Stabilizing Relativistic Fluids on Spacetimes with Non-Accelerated Expansion

Abstract

We establish global regularity and stability for the irrotational relativistic Euler equations with equation of state $\bar{p}{}=K\bar{\rho }{}$, where $0<K<1/3$, for small initial data in the expanding direction of FLRW spacetimes of the form $(\mathbb R\times \mathbb T^3,-d\bar{t}{}^2+\bar{t}{}^2\delta _{ij} dx^i dx^j$). This provides the first case of non-dust fluid stabilization by spacetime expansion where the expansion rate is of power law type but non-accelerated. In particular, the time integral of the inverse scale factor diverges as $t\rightarrow \infty$. The result implies that structure formation in cosmological evolution associated with the development of shocks in fluids necessarily requires a phase of deccelerating expansion of the Universe to occur in the case that the matter is massive.

Calculus Inequalities

In this appendix, we collect, for the convenience of the reader, a number of calculus inequalities that we employ. The proof of the following inequalities are well known and may be found, for example, in the books [1, 8] and [22].

Theorem A.1

(Hölder’s inequality) If $0< p,q,r \le \infty$ satisfy $1/p+1/q = 1/r$, then

$$\begin{aligned} \Vert uv\Vert _{L^r} \le \Vert u\Vert _{L^p}\Vert v\Vert _{L^q} \end{aligned}$$

for all $u\in L^p(\mathbb {T}{}^{n})$ and $v\in L^q(\mathbb {T}{}^{n})$.

Theorem A.2

(Sobolev’s inequality) Suppose $1\le p < \infty$ and $s\in \mathbb {Z}{}_{> n/p}$. Then

$$\begin{aligned} \Vert u\Vert _{L^\infty } \lesssim \Vert u\Vert _{W^{s,p}} \end{aligned}$$

for all $u\in W^{s,p}(\mathbb {T}{}^{n})$.

Theorem A.3

(Product and commutator estimates)  

• (i) Suppose $1\le p_1,p_2,q_1,q_2\le \infty$, $s\in \mathbb {Z}{}_{\ge 1}$, $|\alpha |=s$ and

  $$\begin{aligned} \frac{1}{p_1}+\frac{1}{p_2} = \frac{1}{q_1} + \frac{1}{q_2} = \frac{1}{r}. \end{aligned}$$

  Then

  $$\begin{aligned} \Vert D^\alpha (uv)\Vert _{L^r} \lesssim \Vert u\Vert _{W^{s,p_1}}\Vert v\Vert _{L^{q_1}} + \Vert u\Vert _{L^{p_2}}\Vert v\Vert _{W^{s,q_2}} \end{aligned}$$

  and

  $$\begin{aligned} \Vert [D^\alpha ,u]v\Vert _{L^r} \lesssim \Vert D u\Vert _{L^{p_1}}\Vert v\Vert _{W^{s-1,q_1}} + \Vert D u\Vert _{ W^{s-1,p_2}}\Vert v\Vert _{L^{q_2}} \end{aligned}$$

  for all $u,v \in C^\infty (\mathbb {T}{}^{n})$.

• (ii) Suppose $s_1,s_2,s_3\in \mathbb {Z}{}_{\ge 0}$, $\;s_1,s_2\ge s_3$, $1\le p \le \infty$, and $s_1+s_2-s_3 > n/p$. Then

  $$\begin{aligned} \Vert uv\Vert _{W^{s_3,p}} \lesssim \Vert u\Vert _{W^{s_1,p}}\Vert v\Vert _{W^{s_2,p}} \end{aligned}$$

  for all $u\in W^{s_1,p}(\mathbb {T}{}^{n})$ and $v\in W^{s_2,p}(\mathbb {T}{}^{n})$.

Theorem A.4

(Moser’s estimates) Suppose $1\le p \le \infty$, $s\in \mathbb {Z}{}_{\ge 1}$, $0\le k\le s$, $|\alpha |=k$ and $f\in C^s(U)$, where U is open and bounded in $\mathbb {R}{}$ and contains 0, and $f(0)=0$. Then

$$\begin{aligned} \Vert D^\alpha f(u)\Vert _{L^{p}} \le C\bigl (\Vert f\Vert _{C^s(\overline{U})}\bigr )(1+\Vert u\Vert ^{s-1}_{L^\infty })\Vert u\Vert _{W^{s,p}} \end{aligned}$$

for all $u \in C^0(\mathbb {T}{}^{n})\cap L^\infty (\mathbb {T}{}^{n})\cap W^{s,p}(\mathbb {T}{}^{n})$ with $u(x) \in U$ for all $x\in \mathbb {T}{}^{n}$.

Lemma A.5

(Ehrling’s lemma) Suppose $1\le p < \infty$, $s_0,s,s_1\in \mathbb {Z}{}_{\ge 0}$, and $s_0< s < s_1$. Then for any $\epsilon >0$ there exists a constant $C=C(\epsilon ^{-1})$ such that

$$\begin{aligned} \Vert u\Vert _{W^{s,p}} \le \epsilon \Vert u\Vert _{W^{s_1,p}} + C\Vert u\Vert _{W^{s_0,p}} \end{aligned}$$

for all $u\in W^{s_1,p}(\mathbb {T}{}^{n})$.