ i (, t, U) B   3 ˚ zi i ˚ B (, t, U) = −2 il Λ E δ and

 i

˚ = B (0, t, U)

3 Λ



E −2 δ im −1 −2 lm i K E δ ˚ z ˚ zi ˚−2 δ il E



2233

˚ + 2 S i (, t, U)

˚−2 δ im E . ˚−2 δ lm˚ K −1 E zi

From the above expressions, (2.10) and the calculus inequalities, we then obtain the estimate ˚ −B ˚ i (t, U) ˚ H s−1 ≤ C(U ˚ L∞ ((t,1],H s ) ), Bi (, t, U)

T3 < t ≤ 1.

Next, using (2.60), (2.61), (2.67), (2.70), (2.72), (2.74) and (7.22), we compute ˚ see (2.107), as follows: the components of H(, t, U),   T 2 2 Λ k μ ˚ ˚ 0, 0 , −2(1 + 2 K)E 2 t1+3 K e(1+ K)(ζH +δζ) ˚ z δk + S μ (, t, U), 3  ij T  ˚ ˚ ˚ ˚ 0, 0 T , ˜ ˜ G2 (, t, U) = S (, t, U), 0, 0 , G3 (, t, U) = S(, t, U), ˚ = ˜ 1 (, t, U) G

˚ = (0, 0)T G(, t, U)

and

˚ = 0, ´ t, U) G(,

˚ = 0. It follows immediately from these where S μ , S ij and S all vanish for U expressions and the definitions (2.107) and (7.10) that   T Λ k μ 2 ˚ ζ ζH +δ˚ ˚ ˚ ˚ H(t, U) = −2E te ˚ z δk , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 3 and, with the help of the calculus inequalities and (2.9)–(2.10), that ˚ − H(t, ˚ U) ˚ H s−1 ≤ C(U ˚ L∞ ((t,1],H s ) )U ˚ H s−1 , H(, t, U)

T3 < t ≤ 1. (7.47)

To proceed, we define Z=

1 ˚ − V), (U − U 

(7.48)

and set z(tˆ, x) = Z(−tˆ, x).

(7.49)

In view of the definitions (2.107) and (7.11), we see that the estimate F(, t, ·) − ˚ F(t, ·)H s−1   ≤ C UL∞ ((t,1],H s ) UH s−1 + ∂k ΦH s−1 + ∂k ∂l ΦH s−1 + ∂0 ∂l ΦH s−1  ˚−2 ˚ ˚−2 ˚ Φ)H s−1 + ∂0 ∂l (E −2 Φ − E Φ)H s−1 + ∂k (E −2 Φ − E   ˚−2 ˚ Φ)H s−1 ≤ C UL∞ ((t,1],H s ) UH s + ∂k (E −2 Φ − E  ˚−2 ˚ Φ)H s−1 , + ∂0 ∂l (E −2 Φ − E (7.50)

which holds for T3 < t ≤ 1, follows from (2.9)–(2.10), the estimates (7.41), (7.44) and (7.45), the calculus inequalities, and the estimate

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´ H s−1   1, S  SL2 ≤ C(UL∞ ((t,1],H s ) )UH 1 S ≤ C(UL∞ ((t,1],H s ) )UH s−1 . By (1.42), (1.47), (2.9), (2.63), (7.26), (7.41), (7.48), the invertibility of the Laplacian and the calculus inequalities, we see also that ˚−2 ˚ Φ)H s−1 ∂k (E −2 Φ − E ˚

˚

˚−2 ˚ ΦH s  eζH Πeδζ − eζH Πeδζ H s−2 = E −2 Φ − E ˚

˚

˚

 |eζH − eζH |Π(eδζ − 1)H s−2 + eζH Π(eδζ − eδζ )H s−2   ≤ C δζL∞ ((t,1],H s ) , δ˚ ζL∞ ((t,1],H s ) |ζH − ˚ ζH |eδζ − 1H s−2 + δζ − δ˚ ζH s−1   ≤ C δζL∞ ((t,1],H s ) , δ˚ ζL∞ ((t,1],H s ) δζH s−1 + ZH s−1 + VH s−1   ˚ L∞ ((t,1],H s ) UH s + ZH s−1 + U ˚ Hs ≤ C UL∞ ((t,1],H s ) , U (7.51) for T3 < t ≤ 1, while similar calculations using (2.7), (2.8) and (2.10) show that ˚−2 ˚ Φ)H s−1 ∂0 ∂l (E −2 Φ − E ˚

˚

˚−2 ˚ Φ)H s  ∂t (eζH Πeδζ − eζH Πeδζ )H s−2 = ∂0 (E −2 Φ − E  ≤ C δζL∞ ((t,1],H s ) , δ˚ ζL∞ ((t,1],H s )  × δζH s−1 + 2 ∂t δζH s−1 + δ˚ ζ − δζH s−1  + eδζ ∂t (δζ − δ˚ ζ)H s−2 + δζ − δ˚ ζH s−1 ∂t δ˚ ζH s−1  ≤ C δζL∞ ((t,1],H s ) , δ˚ ζL∞ ((t,1],H s )  × δζH s−1 + 2 ∂t δζH s−1 + ZH s−1 + VH s−1 + eδζ ∂t (δζ − δ˚ ζ)H s−2 + (ZH s−1 + VH s−1 )∂t δ˚ ζH s−1



for T3 < t ≤ 1. Next, by (2.8), it is easy to see that (1.55) is equivalent to  3 j ˚ ˚ ˚ z ∂j δ ζ + ∂ j ˚ z j = 0. ∂t δ ζ + Λ Using this, we derive the estimate  ∂t δ˚ ζH s−1 ≤C(˚ zj L∞ ((t,1],H s ) ) δ˚ ζH s + ˚ zj H s , T3 < t ≤ 1,

(7.52)

(7.53)

while we see from (2.94) and (7.44) that ∂t δζH s−1 ≤ C(UL∞ ((t,1],H s ) , ∂k ΦL∞ ((t,1],H s ) )  δζH s + zj H s + (UH s−1 + ∂k ΦH s−1 ) ≤ C(UL∞ ((t,1],H s ) )UH s

(7.54)

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for T3 < t ≤ 1. We also observe that z k ∂k (δζ − δ˚ ζ)H s−2 eδζ ˚ ! δζ k ! " " ζ)H s−2 ≤ ∂k e ˚ z (δζ − δ˚ ζ) H s−2 + ∂k eδζ ˚ z k (δζ − δ˚ ≤ C(δζL∞ ((t,1],H s ) , ˚ z k L∞ ((t,1],H s ) )δζ − δ˚ ζH s−1

(7.55)

and zm )H s−2 ≤ C(δζL∞ ((t,1],H s ) )zm − ˚ zm H s−1 eδζ ∂k (zm − ˚

(7.56)

hold for T3 < t ≤ 1. Furthermore, by (2.25), (2.27), (2.42), (2.44), (7.23) and (7.48), we see that  z k − ˚ z k H s−1 ≤ C(UL∞ ((t,1],H s ) ) ZH s−1 + VH s−1 + zj H s−1 (7.57) and, with the help of (2.10), (7.41), (7.44) and (7.55)–(7.57), that ζ)H s−2  eδζ (z k ∂k δζ − ˚ z k ∂k δ ˚ ζ)H s−2 eδζ ∂t (δζ − δ˚ ˚−2 ∂k˚ + eδζ (E −2 ∂k zm − E zm )H s−2 + eδζ SH s−2  eδζ (z k − ˚ z k )∂k δζH s−2 + eδζ ˚ z k ∂k (δζ − δ˚ ζ)H s−2 ˚−2 )∂k zm H s−2 + eδζ ∂k (zm − ˚ + eδζ (E −2 − E zm )H s−2 + eδζ SH s−1 ˚ L∞ ((t,1],H s ) )(ZH s−1 + U ˚ H s + UH s ), ≤ C(UL∞ ((t,1],H s ) , U (7.58) where both estimates hold for T3 < t ≤ 1. We also observe that (7.41) and (7.52)–(7.58) imply ˚−2 ˚ Φ)H s−1 ∂0 ∂l (E −2 Φ − E  ˚ L∞ ((t,1],H s ) ) UH s + ZH s−1 + U ˚ Hs ≤ C(UL∞ ((t,1],H s ) , U (7.59) for T3 < t ≤ 1. Gathering (7.50), (7.51) and (7.59) together, we obtain the estimate F(, t, ·) − ˚ F(t, ·)H s−1 ˚ L∞ ((t,1],H s ) )(UH s + ZH s−1 + U ˚ H s ), ≤ C(UL∞ ((t,1],H s ) , U T3 < t ≤ 1.

(7.60) ˚ ˚ The estimates (7.47) and (7.60) show that source terms {H1 , H1 , F1 , F1 }, as defined by (7.3) and (7.13), and z, defined by (7.49), verify the Lipschitz estimate (5.57) from Theorem 5.10 for times −1 ≤ tˆ < −T3 . Having verified that all of the hypotheses of Theorem 5.10 are satisfied, we conclude, with the help of Lemma 6.7, that there exists a constant σ > 0, independent of  ∈ (0, 0 ), such that if the free initial data are chosen so that ˘ uij H s+1 + ˘ uij ρ0 H s + ˘ ν i H s ≤ σ, 0 H s + ˘ then the estimates UL∞ ((T3 ,1],H s ) ≤ Cσ,

˚ L∞ ((T ,1],H s ) ≤ Cσ U 3

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˚ L∞ ((T ,1],H s−1 ) ≤ Cσ and U − U 3

(7.61)

hold for some constant C > 0, independent of T3 ∈ (0, 1) and  ∈ (0, 0 ). Furthermore, from the continuation criterion discussed in Sect. 7.3, it is clear ˚ exist globally on that the bounds (7.61) imply that the solutions U and U M = (0, 1] × T3 and satisfy the estimates (7.61) with T3 = 0 and uniformly for  ∈ (0, 0 ). In particular, this implies via the definitions (2.102) and (7.7) ˚ that of U and U δζ(t) − δ˚ ζ(t)H s−1 ≤ Cσ, uμν 0 (t)H s−1 u0 (t)H s−1

zj (t) − ˚ zj (t)H s−1 ≤ Cσ, μ ν Φ(t)H s−1 ≤ Cσ, − δ δ 0 ∂k ˚

uμν k (t)

≤ Cσ, 0 uμν (t)H s−1 ≤ Cσ, ≤ Cσ, uk (t)H s−1 ≤ Cσ and

u(t)H s−1 ≤ Cσ

for 0 < t ≤ 1, while, from (2.43), we see that 1  1 1 Λ1 1 0 1 ≤ Cσ 1v¯ (t) − 1 1 3 1 s−1 H

holds for 0 < t ≤ 1. This concludes the proof of Theorem 1.7.

Acknowledgements This work was partially supported by the ARC grant FT120100045. Part of this work was completed during a visit by the authors to the IHP as part of the Mathematical Relativity Program in 2015. We are grateful to the Institute for its support and hospitality during our stay. We also thank the referee for their comments and criticisms, which have served to improve the content and exposition of this article.

Appendix A. Calculus Inequalities We use the following Sobolev–Moser inequalities throughout this article. The proofs can be found in [64]. Theorem A.1 (Sobolev’s inequality), If s ∈ Z>n/2 , then f L∞  f H s for all f ∈ H s (Tn ). Lemma A.2. Suppose s ∈ Z≥1 , l ∈ Z≥2 , fi ∈ L∞ (Tn ) for 1 ≤ i ≤ l, fl ∈ H s (Tn ), and Dfi ∈ H s−1 (Tn ) for 1 ≤ i ≤ l − 1. Then there exists a constant C > 0, depending on s and l, such that  l−1 l−1 2  2 fi L∞ + Dfi H s−1 fj L∞ . f1 . . . fl H s ≤ C fl H s i=1

i=1

i =j

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Lemma A.3. Suppose s ∈ Z≥1 , f ∈ L∞ (Tn , V ) ∩ H s (Tn , V ) ∩ C 0 (Tn , V ), W, U ⊂ V are open with U bounded and U ⊂ W , f (x) ∈ U for all x ∈ Tn and F ∈ C s (W ). Then there exists a constant C > 0, depending on s, such that ⎛ ⎞ 12  ⎝ Dβ f L2 ⎠ Dα (F ◦ f )L2 ≤ CDF W s−1,∞ (U ) f s−1 L∞ |β|=s

for any multi-index α satisfying |α| = s. Lemma A.4. If s ∈ Z≥1 and |α| ≤ s, then Dα (f g) − f Dα gL2  Df H s−1 gL∞ + Df L∞ gH s−1 for all f, g satisfying Df, g ∈ L∞ (Tn ) ∩ H s−1 (Tn ).

Appendix B. Matrix Relations Lemma B.1. Suppose

 A=

a bT

b c

is an (n + 1) × (n + 1) symmetric matrix, where a is an 1 × 1 matrix, b is an 1 × n matrix and c is an n × n symmetric matrix. Then ⎛   −1   −1 ⎞  −1 1 1 T 1 1 T T −ab c − ab b b a b ⎟ ⎜a 1 + b c − ab b   −1 −1 ⎠ A−1 = T =⎝ b c 1 T 1 T 1 T c − ab b − c − ab b ab 

Proof. Follows from direct computation. We also recall the well-known Neumann series expansion.

Lemma B.2. If A and B are n × n matrices with A invertible, then there exists an 0 > 0 such that the map (−0 , 0 )   −→ (A + B)−1 ∈ Mn×n is analytic and can be expanded as ∞  (A + B)−1 = A−1 + (−1)n n (A−1 B)n A−1 ,

|| < 0 .

n=1

Appendix C. Analyticity We list some well-known properties of analytic maps that will be used throughout this article. We refer interested readers to [25] or [47] for the proofs. Lemma C.1. Let X, Y and Z be Banach spaces with U ⊂ X and V ⊂ Y open. 1. If L : X → Y is a continuous linear map, then L ∈ C ω (X, Y ); 2. If B : X × Y → Z is a continuous bilinear map, then B ∈ C ω (X × Y, Z); 3. If f ∈ C ω (U, Y ), g ∈ C ω (V, Z) and ran(f ) ⊂ V , then g ◦ f ∈ C ω (U, Z).

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 Lemma C.2. Suppose s ∈ Z>n/2 , F ∈ C ω BR (RN ), R , and that  αN F (y1 , . . . , yN ) = F0 + cα y1α1 . . . yN |α|≥1

is the powerseries expansion for F (y) about 0. Then there exists a constant Cs such that the map   N BR/Cs H s (Tn  (ψ1 , ψ2 , . . . , ψN ) → F (ψ1 , ψ2 , . . . , ψN ) ∈ H s (Tn )   N s n , H (T ) , and is in C ω BR/Cs H s (Tn  αN F (ψ1 , . . . , ψN ) = F0 + cα ψ1α1 ψ2α2 . . . ψN 



|α|≥1

s

n

for all (ψ1 , . . . , ψN ) ∈ BR/Cs H (T

N

.

Appendix D. Index of Notation g˜μν v˜μ ρ¯ p¯ = 2 K ρ¯  = vcT M = (0, 1] × T3 M = M1 a(t) v˜H (t) ρH (t) ¯i ) (¯ xμ ) = (t, x μ (x ) = (t, xi ) ˚ a(t) ˚ ρH (t) f (t, xi ) Xs0 ,r (T3 ) S(, t, ξ), T (, t, ξ), . . . S(, t, ξ), T(, t, ξ), . . . g¯μν v¯μ Ψ ¯ h E(t) Ω(t) 0 i , γ¯j0 γ¯ij

Physical spacetime metric; Sect. 1 Physical fluid four-velocity; Sect. 1 Fluid proper energy density; Sect. 1 Fluid pressure; Sect. 1 Newtonian limit parameter; Sect. 1 Relativistic spacetime manifold; Sect. 1 Newtonian spacetime manifold; Sect. 1 FLRW scale factor; Sect. 1, Eqs. (1.4) and (1.7) FLRW fluid four-velocity; Sect. 1, Eq. (1.5) FLRW proper energy density; Sect. 1, Eq. (1.6) (see also (2.3) and (2.40)) Relativistic coordinates; Sect. 1 Newtonian coordinates; Sect. 1, Eq. (1.8) Newtonian limit of a(t); Sect. 1, Eq. (1.10) Newtonian limit of ρH (t); Sect. 1, Eq. (1.10) Evaluation in Newtonian coordinates; Sect. 1.1.2, Eq. (1.11) Free initial data function space; Sect. 1.1.4 Remainder terms that are elements of E 0 , Sect. 1.1.6 Remainder terms that are elements of E 1 , Sect. 1.1.6 Conformal metric; Sect. 1.2, Eq. (1.12) Conformal four-velocity; Sect. 1.2, Eq. (1.13) Conformal factor; Sect. 1.3, Eq. (1.18) Conformal FLRW metric; Sect. 1.3, Eq. (1.19) Modified scale factor; Sect. 1.3, Eq. (1.20) (see also (2.4)) Modified density; Sect. 1.3, Eq. (1.22) (see also (2.2)) ¯ Sect. 1.3, Non-vanishing Christoffel symbols of h; Eq. (1.23)

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γ¯ σ Z¯ μ ¯μ X ¯ Yμ uμν , u uμν γ zi ζ δζ ¯ij g α gˇij ¯q η¯ ζH (t) C0 ˚ ζH (t) zi ˚ ρ ˚ zj ˚ Φ Π ˚ E(t) ˚ Ω(t) ˚ ζ ρ δρ wk0μ Φ φ U1

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¯ Sect. 1.3, Contracted Christoffel symbols of h; Eq. (1.24) Wave gauge vector field; Sect. 1.4, Eq. (1.26) Contracted Christoffel symbols; Sect. 1.4, Eq. (1.27) Gauge source vector field; Sect. 1.4, Eq. (1.28) Modified conformal metric variables; Sect. 1.5, Eqs. (1.29), (1.32) and (1.34) First-order metric field variables; Sect. 1.5, Eqs. (1.30), (1.31), (1.33) and (1.35) Modified lower conformal fluid 3-velocity; Sect. 1.5, Eq. (1.36) Modified density; Sect. 1.5, Eq. (1.37) Difference between ζ and ζH ; Sect. 1.5, Eq. (1.38) Densitized conformal 3-metric; Sect. 1.5, Eq. (1.39) Cube root of conformal 3-metric determinant; Sect. 1.5, Eq. (1.39) Inverse of the conformal 3-metric g¯ij ; Sect. 1.5, Eq. (1.39) Modified conformal 3-metric determinant; Sect. 1.5, Eq. (1.40) Background Minkowski metric; Sect. 1.5, Eq. (1.41) FLRW modified density; Sect. 1.5, Eqs. (1.42) and (1.43) FLRW constant; Sect. 1.5, Eq. (1.44) Newtonian limit of ζH (t), Sect. 1.5, Eqs. (1.46) and (1.47) (see also (4.8)) Modified upper conformal fluid 3-velocity; Sect. 1.5, Eq. (1.48) Newtonian fluid density; Sect. 1.6 Newtonian fluid 3-velocity; Sect. 1.6 Newtonian potential; Sect. 1.6 Projection operator; Sect. 1.6, Eq. (1.52) Newtonian limit of E(t); Sect. 1.6, Eq. (1.53) Newtonian limit of Ω(t); Sect. 1.6, Eq. (1.54) Modified Newtonian fluid density; Sect. 1.6 Fluid proper energy density in Newtonian coordinates; Sect. 2.3, Eq. (2.39) Difference between ρ and ρH ; Sect. 2.3, Eq. (2.41) Shifted first-order gravitational variable; Sect. 2.4, Eq. (2.62) Gravitational potential; Sect. 2.4, Eq. 2.63 Renormalized spatially average density; Sect. 2.4, Eq. 2.65 Gravitational field vector; Sect. 2.6, Eq. (2.101)

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U U2 δ˚ ζ |||·|||a,H k , |||·|||H k , · MP∞a ,k ([T0 ,T )×Tn ) Q(ξ), R(ξ), S (ξ), . . .

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Combined gravitational and matter field vector; Sect. 2.6, Eq. (2.102) Matter field vector; Sect. 2.6, Eq. (2.103) Difference between ˚ ζ and ˚ ζH ; Sect. 4, Eq. (4.7) Energy norms; Sect. 5.1, Definition 5.5 Analytic remainder terms; Sect. 6

References [1] Blanchet, L.: Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev. Relativ. 17, 2 (2014) [2] Blanchet, L., Faye, G., Nissanke, S.: On the structure of the post-Newtonian expansion in general relativity. Phys. Rev. D 72, 44024 (2005) [3] Brauer, U., Karp, L.: Local existence of solutions of self gravitating relativistic perfect fluids. Commun. Math. Phys. 325, 105–141 (2014) [4] Brauer, U., Rendall, A., Reula, O.: The cosmic no-hair theorem and the nonlinear stability of homogeneous Newtonian cosmological models. Class. Quantum Gravity 11, 2283–2296 (1994) [5] Browning, G., Kreiss, H.O.: Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math. 42, 704–718 (1982) [6] Buchert, T., R¨ as¨ anen, S.: Backreaction in late-time cosmology. Ann. Rev. Nucl. Part. Sci. 62, 57–79 (2012) [7] Chandrasekhar, S.: The post-Newtonian equations of hydrodynamics in general relativity. Astrophys. J. 142, 1488–1512 (1965) [8] Clarkson, C., Ellis, G., Larena, J., Umeh, O.: Does the growth of structure affect our dynamical models of the universe? the averaging, backreaction and fitting problems in cosmology. Rept. Prog. Phys. 74, 112901 (2011) [9] Crocce, M., et al.: Simulating the universe with MICE: the abundance of massive clusters. Mon. Not. R. Astron. Soc. 403, 1253–1267 (2010) [10] Dautcourt, G.: Die Newtonsche gravitationstheorie als strenger grenzfall der allgemeinen relativit¨ atstheorie. Acta Phys. Pol. 25, 637–646 (1964) [11] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1998) [12] Ehlers, J.: On limit relations between, and approximative explanations of, physical theories VII. In: Marcus, B., Dorn, G.J.W., Weingartner, P. (eds.) Logic, Methodology and Philosophy of Science, vol. 114, pp. 387–403. Elsevier, Amsterdam (1986) [13] Einstein, A., Infeld, L., Hoffmann, B.: The gravitational equations and the problem of motion. Ann. Math. 39, 65–100 (1938) [14] Ellis, G.: Inhomogeneity effects in cosmology. Class. Qauntum Gravity 28, 164001 (2011) [15] Evrard, A., et al.: Galaxy clusters in Hubble volume simulations: cosmological constraints from sky survey populations. Astrophys. J. 573, 7 (2002) [16] Frauendiener, J.: A note on the relativistic Euler equations. Class. Quantum Gravity 20, L193–6 (2003)

Vol. 19 (2018)

Newtonian Limits on Long Time Scales

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[17] Friedrich, H.: On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587–609 (1986) [18] Friedrich, H.: On the global existence and the asymptotic behavior of solutions to the Einstein–Maxwell–Yang–Mills equations. J. Differ. Geom. 34, 275–345 (1991) [19] Friedrich, H.: Sharp asymptotics for Einstein-λ-dust flows. Commun. Math. Phys. 350, 1–42 (2016) [20] Friedrich, H., Rendall, A.: The Cauchy problem for the Einstein equations, Einstein’s field equations and their physical implications. Lect. Notes Phys. 540, 127–223 (2000) [21] Futamase, T., Itoh, Y.: The post-Newtonian approximation for relativistic compact binaries. Living Rev. Relativ. 10, 2 (2007) [22] Green, S., Wald, R.: A new framework for analyzing the effects of small scale inhomogeneities in cosmology. Phys. Rev. D 83, 084020 (2011) [23] Green, S., Wald, R.: Newtonian and relativistic cosmologies. Phys. Rev. D 85, 063512 (2012) [24] Hadˇzi´c, M., Speck, J.: The global future stability of the FLRW solutions to the dust-Einstein system with a positive cosmological constant. J. Hyper. Differ. Equ. 12, 87–188 (2015) [25] Heilig, U.: On the existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995) [26] Hwang, J., Noh, H.: Newtonian limit of fully nonlinear cosmological perturbations in Einstein’s gravity. JCAP 04, 035 (2013) [27] Hwang, J., Noh, H.: Newtonian, post-Newtonian and relativistic cosmological perturbation theory. Nuc. Phys. B Proc. Suppl. 246, 191–195 (2014) [28] Hwang, J., Noh, H., Puetzfeld, D.: Cosmological non-linear hydrodynamics with post-Newtonian corrections. JCAP 03, 010 (2008) [29] Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981) [30] Klainerman, S., Majda, A.: Compressible and incompressible fluids. Commun. Pure Appl. Math. 35, 629–651 (1982) [31] Kopeikin, S., Petrov, A.: Post-Newtonian celestial dynamics in cosmology: field equations. Phys. Rev. D 87, 044029 (2013) [32] Kopeikin, S., Petrov, A.: Dynamic field theory and equations of motion in cosmology. Ann. Phys. 350, 379–440 (2014) [33] Kreiss, H.O.: Problems with different time scales for partial differential equations. Commun. Pure Appl. Math. 33, 399–439 (1980) [34] K¨ unzle, H.: Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. Ann. Inst. Henri Poincar´e 17, 337–362 (1972) [35] K¨ unzle, H.: Covariant Newtonian limit of Lorentz space-times. Gen. Relativ. Gravity 7, 445–457 (1976) [36] K¨ unzle, H., Duval, C.: Relativistic and non-relativistic classical field theory on five-dimensional spacetime. Class. Quantum Gravity 3, 957–974 (1986)

2242

C. Liu, T. A. Oliynyk

Ann. Henri Poincar´e

[37] LeFloch, P.G., Wei, C.: The global nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FRW geometry. Preprint [arXiv:1512.03754] [38] Liu, C., Oliynyk, T.: Cosmological Newtonian limits on large spacetime scale. Commun. Math. Phys. (to appear) [39] Lottermoser, M.: A convergent post-Newtonian approximation for the constraint equations in general relativity. Annales de l’institut Henri Poincar´e (A) Physique th´eorique 57, 279–317 (1992) [40] L¨ ubbe, C., Valiente-Kroon, J.: A conformal approach for the analysis of the non-linear stability of radiation cosmologies. Ann. Phys. 328, 1–25 (2013) [41] Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984) [42] Matarrese, S., Terranova, D.: Post-Newtonian cosmological dynamics in Lagrangian coordinates. Mon. Not. R. Astron. Soc. 283, 400–418 (1996) [43] Milillo, I., et al.: Missing link: a nonlinear post-Friedmann framework for small and large scales. Phys. Rev. D 92, 023519 (2015) [44] Noh, H., Hwang, J.: Cosmological post-Newtonian approximation compared with perturbation theory. Astrophys. J. 757, 145 (2012) [45] Oliynyk, T.: The Newtonian limit for perfect fluids. Commun. Math. Phys. 276, 131–188 (2007) [46] Oliynyk, T.: Post-Newtonian expansions for perfect fluids. Commun. Math. Phys. 288, 847–886 (2009) [47] Oliynyk, T.: Cosmological post-Newtonian expansions to arbitrary order. Commun. Math. Phys. 296, 431–463 (2010) [48] Oliynyk, T.: A rigorous formulation of the cosmological Newtonian limit without averaging. J. Hyperbolic Differ. Equ. 7, 405–431 (2010) [49] Oliynyk, T.: Cosmological Newtonian limit. Phys. Rev. D 89, 124002 (2014) [50] Oliynyk, T.: The Newtonian limit on cosmological scales. Commun. Math. Phys. 339, 455–512 (2015) [51] Oliynyk, T.: Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant. Commun. Math. Phys. 346, 293–312 (2016) [52] Oliynyk, T., Robertson, C.: Linear cosmological perturbations on large scales via post-Newtonian expansions (in preparation) [53] R¨ as¨ anen, S.: Applicability of the linearly perturbed FRW metric and Newtonian cosmology. Phys. Rev. D 81, 103512 (2010) [54] Rendall, A.: The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys. 33, 1047–1053 (1992) [55] Rendall, A.: The Newtonian limit for asymptotically flat solutions of the Vlasov– Einstein system. Commun. Math. Phys. 163, 89–112 (1994) [56] Ringstr¨ om, H.: Future stability of the Einstein-non-linear scalar field system. Invent. Math. 173, 123–208 (2008) [57] Ringstr¨ om, H.: The Cauchy Problem in General Relativity. European Mathematical Society, Z¨ urich (2009) [58] Rodnianski, I., Speck, J.: The stability of the irrotational Euler–Einstein system with a positive cosmological constant. J. Eur. Math. Soc. 15, 2369–2462 (2013) [59] Schochet, S.: Asymptotics for symmetric hyperbolic systems with a large parameter. J. Differ. Equ. 75, 1–27 (1988)

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[60] Schochet, S.: Symmetric hyperbolic systems with a large parameter. Commun. Partial Differ. Equ. 11, 1627–1651 (1986) [61] Speck, J.: The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant. Sel. Math. New Ser. 18, 633–715 (2013) [62] Springel, V.: The cosmological simulation code GADGET-2. Mon. Not. R. Astron. Soc. 364, 1105–1134 (2005) [63] Springel, V., et al.: Simulations of the formation, evolution and clustering of galaxies and quasars. Nature 435, 629–636 (2005) [64] Taylor, M.E.: Partial Differential Equations III, Nonlinear Equations. Springer, New York (1996) [65] Walton, R.: A symmetric hyperbolic structure for isentropic relativistic perfect Fluids. Hous. J. Math. 31, 145–160 (2005) [66] Yamamoto, K., et al.: Perturbed Newtonian description of the Lemaˆıtre model with non-negligible pressure. JCAP 2016, 030 (2016) Chao Liu and Todd A. Oliynyk School of Mathematical Sciences Monash University 9 Rainforest Walk Clayton VIC 3800 Australia e-mail: [email protected] Todd A. Oliynyk e-mail: [email protected] Communicated by James A. Isenberg. Received: January 15, 2017. Accepted: March 19, 2018.