Future Stability of the FLRW Spacetime for a Large Class of Perfect Fluids

Abstract

We establish the future nonlinear stability of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein–Euler equations of the universe filled with a large class of perfect fluids (the equations of state are allowed to be certain nonlinear or linear types both). Several previous results as specific examples can be covered in the results of this article. We emphasize that the future stability of FLRW metric for polytropic fluids with positive cosmological constant has been a difficult problem and cannot be directly generalized from the previous known results. Our result in this article has not only covered this difficult case for the polytropic fluids, but also unified more types of fluids in a same scheme of proofs.

Appendix A. A class of symmetric hyperbolic systems

In this Appendix, we introduce the main tool for this article which is a variation of the theorem originally established in [37, Appendix B]. The proof of it has been omit, but readers can see the details⁵ in [37], and its generalizations in [25, 26].

Consider the following symmetric hyperbolic system.

$$\begin{aligned} B^{\mu }\partial _{\mu }u =&\frac{1}{t}\mathbf {BP}u+H\quad \text {in}\;[T_{0},T_{1}]\times \mathbb {T}^{n}, \end{aligned}$$

(A.1)

$$\begin{aligned} u =&u_{0}\quad \quad \quad \qquad \text {in}\;{T_{0}}\times \mathbb T^{n}, \end{aligned}$$

(A.2)

where we require the following Conditions:

• (I) $T_{0}<T_{1}\le 0$.

• (II) $\mathbf {P}$ is a constant, symmetric projection operator, i.e., $\mathbf {P}^{2}=\mathbf {P}$, $\mathbf {P}^{T}=\mathbf {P}$ and $\partial _\mu \mathbf {P}=0$.

• (III) $u=u(t,x)$ and H(t, u) are $\mathbb R^{N}$-valued maps, $H\in C^{0}([T_{0},0],C^{\infty }(\mathbb R^{N}))$ and satisfies $H(t,0)=0$.

• (IV) $B^{\mu }=B^{\mu }(t,u)$ and $\mathbf {B}=\mathbf {B}(t,u)$ are $\mathbb M_{N\times N}$-valued maps, and $B^{\mu },\,\mathbf {B}\in C^{0}([T_{0},0],C^{\infty }( \mathbb R^{N}))$, $B^{0}\in C^{1}([T_{0},0],C^{\infty }( \mathbb R^{N}))$, and they satisfy

  $$\begin{aligned} (B^{\mu })^{T}=B^{\mu },\quad [\mathbf {P}, \mathbf {B}]=\mathbf {PB}-\mathbf {BP}=0. \end{aligned}$$

  (A.3)

• (V) Suppose⁶

  $$\begin{aligned} B^0=&\mathring{B}^0(t)+\tilde{B}^0 (t,u) \end{aligned}$$

  (A.4)

  and

  $$\begin{aligned} \mathbf {B}= & {} \mathring{\mathbf {B}}(t) +\tilde{\mathbf {B}} (t,u) \end{aligned}$$

  (A.5)

  where $\tilde{B}^0 (t,0)=0$ and $\tilde{\mathbf {B}} (t,0)=0$. There exists constants $\kappa ,\,\gamma _{1},\,\gamma _{2}$ such that

  $$\begin{aligned} \frac{1}{\gamma _{1}}\mathbb I\le \mathring{B}^{0}\le \frac{1}{\kappa }\mathring{\mathbf {B}}\le \gamma _{2}\mathbb I \end{aligned}$$

  (A.6)

  for all $t \in [T_{0},0]$.

• (VI) For all $(t,u)\in [T_{0},0]\times \mathbb R^{N}$, we have

  $$\begin{aligned} \mathbf {P}^{\bot }B^{0}(t,\mathbf {P}^\perp u)\mathbf {P}=\mathbf {P}B^{0}(t,\mathbf {P}^\perp u)\mathbf {P}^{\bot }=0, \end{aligned}$$

  where $\mathbf {P}^{\bot }=\mathbb I-\mathbf {P}$ is the complementary projection operator.

• (VII) There exists constants $\varsigma ,\,\beta _{1}$ and $\varpi >0$ such that

  $$\begin{aligned} |\mathbf {P}^{\bot }[D_{u}B^{0}(t,u)(B^{0})^{-1}\mathbf {BP}u]\mathbf {P}^{\bot }|_{op} \le&|t|\varsigma +\frac{2\beta _{1}}{\varpi +|P^{\bot }u|^2}|\mathbf {P}u|^{2}. \end{aligned}$$

  (A.7)

Theorem 5.4

Suppose that $k\ge \frac{n}{2}+1$, $u_{0}\in H^{k}(\mathbb T^{n})$ and conditions (I)–(VII) are fulfilled. Then there exists a $T_{*}\in (T_{0},0)$, and a unique classical solution $u\in C^{1}([T_{0},T_{*}]\times \mathbb T^{n})$ that satisfies $u\in C^{0}([T_{0},T_{*}],H^{k})\cap C^{1}([T_{0},T_{*}],H^{k-1})$ and the energy estimate

$$\begin{aligned} \Vert u(t)\Vert _{H^{k}}^{2}-\int _{T_{0}}^{t}\frac{1}{\tau }\Vert \mathbf {P}u\Vert _{H^{k}}^{2}\mathrm{d}\tau \le Ce^{C(t-T_{0})}(\Vert u(T_{0})\Vert _{H^{k}}^{2}) \end{aligned}$$

for all $T_{0}\le t<T_{*}$, where $C=C(\Vert u\Vert _{L^{\infty }([T_{0},T_{*}),H^{k})},\gamma _{1},\gamma _{2},\kappa ),$ and can be uniquely continued to a larger time interval $[T_{0},T^{*})$ for all $T^{*}\in (T_{*},0]$ provided $\Vert u\Vert _{L^{\infty }([T_{0},T_{*}),W^{1,\infty })}<\infty$.

Let us end this Appendix with a remark of the proof although we omit the detailed proof.

Remark 5.5

We give the key revision in the proof due to the changing of Condition (V), (i.e., (A.4)–(A.6)). As in [26, Page 2203], we rewrite $\mathring{B}^0$ as $\mathring{B}^0 =(\mathring{B} ^0 )^{\frac{1}{2}}(\mathring{B}^0 )^{\frac{1}{2}}$, which can be done since $\mathring{B}^0$ is a real symmetric and positive-definite. We see from (A.6) that

$$\begin{aligned} (\mathring{B} ^0)^{-\frac{1}{2}}\mathring{\mathbf {B}}(\mathring{B}^0)^{-\frac{1}{2}}\ge \kappa \mathbb {1}. \end{aligned}$$

(A.8)

Since, by (A.3)–(A.5)

$$\begin{aligned} \frac{2}{t} \langle D^\alpha u, \mathbf {B} D^\alpha \mathbf {P} u\rangle =&\frac{2}{t} \langle D^\alpha \mathbf {P} u, (\mathring{B}^0)^{\frac{1}{2}}[(\mathring{B}^0)^{-\frac{1}{2}}\mathring{\mathbf {B}} (\mathring{B}^0)^{-\frac{1}{2}}](\mathring{B}^0)^{\frac{1}{2}} D^\alpha \mathbf {P} u\rangle \\&+\frac{2}{t} \langle D^\alpha \mathbf {P} u, \tilde{\mathbf {B}} D^\alpha \mathbf {P} u\rangle \\ \le&\frac{2\kappa }{t} \langle D^\alpha \mathbf {P} u, \mathring{B}^0 D^\alpha \mathbf {P} u\rangle +\frac{2}{t} \langle D^\alpha \mathbf {P} u, \tilde{\mathbf {B}} D^\alpha \mathbf {P} u\rangle \\ =&\frac{2\kappa }{t} \langle D^\alpha \mathbf {P} u, B^0 D^\alpha \mathbf {P} u\rangle -\frac{2\kappa }{t} \langle D^\alpha \mathbf {P} u, \tilde{B}^0 D^\alpha \mathbf {P} u\rangle \\&+\frac{2}{t} \langle D^\alpha \mathbf {P} u, \tilde{\mathbf {B}} D^\alpha \mathbf {P} u\rangle . \end{aligned}$$

It follows immediately from (A.8) that

$$\begin{aligned} \frac{2}{t}\sum _{0\le |\alpha |\le s-1}\langle D^\alpha u,\mathbf {B} D^\alpha \mathbf {P} u\rangle \le \underbrace{\frac{2\kappa }{t}{\left| \left| \left| \mathbf {P} u \right| \right| \right| }^2_{H^{s-1}}}_{\text {I}}-\underbrace{\frac{1}{t} C \Vert u\Vert _{H^{s-1}} \Vert \mathbf {P}u\Vert ^2_{H^{s-1}}}_{\text {II}}, \end{aligned}$$

(A.9)

where

$$\begin{aligned} {\left| \left| \left| u \right| \right| \right| }^2_{H^k}:=\sum _{0\le |\alpha |\le k}\langle D^\alpha u, B^0\bigl (t,u(t,\cdot )\bigr ) D^\alpha u\rangle . \end{aligned}$$

Then, the following proof are the same as [37, Appendix B] or [26, §5] just noting that the term II in (A.9) can be absorbed by I in the rest of estimates provided the data is small enough.