arXiv:1201.1963v1 [math.AP] 10 Jan 2012 ispto;ds qaino tt;eeg urn;globa current; energy state; formation. shock of equation dust dissipation; see yteMteaia cecsRsac nttt th Institute Research Sciences All-Insti Mathematical NSF the an by by istered and Technology, of Institute sachusetts H TBLZN FETO PCTM XASO ON EXPANSION SPACETIME OF EFFECT STABILIZING THE ahmtc ujc lsicto 2010. Classification Subject Mathematics grant Buchsbaum Solomon a by part in supported was author The ∗ ..Codnt ytm n ieeta prtr 12 1 4 operators differential and systems Coordinate conventions 2.2. Index 2.1. Notation article the of 2. Outline analysis the work of 1.3. previous Outline to connections Results and 1.2. Main Motivation of Summary and 1.1. Introduction 1. e od n phrases. and words Key Date EAIITCFUD IHSAPRSLSFRTHE FOR RESULTS SHARP WITH FLUIDS RELATIVISTIC ascuet nttt fTcnlg,Dprmn fMath of Department Technology, of Institute Massachusetts eso fOtbr3,2018. 30, October of Version : ntecnomlivrac fterltvsi ue equat Euler formatio relativistic shock the The of invariance time. conformal finite the in on shocks form conditio initial that arbitr solutions’ solutions that explicit t show the hold, we of to precisely, fails perturbations More factor unstable. scale are the solutions of inverse the for condition Abstract. hc losfrardcint elkonrsl fChrist of result well-known a to reduction a for allows which on.W loasm htteivreo h cl atrass factor scale the of a inverse verifies the metric that spacetime assume expanding also We sound. od hntesaeiei xoetal xadn.I the In state expanding. of exponentially equation analogo is ation an spacetime solutio that the explicit showed when s the which holds to results, perturbed close previous the remain of that and extension times shows future expansion, all ex the for which Fri by analysis, flat generated nonlinear spatially terms well-known Our the family. of Robertson-Walker analogs solution are explicit factor The scale conditions. future-sta initial spatially globally their of is motivated, bations solutions physically fluid of isotropic energy family spatially vectorfield explicit the use an we that assumptions, these Under tion. ihsailsie htaedffoopi to diffeomorphic are spac that expanding slices flat spatial conformally with pre-specified a on equations fisteeuto fstate of equation the ifies AITO QAINO STATE OF EQUATION RADIATION nti ril,w td h iesoa eaiitcE relativistic dimensional 3 + 1 the study we article, this In ceeae xaso;cnomlivrac;cosmologic invariance; conformal expansion; accelerated p (1 = al fContents of Table p / = AE SPECK JARED 3) c ρ, s 2 ρ, eas hwta ftetime-integrability the if that show also we rmr:3A1 eodr:3Q1 60,83F05. 76Y05, 35Q31, Secondary: 35A01; Primary: hr 0 where c i s − eedn ieitgaiiycondi- time-integrability dependent R 3 ∗ ≤ . xsec;rdaineuto fstate; of equation radiation existence; l easm httefli ver- fluid the that assume We og t oegatDMS-0441170. grant core its rough c ue ototrlFlosi admin- Fellowship Postdoctoral tutes s ≤ scnluc perturbed launch can ns l ne ml pertur- small under ble orsodn oeach to corresponding s mtc,Cmrde A USA. MA, Cambridge, ematics, p e h xlctfluid explicit the hen oswhen ions 1 s hswr san is work This ns. rl ml smooth small arily ooeeu,and homogeneous, / edmann-Lemaˆıtre- tm background etime ssaiiyresult stability us stesedof speed the is 3 lisdissipative ploits odoulou. ehdt prove to method aeo h radi- the of case diitrdb h Mas- the by administered ro sbased is proof n cae othe to ociated ltosexist olutions c s 2 1 = uler / lconstant; al 3 , 11 11 11 6 The Stabilizing Effect of Spacetime Expansion ii
2.3. Norms 12 2.4. Numerical constants 13 3. Alternative Formulation of the Relativistic Euler Equations 13 3.1. Alternative formulation of the relativistic Euler equations 13 3.2. Local existence and the continuation principle 16 3.3. Matrix-vector notation 17 4. Norms,Energies,andtheEquationsofVariation 18 4.1. Norms 18 4.2. The equations of variation and energies for the fluid variables 19 5. Sobolev Estimates 24 6. The Energy Norm vs. Sobolev Norms ComparisonProposition 30 7. IntegralInequalitiesfortheEnergyandNorms 31 8. The Future Stability Theorem 32 9. The Sharpness of the Hypotheses for the Radiation Equation of State 34 Acknowledgments 37 A. Sobolev-Moser Inequalities 37 References 38 The Stabilizing Effect of Spacetime Expansion 1
1. Introduction and Summary of Main Results In this article, we study the future-global behavior of small perturbations of a family of physically motivated explicit solutions to the 1+3 dimensional relativistic Euler equations. We assume throughout that the fluid equation of state is
2 (1.1) p = csρ, where p 0 denotes the pressure, ρ 0 denotes the energy density, and the ≥ ≥ constant cs 0 is known as the speed of sound. The explicit solutions model a spatially homogeneous≥ isotropic fluid of strictly positive energy density evolving in a pre-specified spacetime-with-boundary ([1, ) R3,g) which is undergoing expansion. Here, g is an expanding Lorentzian metric∞ × of the form (1.2). As is dis- cussed in Section 1.1 in more detail, such solutions play a central role in cosmology, where much of the “normal matter”1 content of the universe is often assumed to be effectively modeled by a fluid with the aforementioned properties. 2 Although all speeds verifying 0 cs 1 are studied in the cosmology literature, 2 2 ≤ ≤ the cases cs = 0 and cs = 1/3 are of special significance. The former is known as the case of the “pressureless dust,” and is often assumed to be a good model for the normal matter in the present-day universe. The latter is known as the case of “pure radiation,” and is often assumed to be a good model of matter in the early universe. For an introductory discussion of these issues, see [Wal84, Section 5.2]. 2 As we shall see, in order to prove our main results, we will assume that cs [0, 1/3]. 2 ∈ We simultaneously analyze the cases cs (0, 1/3) using the same techniques, while 2 2 ∈ 2 the endpoint cases cs = 0 and cs = 1/3 require some modifications; when cs = 0, ρ loses one degree of differentiability relative to the other cases, while the analysis 2 of the case cs = 1/3 is based on the fact that the relativistic Euler equations are conformally invariant in this case (see Section 9). We will also provide some 2 heuristic evidence of fluid instability when cs > 1/3; (see Section 1.2.3). We assume throughout that the pre-specified spacetime metric g is of the form
3 (1.2) g = dt2 + e2Ω(t) (dxj )2, Ω(1) = 0, − j=1 X where eΩ(t) > 0 is known as the scale factor, t [1, ) is a time coordinate, and (x1, x2, x3) are standard coordinates on R3. A major∈ ∞ goal of this article is to study the influence of the scale factor on the long-time behavior of the fluid. Our analysis will heavily depend on the assumptions we make on eΩ(t), which are stated later in this section. Metrics of the form (1.2) are of particular importance because exper- imental evidence indicates that universe is expanding and approximately spatially flat [EBW92]. According to Steven Weinberg [Wei08, pg. 1], “Almost all of mod- ern cosmology is based on [metrics of the form (1.2)].” The physical significance of these metrics is discussed more fully in Section 1.1. 2 Under equations of state of the form p = csρ, the relativistic Euler equations in the spacetime-with-boundary ([1, ) R3,g) are the four equations ∞ ×
1In cosmological models, “normal matter” is distinguished from “dark matter;” the latter interacts with normal matter indirectly, i.e., only through its gravitational influence. The Stabilizing Effect of Spacetime Expansion 2
αµ (1.3) DαT =0, (µ =0, 1, 2, 3), where
(1.4) T µν = (ρ + p)uµuν + p(g−1)µν is the energy-momentum tensor of a perfect fluid, D is the Levi-Civita connection of g, and uµ is the four-velocity, a future-directed (u0 > 0) vectorfield normalized by
(1.5) g uαuβ = 1. αβ − Equivalently, the following equation for u0 holds in our coordinate system:
0 a b 1/2 (1.6) u =(1+ gabu u ) .
Under the above assumptions, it is well-known (see e.g. [Spe11]) that when ρ> 0, the relativistic Euler equations (1.3) can be written in the equivalent form
α 2 α (1.7a) u Dα ln ρ +(1+ cs)Dαu =0, 2 α µ cs αµ (1.7b) u Dαu + 2 Π Dα ln ρ =0, (µ =0, 1, 2, 3), 1+ cs (1.7c) Πµν def= uµuν + (g−1)µν , (µ,ν =0, 1, 2, 3).
The tensorfield Πµν projects onto the g orthogonal complement of uµ. We remark that the system (1.7a) - (1.7b) is consistently− overdetermined in the sense that the 0 component of (1.7b) is a consequence of the remaining (including (1.6)) equations. Other important experimentally determined facts are: on large scales, the matter content of the universe appears to be approximately spatially homogeneous (see e.g. [YBPS05]) and isotropic (see e.g. [WLR99]). These conditions can be modeled by the following explicit homogeneous/isotropic fluid “background” solution:
def 2 def −3(1+cs)Ω(t) µ µ (1.8) ρ =ρe ¯ , u = δ0 , whereρ> ¯ 0 is a constant.e It follows easily frome the discussion in Section 3.1 that (1.8) are solutions to (1.7a) - (1.7b) (where (t, x1, x2, x3) are coordinates such that the metric g has the form (1.2)); see Remark 3.1. The main goal of this article is to understand the future-global dynamics of solutions to (1.7a) - (1.7b) that are launched by small perturbations of the initial data corresponding the explicit solutions (1.8). In order to prove our main stability results, we make the following assumptions on the scale factor. Assumptions on eΩ(t) The Stabilizing Effect of Spacetime Expansion 3
A1 eΩ(·) C1 [1, ), [1, ) and increases without bound as t ∈ ∞ ∞ → ∞ ∞ −2Ω(s) 2 s=1 e ds < , cs =0, ∞ −Ω(s) ∞ 2 A2 e B(Ω(s)) ds < , 0
qΩ 2 (choose B(Ω) = e for some small constant q > 0 when 0 < cs < 1/3). The significance of these expansion rates will be discussed in more detail in Section 1.1.
1.1. Motivation and connections to previous work. We are interested in the system (1.7a) - (1.7b) because it plays a fundamental role in the standard model of cosmology. More precisely, physicists/cosmologists often couple the Einstein equations of general relativity to the relativistic Euler equations and sometimes also to additional matter models. They then study explicit solutions of the resulting coupled system and make predictions about the long-time behavior of the universe based on the properties of the explicit solutions. Now a basic requirement for the explicit solutions to have any long-time predictive value is that they be future- stable under small perturbations of their data; this basic requirement is the primary motivation behind our investigation. In the coupled problem mentioned above, the spacetime metric g is not pre- specified, but is instead one of the dynamic variables. In this article, we don’t address the coupled problem, but rather the behavior of the fluid matter under the influence of a large class of pre-specified spacetime metrics g. Our hypotheses A1 - A3 on g are meant to roughly capture the main features of the kinds of metrics that may arise in a large class of physically motivated expanding solutions to the coupled problem. Readers may consult e.g. [Ren05b], [Ren06] for background information on the coupled Einstein-matter problem in the presence of expansion. The physical relevance of these expanding metrics is the following: experimental evidence sug- gests that our own universe seems to be undergoing accelerated expansion. There are many experimental references available on the issue of accelerated spacetime ex- pansion; [RFC+98], [PR99] are two examples. However, as noted in e.g. [Ren05a], [Ren06], [Rin09], the precise rate and mechanism of the expansion of our universe have yet to be determined. This uncertainty has motivated us to make the very general hypotheses A1 - A3. We expect that these hypotheses will be roughly consistent with the behavior of most of the physically relevant metrics g that arise in the study of accelerated expansion in the coupled problem. Now d Ω(t) d2 Ω(t) for metrics of the form (1.2), accelerated expansion (i.e. dt e , dt2 e > 0) is def d equivalent to (where ω(t) = dt Ω(t)) (1) ω > 0 d 2 (2) dt ω + ω > 0. We note that these conditions are not equivalent to A1 - A3; it is not difficult to see that (1) (2) may hold even if A2 - A3 fail (for all functions B( )), and conversely A1 - A3−may hold without the expansion being accelerated for· all time (we don’t even need ω to be differentiable in order to prove our results). Nevertheless, many of the scale factors appearing in the cosmology literature verify (1) (2) and also A1 - A3. Later in this section we will further discuss the two previously− mentioned examples of exponential expansion and accelerated power-law expansion. Our main results are further motivated by the following: based on our previous experience [RS09], [Spe11], in certain accelerated expanding regimes, we expect that a nearly quiet fluid’s influence on g in the coupled problem is often lower-order in comparison to the other influences (including the primary influences driving the expansion). In other words, as long as the fluid behavior is tame, we expect that the coupled problem effectively behaves as a partially decoupled system in that g The Stabilizing Effect of Spacetime Expansion 5 can highly influence the fluid, but not the other way around2. Thus, we have the following moral consequence of our Main Results: we expect that the addition of 2 2 a perfect fluid (verifying p = csρ, 0 cs 1/3) would not destroy the stability of a stable solution to a coupled Einstein-matter≤ ≤ system, as long as the stable metrics are expanding at rates in the spirit of A1 - A3. As we will soon discuss, this expectation has previously been confirmed when the expansion is generated by the inclusion of a positive cosmological constant in the Einstein equations; in this case the expansion is very rapid and verifies eΩ(t) eHt, where H > 0 is a constant. The key question from the previous paragraph∼ (which is addressed by our main results) is whether or not the fluid behavior remains tame if it is initially tame. This is a highly non-trivial question whose answer depends on the expansion prop- erties of g. In the case of Minkowski spacetime (where eΩ(t) 1), Christodoulou’s monograph [Chr07] demonstrated that under a general physical≡ equation of state, µ µ the constant solutions (i.e., ρ ρ>¯ 0, u δ0 ) to the relativistic Euler equations are unstable3. More precisely,≡ there exists≡ an open family of initial data whose corresponding solutions develop shock singularities in finite time. Furthermore, the family contains arbitrarily small (non-zero) perturbations of the constant state, in- cluding irrotational data. We state some of Christodoulou’s main results in more detail in Theorem 9.1 below. One very important aspect of his monograph is that it provides a complete description of the nature of the shock (we do not recall the fully detailed picture in Theorem 9.1). His work can be therefore be viewed as a major extension of the well-known article [Sid85], in which Sideris showed that singularities can form in the non-relativistic Euler equations under the polytropic equations of state for arbitrarily small perturbations of constant background solu- tions; in [Sid85], a detailed picture of the singularity formation was not obtained because the argument involved the analysis of averaged quantities. In contrast to Christodoulou’s shock formation result, the works [RS09], [Spe11] of Rodnianski and the author showed the following: when the scale factor verifies Ω(t) Ht 2 e e for some “Hubble constant” H > 0, the equation of state is p = csρ, and ∼2 0 < cs < 1/3, then (g, ρ, u) (defined in (1.2) and (1.8)) is a future-stable solution to the coupled Euler-Einstein equations. The exponential expansion eΩ(t) eHt of ∼ g in these solutions is drivene e by the inclusion of a positive cosmological constant Λ in the Einstein equations4, where H = Λ/3. These special “Friedmann-Lemaˆıtre- Robertson-Walker” solutions (g, ρ, u) lie at the heart of many cosmological predic- 2 p tions. Furthermore, the case cs = 1/3 was recently addressed in [LV11] using an extension of Friedrich’s conformale methode (see e.g. [Fri02]), which is closely related to the methods we use in Section 9. Roughly speaking, the conformal method is a collection of techniques for transforming the question of global-in-time existence into a question of local-in-time existence for the conformal field equations through changes of variables. This method often works when the energy-momentum tensor of the matter model is trace-free (as is the case of the fluid energy-momentum tensor
2This expectation is false for the Friedmann-Lemaˆıtre-Robertson-Walker family of solutions to the Euler-Einstein system with no cosmological constant or additional matter model. In this case, the fluid itself is the primary influence driving the expansion, which is not accelerated. 3There was one exceptional equation of state which features small-data global existence when the fluid is irrotational; this equation of state leads (in the irrotational case) to the quasilinear wave equation for minimal surface graphs. 4In the coupled problem, the scale factor corresponding to the explicit solution metric g verifies Friedmann’s ODE, which leads to the asymptotic behavior eΩ(t) ∼ eHt. The Stabilizing Effect of Spacetime Expansion 6
2 (1.4) when cs = 1/3.) In summary, in the mild initial condition regime, previous results show that exponential expansion suppresses the formation of fluid shocks 2 when 0
(1.9) (t) ǫ SN ≤ holds on a time interval [1,T ) of existence, where ǫ is a sufficiently small positive number; by standard local existence theory, this assumption holds for short times when the data are sufficiently small. Based on this assumption, we will then derive inequalities for (t) leading to an improvement on the assumption (1.9), as long SN as the data and ǫ are sufficiently small. Consequently, since N (t) is continuous, it can never exceed a certain size, and future stability followsS from the continuation principle. The complete argument is provided in the proof of Theorem 8.1. 1.2.2. Energy currents. The only general method we know of for estimating so- lutions to quasilinear hyperbolic PDEs (such as the relativistic Euler equations) involves the derivation of coercive integral identities. This framework is often re- ferred to as the energy method, and its raison d’ˆetre is connected to the following The Stabilizing Effect of Spacetime Expansion 7
basic difficulty: the time derivative of N (t) cannot be controlled in terms of the so- lution itself, for it is a crudely definedS Sobolev norm which has no special structure tying it to the evolution of the fluid solution. To estimate N (t), we will construct S 2 coercive quantities N (t) called energies. N (t) will be used to control the L norms of the up-to-top-orderE (i.e., up-to-N th order)E spatial derivatives of the solution via − the divergence theorem (see (1.11)). We remark that the coercive nature of N (t) 2 E is established in Proposition 6.1. We also remark that in the case cs = 0, we will define separate energies N;velocity and N−1;density for the four-velocity and the density. The reason is thatE in this case, theE evolution of the four velocity decouples from that of the density, and furthermore, the density is necessarily one degree less differentiable than the four-velocity. To construct the energies, we will use a version of the method of vectorfield mul- tipliers. Over the last few decades, this fundamental method has been applied in many different contexts to a large variety of hyperbolic PDEs. All applications of this method are essentially extensions of Noether’s theorem [Noe71]. In particular, we make use of the multiplier method framework developed in [Chr00], a work that can be viewed as a geometric extension of Noether’s theorem to handle regularly hyperbolic (see [Chr00] for Christodoulou’s definition of regular hyperbolicity) sys- tems/solutions that are derivable from a Lagrangian but perhaps lack symmetry (and hence the energy estimates contain error terms). The multiplier method allows us to construct compatible “energy” currents with the help of an auxiliary multi- plier vectorfield; we will further discuss the specific multiplier used in this article later in this section. The energy currents are special solution-dependent vectorfields µ J˙ [∂~αW, ∂~αW], and the energies are
1/2 def 0 3 (1.10) N (t) = J˙ [∂~αW, ∂~αW] d x . E R3 |~αX|≤N Z 2 Here, ∂ is an order N spatial differential operator, W = (ln(e3(1+cs)Ωρ/ρ¯),u1,u2,u3) ~α ≤ is an array of fluid variables, and the precise definition of J˙µ is given in Definition µ 4.2. As is suggested by the notation, J˙ [∂~αW, ∂~αW] depends quadratically on ∂~αW. Furthermore, the coefficients of the quadratic terms depend on the solution W itself. In this article, we will not fully explain our construction of the currents J˙µ. Roughly speaking, these currents exist because the relativistic Euler equations are the Euler-Lagrange equations corresponding to a Lagrangian. For such PDEs, Christodoulou’s aforementioned framework [Chr00] explains how one can construct suitable currents J˙µ using a version of the multiplier method. In fact, energy currents for the relativistic Euler equations were first derived by Christodoulou in the Lagrangian-coordinate framework5 in [Chr00], and later in the Eulerian- coordinate framework in [Chr07]. The Eulerian-coordinate fluid energy current framework has since been applied by the author and his collaborator in various contexts; see [Spe09a], [Spe09b], [Spe11], and [SS11].
5Interestingly, the relativistic Euler equations fall just outside of the scope of the methods of [Chr07]; in the Lagrangian-coordinate framework, the methods of [Chr07] lead to semi-coercive estimates rather than fully coercive estimates. Nevertheless, we are able to derive fully coercive estimates in the Eulerian-coordinate framework. The Stabilizing Effect of Spacetime Expansion 8
Assuming that 0 d 2 def ˙µ 3 (1.11) N (t) = ∂µ J [∂~αW, ∂~αW] d x dtE R3 |~α|≤N Z X (2) (3) (N) 3 = FN (Ω(t),ω(t); W, ∂W, ∂ W, ∂ W, , ∂ W) d x. R3 ··· Z Above, FN is a smooth function of its arguments. By (1), the right-hand side of (1.11) is controllable in terms of N (t). Thus, N (t) is a norm-like quantity whose time evolution can be estimated.E Roughly speaking,E in order to apply the continuation principle, we have to show that the identity (1.11) prevents N (t) and (t) from blowing up in finite time. In particular, we have to carefullyE SN investigate the structure of FN ( ); see the extended discussion in Section 1.2.4. ···2 ˙µ We also remark that in the case cs =0, we will utilize two distinct currents Jvelocity ˙µ and Jdensity in order to handle the partially decoupled nature of the relativistic Euler equations under the dust equation of state; this is of course connected to our previously mentioned use of separate energies N;velocity and N−1;density when 2 E E cs =0. It is interesting to note that Christodoulou showed [Chr00] that the Lagrangian for relativistic fluid mechanics is generic, which means that up to null currents6 (which by themselves lead to non-coercive integral identities) the method of vector- field multipliers generates all possible integral identities that a solution can verify. Furthermore, he showed that a certain widely used subclass of compatible cur- rents, namely those generated by domain multiplier vectorfields, have the following property: the only useful members of this class (i.e., generating an L2 coercive compatible “energy” current) are generated by domain multiplier vectorfields− that belong to the inner core of the characteristic subset of the tangent space. Now for the relativistic Euler equations, the inner core is the degenerate one-dimensional set span u (see [Spe09a]); this is connected to the fact that the fluid vorticity two-form{ verifies} a transport equation in the direction of u. Thus, it can be shown that up to multiplication by a scalar function, the only coercive compatible energy current generated by the domain multiplier vectorfields is the one defined below in Definition 4.2. 6A null current is a solution-dependent vectorfield whose divergence can be shown to be in- dependent of the solution’s derivatives without having to use the equations of motion (which are the relativistic Euler equations in the present article). Such null currents have a special algebraic structure and have been completely classified [Chr00]. The Stabilizing Effect of Spacetime Expansion 9 2 1.2.3. The analytic effect of expansion and the restriction 0 cs 1/3. From the analytic standpoint, the stabilizing effect of spacetime expansion≤ ca≤n be summarized Ω(t) 2 j as follows: the scale factor e generates the dissipative ω(t)(3cs 2)u term on the right-hand side of (3.7c). The sources of these dissipative terms− are the Christoffel symbols of g relative to the coordinate system (t, x1, x2, x3), which are provided in 2 Lemma 3.3. In the cases 0 ′ Ω j 2 B (Ω) Ω j Ω ′j (1.12) ∂t e B(Ω)u = ω 3c 1+ e B(Ω)u + e B(Ω) , s − B(Ω) △ n o def d where B is the function from the hypotheses A2 - A3 and ω(t) = dt Ω(t). If we assume that the right-hand side of (1.12) is negligible (we will in fact establish 2 this assumption when 0 < cs < 1/3 and (1.9) holds), then by integrating the “ODE” (1.12), we deduce that uj . ǫe−Ω(t)B−1(Ω(t)). Therefore, the size of the | | a b 2Ω a b 2 −2 spatial part of u as measured by g is gabu u = e δabu u . ǫ B (Ω(t)). Using | | 2 a b hypotheses A1 - A3, it follows that when 0 1.2.4. Comments on the analysis of (1.11). Much of our work goes into estimating (2) (3) (N) 3 the 3 F (Ω(t),ω(t); W, ∂W, ∂ W, ∂ W, , ∂ W) d x term on the right- R N ··· hand side of (1.11) in terms of the Sobolev norm N under the bootstrap assump- R S 2 tion (1.9). The end result of these estimates in the cases 0 µ 3 upper bound for 3 ∂ (J˙ [∂ W, ∂ W]) d x in terms of the norm . A |~α|≤N R µ ~α ~α SN very important aspect of our argument in these cases is the following: in order P R to derive (5.10) and to close our global existence argument, we need to prove that the lower-order spatial derivatives of uj decay faster than its top-order derivatives. To see why this is the case, we will now show that without this improved decay, there would be an obstruction to global existence. We illustrate this obstruction by closely investigating a specific term from the right-hand side of (1.11). In particular, 2 2 2csF (1+cs) a b the 2 L term from the identity (4.18) below, where F = ω 0 gabu u is the (1+cs) − u term on the right-hand side of (3.2a), leads to the identity 2Ω e δab d 2 2 1 a b 3 (1.13) N (t)= 2cs ω 0 gab u u L d x + , dtE − R3 u ··· Z z}|{ def 2 3 3(1+cs)Ω 2 2 2Ω j 2 where L = ln e ρ/ρ¯ , and (t) L N + e u N . Using EN ≈ k kH j=1 k kH j j −Ω (1.9), (1.13), the Sobolev embedding result u L∞ . u HP2 . e N , and the estimate u0 1, we deduce that k k k k E ≥ d (1.14) 2 (t) . ω 3 (t)+ . dtEN EN ··· 2 Inequality (1.14) allows for the dangerous possibility that N (t) grows unabatedly, and is therefore not sufficient to prove our main stability theorem.E def To remedy this difficulty, we introduce the lower-order Sobolev norm N−1 = 1/2 U Ω 3 j 2 e B(Ω) u N−1 (see Definition 4.1). Here, B is the function from the j=1 k kH hypotheses A2 - A3. The main point is the following: if we knew that N−1 ǫ, P j j −Ω −1 U ≤ then the Sobolev embedding estimate u ∞ . u 2 . e B (Ω) . k kL k kH UN−1 ǫe−ΩB−1(Ω) would allow us to upgrade (1.14) to d (1.15) 2 (t) . ǫω(t)B−1(Ω(t)) 2 (t)+ . dtEN EN ··· Because of our assumptions A1 - A3 on the scale factor, we have that ω(t)B−1(Ω(t)) Ω(t) = d B−1(Ω) dΩ L1([1, )). Therefore, the first term on the right-hand dt 1 ∈ t ∞ siden ofR (1.15) is amenableo to Gronwall’s inequality. Assuming that the remaining terms could bee treatede similarly, we would therefore be able to derive an a priori ··· 2 estimate for N (t) guaranteeing that it remains uniformly small for all time; thanks to the continuationE principle, this is the main step in proving global existence. At its core, the proof of our main stability theorem is essentially a more elaborate version of the estimate (1.15). It remains to discuss how we obtain the desired control over the norm N−1. Roughly speaking, we will use the improved decay for the uj suggested by equationU (1.12). It turns out that the error terms corresponding to this equation are small enough that we can effectively control the time derivative of N−1(t) even if we 3 Ω j U only have control over the weaker norm e ∂ u 2 of the top-order j=1 |~α|=N k ~α kL derivatives (which will follow from the bootstrap assumption (1.9)). The precise P P estimates corresponding to this fact are provided in inequalities (4.23) and (5.4d) The Stabilizing Effect of Spacetime Expansion 11 below. We emphasize that in order to derive the improved decay (improved rela- tive to the rate predicted by the Sobolev norm bootstrap assumption (1.9)), what matters is the interaction between the hypotheses A1 - A3 on the scale factor and the structure of the error terms. Note that (1.12) is not literally an ODE in uj since spatial derivatives of the solution are present on the right-hand side as “error terms.” Therefore, the approach we have described only allows us to derive improved decay estimates for the below-top-order derivatives of the solution. We remark that we do not use the norm in our analysis of the cases c2 =0, 1/3. UN−1 s 2 2 1.2.5. Sharp results in case cs = 1/3. Our sharp analysis in the case cs = 1/3 is based on the well-known conformal invariance of the relativistic Euler equations 2 in this case; see Section 9 for more details. Roughly speaking, when cs =1/3, one dτ −Ω(t) can perform a change-of-time-variable dt = e , τ(1) = 1, and also a rescaling of the fluid variables in order to translate the problem of interest into an equivalent ∞ −Ω(s) problem in Minkowski spacetime. Now if assumption A1 holds but s=1 e ds = , then the analysis of Section 9 shows that the instability of the background fluid∞ solutions follows easily from the instability results derivedR in [Chr07]; see Corollary 9.2. Furthermore, the arguments given in Section 9 could easily have been extended to show the future stability of the background fluid solutions when 2 ∞ −Ω(s) cs = 1/3, hypothesis A1 holds, and s=1 e ds < . However, we instead chose to use the energy-method/continuation principle framewor∞ k to prove future R stability (see Theorem 8.1) since this framework is stable and since there is no known alternative to this framework in the cases c2 [0, 1/3). s ∈ 1.3. Outline of the article. The remainder of the article can be summarized as follows. In Section 2 we introduce some standard notation and conventions. • In Section 3, we derive an equivalent version of the relativistic Euler equa- • tions that is useful for our ensuing analysis. We also introduce some stan- dard PDE matrix-vector notation. In Section 4 we introduce the Sobolev norms and the related energies that • we use to analyze solutions. In Section 5, we derive the Sobolev estimates that play a major role in our • derivation of differential inequalities for the fluid norms and energies. In Section 6 we prove a simple comparison proposition, which shows that • the energies we have defined can be used to control the norms. In Section 7, we use the estimates of Section 5 to derive differential inequal- • ities for the norms and energies. In Section 8, we use the differential inequalities to prove our main future • stability theorem. In Section 9, we use a well-known result of Christodoulou to show that in • 2 ∞ −Ω(s) the case cs =1/3, the non-integrability condition s=1 e ds = leads to the nonlinear instability of the explicit fluid solutions. ∞ R 2. Notation 2.1. Index conventions. Greek “spacetime” indices α, β, take on the values 0, 1, 2, 3, while Latin “spatial” indices a, b, take on the··· values 1, 2, 3. Pairs of repeated indices are summed over their respective··· ranges. We lower spacetime −1 µν indices with gµν and raise them with (g ) . The Stabilizing Effect of Spacetime Expansion 12 2.2. Coordinate systems and differential operators. We perform most of our computations relative to the standard rectangular coordinate system (x0, x1, x2, x3) on R4. We often write t instead of x0. In this coordinate system, the metric g is of ∂ the form (1.2). We often use the symbol ∂µ to denote the coordinate derivative ∂xµ , and we often write ∂t instead of ∂0. Occasionally, (see e.g. the derivation of (5.27)) we also use the rescaled time variable τ introduced in Section 9. The corresponding rescaled partial time derivative is denoted by ∂τ . All spatial derivatives are always taken with respect to the coordinates (x1, x2, x3). If ~α = (n1,n2,n3) is a triplet of non-negative integers, then we define the spatial def n1 n2 n3 multi-index coordinate differential operator ∂~α by ∂~α = ∂1 ∂2 ∂3 . We use the notation ~α def= n + n + n to denote the order of ~α. | | 1 2 3 (2.1) r s D T ν1···νr = ∂ T ν1···νr + Γ νa T ν1···νa−1ανa+1νr Γ α T ν1···νr µ µ1···µs µ µ1···µs µ α µ1···µs − µ µa µ1···µa−1αµa+1µs a=1 a=1 X X denotes the components of the covariant derivative of a spacetime tensorfield T. α The Christoffel symbols Γµ ν are defined in (3.11). (N) ν1···νr th ∂ Tµ1···µs denotes the array containing of all of the N order spacetime coordi- ν1···νr (N) ν1···νr nate derivatives (including time derivatives) of the component Tµ1···µs . ∂ Tµ1···µs denotes the array of all N th order spatial coordinate derivatives of the component ν1···νr Tµ1···µs . We omit the superscript when N =1. 2.3. Norms. We define the standard Sobolev norm of a function f HN as follows: 1/ 2 def 1 2 3 2 3 (2.2) f HN = ∂~αf(t, x , x , x ) d x . R3 |~α|≤N Z X The above volume form d3x corresponds to the standard flat metric on R3. We denote the N th order homogeneous Sobolev norm of f by (N) def (2.3) ∂ f L2 = ∂~αf L2 . |~α|=N X Rn Tn N If K or K , then Cb (K) denotes the set of N times continuously differentiable⊂ functions⊂ (either scalar or array-valued, depending− on context) on the interior of K with bounded derivatives up to order N that extend continuously to the closure of K. We define the norm corresponding to this function space by def (2.4) F N,K = ess sup ∂I~F ( ) , | | ·∈K | · | |I~X|≤N where ∂I~ is a multi-indexed differential operator representing repeated partial dif- ferentiation with respect to the arguments of F, which may be either spacetime coordinates or metric/fluid components depending· on context. When N = 0, we use the slightly more streamlined notation The Stabilizing Effect of Spacetime Expansion 13 def (2.5) F K = esssup F ( ) . | | ·∈K | · | Furthermore, we define (N) def (2.6) F K = ∂ F K. | | | I~ | |I~X|=N When K = R3, we sometimes use the more familiar notation def (2.7) F L∞ = ess sup F (x) , k k x∈R3 | | def ∞ (2.8) F CN = ∂~αF L . k k b k |~α|≤N X If A is an m n (often 4 4 in this article) array-valued function with entries × × A , (1 j m, 1 k n), then in Section 5, we write e.g. A N to denote the jk ≤ ≤ ≤ ≤ k kH m n array whose entries are Ajk HN . We use similar notation for other norms of ×A. k k If I R is an interval and X is a normed function space, then CN (I,X) denotes the set⊂ of N-times continuously differentiable maps from I into X. 2.4. Numerical constants. C denotes a numerical constant that is free to vary from line to line. We sometimes write e.g. C(N) when we want to explicitly indicate the dependence of C on quantities. We use symbols such as c, C∗, etc., to denote constants that play a distinguished role in the discussion. We write X . Y when there exists a constant C > 0 such that X CY. We write X Y when X . Y and Y . X. ≤ ≈ 3. Alternative Formulation of the Relativistic Euler Equations In this section, we derive an equivalent version the relativistic Euler equations 2 (1.7a) - (1.7b) under the equation of state p = csρ; we will work with this version for most of the remainder of the article. We then briefly discuss a classical local existence result and a continuation principle. The latter provides sufficient criteria for the solution to avoid forming a singularity. 3.1. Alternative formulation of the relativistic Euler equations. Our al- ternative formulation of the relativistic Euler equations is captured it the next proposition. Proposition 3.1 (Alternative formulation of the Euler equations). Assume 2 that the fluid verifies the equation of state p = csρ. Let def 2 (3.1) L = ln e3(1+cs)Ωρ/ρ¯ be a normalized energy density variable, where ρ>¯ 0 is the constant corresponding 2 to the background density variable ρ =ρe ¯ −3(1+cs)Ω. Then in the coordinate system e The Stabilizing Effect of Spacetime Expansion 14 (t, x1, x2, x3), the relativistic Euler equations (1.7a) - (1.7b) are equivalent to the following system of equations in the unknowns (L,u1,u2,u3) (for j =1, 2, 3): 1 (1 + c2) (3.2a) uα∂ L +(1+ c2) u ∂ ua +(1+ c2)∂ ua = ω s g uaub, α s u0 a t s a − u0 ab 2 α j cs jα 2 0 j (3.2b) u ∂αu + 2 Π ∂αL = ω(3cs 2)u u . (1 + cs) − Above, def def def d u0 = (1+ g uaub)1/2, Πµν = uµuν + (g−1)µν , ω(t) = Ω(t). ab dt Furthermore, u0 is a solution to the following equation: 2 α 0 cs 0α 2 a b (3.3) u ∂αu + 2 Π ∂αL = ω(3cs 1)gabu u . (1 + cs) − Proof. To obtain (3.2a), we first expand the covariant differentiation in (1.7a) to deduce the following equation: (3.4) uα∂ ln ρ +(1+ c2)∂ uα = (1 + c2)Γ α uβ. α s α − s α β α β 0 0 Lemma 3.3 implies that Γα βu = 3ωu , while equation (1.6) implies that ∂tu = 1 a a b u0 ua∂tu + ωgabu u . Equation (3.2a) now follows from these identities and the identity ∂ ln ρ = ∂ ln L 3(1 + c2)ω. t t s Similarly, to obtain (3.2b),− we first expand the covariant differentiation in (1.7b) to deduce the following equation: 2 α j cs jα j α β (3.5) u ∂αu + 2 Π ∂α ln ρ = Γα βu u . (1 + cs) − 2 cs jα Equation (3.2b) now follows from (3.5), Lemma 3.3, and the identity 2 Π ∂α ln ρ = (1+cs) 2 cs jα 2 0 j 2 Π ∂αL 3csωu u . The proof of (3.3) is similar, and we omit the details. (1+cs) − Remark 3.1. Note that in terms of the variables (L,u1,u2,u3), the background solution (1.8) takes the form (3.6) (L, u1, u2, u3)=(0, 0, 0, 0). Part of our analysis involves solving for ∂ L and ∂ uµ and treating spatial deriva- e e e e t t tives as inhomogeneous error terms. This approach can also be used to analyze ∂~αL µ and ∂~αu , for ~α N 1. As a preliminary step in this analysis, we solve for ∂tL µ | |≤ − and ∂tu . The Stabilizing Effect of Spacetime Expansion 15 Corollary 3.2 (Isolated Time Derivatives). Let (L,u1,u2,u3) be a solution to the relativistic Euler equations (3.2a) - (3.2b), where u0 is defined by (1.6). Then the time derivatives of the fluid variables can be expressed as follows: (3.7a) ∂ L = ′, t △ (3.7b) ∂ u0 = ′0, t △ (3.7c) ∂ uj = ω(3c2 2)uj + ′j , t s − △ where the error terms ′, ′0, and ′j are defined by △ △ △ (3.8a) a b gabu u ′ 2 2 (u0)2 = ω(1 + cs)(1 3cs) a b 2 gabu u △ − 1 c 0 2 − s (u ) g uaub −1 ua ∂ ua (1 + c2) + 1 c2 ab (c2 1) ∂ L (1 + c2) a + s g uauk∂ ub , − s (u0)2 s − u0 a − s u0 (u0)3 ab k (3.8b) n o a b a b gabu u gabu u 2 1 0 2 ′0 2 u0 cs − (u ) a = ω(3cs 1) a b a b u ∂aL 2 gabu u 2 2 gabu u △ − 1 c 0 2 − (1 + cs) 1 c 0 2 − s (u ) − s (u ) a b a k b 2 gabu u gabu u ∂k u c 0 2 0 2 s (u ) a (u ) + a b ∂au a b , 2 gabu u 2 gabu u 1 c 0 2 − 1 c 0 2 − s (u ) − s (u ) (3.8c) a b a a k b gabu u ∂au gabu u ∂k u 0 2 0 0 3 ′j 2 2 j (u ) 2 j u − (u ) = ωcs(3cs 1)u a b + csu a b 2 gabu u 2 gabu u △ − 1 c 0 2 1 c 0 2 − s (u ) − s (u ) a b gabu u 4 1 0 2 a a 2 −1 aj cs j − (u ) u u j cs (g ) ∂aL u a b ∂aL ∂au . 2 2 gabu u 0 0 2 0 − (1 + cs) 1 c 0 2 u − u − (1 + cs) u − s (u ) Proof. The proof consists of tedious but simple calculations; we omit the details. Remark 3.2. Note that certain terms on the right-hand side of (3.8a) - (3.8c) 2 2 vanish when cs = 0 or cs = 1/3. This vanishing is important in our analysis of these cases. In the next lemma, we provide the Christoffel symbols of the metric g relative to our coordinate system. The lemma was used in the proof of Proposition 3.1. Lemma 3.3. The non-zero Christoffel symbols of the metric g = dt2 + e2Ω(t) 3 (dxj )2 are − j=1 P 0 0 j j j (3.9) Γj k =Γk j = ωgjk, Γk 0 =Γ0 k = ωδk, (j, k =1, 2, 3), where def d (3.10) ω = Ω. dt The Stabilizing Effect of Spacetime Expansion 16 Proof. The lemma follows via simple computations from the definition 1 (3.11) Γ α def= (g−1)αλ(∂ g + ∂ g ∂ g ). µ ν 2 µ λν ν µλ − λ µν 3.2. Local existence and the continuation principle. In this section, we state a standard local existence result for the system (3.2a) - (3.2b). Theorem 3.1 (Local Existence). Let N 3 be an integer. Let ˚L = L t=1 = j j ≥ | ln ρ/ρ¯ t=1, ˚u = u t=1, (j = 1, 2, 3), be initial data for the relativistic Euler equations| (3.2a) - (3.2b)| satisfying HN−1, c2 =0, (3.12a) ˚L s ∈ HN , 0 0 R3 2 ˚ C ([1,T+) ), cs =0, (3.13a) L 1 × 3 2 ∈ C ([1,T+) R ), 0 0 N−1 2 ˚ C ([1,T+),H ), cs =0, (3.14a) L 0 N 2 ∈ C ([1,T+),H ), 0 3 j (3.15) lim sup L(s, ) L∞ + u (s, ) C1 = . − k · k k · k b ∞ t→Tmax 0≤s≤t ( j=1 ) X Furthermore, in the cases 0 3 j 1 1 (3.16) lim− sup L(s, ) C + u (s, ) C = . t→T k · k b k · k b ∞ max 0≤s≤t ( j=1 ) X 2 Proof. See e.g. [H¨or97], [Spe09a] for the ideas behind a proof. The case cs = 0 is special because in this case, the evolution of the uj decouples from that of L, and furthermore, the evolution equation (3.2a) is linear in L. 3.3. Matrix-vector notation. It is convenient to write the system (3.2a) - (3.2b) in abbreviated form using standard PDE matrix-vector notation: β (3.17) A ∂βW = b. Here, W is a fluid variable column array, and b is a column array of inhomogeneous terms: 2 L (1+cs) a b ω u0 gabu u 1 − 2 0 1 def u def ω(3cs 2)u u (3.18) W = 2 , b = 2 − 0 2 . u ω(3cs 2)u u u3 ω(3c2 − 2)u0u3 s − The Aµ are 4 4 matrices defined by × 0 2 u1 2 u2 2 u3 u (1 + cs) u0 (1 + cs) u0 (1 + cs) u0 2 cs 10 0 2 Π u 0 0 (1+cs) 0 2 (3.19a) A = cs 20 0 , 2 Π 0 u 0 (1+cs) 2 cs 30 0 (1+c2) Π 0 0 u s 1 2 u (1 + cs) 0 0 2 cs 11 1 2 Π u 0 0 (1+cs) 1 2 (3.19b) A = cs 21 1 , 2 Π 0 u 0 (1+cs) 2 cs 31 1 2 Π 0 0 u (1+cs) and analogously for A2, A3. Furthermore, for later use, we calculate that det(A0)= 0 2 2 00 (u ) −csΠ (u0)2 (u0)2(1 c2)+ c2 , − s s n z }| { o The Stabilizing Effect of Spacetime Expansion 18 −1 (3.20) (A0)−1 = (u0)2 c2Π00 − s 0 2 u1 2 u2 2 u3 u (1 + cs) u0 (1 + cs) u0 (1 + cs) u0 2 − 2 2 cs 10 0 cs 10 cs 10 2 Π u d1 0 2 Π u2 0 2 Π u3 (1+cs) (u ) (u ) − 2 2 − 2 cs 20 cs 20 0 cs 20 , × 2 Π 0 2 Π u1 u d2 0 2 Π u3 (1+cs) (u ) (u ) − 2 2 2 − cs 30 cs 30 cs 30 0 2 Π 0 2 Π u1 0 2 Π u2 u d3 − (1+cs) (u ) (u ) − 2 2 2 def cs 20 30 def cs 10 30 def cs 10 and d1 = (u0)2 Π u2 + Π u3 , d2 = (u0)2 Π u1 + Π u3 , d3 = (u0)2 Π u1 + 20 Π u2 . 4. Norms, Energies, and the Equations of Variation In this section, we define some Sobolev norms and related energies, all of which are used in the proof of our main stability theorem. Our choice of norms is mo- tivated in part by the continuation principle (Proposition 3.4); with the help of Sobolev embedding, our norms will control the quantities appearing in the state- ment of the continuation principle. The energies are introduced in order to control the up-to-top-order spatial derivatives of the solution. Unlike the Sobolev norm N , the energy’s time derivative can be controlled in terms of the energy itself withS the help of the divergence theorem. This is because our energy is defined with the help of energy currents J˙µ, which are solution-dependent, coercive vectorfields whose divergence can be controlled; see the discussion in Section 1.2.2. We also introduce the equations of variation, which are the PDEs verified by the spatial derivatives of the solution. The structure of the equations of variation plays an important role in our derivation of estimates for the spatial derivatives. 4.1. Norms. Definition 4.1 (Norms). Let N be a positive integer, let W def= (L,u1,u2,u3)T 2 be the array of fluid variables. In the cases 0 3 − 2Ω j 2 L HN 1 + e j=1 u HN , cs =0, def k k Ω 3 jk k 2 (4.2) = L N + e u N + , 0 3 def 2Ω j (4.3) = e u N . SN;velocity k kH j=1 X The Stabilizing Effect of Spacetime Expansion 19 Remark 4.1. The proof of our main stability theorem will show that all of the norms remain uniformly small for all future times if they are initially small. Fur- 2 thermore, in the cases 0 < cs < 1/3, the boundedness of N−1 shows that the lower-order derivatives of uj have an L2 norm that decaysU at least as fast as −Ω −1 e B (Ω), while the boundedness of N shows only that the top-order deriva- tives (i.e. ~α = N) of uj decay at leastS as fast as e−Ω; see Section 1.2.4 for a discussion of| | the importance of the improved decay for the lower-order derivatives j 2 of the u . We remark that in the special case cs =0, the boundedness of N shows that all spatial derivatives of uj decay in L2 at least as fast as e−2Ω. NoteS also that 2 in the case cs = 1/3, we do not prove an improved decay rate for the lower-order j derivatives of u (and hence the N−1 norm is not needed for the analysis in this case). Our inability to show improvedU decay in this case is intimately connected 2 2 j to the fact that when cs =1/3, the (3cs 2)ωu term on the right-hand side of equation (3.7c) suggests that we can− only prove− that the lower-order derivatives of uj decay like e−Ω (i.e., the same decay as for the top-order derivatives). 4.2. The equations of variation and energies for the fluid variables. In this section, we define the fluid energy. This energy, which depends on the up-to-top- 1 2 3 order spatial derivatives (∂~αL, ∂~αu , ∂~αu , ∂~αu ) ~α N, plays a central role in the proof of our main stability theorem. We note the| |≤ following previously mentioned 2 exception to the preceding sentence: when cs = 0 we will only be able to control ∂~αL for ~α N 1. In order to study the evolution of the solution’s derivatives, we will have| | ≤ to commute− the equations (3.2a) - (3.2b) (or equivalently equation 1 2 3 (3.17)) with the operator ∂~α. The quantities (∂~αL, ∂~αu , ∂~αu , ∂~αu ) verify linear 1 2 3 (in (∂~αL, ∂~αu , ∂~αu , ∂~αu )) PDEs with principal coefficients that depend on the so- lution and inhomogeneous terms that depend on (∂ L, ∂ u1, ∂ u2, ∂ u3), β~ ~α ; β~ β~ β~ β~ | | ≤ | | see Lemma 4.2 for the details. We refer to this system of PDEs as the equations of 1 2 3 def 1 2 3 variation, while the unknowns (L,˙ u˙ , u˙ , u˙ ) = (∂~αL, ∂~αu , ∂~αu , ∂~αu ) are called the variations. More specifically, we define the equations of variation in the un- knowns (L,˙ u˙ 1, u˙ 2, u˙ 3) corresponding to (L,u1,u2,u3) as follows: Equations of Variation 1 (4.4a) uα∂ L˙ +(1+ c2) u ∂ u˙ a +(1+ c2)∂ u˙ a = F, α s u0 a t s a c2 α j s jα ˙ 2 0 j Gj (4.4b) u ∂αu˙ + 2 Π ∂αL = ω(3cs 2)u u˙ + . (1 + cs) − The terms F, ω(3c2 2)u0u˙ j , and Gj denote the inhomogeneous terms that arise s − from commuting (3.2a) - (3.2b) with ∂~α. Note that we have split the inhomogeneous 2 0 j term in (4.4b) into two parts; the ω(3cs 2)u u˙ term is primarily responsible for creating decay inu ˙ j, while Gj will be shown− to be an error term. Using matrix-vector notation, we can rewrite (4.4a)- (4.4b) as β (4.5) A ∂βW˙ = I, where them matrices Aµ are defined in (3.19a) - (3.19b), and The Stabilizing Effect of Spacetime Expansion 20 (4.6) W˙ def= (L,˙ u˙ 1, u˙ 2, u˙ 3)T , T (4.7) I def= ω(3c2 2)u0 0, u˙ 1, u˙ 2, u˙ 3 + (F, G1, G2, G3)T . s − To each variation (u ˙ 1, u˙ 2, u˙ 3), we associate a quantityu ˙ 0 defined by 1 (4.8) u˙ 0 def= g uau˙ b. u0 ab This quantity appears below in the expression (4.16), which defines our fluid energy current. The importance of the definition (4.8) is that it leads to the identity α β gαβu u˙ =0, which is essential for the derivation of the divergence identity (4.18) below. In our analysis, we will need the following lemma, which essentially states thatu ˙ 0 is a solution to a linearization of (3.3) around (L,u1,u2,u3). Lemma 4.1. Assume that (L,˙ u˙ 1, u˙ 2, u˙ 3) is a solution to the equations of variation 1 2 3 0 def 1 a (4.4a) - (4.4b) corresponding to (L,u ,u ,u ), and let u˙ = u0 uau˙ be as defined in (4.8). Then u˙ 0 verifies the following equation: c2 α 0 s 0α ˙ G0 (4.9) u ∂αu˙ + 2 Π ∂αL = , (1 + cs) where ua 1 (4.10) G0 = g uν ∂ u˙ b +3c2ωu˙ 0 +2ω(u0 1)u ˙ 0 + g uaGb. ab ν u0 s − u0 ab Proof. By the definition ofu ˙ 0, the left-hand side of (4.9) is equal to 2 a ua α a cs 0α ˙ α u b a b (4.11) 0 u ∂αu˙ + 2 Π ∂αL + gab u ∂α 0 u˙ +2ωgabu u˙ . u (1 + cs) u On the other hand, contracting equation (4.4b) against uj and using the identity u Πaα = u Π0α = u0Π0α, we conclude that a − 0 c2 α a 0 s 0α ˙ 2 a b aGb (4.12) uau ∂αu˙ + u 2 Π ∂αL = (3cs 2)ωgabu u˙ + gabu . (1 + cs) − 1 Multiplying (4.12) by u0 and using (4.11), we arrive at (4.9). In the next lemma, we investigate the structure of the inhomogeneous terms in 1 2 3 T the equations of variation verified by a solution’s derivatives (∂~αL, ∂~αu , ∂~αu , ∂~αu ) . We again split the inhomogeneous terms into a decay-inducing piece b~α and a small error term b△~α. def Lemma 4.2. Let W = (L,u1,u2,u3)T be a solution to the relativistic Euler equa- β tions (3.17), i.e., A ∂βW = b, def 2 T (1+cs) a b 2 0 1 2 0 2 2 0 3 1 2 3 T b = ω 0 g u u , (3c 2)u u , (3c 2)u u , (3c 2)u u . Then (L,˙ u˙ , u˙ , u˙ ) − u ab s− s− s− def 1 2 3 T = (∂~αL, ∂~αu , ∂~αu , ∂~αu ) is a solution to the equations of variation (4.5) with an inhomogeneous term I that can be expressed as follows: The Stabilizing Effect of Spacetime Expansion 21 def (4.13) I = b~α + b△~α, where def T (4.14a) b = ω(3c2 2)u0 0, ∂ u1, ∂ u2, ∂ u3 , ~α s − ~α ~α ~α def T (4.14b) b = F , G1 , G2 , G3 = ∂ b b + A0∂ (A0)−1b ∂ b △~α ~α ~α ~α ~α ~α − ~α ~α − ~α n o + A0 (A0)−1Aa∂ ∂ W ∂ (A0 )−1Aa∂ W . a ~α − ~α a n o 0 def 1 a Furthermore, u˙ = u0 uau˙ is a solution to equation (4.9) with an inhomogeneous 0 term G~α defined by (4.15) ua 3c2 2+2u0 1 G0 = g uν ∂ ∂ ub + ω s − g ua∂ ub + g uaGb . ~α ab ν u0 ~α u0 ab ~α u0 ab ~α Proof. Equations (4.13) - (4.14b) are a straightforward decomposition of the inho- 0 0 −1 µ 0 0 −1 µ mogeneous term I = A ∂~α (A ) A ∂µW + A [(A ) A ∂µ, ∂~α]W = A0∂ (A0)−1b + A0[(A0)−1Aa∂ , ∂ ]W, where [ , ] denotes the commutator. ~α a ~α The relation (4.15) follows directly from (4.10). · · 4.2.1. The fluid energy currents. Our energy, which will control the up-to-top-order spatial derivatives of the solution, will be defined with the help of energy current vectorfields J˙µ. Definition 4.2 (Currents). In the cases 0 < c2 1/3, to each variation W˙ = s ≤ ˙ 1 2 3 T 0 def 1 a (L, u˙ , u˙ , u˙ ) , we associate the following energy current, whereu ˙ = u0 uau˙ : 2 µ ˙µ def csu ˙ 2 2 µ ˙ 2 µ α β (4.16) J = 2 L +2csu˙ L +(1+ cs)u gαβu˙ u˙ . (1 + cs) 2 In the case cs = 0, we associate both a velocity energy current and a density energy current: ˙µ 1 2 3 1 2 3 def 2Ω µ α β (4.17a) Jvelocity[(u ˙ , u˙ , u˙ ), (u ˙ , u˙ , u˙ )] = e u gαβu˙ u˙ , ˙µ ˙ ˙ def µ ˙ 2 (4.17b) Jdensity[L, L] = u L . Similar currents have been used in [Chr07], [Spe09a], [Spe09b], and [Spe11]. See the discussion in Section 1.2.2. Remark 4.2. We use the notation J˙µ[ , ] when we want to emphasize that J˙µ depends quadratically on the variations. · · The Stabilizing Effect of Spacetime Expansion 22 4.2.2. The coordinate divergence of the currents. In this section, we provide expres- sions for the coordinate divergence of the currents. These expressions will play a key role in our analysis of the evolution of the energies , , and . EN EN;velocity EN−1;density Lemma 4.3 (Divergence of the Currents). Let W˙ = (L,˙ u˙ 1, u˙ 2, u˙ 3)T be a solution to the equations of variation (4.4a) - (4.4b). Then in the cases 0 < c2 s ≤ 1/3, the coordinate divergence of J˙µ can be expressed as follows: 2 µ ˙µ ˙ ˙ cs(∂µu ) ˙ 2 2 µ 0 2 a b (4.18) ∂µ J [W, W] = 2 L +(1+ cs)(∂µu ) (u ˙ ) + gabu˙ u˙ (1 + cs) − ua g uau˙ b +2c2g ∂ u˙ bL˙ +4c2ω ab L˙ s ab t u0 s u0 h i +2(1+ c2)(3c2 1)ωg u˙ au˙ b s s − ab 0 if c2 1/3 ≤ s ≤ | 2c2F {z } s ˙ 2 G0 0 2 Ga b + 2 L 2(1 + cs) u˙ +2(1+ cs)gab u˙ . (1 + cs) − 2 ˙µ ˙µ In the case cs = 0, the coordinate divergences of Jvelocity and Jdensity can be respectively expressed as (4.19a) ∂ J˙µ [(u ˙ 1, u˙ 2, u˙ 3), (u ˙ 1, u˙ 2, u˙ 3)] = e2Ω(∂ uµ) (u ˙ 0)2 + g u˙ au˙ b µ velocity µ − ab +2ωe2Ω(u0 1)g u˙ au˙ b 2ωe2Ω u0(u ˙ 0)2 − ab − 2e2ΩG0u˙ 0 +2e2Ωg Gau˙ b, − ab 4 (4.19b) ∂ J˙µ [L,˙ L˙ ] = (∂ uµ)L˙ 2 + ωg uau˙ bL˙ µ density µ u0 ab 2 2 g uaGbL˙ + g ua(uk∂ u˙ b)L˙ − (u0)2 ab (u0)2 ab k 2(∂ u˙ a)L˙ +2FL.˙ − a Remark 4.3. We stress that the right-hand sides of (4.18) and (4.19a) - (4.19b) do not depend on the derivatives of the variations. This property is essential for our analysis of the energies. We have organized the terms on the right-hand sides of (4.18) and (4.19a) - (4.19b) in order to help us deduce the estimates (5.8) - (5.9) and (5.10). Proof. To deduce (4.18), one can take the divergence of (4.16) and then use the equations (4.4a) - (4.4b) and (4.9) to replace the derivatives of W˙ with inhomoge- neous terms. We leave the tedious details up to the reader. The proofs of (4.19a) - (4.19b) are similar but simpler. 4.2.3. The definition of the fluid energies. We are now ready to define our fluid en- ergies, which will be used to control all spatial derivatives of the solution, including those of top order. Definition 4.3 (Energies). Let W def= (L,u1,u2,u3)T be the array of fluid vari- ables, and let J˙µ[ , ], J˙µ [ , ], J˙µ [ , ] be the currents defined in (4.16), · · velocity · · density · · The Stabilizing Effect of Spacetime Expansion 23 2 (4.17a) - (4.17b). In the cases 0 < cs 1/3, we define the fluid energy N (t) 0 (see inequality (6.4)) by ≤ E ≥ 2 def ˙0 3 (4.20) N (t) = J [∂~αW, ∂~αW] d x. E R3 |~αX|≤N Z 2 In the case cs =0, we define the energies N;velocity(t) 0 and N−1;density(t) 0 by E ≥ E ≥ 2 def ˙0 1 2 3 1 2 3 3 (4.21a) N;velocity(t) = Jvelocity [∂~α(u ,u ,u ), ∂~α(u ,u ,u )] d x, E R3 |~αX|≤N Z 2 def ˙0 3 (4.21b) N−1;density(t) = Jdensity[∂~αL, ∂~αL] d x. E R3 |~α|≤XN−1 Z In the next corollary, we provide a preliminary estimate for the time derivatives of some of the fluid-controlling quantities. These estimates are the starting point for our proof of Proposition 7.1, which provides the integral inequalities that form the crux of our future stability theorem. def Corollary 4.4 (Preliminary Norm and Energy Inequalities). Let W = (L,u1,u2,u3)T be a classical solution to the relativistic Euler equations (3.17). Let N−1(t), N (t), N;velocity, and N−1;density be the norm and energies defined in U E E E 2 Definitions 4.1 and 4.3. Then in the cases 0 < cs 1/3, the following identity holds: ≤ d 2 ˙µ 3 (4.22) N = ∂µ J [∂~αW, ∂~αW] d x. dt E R3 |~α|≤N Z X 2 In the cases 0 (4.23) ′ 3 d 2 2 B (Ω) 2 Ω ′a 2 3c 1+ ω +2 N−1 e B(Ω) N−1 . dt UN−1 ≤ s − B(Ω) UN−1 U k△ kH a=1 n o X 2 In the case cs =0, the following identities hold: d 2 ˙µ 1 2 3 1 2 3 3 (4.24a) N;velocity = ∂µ J [∂~α(u ,u ,u ), ∂~α(u ,u ,u )] d x, dt E R3 |~α|≤N Z X d 2 ˙µ 3 (4.24b) N−1;density = ∂µ Jdensity[∂~αL, ∂~αL] d x. dt E R3 |~α|≤N−1 Z X Proof. To prove (4.23), we use the definition (4.1) of and equation (3.7c) UN−1 (differentiated with ∂~α) to conclude that (4.25) ′ 3 d 2 2 B (Ω) 2 2Ω 2 a ′a 3 N−1 =2 3cs 1+ ω N−1 +2e B (Ω) (∂~αu )∂~α d x. dt U − B(Ω) U R3 △ |~α|≤N−1 a=1 Z n o X X The Stabilizing Effect of Spacetime Expansion 24 Inequality (4.23) now follows from (4.25), the definition of N−1, and the Cauchy- Schwarz inequality for integrals. U (4.24b) and (4.24a) - (4.24b) follow easily from the definitions (4.20), (4.21a) - (4.21b), and the divergence theorem. 5. Sobolev Estimates In this section, we derive Sobolev estimates for the inhomogeneous terms appear- ing in equations (3.2a) - (3.2b), (3.3), and (3.7a) - (3.7c), for the terms in Lemma 4.2, and for the right-hand sides of (4.23) - (4.25). All of the bounds are derived under a smallness assumption on the norm N . These bounds form the backbone of Section 7, where they are used to help deriveS integral inequalities for and UN−1 N , N;velocity, and N−1;density. We collect together all of the estimates in the next Eproposition.E E Remark 5.1. Note that in following proposition, we establish different estimates 2 2 for the special cases cs = 0 and cs =1/3 compared to the other cases. 2 1 2 3 Proposition 5.1 (Sobolev Estimates). Assume that 0 cs 1/3. Let (L,u ,u ,u ) be a classical solution to the relativistic Euler equations (3.2a)≤ -≤(3.2b) on the space- time slab [1,T ) R3, and let be the solution norm defined in Definition 4.1. × SN Assume that N (t) ǫ on [1,T ). Then if ǫ is sufficiently small, the following esti- mate for u0 holdsS on≤[1,T ): −2Ω 2 e N;velocity, cs =0, 0 −1 S 2 (5.1) u 1 N . B (Ω) , 0 −2Ω 2 (5.2a) ∂ L N−2 . (ω + 1)e , c =0, k t kH SN;velocity s −1 −Ω 2 ωB (Ω) N + e N , 0 2 (1+cs)L a b 2 0 j For the inhomogeneous terms ω u0 gabu u and (3cs 2)ωu u from Propo- − 2 sition 3.1, we have the following estimates on [1,T ) in the cases 0 cs < 1/3 (the 2 ≤ case cs =1/3 is not directly needed for the remaining estimates): 2 (1 + cs) a b −1 (5.3a) ω gabu u . ωB (Ω) N , u0 HN S 2 0 j −Ω (5.3b) (3c 2)ωu u HN . ωe N , k s − k S 0 j 0 j −Ω −1 (5.3c) ∂ (ωu u ) ωu ∂ u 2 . ωe B (Ω) , ( ~α N). k ~α − ~α kL SN | |≤ For the inhomogeneous terms ′, ′0, and ′j from Corollary 3.2, we have the following estimates on [1,T ): △ △ △ ′ −2Ω 2 (5.4a) N−2 . (ω + 1)e , c =0, k△ kH SN;velocity s −1 −Ω 2 ′ ωB (Ω) N + e N , 0 3 a 2 N;velocity a=1 u˙ L2 , cs =0, 0 S 3k k 2 Ω −1 a 2 (5.5) u˙ L . e B (Ω) N a=1 u˙ L2 , 0 −2Ω (5.6a) F 2 . (ω + 1)e , (0 ~α N 1), k ~αkL SN ≤ | |≤ − 1 −2Ω G~α e 2 −2Ω (5.6b) G~α . N;velocity e , (0 ~α N), 3 S −2Ω ≤ | |≤ G 2 e ~α L 0 −2Ω (5.6c) G L2 . (ω + 1)e N;velocity, (0 ~α N). k ~α k S ≤ | |≤ 2 Furthermore, we have the following estimates on [1,T ) in the cases 0 −1 −Ω F~α B (Ω) e 1 −Ω −1 −2Ω G~α e B (Ω) e (5.7a) 2 . ω N −Ω −1 + N −2Ω , (0 ~α N), G~α S e B (Ω) S e ≤ | |≤ 3 −Ω −1 −2Ω G 2 e B (Ω) e ~α L 0 −1 −Ω (5.7b) G~α L2 . ωB (Ω) N + e N , (0 ~α N). k k S S ≤ | |≤ The Stabilizing Effect of Spacetime Expansion 26 In the case c2 =0, for the currents J˙µ [ , ] and J˙µ [ , ] defined in (4.17a) s velocity · · density · · - (4.17b), we have the following estimates on [1,T ): (5.8) ˙µ 1 2 3 1 2 3 3 −2Ω 2 ∂µ Jvelocity[∂~α(u ,u ,u ), ∂~α(u ,u ,u )] d x . (ω + 1)e N;velocity, R3 S |~α|≤N Z X ˙µ 3 −2Ω 2 (5.9) ∂µ Jdensity[∂~αL, ∂~αL] d x . (ω + 1)e N . R3 S |~α|≤N−1 Z X 2 ˙µ In the cases 0 (5.10) −1 2 −Ω 2 2 ˙µ 3 ωB (Ω) N + e N , 0 2 Proof. Proofs of (5.1) - (5.4d): We first consider the cases 0 0 a b 2Ω a b −1 −1 u 1 N . g u u N . e δ u ∞ u N . B (Ω) . B (Ω) , k − kH k ab kH abk kL k kH UN−1 SN where δab is the Kronecker delta. The estimates (5.3a) - (5.3c) and (5.4a) - (5.4d) follow similarly (use Proposition A-5 to deduce (5.3c)). The estimates (5.2a) - (5.2d) then follow trivially with the help of equations (3.7a) - (3.7c). The cases 2 cs =0, 1/3 follow similarly. 2 Proof of (5.5): To prove (5.5) in the cases 0 < cs < 1/3, we use equation (4.8), (5.1), and Sobolev embedding to deduce 0 1 a Ω −1 a (5.11) u˙ L2 . ua L∞ u˙ L2 . e B (Ω) N u˙ L2 . k k u0 L∞ k k k k S k k The cases c2 =0, 1/3 follow similarly. s 2 Proofs of (5.6a) - (5.7b): We first prove (5.7a) (in the cases 0 < cs < 1/3). We begin by using equation (4.14b) to deduce that F~α 1 G~α 0 0 −1 (5.12) 2 ∂~αb b~α L2 + A ∂~α (A ) b ∂~αb L2 G~α ≤ − − 3 G 2 ~α L 0 0 −1 a 0 −1 a + A (A ) A ∂a∂~αW ∂~α (A ) A ∂aW . − L2 n o The Stabilizing Effect of Spacetime Expansion 27 See the remarks at the end of Section 2.3 concerning our use of notation for the norms of array-valued functions. We now deduce the following preliminary estimates for W def= (L,u1,u2,u3)T , 2 T (1+cs) a b 2 0 1 2 0 2 2 0 3 def 2 b = ω 0 g u u , (3c 2)u u , (3c 2)u u , (3c 2)u u , b = ω(3c − u ab s − s − s − ~α s − T 2)u0 0, ∂ u1, ∂ u2, ∂ u3 , ~α N, Aµ, and (A0)−1 (explicit expressions for the ~α ~α ~α | |≤ matrices are provided in (3.19a) - (3.20)): (5.13) B−1(Ω) B−1(Ω) e−ΩB−1(Ω) e−ΩB−1(Ω) b N−1 . ω N , ∂~αb b~α 2 . ω N , k kH S e−ΩB−1(Ω) k − kL S e−ΩB−1(Ω) e−ΩB−1(Ω) e−ΩB−1(Ω) (5.14) Ω −1 Ω −1 Ω −1 1 e B (Ω) N e B (Ω) N e B (Ω) N −Ω −1 S S S 0 e B (Ω) N 1 0 0 A L∞ . −Ω −1 S , k k e B (Ω) N 0 1 0 S e−ΩB−1(Ω) 0 0 1 SN (5.15) Ω −1 Ω −1 Ω −1 1 e B (Ω) N e B (Ω) N e B (Ω) N −Ω −1 S −1 S −1 S 0 −1 e B (Ω) N 1 B (Ω) N B (Ω) N (A ) ∞ . S S S , L e−ΩB−1(Ω) B−1(Ω) 1 B−1(Ω) SN SN SN e−ΩB−1(Ω) B−1(Ω) B−1(Ω) 1 SN SN SN B−1(Ω) eΩ eΩ eΩ −Ω −1 −1 −1 0 −1 e B (Ω) B (Ω) B (Ω) (5.16) ∂(A ) N−1 . N , H S e−Ω B−1(Ω) B−1(Ω) B−1(Ω) e−Ω B−1(Ω) B−1(Ω) B−1(Ω) 1 e−Ω (5.17) W N . N , k kH S e−Ω e−Ω (5.18) e−ΩB−1(Ω) 1 0 0 SN e−2Ω e−ΩB−1(Ω) 0 0 1 ∞ N A L . −2Ω S −Ω −1 , k k e 0 e B (Ω) N 0 −2Ω S −Ω −1 e 0 0 e B (Ω) N S e−Ω 0 0 0 −2Ω −1 −Ω 1 e B (Ω) e 0 0 (5.19) ∂A N−1 . N , k kH S e−2ΩB−1(Ω) 0 e−Ω 0 e−2ΩB−1(Ω) 0 0 e−Ω The Stabilizing Effect of Spacetime Expansion 28 and analogously for the matrices A2, A3. The above preliminary estimates follow from repeated applications of Proposition A-2, Proposition A-4, the definition (4.2) of N , and the estimates (5.1) - (5.3c). SWe now estimate the second term on the right-hand side of (5.12). To this end, we use Propositions A-4 and A-5, together with the above estimates and Sobolev embedding to conclude that for 0 ~α N ≤ | |≤ 0 0 −1 0 ∞ 0 −1 − (5.20) A ∂~α (A ) b ∂~αb . A L ∂(A ) N−1 b HN 1 − L2 k k ∗ H ∗k k −1(Ω) B e−ΩB−1(Ω) . ω N , S e−ΩB−1(Ω) e−ΩB−1(Ω) where we write to indicate that we are performing matrix multiplication on the matrices of norms.∗ We similarly estimate the third term on the right-hand side of (5.12) (using 0 −1 a 0 −1 a Proposition A-1 to estimate ∂[(A ) A ] HN−1 . (A ) L∞ ∂A HN−1 + 0 −1 a k k k k ∗k k ∂(A ) N−1 A ∞ ), thus arriving at the following bound: k kH ∗k kL (5.21) 0 0 −1 a 0 −1 a A (A ) A ∂a∂~αW ∂~α (A ) A ∂aW − L2 n 0 0 −1 a o 0 −1 a . A ∞ (A ) ∞ ∂A N−1 + ∂( A ) N−1 A ∞ ∂ W N−1 k kL ∗ k kL ∗k kH k kH ∗k kL ∗k a kH e−Ω n o e−2Ω . N . S e−2Ω e−2Ω Finally, adding the second estimate in (5.13) and (5.20) - (5.21), we deduce 2 (5.7a). The proofs of (5.6a) - (5.6b) (for cs = 0) are similar, and we omit the details. To prove (5.7b), we use equation (4.15), (5.1), (5.2c) - (5.2d), and (5.7a) to conclude that a 2 0 0 ν u b (3cs 2+2u )ua a (5.22) G~α L2 gab u ∂ν u HN + ω − u HN k k ≤ u0 L∞ k k u0 L∞ k k 1 ∞ a + ua L G~α L2 u 0 L∞ k k k k −1 −Ω . ω B (Ω) N + e N . S S 2 This completes the proof of (5.7b). The proof of (5.6c) (for cs = 0) is similar, and we omit the details. 2 Proofs of (5.9) - (5.10): To prove (5.10) in the cases 0 < cs < 1/3, we first recall 1 2 3 T def 1 2 3 T equation (4.18), where W˙ = (L,˙ u˙ , u˙ , u˙ ) = ∂~αW = (∂~αL, ∂~αu , ∂~αu , ∂~αu ) , 0 def 1 a and as in (4.8),u ˙ = u0 uau˙ : The Stabilizing Effect of Spacetime Expansion 29 2 µ ˙µ ˙ ˙ cs(∂µu ) ˙ 2 2 µ 0 2 a b (5.23) ∂µ J [W, W] = 2 L +(1+ cs)(∂µu ) (u ˙ ) + gabu˙ u˙ (1 + cs) − ua g uau˙ b +2c2g ∂ u˙ bL˙ +4c2ω ab L˙ s ab t u0 s u0 h i +2(1+ c2)(3c2 1)ωg u˙ au˙ b s s − ab 0 if c2 1/3 ≤ s ≤ | 2c2F {z } s ˙ 2 G0 0 2 Ga b + 2 L 2(1 + cs) u˙ +2(1+ cs)gab u˙ . (1 + cs) − def def µ Above, the terms F = F~α and G = G~α, are defined in (4.14b) and (4.15). We now deduce (5.10) using the following four steps: i) we ignore the non-positive 2 2 a b term 2(1 + cs)(3cs 1)ωgabu˙ u˙ on the right-hand side of (5.23); ii) we bound ˙ µ− µ 2 each variation L, u˙ and F~α, G~α in L , and use the estimates (5.5), (5.7a), and (5.7b); iii) we bound all of the remaining terms in L∞ and use Sobolev embedding together with the estimates (5.1) and (5.2c) - (5.2d); iv) we make repeated use of the estimate v v v 1 v ∞ v 2 v 2 , where v and v are terms k 1 2 3kL ≤ k 1kL k 2kL k 3kL 2 3 estimated in step ii), and v1 is estimated in step iii). 2 The proofs of (5.8) - (5.9) (in the case cs = 0) are similar but simpler (use the 2 expressions (4.19a) - (4.19b)). The proof of (5.10) in the case cs =1/3 is provided in the next paragraph. 2 2 Details for the special case cs = 1/3: We now address the case cs = 1/3. Our goal is to show that in this case, many of the estimates (5.1) - (5.10) are valid without the ω terms on the right-hand side. The fact that these estimates hold without these··· terms is a consequence of special algebraic cancellation. This cancellation, which is connected to the conformal invariance of the relativistic Eu- 2 ler equations when cs = 1/3, is discussed in Section 9 in more detail. We will only demonstrate the cancellation for the estimate (5.10); this is the only esti- 2 mate needed to derive the crucial inequality (7.2) below in the case cs = 1/3. 2 In Section 9, we will show that when cs = 1/3, (ρ,u) verify the relativistic Eu- ler equations (1.7a) - (1.7b) corresponding to the metric g if and only if the rescaled variables (ρ′,U) def= (e4Ωρ,eΩu) verify the relativistic Euler equations (1.7a) - (1.7b) corresponding to the Minkowski metric m = e−2Ωg. Now if we consider dτ −Ω(t) the change-of-time-variable dt = e , τ(t =1)=1, then (9.2) shows that we have m = dτ 2 + 3 (dxj )2. Therefore, it follows that L def= ln e4Ω(t◦τ)ρ/ρ¯ , − j=1 j def Ω(t◦τ) j U = e u , verifyP the relativistic Euler equations (3.2a) - (3.2b) corresponding to the Minkowski metric m = dτ 2+ 3 (dxj )2 and the coordinates (τ, x1, x2, x3). − j=1 In particular, in these new fluid and time variables, the right-hand sides of (3.2a) τ aPb 1/2 0 - (3.2b) vanish, and U = (1+ δabU U ) = u . Here, δab is the Kronecker delta and the index “τ” is meant to emphasize that we are referring to the time com- ponent of U relative to the new time coordinate system (while u0 refers to the time component of u relative to the original “t” coordinate system). Furthermore, The Stabilizing Effect of Spacetime Expansion 30 relative to these new variables, we can define a current analogous to the current J˙µ defined in (4.16): 2 τ ˙τ def csU ˙ 2 2 ˙ τ ˙ 2 τ ˙ τ 2 ˙ a ˙ b (5.24a) J = 2 L +2csU L +(1+ cs)U [ (U ) + δabU U ], (1 + cs) − 2 j ˙j def csU ˙ 2 2 ˙ j ˙ 2 j ˙ τ 2 ˙ a ˙ b (5.24b) J = 2 L +2csU L +(1+ cs)U [ (U ) + δabU U ], (1 + cs) − ˙ j Ω j ˙ τ 1 a ˙ b 1 a b 0 ˙τ ˙0 where U = e u˙ , U = U τ δabU U = u0 gabu u˙ =u ˙ . Note that J = J , J˙j = eΩJ˙j , and τ a Ω 0 a def Ω µ (5.25) ∂τ J˙ + ∂aJ˙ = e ∂tJ˙ + ∂aJ˙ = e ∂µJ˙ . Setting W def= (L,U 1,U 2,U 3), using the vanishing of the right-hand sides of (3.2a) - (3.2b) in the new coordinate + fluid variables, and noting definition (4.2), our prior proof of (5.10) (under the assumption that the metric is the standard Minkowski metric) yields ˙τ ˙a 3 (5.26) ∂τ J [∂~αW, ∂~αW] + ∂a J [∂~αW, ∂~αW] d x R3 |~α|≤N Z X 3 2 j 2 . L N + U N . k kH k kH j=1 X Using (5.25), (5.26), and definition (4.2), we deduce Ω µ 3 (5.27) e ∂µ J˙ [∂~αW, ∂~αW] d x R3 |~α|≤N Z X τ a 3 = ∂τ J˙ [∂~αW, ∂~αW] + ∂a J˙ [∂~αW, ∂~αW] d x R3 |~α|≤N Z X 3 3 2 j 2 def 2 Ω(t) j 2 def 2 . L N + U N = L N + e u N = , k kH k kH k kH k kH SN j=1 j=1 X X which is the desired estimate. 6. The Energy Norm vs. Sobolev Norms Comparison Proposition In this section, we prove a proposition that compares the coercive properties of the norms N−1, N , and N;velocity to the coercive properties of the energies N , ,Uand S S. E EN;velocity EN−1;density Proposition 6.1 (Equivalence of Sobolev Norms and Energy Norms). Let N 3 be an integer, and assume that N (t) ǫ on [1,T ). Then there exist implicit constants≥ such that if ǫ is sufficientlyS small,≤ then the following estimates hold for the norms N−1, N , and N;velocity defined in Definition 4.1 and the energies N , ,Uand S S defined in Definition 4.3: E EN;velocity EN−1;density The Stabilizing Effect of Spacetime Expansion 31 2 2 2 2 N;velocity + N−1;density N , cs =0, E2 E 2 ≈ S 2 N;velocity N;velocity, cs =0, (6.1) E 2 2≈ S 2 2 N−1 + N N , 0 3 2 2Ω a 2 ˙0 3 (6.2) ∂~αL L2 + e ∂~αu L2 J [∂~αW, ∂~αW] d x. k k k k ≈ R3 a=1 |~αX|≤Nn X o |~αX|≤N Z The desired inequalities would then follow easily from (6.2) and Definitions 4.1 and 4.3. To prove (6.2), for notational convenience, we set W def= (L,u1,u2,u3)T , j def j ˙ def 0 def 1 a u˙ = ∂~αu , L = ∂~αL, and as in (4.8), we defineu ˙ = u0 uau˙ . We now recall definition (4.16): 2 0 ˙0 def csu ˙ 2 2 0 ˙ 2 0 α β (6.3) J = 2 L +2csu˙ L +(1+ cs)u gαβu˙ u˙ . (1 + cs) 2 1 2 0 ˙ Using 0 < cs 3 , (5.1), Sobolev embedding, and the simple estimate 2csu˙ L 2 ≤ 2 | | ≤ cs ˙ 2 2 2 1 a 2 L +2c (1 + c ) 0 uau˙ , it follows that 2(1+cs) s s u 2 0 2 ˙0 cs(u 1/2) ˙ 2 2 0 a b 2 0 2 1 a (6.4) J − 2 L +(1+ cs)u gabu˙ u˙ (1 + cs)(u +2cs) 0 uau˙ ≥ (1 + cs) − u C (L˙ 2 + e2Ωδ u˙ au˙ b) C ǫ2e2ΩB−2(Ω)δ u˙ au˙ b & L˙ 2 + e2Ωδ u˙ au˙ b. ≥ 1 ab − 2 ab ab A reverse inequality can similarly be shown. Integrating these inequalities over 3 R and summing over all derivatives ∂~αW with ~α N, we deduce (6.2). The 2 | | ≤ remaining estimates in (6.1) (in the cases cs =0, 1/3) can be proved similarly. 7. Integral Inequalities for the Energy and Norms In this section, we use the Sobolev estimates of Section 5 to derive energy and norm integral inequalities. These inequalities form the backbone of our future stability proof. Proposition 7.1 (Integral Inequalities). Let (L,u1,u2,u3) be a classical solu- tion to the relativistic Euler equations (3.2a) - (3.2b) on the spacetime slab [1,T ) R3. Let , , and be the norms in Definition 4.1, and let ×, UN−1 SN SN;velocity EN N;velocity, and N−1;density be the energies in Definition 4.3. Let N 3 be an E E 2 ≥ integer, and assume that N (t) ǫ on [1,T ). If cs =0 and ǫ is sufficiently small, then the following integralS inequalities≤ are verified for 1 t t 2 If 0 (7.4) 0 for large Ω ≤ ′ 3 d 2 2 B (Ω) 2 Ω ′a 2 3c 1+ ω +2e B(Ω) N−1 N−1 dt UN−1 ≤ s − B(Ω) UN−1 U k△ kH z }| { a=1 n o X 0 for large Ω ≤ ′ 2 B (Ω) 2 −1 2 −Ω 2 2 3cs 1+ ω N−1 + CωB (Ω) N−1 + Ce B(Ω) N . ≤ z − }| B(Ω){ U U S n o Integrating (7.4) from t1 to t, we deduce (7.3). Inequalities (7.1a) - (7.1b) and (7.2) follow similarly from (4.24a) - (4.24b), (4.22) and (5.8) - (5.9), (5.10). 8. The Future Stability Theorem In this section, we state and prove our future stability theorem. The proof is through a standard bootstrap argument based on the continuation principle (Propo- sition 3.4) and the integral inequalities of Proposition 7.1. Theorem 8.1 (Future Stability of the Explicit Fluid Solutions). Assume 2 A1 A3 that 0 cs 1/3, that the metric scale factor verifies the hypotheses - from ≤ ≤ def def j def j Section 1, and that N 3 is an integer. Let ˚L = L t=1 = ln ρ/ρ¯ t=1 ˚u = u t=1 (j = 1, 2, 3) be initial data≥ for the relativistic Euler| equations (3.2a)| - (3.2b),| and let N (t) be the norm defined in Definition 4.1. There exist a small constant ǫ0 S −1 with 0 <ǫ0 < 1 and a large constant C∗ such that if ǫ ǫ0 and N (1) = C∗ ǫ, then 2 ≤ S the classical solution L = ln e3(1+cs)Ωρ/ρ¯ ,u provided by Theorem 3.1 exists for (t, x1, x2, x3) [1, ) R3 and furthermore, the following estimate holds for t 1 : ∈ ∞ × ≥ (8.1) (t) ǫ. SN ≤ The Stabilizing Effect of Spacetime Expansion 33 In addition, Tmax = , where Tmax is the time from the hypotheses of Proposition 3.4. ∞ 2 2 Proof. We provide full details in the cases 0 (8.2) (t) ǫ. SN ≤ We define (8.3) T def= sup T 1 The solution exists on [1,T ) and (8.2) holds on [1,T ) . local ≥ | local local We will show that if C∗ is sufficiently large and ǫ is sufficiently small, then T = . Throughout the remainder of the proof, we use the notation ∞ (8.4) ˚ǫ def= (1) = C−1ǫ. SN ∗ We first use the bootstrap assumption (8.2) (under the assumption that ǫ is sufficiently small), Proposition 6.1, the inequalities (7.2) - (7.3), and the hypotheses A1 - A3 from Section 1 to derive the following inequality, which is valid for all sufficiently large t and t verifying t t t 2 2 −1 −Ω (8.5) N (t) C N (t1)+ Cǫ ωB (Ω) +e B(Ω) ds. S ≤ S s=t1 Z 2 absent if cs =1/3 Now our hypotheses A1 - A3 on eΩ imply| that{z the} integrand on the right-hand side of (8.5) is integrable in s over the interval s [1, ). Therefore, if t 1 (8.6) 2 (t) < C (t )+ ǫ2, t [t ,T ). SN SN 1 2 ∈ 1 It remains to derive a suitable bound for (t ); this will be a standard local- SN 1 existence-type estimate. To this end, for this fixed value of t1, we again use the bootstrap assumption (8.2), Proposition 6.1, and (7.2) - (7.3) to derive t (8.7) 2 (t) C˚ǫ2 + c(t ) 2 (s) ds, t [1,t ). SN ≤ 1 SN ∈ 1 Zs=1 Applying Gronwall’s inequality to (8.7), we deduce that (8.8) 2 (t) C˚ǫ2ec(t1)t, t [1,t ). SN ≤ ∈ 1 Furthermore, we note that by Proposition 3.4, Sobolev embedding, and the conti- nuity of (t), it follows from (8.8) and definition (8.4) that if ǫ is sufficiently small SN and C∗ is sufficiently large, then The Stabilizing Effect of Spacetime Expansion 34 (8.9) t1 < T. Combining (8.6) and (8.8) and using definition (8.4), we deduce that 2 2 c(t1) ǫ 1 2 (8.10) N (t) Ce 2 + ǫ , t [1,T ). S ≤ C∗ 2 ∈ Therefore, if ǫ is sufficiently small and C∗ is sufficiently large, it follows from (8.10) that (8.11) 2 (t) <ǫ2, t [1,T ). SN ∈ Note that (8.11) is a strict improvement over the bootstrap assumption (8.2). Thus, using Proposition 3.4, Sobolev embedding, and the continuity of N (t), it follows that T = and that (8.11) holds for t [1, ). S ∞ 2 ∈ ∞ The case cs = 0 can be handled similarly using inequalities (7.1a) - (7.1b). 9. The Sharpness of the Hypotheses for the Radiation Equation of State 2 In this section, we show that if cs = 1/3, hypothesis A1 from Section 1 holds, ∞ −Ω(s) −4Ω µ µ and s=1 e ds = , then the background solution ρ =ρe ¯ , u = δ0 to the relativistic Euler equations∞ (1.7a) - (1.7b) on the spacetime-with-boundary ([1, ) R3 R ∞ × ,g) is nonlinearly unstable. The main idea of the proofe is to usee the special 2 conformally invariant structure of the fluid equations when cs =1/3 to reduce the problem to the case in which the spacetime is Minkowskian; we can then quote Christodoulou’s results [Chr07] to deduce the instability. We begin by noting a standard result: that the change of time variable dτ (9.1) = e−Ω(t) dt allows us to write the metric (1.2) in the following form: 3 (9.2) g = e2Ω(t◦τ) dτ 2 + (dxj )2 = e2Ω(t◦τ)m, − j=1 X where m = dτ 2 + 3 (dxj )2 is the Minkowski metric on [1, ) R3 equipped − i=j ∞ × with standard rectangular coordinates. The following elementary but important observation play a fundamentalP role in the discussion in this section: if the hypoth- ∞ −Ω(s) esis A1 is verified but s=1 e ds = , then after translating τ by a constant if necessary, it follows that ∞ R (9.3) τ : [1, ) [1, ), t τ(t) ∞ → ∞ → is an autodiffeomorphism of [1, ). ∞ 2 We now prove the conformal invariance of the fluid equations when cs = 1/3; this is a standard result, and we provide the short proof only for the convenience of the reader. The Stabilizing Effect of Spacetime Expansion 35 Proposition 9.1 (Conformal Invariance of the Relativistic Euler Equa- 2 tions When cs =1/3). Let g be the conformally flat metric defined in (9.2). Then (ρ,u) is a solution to the relativistic Euler equations (1.7a) - (1.7b) corresponding to the metric g if and only if (ρ′,U) is a solution to the relativistic Euler equations (1.7a) - (1.7b) corresponding to the Minkowski metric m. Here, the rescaled fluid variables (ρ′,U) are defined by def def (9.4) ρ′ = e4Ωρ, U = eΩu. Proof. Throughout this proof, denotes the Levi-Civita connection corresponding to the Minkowski metric m and∇D denotes the Levi-Civita connection corresponding to g = e2Ωm. As discussed in the beginning of the article, the relativistic Euler αµ equations (1.7a) - (1.7b) are equivalent to the equations DαT = 0 plus the α β normalization condition gαβu u = 1 (see (1.3) and (1.5)). In the case p = 2 − 1/3csρ, we note that 4 1 4 1 (9.5) T µν = ρuµuν + ρ(g−1)µν = ρuµuν + ρe−2Ω(m−1)µν , 3 3 3 3 and we define the following rescaled “Minkowskian” energy-momentum tensor: 4 1 (9.6) T µν def= e6ΩT µν = ρ′U µU ν + ρ′(m−1)µν , (m) 3 3 where ρ′ and U µ are defined in (9.4). We also note that by (9.2), (9.7) m U αU β = 1. αβ − The key step is the following identity, whose simple proof we omit: (9.8) D T αµ = e−6Ω (e6ΩT αµ)= e−6Ω T αµ . α ∇α ∇α (m) We remark that the proof of the identity (9.8) heavily leans on the fact that αβ gαβT = 0 for the equation of state p = (1/3)ρ. It now follows from (9.8) that (9.9) D T αµ T αµ =0. α ⇐⇒ ∇α (m) According to the remarks made at the beginning of the proof, this completes the proof of the proposition. In the next theorem, we will recall some important aspects of Christodoulou’s shock formation result [Chr07]. To this end, we need to introduce some notation. In order to emphasize the connections between Proposition 9.1, Theorem 9.1, and Corollary 9.2, we will denote the energy density by ρ′, the four velocity by U, and the spacetime coordinates by (τ, x1, x2, x3). Let (˚ρ′, U˚1, U˚2, U˚3) be initial data (at time τ = 1) for the relativistic Euler equations (1.7a) - (1.7b) on the spacetime- with-boundary [1, ) R3,m , where m = dτ 2 + 3 (dxj )2 is the standard ∞ × − j=1 Minkowski metric on R4. Letρ> ¯ 0 be a constant background density. Let B R3 P r denote a solid ball of radius 2 r < 1 centered at the origin in the Cauchy⊂ 3 ≤ hypersurface (τ, x1, x2, x3) τ = 1 R3 (embedded in Minkowski spacetime) { | } ≃ The Stabilizing Effect of Spacetime Expansion 36 and let ∂B denote its boundary. We define the order-1 Sobolev norm 0 r SB1\Br ≥ of the data over the annular region in between Br and the unit ball as follows: 3 def ′ j (9.10) = ˚ρ ρ¯ 1 + U˚ 1 . SB1\Br k − kH (B1\Br ) k kH (B1\Br ) j=1 X We define the standard Sobolev norm of the data as follows: 3 def ′ j (9.11) = ˚ρ ρ¯ M + U˚ M . DM k − kH k kH j=1 X We also define the following surface + annular integrals of the data: (9.12) def ′ 4 i ′ 4 i 3 Q(r) = r (˚ρ ρ¯)+ ρ¯U˚ Nˆi dσ + 2(˚ρ ρ¯)+ ρ¯U˚ Nˆi d x. ∂B − √3 (B1\B ) − √3 Z r n o Z r def 3 j 2 R3 ˆ Above, r = j=1(x ) denotes the standard radial coordinate on , and N denotes the outward unit normal to ∂B . qP r Theorem 9.1 ([Chr07] Theorem 14.2, pg. 925). Consider the relativistic Euler equations (1.7a) - (1.7b) on the manifold-with-boundary [1, ) R3 equipped with the Minkowski metric and rectangular coordinates (τ, x1, x∞2, x×3): m = dτ 2 + 3 j 2 − j=1(dx ) . Let ρ>¯ 0 be a constant background density. Then the explicit solution ′ τ 1 2 3 Pρ =ρ, ¯ (U , U , U , U ) = (1, 0, 0, 0) is unstable. More specifically, there exists an open family of arbitrarily small (non-zero) smooth perturbations of the explicit solution’se e datae whiche e launch solutions that form shocks in finite time. Here, U τ denotes the time component of U relative to the coordinate system (τ, x1, x2, x3). Even more specifically, there exists a large integer M, a large constant C >e0, and a small constant ǫ > 0 suche that the following three conditions on the initial data together guarantee finite-time shock formation in the corresponding solution: (9.13a) ǫ, DM ≤ (9.13b) Q(r) C + √1 r , ≥ DM DM − S(B1\Br) 2 (9.13c) r C′(1 r) (9.14) τ (r) = exp − . max Q(r) Remark 9.1. Note that if we are given any data such that Q(r) > 0, then if we rescale its amplitude (more precisely, the amplitude of its deviation from the background constant solution) by a sufficiently small constant factor, then (9.13a) and (9.13b) are both verified by the rescaled data. This follows from the fact that The Stabilizing Effect of Spacetime Expansion 37 for a fixed r, M , (B1\Br ), and Q(r) shrink linearly with the scaling factor, while the right-handD sideS of (9.13b) shrinks like the scaling factor to the power 3/2. The following corollary follows easily from (9.1) - (9.2), Proposition 9.1, and Theorem 9.1. 2 Corollary 9.2 (Nonlinear Instability of the Explicit Solutions When cs =1/3). 2 Assume that cs =1/3. Consider the relativistic Euler equations (1.7a) - (1.7b) on the manifold-with-boundary [1, ) R3 equipped with a Lorentzian metric g = ∞ × dt2 + e2Ω(t) 3 (dxj )2, Ω(1) = 0 (as in (1.2)). Assume that hypothesis A1 of − j=1 Section 1 holds, and that ∞ e−Ω(s) ds = . Then the explicit solution (1.8) is P s=1 unstable. More specifically, there exist arbitrarily∞ small (non-zero) perturbations of R the explicit solution’s initial data which launch perturbed solutions that form shocks in finite time. More precisely, if the open conditions (9.13a) - (9.13c) are verified by the data, then a shock will form before the conformal τ coordinate time − C′(1 r) (9.15) τ (r) = exp − . max Q(r) In the original time coordinate t, the shock will form before tmax(r), which is im- plicitly determined in terms of τmax(r) via the relation tmax(r) −Ω(s) (9.16) τmax(r)= e ds. Z1 Remark 9.2. Unlike the proof of Theorem 8.1, the proof of Corollary 9.2 is not easily seen to be stable under small perturbations of the metric g. In order to show that the proof is stable, one would need to show that Christodoulou’s proof of Theorem 9.1 is stable under small perturbations of the Minkowski metric. Acknowledgments This research was supported in part by a Solomon Buchsbaum grant adminis- tered by the Massachusetts Institute of Technology, and in part by an NSF All- Institutes Postdoctoral Fellowship administered by the Mathematical Sciences Re- search Institute through its core grant DMS-0441170. I would like to thank John Barrow for offering his many insights, which propelled me to closely investigate the 2 2 cases cs = 0 and cs =1/3. Appendices A. Sobolev-Moser Inequalities In this Appendix, we provide some Sobolev-Moser estimates. The propositions and corollaries stated below are standard results that can be proved using meth- ods similar to those used in [H¨or97, Chapter 6] and in the Appendix of [KM81]. Throughout this Appendix, Lp = Lp(R3) and HM = HM (R3). Proposition A-1. Let M 0 be an integer. If va 1≤a≤l are functions such that ∞ (M) ≥ { } va L , ∂ va L2 < for 1 a l, and ~α1, , ~αl are spatial derivative multi-indices∈ k with k~α + ∞+ ~α =≤M,≤then ··· | 1| ··· | l| The Stabilizing Effect of Spacetime Expansion 38 l (M) (A.1) (∂ v )(∂ v ) (∂ v ) 2 C(l,M) ∂ v 2 v ∞ . k ~α1 1 ~α2 2 ··· ~αl l kL ≤ k akL k bkL a=1 X bY6=a Proposition A-2. Let M 1 be an integer, let K be a compact set, and let F CM (K) be a function. Assume≥ that v is a function such that v(R3) K and ∈ b ⊂ ∂v HM−1. Then ∂(F v) HM−1, and ∈ ◦ ∈ M (l) l−1 (A.2) ∂(F v) M−1 C(M) ∂v M−1 F K v ∞ . k ◦ kH ≤ k kH | | k kL Xl=1 Proposition A-3. Let M 1 be an integer, let K be a compact, convex set, and let F CM (K) be a function.≥ Assume that v is a function such that v(R3) K ∈ b ⊂ and v v¯ HM , where v¯ K is a constant. Then F v F v¯ HM , and − ∈ ∈ ◦ − ◦ ∈ (A.3) M (1) (l) l−1 F v F v¯ M C(M) F K v v¯ 2 + ∂v M−1 F K v ∞ . k ◦ − ◦ kH ≤ | | k − kL k kH | | k kL n Xl=1 o Proposition A-4. Let M 1,l 2 be integers. Suppose that va 1≤a≤l are ∞≥ ≥ M { } M−1 functions such that va L for 1 a l, that vl H , and that ∂va H for 1 a l 1. Then∈ ≤ ≤ ∈ ∈ ≤ ≤ − (A.4) l−1 l−1 v v v M C(l,M) v M v ∞ + ∂v M−1 v ∞ . k 1 2 ··· lkH ≤ k lkH k akL k akH k bkL a=1 a=1 n Y X bY6=a o Remark A.1. The significance of this proposition is that only one of the functions, 2 namely vl, is estimated in L . Proposition A-5. 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