On the stability of relativistic perfect fluids with linear equations of state $p=K\rho$ where $1/3<K<1$

Abstract

For $1/3<K<1$, we consider the stability of two distinct families of spatially homogeneous solutions to the relativistic Euler equations with a linear equation of state $p=K\rho$ on exponentially expanding FLRW spacetimes. The two families are distinguished by one being spatially isotropic while the other is not. We establish the future stability of nonlinear perturbations of the non-isotropic family for the full range of parameter values $1/3<K<1$, which improves a previous stability result established by the second author that required K to lie in the restricted range (1/3, 1/2). As a first step towards understanding the behaviour of nonlinear perturbations of the isotropic family, we construct numerical solutions to the relativistic Euler equations under a $\mathbb {T}{}^2$-symmetry assumption. These solutions are generated from initial data at a fixed time that is chosen to be suitably close to the initial data of an isotropic solution. Our numerical results reveal that, for the full parameter range $1/3<K<1$, the density gradient $\frac{\partial _{x}\rho }{\rho }$ associated to a nonlinear perturbation of an isotropic solution develops steep gradients near a finite number of spatial points where it becomes unbounded at future timelike infinity. This behavior of the density gradient was anticipated by Rendall (Ann Henri Poincaré 5(6):1041–1064, 2004), and our numerical results confirm his expectations.

1 Introduction

Relativistic perfect fluids with a linear equation of state on a prescribed spacetime $(M,\tilde{g}{})$ are governed by the relativistic Euler equations¹

$$\begin{aligned} \tilde{\nabla }{}_i \tilde{T}{}^{ij}=0 \end{aligned}$$

(1.1)

where

$$\begin{aligned} \tilde{T}{}^{ij} = (\rho +p)\tilde{v}{}^i \tilde{v}{}^j + p \tilde{g}{}^{ij} \end{aligned}$$

is the stress energy tensor, $\tilde{v}{}^{i}$ is the fluid four-velocity normalized by $\tilde{g}{}_{ij}\tilde{v}{}^i \bar{v}{}^j=-1$, and the fluid’s proper energy density $\rho$ and pressure p are related by

$$\begin{aligned} p = K \rho . \end{aligned}$$

Since $K=\frac{dp}{d\rho }$ is the square of the sound speed, we will always assume² that $0\le K \le 1$ in order to ensure that the speed of sound is less than or equal to the speed of light. We further restrict our attention to exponentially expanding Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes $(M,\tilde{g}{})$ where $M = (0,1]\times \mathbb {T}{}^3$ and³

$$\begin{aligned} \tilde{g}{}= \frac{1}{t^2} g \end{aligned}$$

(1.2)

with

$$\begin{aligned} g = -dt\otimes dt + \delta _{IJ}dx^I \otimes dx^J. \end{aligned}$$

(1.3)

It is important to note that, due to our conventions, the future is located in the direction of decreasing t and future timelike infinity is located at $t=0$. Consequently, we require that $\tilde{v}{}^0<0$ holds in order to guarantee that the four-velocity is future directed. For use below, we find it convenient to introduce the conformal four-velocity via

$$\begin{aligned} v^i = \frac{1}{t}\tilde{v}{}^i. \end{aligned}$$

(1.4)

1.1 Stability for $0\le K\le 1/3$

It can be verified by a straightforward calculation that

$$\begin{aligned} (\rho _*,v_*^i) = (t^{3(1+K)}\rho _c,-\delta ^i_0), \quad t\in (0,1], \end{aligned}$$

(1.5)

defines a spatially homogeneous solution of the relativistic Euler equations (1.1) on the exponentially expanding FLRW spacetimes $(M,\tilde{g}{})$ for any choice of the parameter $0\le K\le 1$ and constant $\rho _c\in (0,\infty )$. From a cosmological perspective, these solutions are, in a sense, the most natural since they are also spatially isotropic and hence do not determine a preferred direction.

The future, nonlinear stability of the solutions (1.5) on the exponentially expanding FLRW spacetimes was first established in the articles⁴ articles [24, 27] for the parameter values $0<K<1/3$. Stability results for the end points $K=1/3$ and $K=0$ were established later⁵ in [17] and [11], respectively. See also [7, 14, 15, 18] for different proofs and perspectives, the articles [12, 16] for related stability results for fluids with nonlinear equations of state on the exponentially expanding FLRW spacetimes, the articles [5, 28, 30] for analogous stability results on other classes of expanding cosmological spacetimes, and [23] for related, early stability results for the Einstein-scalar field system. One of the important aspects of these works is they demonstrate that spacetime expansion can suppress shock formation in fluids, which was first discovered in the Newtonian cosmological setting [25]. This is in stark contrast to fluids on Minkowski space where arbitrary small perturbations of a class of homogeneous solutions to the relativistic Euler equations form shocks in finite time [4].

A consequence of these stability proofs is that the spatial components of the conformal four-velocity $v^i$ of small, nonlinear perturbations of the homogeneous solution (1.5) decay to zero at future timelike infinity, that is,

$$\begin{aligned} \lim _{t\searrow 0} v^I = 0, \end{aligned}$$

for the parameter values $0\le K < 1/3$. This behaviour is, of course, consistent with the isotropic homogeneous solutions (1.5). On the other hand, when $K=1/3$, the spatial components of the conformal four-velocity $v^i$ for perturbed solutions do not, in general, decay to zero at timelike infinity, and instead limit to a spatial vector field $\xi ^I$ on $\mathbb {T}{}^3$, that is,

$$\begin{aligned} \lim _{t\searrow 0} v^I = \xi ^I. \end{aligned}$$

This behaviour is consistent with a family of non-isotropic homogeneous solutions defined by⁶

$$\begin{aligned} (\rho _\bullet ,v_\bullet ^i) = (t^{3(1+K)}\rho _c,-\sqrt{1+\nu _c^2}\delta ^i_0 + \nu _c\delta ^i_1), \quad t\in (0,1], \end{aligned}$$

(1.6)

where $(\rho _c,\nu _c)\in (0,\infty )\times (0,\infty )$, that satisfy the relativistic Euler equations for $K=1/3$. The known stability results, discussed above, for $K=1/3$ imply the future stability of nonlinear perturbations of these solutions.

Noting that, for $K=1/3$, solutions of the type (1.6) can be made arbitrarily close to solutions of the type (1.5) by choosing $\nu _c$ sufficiently small, from a stability point of view there seems to be little difference between theses two classes of solutions for small $\nu _c$. Indeed, the future nonlinear stability of both types of solutions, provided $\nu _c$ is sufficiently small, can be achieved via a common proof. However, as we will become clear, the crucial distinction between these two types of solutions is, from an initial data point of view, that stable perturbations of solutions of the type (1.5) are generated from initial data $(\rho ,v^I)|_{t=1}$ that is sufficiently close to $(\rho _*,v_*^I)|_{t=1}$ and satisfies

$$\begin{aligned} \min _{x\in \mathbb {T}{}^3} (g_{IJ}v^I v^J)\bigl |_{t=1}=0, \end{aligned}$$

(1.7)

while stable perturbations of solutions of the type (1.6) are generated from initial data $(\rho ,v^I)|_{t=1}$ that is sufficiently close to $(\rho _\bullet ,v_\bullet ^I)|_{t=1}$ and satisfies

$$\begin{aligned} \min _{x\in \mathbb {T}{}^3} (g_{IJ}v^I v^J)\bigl |_{t=1}>0. \end{aligned}$$

(1.8)

1.2 Stability for $1/3<K<1$

In the article [22], Rendall used formal expansion to investigate the asymptotic behaviour of relativistic fluids with a linear equation of state on exponentially expanding spacetimes. He observed that the formal expansions become inconsistent for K in the range $1/3<K<1$ if the leading order term in the expansion of $g_{IJ}v^I v^J$ vanished somewhere at timelike infinity. He speculated that the inconsistent behaviour in the expansions is due to inhomogeneous features developing in the fluid density that would ultimately result in the blow-up of the density gradient $\frac{\partial _{I}\rho }{\rho }$ at future timelike infinity. Speck [28, §1.2.3] added further support to Rendall’s arguments by presenting a heuristic analysis that suggested uninhibited growth would set in for solutions of the relativistic Euler equations for the parameter values $1/3< K<1$.

On the other hand, under the assumption that the leading order term in the expansion of $g_{IJ}v^I v^J$ is non-vanishing everywhere at timelike infinity, Rendall constructed formal power series solutions to the relativistic Euler equations on exponentially expanding spacetimes. These expansions were independently derived in [13], and in that article, the authors noted that number of free functions in the expansions matched the expected number for a generic solution, suggesting these solutions would be stable. It was also noted in [13] that the fluid velocity for these formal solutions becomes null at timelike infinity. This behavior is called extreme tilt and was first shown rigorously, using dynamical system methods, to occur in the homogeneous setting in the articles [9, 10].

For $1/3<K<1$, it was shown in [20] that there exists a family of non-isotropic homogeneous solutions to the relativistic Euler equations with extreme tilt that take the form

$$\begin{aligned} \bigl (\rho _\bullet ,v_\bullet ^i) = \biggl ( \frac{\rho _c t^{\frac{2(1+K)}{1-K}}}{(t^{2\mu }+ e^{2u})^{\frac{1+K}{2}}}, -t^{-\mu }\sqrt{e^{2u}+t^{2 \mu } }\delta _0^i+ t^{-\mu }e^{u}\delta _1^i\biggr ), \quad t\in (0,1],\quad \end{aligned}$$

(1.9)

where

$$\begin{aligned} \mu = \frac{3K-1}{1-K} \end{aligned}$$

(1.10)

and $u=u(t)$ solves the initial value problem (IVP)

$$\begin{aligned} u'\!(t)&=\frac{K\mu t^{2 \mu -1}}{t^{2 \mu }+(1-K) e^{2 u(t)}},\quad 0<t\le 1, \end{aligned}$$

(1.11)

$$\begin{aligned} u(1)&= u_0. \end{aligned}$$

(1.12)

Existence of solutions to this IVP is guaranteed by Proposition 3.1. from [20], which we restate here:

Proposition 1.1

Suppose $1/3<K<1$, $\mu = (3K-1)/(1-K)$, and $u_0 \in \mathbb {R}{}$. Then there exists a unique solution $u \in C^\infty ((0,1]) \cap C^0([0,1])$ to the initial value problem (1.11)–(1.12) that satisfies

$$\begin{aligned} |u(t)-u(0)| \lesssim t^{2\mu } {\quad \text {and}\quad }|u'\!(t)| \lesssim t^{2\mu -1} \end{aligned}$$

(1.13)

for all $t\in (0,1]$. Moreover, for each $\rho _c\in (0,\infty )$, the solution u determines a homogeneous solution of the relativistic Euler equations (1.1) via (1.4) and (1.9).

The nonlinear stability to the future of the family of solutions (1.9) was established in [20] under the assumption that $1/3< K <1/2$, which confirms the expectations of [13], at least, for the parameter range $1/3<K<1/2$. It was also established in [20] that under a $\mathbb {T}{}^2$-symmetry assumption, future stability held for the full parameter range $1/3<K<1$. An important point that is worth emphasising is that the initial data used to generate the perturbed solutions from [20] satisfies the condition (1.8) at $t=1$, and furthermore, this positivity property propagates to the future in the sense that the perturbed solutions satisfy

$$\begin{aligned} \inf _{(t,x)\in M}(t^{2\mu }g_{IJ}v^I v^J)>0. \end{aligned}$$

It is this property of the perturbed solutions from [20] that avoids the problematic scenario identified by Rendall. It is also worth noting that these perturbed solutions exhibit extreme tilt, which again confirms the expectations of [13].

This article has two main aims: the first is to establish the nonlinear stability to the future of the homogeneous solutions (1.9) for the full parameter range $1/3< K<1$ without the $\mathbb {T}{}^2$-symmetry that was required in [20]. The second aim is to provide convincing numerical evidence that shows the density gradient blow-up scenario of Rendall is realized if the condition (1.8) on the initial data is violated. Before stating a precise version of our stability result for the homogeneous solutions (1.9), we first recall two formulations of the relativistic Euler equations from [20]. The first formulation, which was introduced in [19] and subsequently employed in [18] to establish stability for the parameter range $0<K\le 1/3$, involves representing the fluid in terms of the modified fluid density $\zeta$ defined via

$$\begin{aligned} \rho = t^{3(1+K)}\rho _c e^{(1+K)\zeta } \end{aligned}$$

(1.14)

and the spatial components $v_I$ of the conformal fluid four-covelocity⁷$v_i=g_{ij}v^j$. In terms of these variables, the relativistic Euler equations (1.1) can be formulated as the following symmetric hyperbolic system:

$$\begin{aligned} B^k \partial _{k}V = \frac{1}{t}\mathcal {B}{}\pi V \end{aligned}$$

(1.15)

where

$$\begin{aligned} V&= (\zeta , v_J )^{\textrm{tr}} , \end{aligned}$$

(1.16)

$$\begin{aligned} v_0&= \sqrt{|v|^2 +1} , \qquad |v|^2 = \delta ^{IJ}v_I v_J, \end{aligned}$$

(1.17)

$$\begin{aligned} v^i&= \delta ^{iJ}v_J - \delta ^{i}_0 v_0, \end{aligned}$$

(1.18)

$$\begin{aligned} \mathcal {B}{}&= \frac{-1}{v^0}\begin{pmatrix} 1 &{} 0 \\ 0 &{} \frac{1-3K}{v_0}\delta ^{JI} \end{pmatrix}, \end{aligned}$$

(1.19)

$$\begin{aligned} \pi&= \begin{pmatrix} 0 &{} 0 \\ 0 &{} \delta _{I}^J \end{pmatrix}, \end{aligned}$$

(1.20)

$$\begin{aligned} L^k_I&= \delta ^k_J - \frac{v_J}{v_0} \delta ^k_0, \end{aligned}$$

(1.21)

$$\begin{aligned} M_{IJ}&= \delta _{IJ} - \frac{1}{(v_0)^2}v_I v_J, \end{aligned}$$

(1.22)

$$\begin{aligned} B^0&= \begin{pmatrix} K &{} \frac{K}{v^0} L^0_M \delta ^{MJ} \\ \frac{K}{v^0} \delta ^{LI} L^0_L &{} \delta ^{LI} M_{LM} \delta ^{MJ} \end{pmatrix} \end{aligned}$$

(1.23)

and

$$\begin{aligned} B^K&=\frac{1}{v^0}\begin{pmatrix} Kv^K &{}\qquad K L^K_M \delta ^{MJ} \\ K \delta ^{LI} L^K_L &{}\qquad \delta ^{LI} M_{LM} \delta ^{MJ} v^K \end{pmatrix}. \end{aligned}$$

(1.24)

The second formulation of the relativistic Euler equations is obtained by introducing a new density variable $\tilde{\zeta }{}$ via

$$\begin{aligned} \tilde{\zeta }{}= \zeta + \ln (v_0) \end{aligned}$$

(1.25)

and decomposing the spatial components of the conformal fluid four-velocity as

$$\begin{aligned} v_1&= \frac{t^{-\mu } e^{u(t)+w_1}}{\sqrt{t^{2 \mu } \left( (w_2-w_3)^2+(w_2+w_3)^2\right) +1}}, \end{aligned}$$

(1.26)

$$\begin{aligned} v_2&= \frac{(w_2+w_3) e^{u(t)+w_1}}{\sqrt{t^{2 \mu } \left( (w_2-w_3)^2+(w_2+w_3)^2\right) +1}} \end{aligned}$$

(1.27)

$$\begin{aligned} \text {and} v_3&= \frac{(w_2-w_3) e^{u(t)+w_1}}{\sqrt{t^{2 \mu } \left( (w_2-w_3)^2+(w_2+w_3)^2\right) +1}}, \end{aligned}$$

(1.28)

where u(t) solves the IVP (1.11)–(1.12). Then setting

$$\begin{aligned} \breve{w}{}_1&= u+w_1, \end{aligned}$$

(1.29)

$$\begin{aligned} \psi&= t^{2 \mu }+e^{2 \breve{w}{}_1}, \end{aligned}$$

(1.30)

$$\begin{aligned} \chi&= t^{2\mu }-(K-1) e^{2 \breve{w}{}_1}, \end{aligned}$$

(1.31)

$$\begin{aligned} \phi&= 2 t^{2\mu } \left( w_2^2+w_3^2\right) +1, \end{aligned}$$

(1.32)

$$\begin{aligned} \eta _\Lambda&= \left( 2 w_\Lambda t^{2 \mu } (w_2-w_3)+(-1)^\Lambda 1\right) , \quad \Lambda =2,3, \end{aligned}$$

(1.33)

and

$$\begin{aligned} \xi _\Lambda&= \left( 2 w_\Lambda t^{2 \mu } (w_2+w_3)+1\right) , \quad \Lambda =2,3, \end{aligned}$$

(1.34)

it was shown in [20, §3.2] that in terms of the variables

$$\begin{aligned} W = (\tilde{\zeta }{}, w_1, w_2, w_3 )^{\textrm{tr}} \end{aligned}$$

(1.35)

the relativistic Euler equations become

$$\begin{aligned} \partial _{t}W + \mathcal {A}{}^I \partial _{I}W =-\frac{\mu }{t}\Pi W + t^{\mu -1}\mathcal {G}{}\end{aligned}$$

(1.36)

where

$$\begin{aligned} \mathcal {A}{}^1= & {} \frac{1}{\sqrt{\frac{t^{2\mu }}{e^{2\tilde{w}{}_1}}+1}}\begin{pmatrix} -\frac{1}{\sqrt{\phi }} &{} -\frac{t^{2 \mu }}{\psi \sqrt{\phi }} &{} \frac{2 t^{2\mu } w_2}{\phi ^{3/2}} &{} \frac{t^{2\mu } w_3}{\phi ^{3/2}}\\ -\frac{K t^{2 \mu } e^{-2 \breve{w}{}_1} \psi }{\sqrt{\phi }\chi } &{} \frac{(2 K-1) t^{2 \mu }+(K-1) e^{2 \breve{w}{}_1}}{\sqrt{\phi } \chi } &{} -\frac{2 K t^{2\mu } \psi w_2}{\phi ^{3/2}\chi } &{} -\frac{2 K t^{2\mu } \psi w_3}{\phi ^{3/2}\chi } \\ K t^{2\mu }w_2 e^{-2 \breve{w}{}_1} \sqrt{\phi } &{} -\frac{K t^{2\mu } w_2 \sqrt{\phi }}{\psi } &{} -\frac{1}{\sqrt{\phi }} &{} 0\\ K t^{2\mu }w_3 e^{-2 \breve{w}{}_1} \sqrt{\phi } &{} -\frac{K t^{2\mu } w3 \sqrt{\phi }}{ \psi } &{} 0 &{} - \frac{1}{\sqrt{\phi }} \end{pmatrix}, \end{aligned}$$

(1.37)

$$\begin{aligned} \mathcal {A}{}^2= & {} \frac{1}{\sqrt{\frac{t^{2\mu }}{e^{2\tilde{w}{}_1}}+1}}\begin{pmatrix} -\frac{t^{\mu } (w_3+w_2)}{\sqrt{\phi }} &{} -\frac{t^{3 \mu } (w_3+w_2)}{\psi \sqrt{\phi }} &{} \frac{t^{\mu } \eta _3}{\phi ^{3/2}} &{} -\frac{t^\mu \eta _2}{\phi ^{3/2}} \\ -\frac{K t^{3 \mu } (w_2+w_3) e^{-2 \breve{w}{}_1} \psi }{\sqrt{\phi }\chi } &{} \frac{t^{\mu } (w_2+w_3) \left( (2 K-1) t^{2 \mu }+(K-1) e^{2 \breve{w}{}_1}\right) }{\sqrt{\phi } \chi } &{} -\frac{K t^{\mu } \psi \eta _3}{\phi ^{3/2}\chi } &{} \frac{K t^{\mu } \psi \eta _2}{\phi ^{3/2}\chi } \\ -\frac{1}{2} K t^{\mu } e^{-2 \breve{w}{}_1} \sqrt{\phi } &{} \frac{K t^{\mu } \sqrt{\phi }}{2 \psi } &{} -\frac{t^{\mu } (w_3+w_2)}{\sqrt{\phi }} &{} 0 \\ -\frac{1}{2} K t^{\mu } e^{-2 \breve{w}{}_1} \sqrt{\phi } &{} \frac{K t^{\mu } \sqrt{\phi }}{2 \psi } &{} 0 &{} -\frac{t^{\mu } (w_3+w_2)}{\sqrt{\phi }} \end{pmatrix},\nonumber \\ \end{aligned}$$

(1.38)

$$\begin{aligned} \mathcal {A}{}^3= & {} \frac{1}{\sqrt{\frac{t^{2\mu }}{e^{2\tilde{w}{}_1}}+1}}\begin{pmatrix} \frac{t^{\mu } (w_3-w_2)}{\sqrt{\phi }} &{} \frac{t^{3 \mu } (w_3-w_2)}{\psi \sqrt{\phi }} &{} -\frac{t^{\mu } \xi _3}{\phi ^{3/2}} &{} \frac{t^\mu \xi _2}{\phi ^{3/2}} \\ -\frac{K t^{3 \mu } (w_2-w_3) e^{-2 \breve{w}{}_1} \psi }{\sqrt{\phi }\chi } &{} \frac{t^{\mu } (w_2-w_3) \left( (2 K-1) t^{2 \mu }+(K-1) e^{2 \breve{w}{}_1}\right) }{\sqrt{\phi } \chi } &{} \frac{K t^{\mu } \psi \xi _3}{\phi ^{3/2}\chi } &{} -\frac{K t^{\mu } \psi \xi _2}{\phi ^{3/2}\chi } \\ -\frac{1}{2} K t^{\mu } e^{-2 \breve{w}{}_1} \sqrt{\phi } &{} \frac{K t^{\mu } \sqrt{\phi }}{2 \psi } &{} \frac{t^{\mu } (w_3-w_2)}{\sqrt{\phi }} &{} 0 \\ \frac{1}{2} K t^{\mu } e^{-2 \breve{w}{}_1} \sqrt{\phi } &{} -\frac{K t^{\mu } \sqrt{\phi }}{2 \psi } &{} 0 &{} \frac{t^{\mu } (w_3-w_2)}{\sqrt{\phi }} \end{pmatrix},\nonumber \\ \end{aligned}$$

(1.39)

$$\begin{aligned} \mathcal {G}{}= & {} \begin{pmatrix} 0 \\ -\frac{K (3 K-1) \left( e^{2 w_1}-1\right) e^{2 u}}{\left( (K-1) e^{2 u}-t^{2 \mu }\right) \left( (K-1) e^{2 \breve{w}{}_1}-t^{2 \mu }\right) } \\ 0 \\ 0 \end{pmatrix}, \end{aligned}$$

(1.40)

and

$$\begin{aligned} \Pi = {\textrm{diag}}(0,0,1,1). \end{aligned}$$

(1.41)

For later use, we also define

$$\begin{aligned} \Pi ^\perp = \mathord {{\mathrm 1}\hspace{-2.70004pt}{\mathrm I}}\hspace{3.50006pt}- \Pi , \end{aligned}$$

(1.42)

and observe that $\Pi$ and $\Pi ^\perp$ satisfy the relations

$$\begin{aligned} \Pi ^2 = \Pi , \quad (\Pi ^\perp )^2 = \Pi ^\perp , \quad \Pi \Pi ^\perp =\Pi ^\perp \Pi = 0 {\quad \text {and}\quad }\Pi +\Pi ^\perp = \mathord {{\mathrm 1}\hspace{-2.70004pt}{\mathrm I}}\hspace{3.50006pt}. \end{aligned}$$

(1.43)

An important point regarding the formulation (1.36) is that it is symmetrizable. Indeed, as shown in [20], multiplying (1.36) by the positive definite, symmetric matrix

$$\begin{aligned} A^0 = \begin{pmatrix} K &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{t^{2 \mu } e^{2 \breve{w}{}_1}-(K-1) e^{4 \breve{w}{}_1}}{\psi ^2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{2 e^{2 \breve{w}{}_1} \left( 2 w_3^2 t^{2 \mu }+1\right) }{\phi ^2} &{} -\frac{4 w_2 w_3 t^{2 \mu } e^{2 \breve{w}{}_1}}{\phi ^2} \\ 0 &{} 0 &{} -\frac{4 w_2 w_3 t^{2 \mu } e^{2 \breve{w}{}_1}}{\phi ^2} &{} \frac{2 e^{2 \breve{w}{}_1} \left( 2 w_2^2 t^{2 \mu }+1\right) }{\phi ^2} \end{pmatrix} \end{aligned}$$

(1.44)

yields

$$\begin{aligned} A^0\partial _{t}W + A^I \partial _{I}W =-\frac{\mu }{t} A^0\Pi W + t^{\mu -1}A^0\mathcal {G}{}\end{aligned}$$

(1.45)

where it is straightforward to verify from (1.37)–(1.39) that the matrices

$$\begin{aligned} A^I = A^0 \mathcal {A}{}^I \end{aligned}$$

(1.46)

are symmetric, that is,

$$\begin{aligned} (A^I)^{\textrm{tr}}=A^I. \end{aligned}$$

(1.47)

We are now in a position to state the main stability theorem of this article. The proof is presented in Sect. 2. Before stating the theorem, it is important to note that, due to change of variables defined via (1.25)–(1.28) and (1.35), the homogeneous solutions (1.9) correspond to the trivial solution $W=0$ of (1.36).

Theorem 1.2

Suppose $k\in \mathbb {Z}{}_{>3/2+1}$, $1/3<K < 1$, $\mu = (3K-1)/(1-K)$, $\sigma > 0$, $u_0\in \mathbb {R}{}$, $u \in C^\infty ((0,1])\cap C^0([0,1])$ is the unique solution to the IVP (1.11)–(1.12) from Proposition 1.1 and $\tilde{\zeta }{}_0, w^0_J \in H^{k+1}(\mathbb {T}{}^3)$. Then for $\delta >0$ small enough, there exists a unique solution

$$\begin{aligned} W=(\tilde{\zeta }{},w_J)^{\textrm{tr}} \in C^0\bigl ((0,1], H^{k+1}(\mathbb {T}{}^3,\mathbb {R}{}^4)\bigr )\cap C^1\bigl ((0,1],H^{k}(\mathbb {T}{}^3,\mathbb {R}{}^4)\bigr ) \end{aligned}$$

to the initial value problem

$$\begin{aligned} \partial _{t}W + \mathcal {A}{}^I\partial _{I}W&= -\frac{\mu }{t}\Pi W + t^{\mu -1}\mathcal {G}{}\quad \text {in} (0,1]\times \mathbb {T}{}^3, \end{aligned}$$

(1.48)

$$\begin{aligned} W&= (\tilde{\zeta }{}_0, w^0_J)^{\textrm{tr}}\quad \text {in} \{1\}\times \mathbb {T}{}^3, \end{aligned}$$

(1.49)

provided that

$$\begin{aligned} \biggl (\Vert \tilde{\zeta }{}_0\Vert _{H^{k+1}}^2+\sum _{J=1}^3\Vert w^0_J\Vert _{H^{k+1}}^2\biggr )^{\frac{1}{2}}\le \delta . \end{aligned}$$

Moreover,

• (i) $W=(\tilde{\zeta }{},w_J)^{\textrm{tr}}$ satisfies the energy estimate

  $$\begin{aligned} \mathcal {E}{}(t) + \int _t^1 \tau ^{2\mu -1}\bigl (\Vert D\tilde{\zeta }{}(\tau )\Vert _{H^k}^2+\Vert Dw_1(\tau )\Vert _{H^k}^2\bigr )\,d\tau \\ \lesssim \Vert \tilde{\zeta }{}_0\Vert _{H^{k+1}}^2+\sum _{J=1}^3\Vert w^0_J\Vert _{H^{k+1}}^2 \end{aligned}$$

  for all $t\in (0,1]$ where⁸

  $$\begin{aligned} \mathcal {E}{}(t)=\Vert \tilde{\zeta }{}(t)\Vert _{H^k}^2+\Vert w_1(t)\Vert _{H^k}^2+t^{2\mu }\Bigl (\Vert D\tilde{\zeta }{}(t)\Vert _{H^k}^2\\ +\Vert Dw_1(t)\Vert _{H^k}^2+\Vert w_2(t)\Vert _{H^{k+1}}^2+\Vert w_3(t)\Vert _{H^{k+1}}^2\Bigr ), \end{aligned}$$

• (ii) there exists functions $\tilde{\zeta }{}_*, w_1^* \in H^{k-1}(\mathbb {T}{}^3)$ and $\bar{w}{}_2^*,\bar{w}{}_3^* \in H^{k}(\mathbb {T}{}^3)$ such that the estimate

  $$\begin{aligned} \bar{\mathcal {E}{}}(t) \lesssim t^{\mu -\sigma } \end{aligned}$$

  holds for all $t\in (0,1]$ where

  $$\begin{aligned} \bar{\mathcal {E}{}}(t)= & {} \Vert \tilde{\zeta }{}(t) - \tilde{\zeta }{}_*\Vert _{H^{k-1}}+\Vert w_1(t) - w_1^*\Vert _{H^{k-1}}\\{} & {} \quad +\Vert t^\mu w_2(t) - \bar{w}{}_2^*\Vert _{H^{k}}+\Vert t^\mu w_3(t) - \bar{w}{}_3^*\Vert _{H^{k}}, \end{aligned}$$

• (iii) u and $W=(\tilde{\zeta }{},w_J)^{\textrm{tr}}$ determine a unique solution of the relativistic Euler equations (1.1) on the spacetime region $M=(0,1]\times \mathbb {T}{}^3$ via the formulas

  $$\begin{aligned} \rho&= \frac{\rho _c t^{\frac{2(1+K)}{1-K}} e^{(1+K)\tilde{\zeta }{}}}{(t^{2\mu }+ e^{2(u+w_1)})^{\frac{1+K}{2}}}, \end{aligned}$$

  (1.50)
  $$\begin{aligned} \tilde{v}{}^0&= -t^{1-\mu }\sqrt{e^{2 (u+w_1)}+t^{2 \mu } }, \end{aligned}$$

  (1.51)
  $$\begin{aligned} \tilde{v}{}^1&=t^{1-\mu }\biggl ( \frac{e^{u+w_1}}{\sqrt{ (t^{\mu }w_2-t^{\mu }w_3)^2 +(t^{\mu }w_2+t^{\mu }w_3)^2+1}} \biggr ), \end{aligned}$$

  (1.52)
  $$\begin{aligned} \tilde{v}{}^2&= t^{1-\mu }\biggl ( \frac{(t^{\mu }w_2+t^{\mu }w_3) e^{u+w_1}}{\sqrt{ (t^{\mu }w_2-t^{\mu }w_3)^2 +(t^{\mu }w_2+t^{\mu }w_3)^2+1}}\biggr ), \end{aligned}$$

  (1.53)
  $$\begin{aligned} \tilde{v}{}^3&= t^{1-\mu }\biggl ( \frac{(t^{\mu }w_2-t^{\mu }w_3) e^{u+w_1}}{\sqrt{ (t^{\mu }w_2-t^{\mu }w_3)^2 +(t^{\mu }w_2+t^{\mu }w_3)^2+1}}\biggr ), \end{aligned}$$

  (1.54)

• (iv) and the density gradient $\frac{\partial _{I}\rho }{\rho }$ satisfies

  $$\begin{aligned} \lim _{t\searrow 0} \Bigl \Vert \frac{\partial _{I}\rho }{\rho } - (1+K)\partial _{I}(\tilde{\zeta }{}_*-w_1^*) \Bigr \Vert _{H^{k-2}} = 0. \end{aligned}$$

  (1.55)

Remark 1.3

From the formulas (1.52)–(1.53) and the energy and decay estimates from statements (i) and (ii) of Theorem 1.2, it is not difficult, with the help of Sobelev’s inequality, to verify that the fluid four-velocities $\tilde{v}{}^i$ of the perturbed solutions from Theorem 1.2 satisfy

$$\begin{aligned} \lim _{t\searrow 0} \tilde{v}{}^2_{\tilde{g}{}} = 0 \end{aligned}$$

everywhere on $\mathbb {T}{}^3$. This confirms the expectations of [13] and rigorously verifies that, for $1/3<K<1$, every sufficiently small perturbation of a homogeneous solution to the relativistic Euler equations of the type (1.9) exhibits extreme tilt.

1.2.1 Proof summary

The proof of Theorem 1.2 proceeds in three main steps. The first step involves casting the relativistic Euler equations (1.36) into a suitable Fuchsian form following closely the approach of [20] with one critical modification that is responsible for the improvements in the stability results presented here; see below for details. The next step involves verifying that the coefficients of the Fuchsian system satisfy the required properties needed to apply the Fuchsian global existence theory from [3]. In the third and final step, we apply the Fuchsian global existence theory from [3] in conjunction with a uniqueness result and continuation principle for solutions to (1.36) in order to deduce the nonlinear stability of the trivial solution $W=0$. Since the trivial solution corresponds, through an appropriate choice of initial condition for u, to any of the homogeneous solutions (1.9) via (1.25)–(1.28) and (1.35), the stability of the trivial solution implies the stability of the whole family of homogeneous solutions (1.9), which completes the proof.

To describe the technical modifications that lead to the improvement in the range of K where stability holds from $1/3<K<1/2$ in the article [20] to $1/3<K<1$ in this article, we first need to reformulate (1.36). We do this by first applying the projection operator $\Pi$ to (1.36), while noting that $\Pi \mathcal {G}{}= 0$ by (1.40)–(1.41), to get

$$\begin{aligned} \partial _{t}(\Pi W) + \Pi \mathcal {A}{}^I \partial _{I}W =-\frac{\mu }{t}\Pi W. \end{aligned}$$

Multiplying this equation through by $t^{\mu }$ gives

$$\begin{aligned} \partial _{t}(t^{\mu }\Pi W) + t^{\mu }\Pi \mathcal {A}{}^I \partial _{I}W = 0. \end{aligned}$$

(1.56)

Next, applying $\Pi ^\perp$ to (1.36), we further observe, with the help of (1.43), that

$$\begin{aligned} \partial _{t}(\Pi ^\perp W) + \Pi ^\perp \mathcal {A}{}^I \partial _{I}W = t^{2\mu -1}\Pi ^\perp \mathcal {G}{}. \end{aligned}$$

(1.57)

Setting

$$\begin{aligned} \bar{W}{}:= \Pi ^\perp W+t^{\mu }\Pi W = (\tilde{\zeta }{},w_1,t^{\mu }w_2,t^{\mu }w_3)^{\textrm{tr}}, \end{aligned}$$

(1.58)

we then see from adding (1.56) and (1.57) that $\bar{W}{}$ satisfies

$$\begin{aligned} \partial _{t}\bar{W}{}+t^\mu \Pi \mathcal {A}{}^I \partial _{I}W+\Pi ^\perp \mathcal {A}{}^I \partial _{I}W = t^{2\mu -1}\Pi ^\perp \mathcal {G}{}. \end{aligned}$$

(1.59)

Now, in the article [20], this equation was the starting point for deriving a Fuchsian formulation of the relativistic Euler equations. To complete the Fuchsian formulation, (1.59) was then spatially differentiated, see (2.9) below, to obtain a evolution equation for

$$\begin{aligned} \bar{W}{}\!_J:= t^\mu \partial _{J}W = (t^\mu \partial _{J}\tilde{\zeta }{},t^\mu \partial _{J}w_1,t^{\mu }\partial _{J}w_2,t^{\mu } \partial _{J} w_3)^{\textrm{tr}}. \end{aligned}$$

(1.60)

This variable definition was employed in [20] to express (1.57) as an ODE of the form

(1.61)

The analysis carried out in [20] identified the red term in (1.61) as the only potentially problematic singular term in the evolution system for the variables $\bar{W}{}$ and $\bar{W}{}_J$. This term proved to limit the range of K since the arguments employed in [20] required that $0<\mu <1$ in order to establish stability. Recalling (1.10), this lead to the restriction $1/3<K<1/2$.

As we show in Sect. 2, the improvements in this article arise from handling the red term in (1.61) differently. Namely, we replace it in (1.61) using

$$\begin{aligned} t^{-\mu }\Pi ^\perp \mathcal {A}{}^I \Pi ^\perp \bar{W}{}_I = \Pi ^\perp \mathcal {A}{}^I \Pi ^\perp \partial _{I}\bar{W}{}\end{aligned}$$

to get

$$\begin{aligned} \partial _{t}\bar{W}{}+\Pi ^\perp \mathcal {A}{}^I \Pi ^\perp \partial _{I}\bar{W}{}=-\Pi \mathcal {A}{}^I \bar{W}{}_I- t^{-\mu }\Pi ^\perp \mathcal {A}{}^I \Pi \bar{W}{}_I + t^{2\mu -1}\Pi ^\perp \mathcal {G}{}, \end{aligned}$$

which is now a PDE for $\bar{W}{}$. The benefit of using this equation is twofold. First, it is symmetrizable and so it can be used to obtain energy estimates. Second, the problematic singular term no longer appears and because of this we are able to obtain stability for the full parameter range $1/3<K<1$ or equivalently $0<\mu <\infty$.

1.3 Instability for $1/3< K< 1$

It is essential for the stability result stated in Theorem 1.2 to hold that the initial data used to generated the nonlinear perturbations of homogeneous solutions of the type (1.9) satisfies (1.8). This leaves the question of what happens when this condition is violated, which would be guaranteed to happen for some choice of initial data from any given open set of initial data that contains initial data corresponding to an isotropic homogeneous solution (1.5). To investigate this situation, we consider a $\mathbb {T}{}^2$-symmetric reduction of the system (1.15) obtained by the ansatz

$$\begin{aligned} \tilde{\zeta }(t,x^1,x^2,x^3)&=\texttt{z}{}(t,x^1), \end{aligned}$$

(1.62)

$$\begin{aligned} v_{I}(t,x^1,x^2,x^3)&= t^{-\mu }\texttt{w}{}(t,x^1) \delta _I^1, \end{aligned}$$

(1.63)

where $\tilde{\zeta }{}$ is as defined above by (1.25). It is not difficult to verify via a straightforward calculation that the relativistic Euler equations (1.15) will be satisfied provided that $\texttt{z}{}$ and $\texttt{w}{}$ solve⁹

$$\begin{aligned}&\partial _{t}\texttt{z}{}-\frac{\texttt{w}{}}{(t^{2 \mu }+\texttt{w}{}^2)^{\frac{1}{2}}} \partial _{x}\texttt{z}{}- \frac{t^{2\mu }}{(t^{2 \mu }+\texttt{w}{}^2)^{\frac{3}{2}}}\partial _{x}\texttt{w}{}=0, \end{aligned}$$

(1.64)

$$\begin{aligned}&\partial _{t}\texttt{w}{}-\frac{Kt^{2 \mu } (t^{2 \mu }+\texttt{w}{}^2)^{\frac{1}{2}}}{(t^{2 \mu }-(K-1)\texttt{w}{}^2)}\partial _{x}\texttt{z}{}+\frac{\bigl ((2 K-1) t^{2 \mu }+(K-1) \texttt{w}{}^2\bigr )\texttt{w}{}}{(t^{2 \mu }+\texttt{w}{}^2)^{\frac{1}{2}} (t^{2 \mu }-(K-1)\texttt{w}{}^2)}\partial _{x}\texttt{w}{}\nonumber \\&=\frac{t^{2 \mu -1}(-3 K+\mu +1) \texttt{w}{}}{t^{2 \mu }-(K-1)\texttt{w}{}^2} . \end{aligned}$$

(1.65)

In Sect. 3, we numerically solve this system for specific choices of initial data

$$\begin{aligned} (\texttt{z}{},\texttt{w}{})|_{t={t_0}} = (\texttt{z}{}_0,\texttt{w}{}_0) \quad \text {in} \mathbb {T}{}^1. \end{aligned}$$

Importantly, these choices include initial data for which $\texttt{w}{}_0$ crosses zero at two points in $\mathbb {T}{}^1$, and as a consequence, violates (1.8). From our numerical solutions, we observe the following behaviour:

• (1) For all $K\in (1/3,1)$ and all choices of initial data $(\texttt{z}{}_0,\texttt{w}{}_0)$ that are sufficiently close to homogeneous initial data of either family of solutions (1.5) and (1.9), $\texttt{z}{}$ and $\texttt{w}{}$ remain bounded and converge pointwise as $t\searrow 0$.

• (2) For each $K\in (1/3,1)$ and each choice of initial data $(\texttt{z}{}_0,\texttt{w}{}_0)$ that violates (1.8) and is sufficiently close to homogeneous initial data of the family of solutions (1.5), there exists a $\ell =\ell (K)\in \mathbb {Z}{}_{\ge 0}$ such that

  $$\begin{aligned} \sup _{x\in \mathbb {T}{}^1}\bigl (|\partial _{x}^{\ell } \texttt{z}{}(t,x)|+|\partial _{x}^{\ell }\texttt{w}{}(t,x)|\bigr ) \nearrow \infty \quad \text {as} t\searrow 0. \end{aligned}$$

  This indicates an instability in the $H^\ell$-spaces for solutions of (1.64)–(1.65) that is not present, c.f. Theorem 1.2, in solutions generated from initial data satisfying (1.8). We also observe that the integer $\ell$ is a monotonically decreasing function of K with a minimum value of 1. For the initial data we tested, the blow-up at $t=0$ in the derivatives occurs at a finite set of spatial points.

• (3) For all $K\in (1/3,1)$ and all choices of initial data $(\texttt{z}{}_0,\texttt{w}{}_0)$ that are sufficiently close to homogeneous initial data of either family of solutions (1.5) and (1.9), solutions to (1.64)–(1.65) are approximated remarkably well, for times sufficiently close to zero, by solutions to the asymptotic system¹⁰

  $$\begin{aligned} \partial _{t}\tilde{\texttt{z}{}}{}&=0, \end{aligned}$$

  (1.66)
  $$\begin{aligned} \partial _{t}\tilde{\texttt{w}{}}{}&=\frac{t^{2 \mu -1}(-3 K+\mu +1) \tilde{\texttt{w}{}}{}}{t^{2 \mu }-(K-1)\tilde{\texttt{w}{}}{}^2}, \end{aligned}$$

  (1.67)

  everywhere except, possibly, at a finite set of points where steep gradients form in z, which only happens for K large enough and initial data violating (1.8).

• (4) For each $K\in (1/3,1)$ and each choice of initial data $(\texttt{z}{}_0,\texttt{w}{}_0)$ that violates (1.8) and is sufficiently close to homogeneous initial data of the family of solutions (1.5), the density gradient $\frac{\partial _{x}\rho }{\rho }$ develops steep gradients near a finite number of spatial points where it becomes unbounded as $t\searrow 0$. This behaviour was anticipated by Rendall in [22], and it of physical interest because is not the expected behavior for cosmological solutions on spacetimes with a positive cosmological constant (i.e. exponentially expanding) [1, §V].

1.3.1 Coupling to gravity

The most natural and physically relevant extension of the numerical results developed in this article is to consider non-linear perturbations of the FLRW fluid solutions to the Einstein–Euler equations. This would allow one to determine whether the instability observed here, in the fixed background case, persists when coupling to the gravitational field is included. The authors of the current paper along with a collaborator have studied this situation in [1] where the Einstein–Euler equations were numerically evolved under a Gowdy-symmetry assumption. Specifically, numerical solutions were constructed globally to the future for small non-linear perturbations of FLRW initial data in which the spatial velocity of the fluid vanished at a finite number of points on the initial hypersurface. The two main conclusions of this work are as follows:

• For all $K \in (1/3, 1)$ and all choices of initial data that are sufficiently close to FLRW initial data, the numerical solutions are dominated by ODE behaviour near future timelike infinity and are well-approximated by an asymptotic system analogous to (1.66)–(1.67).

• The density gradient blows up at finitely many spatial points as $t \searrow 0$ for all $K \in (1/3, 1)$ and all choices of initial data that are sufficiently close to FLRW initial data and for which the spatial fluid velocity crosses zero somewhere on the initial hypersurface.

These conclusions confirm that the instability observed in this article, see Sect. 3, persists when coupling to the gravitational field is included. In particular, the numerical results of [1] validate the density gradient blow-up scenario conjectured by Rendall in [22]. The next step to take with this line of research would be to remove the symmetry assumptions in [1] and study the full 3+1 system.

1.4 Stability/instability for $K=1$

When the sound speed is equal to the speed of light, i.e. $K=1$, it is well known that the irrotational relativistic Euler equations coincide, under a change of variables, with the linear wave equation. Even though the future global existence of solutions to linear wave equations on exponentially expanding FLRW spacetimes can be inferred from standard existence results for linear wave equations, a corresponding future global existence result for the irrotational relativistic Euler equations does not automatically follow. This is because the change of variables needed to interpret a wave solution as a solution of the relativistic Euler equations requires the gradient of the wave solution to be timelike. Thus an instability in the irrotational relativistic Euler equations can still occur for $K=1$ if the gradient of the wave solution starts out timelike but becomes spacelike somewhere in finite time. This phenomena was shown in [6] to occur in the more difficult case where coupling to Einstein’s equations with a positive cosmological constant was taken into account. In fact, it was shown in [6] that all wave solutions generated from initial data sets that correspond to a sufficiently small perturbation of the FLRW fluid solution (i.e. (1.5) in our setting) become spacelike in finite time. This proves that the self-gravitating version of the isotropic homogeneous (1.5) are unstable, and in the irrotational setting at least, characterizes the cause of the instability. What is not known is if the other family of homogeneous solutions (1.9) or their self-gravitating versions remain stable for $K=1$.

1.5 Future directions

The most natural and physically relevant generalization of the stability result stated in Theorem 1.2 would be an analogous stability result for the coupled Einstein–Euler equations with a positive cosmological constant for K satisfying $1/3<K<1$. We expect that establishing this type of stability result is feasible by adapting the arguments from [18]. This expectation is due to the behaviour of the term $t^{-2}\rho v_i v_j$, which is the only potentially problematic term that could, if it grew too quickly as $t\searrow 0$, prevent the use of the arguments from [18]. However, by Theorem 1.2, we know that $\rho ={{\,\textrm{O}\,}}\bigl (t^{\frac{2(1+K)}{1-K}}\bigr )$ and $v_i = {{\,\textrm{O}\,}}\bigl (t^{\frac{1-3 K}{1-K}}\bigr )$ from which it follows that $t^{-2}\rho v_i v_j = {{\,\textrm{O}\,}}(t^2)$. This shows that $t^{-2}\rho v_i v_j$ decays quickly enough as $t\searrow 0$ to suggest that it will not be problematic. We are currently working on generalizing Theorem 1.2 to include coupling to Einstein’s equations with a positive cosmological constant, and we will report on any progress in this direction in a follow-up article.

2 Proof of Theorem 1.2

2.1 Step 1: Fuchsian formulation

To begin the derivation of the Fuchsian formulation of the relativistic Euler equations that we will use to establish stability, we decompose the term $\Pi ^\perp \mathcal {A}{}^I \partial _{I}W$ in (1.57) as

$$\begin{aligned} \Pi ^\perp \mathcal {A}{}^I \partial _{I}W=\Pi ^\perp \mathcal {A}{}^I \Pi ^\perp \partial _{I}( \Pi ^\perp W) + t^{-\mu }\Pi ^\perp \mathcal {A}{}^I \Pi t^{\mu }\partial _{I} W. \end{aligned}$$

Inserting this into (1.57) and multiplying the resulting equation on the left by $\Pi ^\perp A^0 \Pi ^\perp$ gives

$$\begin{aligned}{} & {} \Pi ^\perp A^0 \Pi ^\perp \partial _{t}(\Pi ^\perp W) +\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi ^\perp \partial _{I}( \Pi ^\perp W) + t^{-\mu }\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi t^{\mu }\partial _{I} W\nonumber \\{} & {} \quad = t^{2\mu -1}\Pi ^\perp A^0\Pi ^\perp \mathcal {G}{}. \end{aligned}$$

(2.1)

Then multiplying (2.1) by

$$\begin{aligned} S = \begin{pmatrix} \frac{e^{-2\breve{w}{}_1}\psi ^2}{\chi } &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{\psi ^2}{t^{2 \mu } e^{2 \breve{w}{}_1}-(K-1) e^{4 \breve{w}{}_1}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \end{pmatrix} \end{aligned}$$

(2.2)

and adding the resulting equation to (1.56) yields

$$\begin{aligned}&\partial _{t}(t^{\mu }\Pi W)+S\Pi ^\perp A^0 \Pi ^\perp \partial _{t}(\Pi ^\perp W)\\&+S\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi ^\perp \partial _{I}( \Pi ^\perp W) =-\Pi \mathcal {A}{}^I t^{\mu }\partial _{I}W\\&- t^{-\mu }S\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi t^{\mu }\partial _{I} W + t^{2\mu -1}S\Pi ^\perp A^0\Pi ^\perp \mathcal {G}{}. \end{aligned}$$

Recalling the definition (1.58) of $\bar{W}{}$ above, it is then straightforward to verify that the above equation can be expressed as

$$\begin{aligned}{} & {} B^0\partial _{t}\bar{W}{}+ B^I \partial _{I}\bar{W}{}\nonumber \\{} & {} \quad = -\Pi \mathcal {A}{}^I t^{\mu }\partial _{I}W - t^{-\mu }S\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi t^{\mu }\partial _{I} W + t^{2\mu -1}S\Pi ^\perp A^0\Pi ^\perp \mathcal {G}{}\nonumber \\ \end{aligned}$$

(2.3)

where

$$\begin{aligned} B^0 =S\Pi ^\perp A^0 \Pi ^\perp + \Pi {\quad \text {and}\quad }B^I =S\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi ^\perp . \end{aligned}$$

(2.4)

Noting from (1.37)–(1.39) that

$$\begin{aligned} \Pi ^\perp \mathcal {A}{}^I \Pi ^\perp = \frac{b^I}{\sqrt{\frac{t^{2\mu }}{e^{2\tilde{w}{}_1}}+1}}\begin{pmatrix} -\frac{1}{\sqrt{\phi }} &{} -\frac{t^{2 \mu }}{\psi \sqrt{\phi }} &{}0 &{}0\\ -\frac{K t^{2 \mu } e^{-2 \breve{w}{}_1} \psi }{\sqrt{\phi }\chi } &{} \frac{(2 K-1) t^{2 \mu }+(K-1) e^{2 \breve{w}{}_1}}{\sqrt{\phi } \chi } &{} 0 &{} 0\\ 0 &{} 0 &{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0 \end{pmatrix} \end{aligned}$$

where

$$\begin{aligned} b^1 = 1, \quad b^2 = t^{\mu }(w_3+w_2) {\quad \text {and}\quad }b^3 = t^\mu (w_2-w_3), \end{aligned}$$

(2.5)

a short calculation using (1.41)–(1.42), (1.44), and (2.2) shows that the matrices (2.4) are given by

$$\begin{aligned} B^0 = \begin{pmatrix} \frac{ K e^{-2\breve{w}{}_1}\psi ^2}{\chi } &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \end{aligned}$$

(2.6)

and

$$\begin{aligned} B^I = \frac{b^I}{\sqrt{\frac{t^{2\mu }}{e^{2\tilde{w}{}_1}}+1}}\begin{pmatrix} -\frac{K e^{-2\breve{w}{}_1}\psi ^2}{\chi \sqrt{\phi }} &{} -\frac{K t^{2 \mu }e^{-2\breve{w}{}_1}\psi }{\sqrt{\phi }\chi } &{}0 &{}0\\ -\frac{K t^{2 \mu } e^{-2 \breve{w}{}_1} \psi }{\sqrt{\phi }\chi } &{} \frac{(2 K-1) t^{2 \mu }+(K-1) e^{2 \breve{w}{}_1}}{\sqrt{\phi } \chi } &{} 0 &{} 0\\ 0 &{} 0 &{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0 \end{pmatrix}. \end{aligned}$$

(2.7)

From these formulas, it is clear that the matrices $B^i$ are symmetric, that is,

$$\begin{aligned} (B^i)^{\textrm{tr}}=B^i. \end{aligned}$$

(2.8)

We proceed by differentiating (1.36) spatially to get

$$\begin{aligned} \partial _{t}\partial _{J}W + \mathcal {A}{}^I \partial _{I}\partial _{J}W + \partial _{J}\mathcal {A}{}^I \partial _{I}W = -\frac{\mu }{t}\Pi \partial _{J}W + t^{2\mu -1}\partial _{J}\mathcal {G}{}. \end{aligned}$$

Using (1.60), we can write this as

$$\begin{aligned} \partial _{t}\bar{W}{}\!_J + \mathcal {A}{}^I \partial _{I}\bar{W}{}\!_J + \partial _{J}\mathcal {A}{}^I \bar{W}{}_I = \frac{\mu }{t}\Pi ^\perp \bar{W}{}\!_J + t^{3\mu -1}\partial _{J}\mathcal {G}{}. \end{aligned}$$

(2.9)

Multiplying the above equation on the left by $A^0$ and recalling the definitions (1.46), we find that $\bar{W}{}\!_J$ satisfies

$$\begin{aligned} A^0\partial _{t}\bar{W}{}\!_J + A^I \partial _{I}\bar{W}{}\!_J = \frac{\mu }{t}A^0\Pi ^\perp \bar{W}{}\!_J + t^{3\mu -1}A^0\partial _{J}\mathcal {G}{}- A^0\partial _{J}\mathcal {A}{}^I \bar{W}{}_I. \end{aligned}$$

(2.10)

Finally, combining (2.3) and (2.10) yields the Fuchsian system

$$\begin{aligned} \mathscr {A}{}^0\partial _{t}\mathscr {W}{}+ \mathscr {A}{}^I \partial _{I}\mathscr {W}{}= \frac{\mu }{t}\mathscr {A}{}^0\mathbb {P}{}\mathscr {W}{}+ \mathscr {F}{}\end{aligned}$$

(2.11)

where

$$\begin{aligned} \mathscr {W}{}&= \begin{pmatrix} \bar{W}{}\\ \bar{W}{}\!_J \end{pmatrix}, \end{aligned}$$

(2.12)

$$\begin{aligned} \mathscr {A}{}^0&= \begin{pmatrix} B^0 &{} 0 \\ 0 &{} A^0 \end{pmatrix}, \end{aligned}$$

(2.13)

$$\begin{aligned} \mathscr {A}{}^I&= \begin{pmatrix} B^I &{} 0 \\ 0 &{} A^I \end{pmatrix}, \end{aligned}$$

(2.14)

$$\begin{aligned} \mathbb {P}{}&= \begin{pmatrix} 0 &{} 0 \\ 0 &{} \Pi ^\perp \end{pmatrix}, \end{aligned}$$

(2.15)

$$\begin{aligned}&\text {and}\nonumber \\ \mathscr {F}{}&=\begin{pmatrix} -\Pi \mathcal {A}{}^I \bar{W}{}_I - t^{-\mu }S\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi \bar{W}{}_I + t^{2\mu -1}S\Pi ^\perp A^0\Pi ^\perp \mathcal {G}{}\\ t^{3\mu -1}A^0\partial _{J}\mathcal {G}{}- A^0\partial _{J}\mathcal {A}{}^I \bar{W}{}_I\end{pmatrix}. \end{aligned}$$

(2.16)

As will be established in Step 2 below, the Fuchsian system (2.11) satisfies assumption needed to apply the Fuchsian global existence theory from [3]; see, in particular, [3, Thm. 3.8.] and [3, §3.4.]. This global existence theory will be used in Step 3 of the proof to establish uniform bounds on solutions to the initial value problem (1.48)–(1.49) under a suitable small initial data assumption. These bounds in conjunction with a continuation principle will then yield the existence solutions to (1.48)–(1.49) on $(0,1]\times \mathbb {T}{}^3$ as well as decay estimates as $t\searrow 0$.

2.2 Step 2: Verification of the coefficient assumptions

In order to apply Theorem 3.8. from [3], see also [3, §3.4.], to the Fuchsian system (2.11), we need to verify that the coefficients of this equations satisfy the assumptions from Section 3.4. of [3], see also [3, §3.1.]. To begin the verification, we set

$$\begin{aligned} \bar{t}{}= t^{2\mu }, {\quad \text {and}\quad }\bar{w}{}_\Lambda = t^\mu w_\Lambda , \quad \Lambda =2,3, \end{aligned}$$

(2.17)

and observe from (1.29)–(1.32), (1.44), (2.6) and (2.13) that the matrix $\mathscr {A}{}^0$ can be treated as a map depending on the variables (2.17), that is,

$$\begin{aligned} \mathscr {A}{}^0 = \mathscr {A}{}^0(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3), \end{aligned}$$

(2.18)

where for each $R>0$ there exists constants $r,\omega >0$ such that $\mathscr {A}{}^0$ is smooth on the domain defined by

$$\begin{aligned} (\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) \in (-r,2) \times (-R,R) \times (-R,R)\times (-R,R), \end{aligned}$$

(2.19)

and satisfies

$$\begin{aligned} \mathscr {A}{}^0(\bar{t}{},\breve{w}{}_1,0,0) \ge \omega \mathord {{\mathrm 1}\hspace{-2.70004pt}{\mathrm I}}\hspace{3.50006pt}\end{aligned}$$

(2.20)

for all $(\bar{t}{},\breve{w}{}_1)\in (-r,2)\times (-R,R)$. In the following, we will always be able to choose $R>0$ and $r>0$ as needed in order to guarantee that the statements we make are valid.

Differentiating $\mathscr {A}{}^0$ with respect to t then shows, with the help of (1.29), (1.58) and (2.17)–(2.18), that

$$\begin{aligned} \partial _{t}\mathscr {A}{}^0&= D\mathscr {A}{}^0(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) \begin{pmatrix} 2\mu t^{2\mu -1} \\ u'(t)+\partial _{t}w_1\\ \partial _{t}\bar{w}{}_2\\ \partial _{t}\bar{w}{}_3 \end{pmatrix}\nonumber \\&= D\mathscr {A}{}^0(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) \left( \begin{pmatrix} 2\mu t^{2\mu -1} \\ u'(t)\\ 0 \\ 0 \end{pmatrix} + \mathcal {P}{}_1\partial _{t}\bar{W}{}\right) \end{aligned}$$

(2.21)

where

$$\begin{aligned} \mathcal {P}{}_1 = {\textrm{diag}}(0,1,1,1), \end{aligned}$$

and $\partial _{t}\bar{W}{}$ can be computed from from (2.3), that is,

$$\begin{aligned} \partial _{t}\bar{W}{}=(B^0)^{-1}\Bigl (- B^I \partial _{I}\bar{W}{}-\Pi \mathcal {A}{}^I \bar{W}{}_I - t^{-\mu }S\Pi ^\perp A^0 \Pi ^\perp \mathcal {A}{}^I \Pi \bar{W}{}_I + t^{2\mu -1}S\Pi ^\perp A^0\Pi ^\perp \mathcal {G}{}\Bigr ).\nonumber \\ \end{aligned}$$

(2.22)

We note from (1.29)–(1.31), (2.2), (2.6) and (2.17) that the matrices

$$\begin{aligned} S = S(\bar{t}{},\breve{w}{}_1) {\quad \text {and}\quad }B^0 = B^0(\bar{t}{},\breve{w}{}_1) \end{aligned}$$

(2.23)

are smooth on the domain $(\bar{t}{},\breve{w}{}_1)\in (-r,2)\times (-R,R)$, and that $B^0$ is bounded below by

$$\begin{aligned} B^0 \ge \omega \mathord {{\mathrm 1}\hspace{-2.70004pt}{\mathrm I}}\hspace{3.50006pt}\end{aligned}$$

(2.24)

for all $(\bar{t}{},\breve{w}{}_1)\in (-r,2)\times (-R,R)$ where $\omega$ can be taken as the same constant as in (2.20). We further note from (1.29)–(1.32), (1.44), (1.46), (2.7), (2.5) and (2.17) that the matrices

$$\begin{aligned} A^i= A^i(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) {\quad \text {and}\quad }B^I= B^I(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) \end{aligned}$$

(2.25)

are smooth on the domain (2.19), while is clear from (1.40) that the vector-valued map

$$\begin{aligned} \mathcal {G}{}= \mathcal {G}{}(\bar{t}{},\breve{w}{}_1,w_1) \end{aligned}$$

(2.26)

is smooth on the domain $(\bar{t}{},\breve{w}{}_1,w_1)\in (-r,2)\times (-R,R) \times (-R,R)$.

Next, setting

$$\begin{aligned} \hat{w}{}_1 = t^\mu e^{-2 \breve{w}{}_1}, \end{aligned}$$

(2.27)

it follows from (1.29)–(1.34), (1.37)–(1.39) and (2.17) that the matrices $\mathcal {A}{}^I$ can be expanded as

$$\begin{aligned} \mathcal {A}{}^I = \mathcal {A}{}^I_1(\hat{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)+ t^\mu \mathcal {A}{}^I_2(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)+ t^{2\mu } \mathcal {A}{}^I_3(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) \end{aligned}$$

(2.28)

where the $\mathcal {A}{}^I_2$, $\tilde{A}{}^I_3$ are smooth on the domain (2.19) and the $\mathcal {A}{}^I_1$ are smooth on the domain defined by

$$\begin{aligned} (\hat{w}{}_1,\bar{w}{}_2,\bar{w}{}_3) \in (-R,R)\times (-R,R)\times (-R,R). \end{aligned}$$

It is also not difficult to verify from (1.37)–(1.39) that the $\mathcal {A}{}^I_1$ satisfy

$$\begin{aligned} \Pi ^\perp \mathcal {A}{}^I_1 \Pi = 0. \end{aligned}$$

(2.29)

Differentiating the matrices $\mathcal {A}{}^I$ spatially, we have by (1.29), (1.60), (2.17), (2.27) and (2.28) that

$$\begin{aligned} \partial _{J}\mathcal {A}{}^I&= D\mathcal {A}{}^I_1(\hat{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)\begin{pmatrix}-2 e^{-2 \breve{w}{}_1} t^\mu \partial _{J}w_1 \\ t^\mu \partial _{J}w_2 \\ t^\mu \partial _{J}w_2 \end{pmatrix}\nonumber \\&\quad + t^\mu D\mathcal {A}{}^I_2(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)\begin{pmatrix} 0 \\ \partial _{J}w_1\\ t^\mu \partial _{J}w_2 \\ t^\mu \partial _{J} w_3 \end{pmatrix} +t^{2\mu } D\mathcal {A}{}^I_3(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)\begin{pmatrix} 0 \\ \partial _{J}w_1\\ t^\mu \partial _{J}w_2 \\ t^\mu \partial _{J} w_3 \end{pmatrix} \nonumber \\&= \Bigl (D\mathcal {A}{}^I_1(\hat{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)\mathcal {P}{}_2 + D\mathcal {A}{}^I_2(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)\mathcal {P}{}_3+t^\mu D\mathcal {A}{}^I_2(\bar{t}{},\breve{w}{}_1,\bar{w}{}_2,\bar{w}{}_3)\mathcal {P}{}_3\Bigr ) \bar{W}{}\!_J, \end{aligned}$$

(2.30)

where

$$\begin{aligned} \mathcal {P}{}_2 = \begin{pmatrix} 0 &{} -2 e^{-2 \breve{w}{}_1} &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} {\quad \text {and}\quad }\mathcal {P}{}_3 = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} t^\mu &{} 0 \\ 0 &{} 0 &{} 0 &{} t^\mu \end{pmatrix}. \end{aligned}$$

By (1.41)–(1.42) and (1.44), we note that the matrix $A^0$ satisfies $[\Pi ^\perp ,A^0] = 0$ and $\Pi ^\perp A^0 \Pi = \Pi A^0 \Pi ^\perp = 0$. Using these identities, it is then follows from the definitions (2.13) and (2.15) that $\mathscr {A}{}^0$ satisfies

$$\begin{aligned}{}[\mathbb {P}{},\mathscr {A}{}^0] = 0 \end{aligned}$$

(2.31)

and

$$\begin{aligned} \mathbb {P}{}^\perp \mathscr {A}{}^0 \mathbb {P}{}= \mathbb {P}{}\mathscr {A}{}^0 \mathbb {P}{}^\perp = 0, \end{aligned}$$

(2.32)

where

$$\begin{aligned} \mathbb {P}{}^\perp = \mathord {{\mathrm 1}\hspace{-2.70004pt}{\mathrm I}}\hspace{3.50006pt}-\mathbb {P}{}. \end{aligned}$$

(2.33)

Additionally, by (1.41)–(1.43), we observe that $\mathbb {P}{}$ satisfies

$$\begin{aligned} \mathbb {P}{}^2 = \mathbb {P}{}, \quad \mathbb {P}{}^{\textrm{tr}} = \mathbb {P}{}, \quad \partial _{t}\mathbb {P}{}= 0 {\quad \text {and}\quad }\partial _{I} \mathbb {P}{}= 0, \end{aligned}$$

(2.34)

while the symmetry of the matrices $\mathscr {A}{}^i$, that is,

$$\begin{aligned} (\mathscr {A}{}^i)^{\textrm{tr}} = \mathscr {A}{}^i, \end{aligned}$$

(2.35)

is obvious from the definitions (1.44) and (2.13)–(2.14), and the relations (1.47) and (2.8).

Now, from the definitions (1.29), (1.58), (1.60), (2.12), (2.17) and (2.27), the formulas (2.21) and (2.22), the estimates (1.13) for u(t) and $u'(t)$, the smoothness properties (2.18), (2.23), (2.25), (2.26) and (2.28) of the matrices $\mathscr {A}{}^0$, S, $A^0$, $B^0$, $A^I$, $B^I$, $\mathcal {A}{}^I$ and the source term $\mathcal {G}{}$, the lower bound (2.24) on $B^0$, and the identity (2.29), it is not difficult to verify that for each $\mu \in (0,\infty )$ that there exists a constant $\theta >0$ such that

$$\begin{aligned} |\partial _{t}\mathscr {A}{}^0| \le \theta (t^{2\mu -1}+1) \end{aligned}$$

(2.36)

for all $(t,\mathscr {W}{},D\mathscr {W}{})\in [0,1]\times B_R(\mathbb {R}{}^{16})\times B_{R}(\mathbb {R}{}^{16\times 3})$, where $D\mathscr {W}{}= (\partial _{I}\mathscr {W}{})$. From (2.16) and similar considerations, it is also not difficult to verify

$$\begin{aligned} |\mathscr {F}{}| \lesssim (t^{2\mu -1}+1)|\mathscr {W}{}| \end{aligned}$$

(2.37)

for all $(t,\mathscr {W}{})\in [0,1]\times B_R(\mathbb {R}{}^{16})$. It is also clear that we can view (2.11) as an equation for the variables $\mathscr {W}{}=(\bar{W}{},\bar{W}{}_J)$, with $\bar{W}{}=(\tilde{\zeta }{},w_1,\bar{w}{}_2,\bar{w}{}_3)$ and $\bar{W}{}_J=(\tilde{\zeta }{}_J, w_{1J},\bar{w}{}_{2J},\bar{w}{}_{3J})$, where the maps $\mathscr {A}{}^i$ and $\mathscr {F}{}$ depend on the variables $(t,\bar{W}{})$ and $(t,\mathscr {W}{})$, respectively.

Taken together, (i) the variable definitions (1.29), (1.58), (1.60), (2.17) and (2.27), (ii) the smoothness properties (2.18), (2.23), (2.25), (2.26) and (2.28) of the matrices $\mathscr {A}{}^0$, S, $A^0$, $B^0$, $A^I$, $B^I$, $\mathcal {A}{}^I$ and the source term $\mathcal {G}{}$, (iii) the identities (2.31)–(2.32) and the lower bound (2.20) satisfied by matrix $\mathscr {A}{}^0$, (iv) the definitions (2.14) and (2.16) of the matrices $\mathscr {A}{}^I$ and the source term $\mathscr {F}{}$, (v) the properties (2.34) of the projection map $\mathbb {P}{}$, and (vi) the bounds (2.36) and (2.37) on $\partial _{t}\mathscr {A}{}^0$ and $\mathscr {F}{}$, respectively, imply that for any¹¹$\mu \in (0,\infty )$ and $R>0$ chosen sufficiently small, there exist constants $\theta ,\gamma _1=\tilde{\gamma }{}_1,\gamma _2=\tilde{\gamma }{}_2>0$ such that the Fuchsian system (2.11) satisfies satisfies all the assumptions from Section 3.4 of [3] for following choice of constants: $\kappa =\tilde{\kappa }{}=\mu$, $\beta _\ell =0$, $1\le \ell \le 7$,

$$\begin{aligned} \quad p={\left\{ \begin{array}{ll}2\mu &{} \hbox { if}\ 0<\mu \le 1/2\\ 1 &{} \hbox { if}\ \mu > 1 \end{array}\right. } \end{aligned}$$

and $\lambda _1=\lambda _2=\lambda _3= \alpha =0$. As discussed in [3, §3.4], under the time transformation¹²$t \mapsto t^p$, the transformed version of (2.11) will satisfy all of the assumptions from Section 3.1 of [3]. Moreover, since the matrices $\mathscr {A}{}^I$ have a regular limit as $t\searrow 0$, the constants $\texttt{b}{}$ and $\tilde{\texttt{b}{}}$ from Theorem 3.8 of [3] vanish. This fact together with $\beta _1=0$ and $\kappa =\tilde{\kappa }{}=\mu$ implies that the constant¹³${\mathfrak {z}}$ from Theorem 3.8 of [3] that determines the decay rate is given by ${\mathfrak {z}}= \mu$.

2.3 Step 3: Existence and uniqueness

By (1.44) and (1.47), we know that the matrices $A^i$ are symmetric. Furthermore, from the analysis carried out in Step 2 above, we know that the matrices $A^i$ and the source term $A^0 \mathscr {G}{}$ depend smoothly on the variables $(t,w_J)$ for $t\in (0,1]$ and $w_J$ in an open neighbourhood of zero, and that the matrix $A^0$ is positive definite on this neighbourhood. As a consequence, the system (1.48) is symmetrizable and can be put in the symmetric hyperbolic form (1.45) by multiplying it on the left by the matrix $A^0$. Since $k\in \mathbb {Z}{}_{>3/2+1}$ and $W_0:=(\tilde{\zeta }{}_0, w^0_J)^{\textrm{tr}}\in H^{k+1}(\mathbb {T}{}^3,\mathbb {R}{}^4)$, we obtain from an application of standard local-in-time existence and uniqueness theorems and the continuation principle for symmetric hyperbolic systems, see Propositions 1.4, 1.5 and 2.1 from [29, Ch. 16], the existence of a unique solution

$$\begin{aligned} W=(\tilde{\zeta }{},w_J) \in C^0\bigl ((T_*,1], H^{k+1}(\mathbb {T}{}^3,\mathbb {R}{}^4)\bigr )\cap C^1\bigl ((T_*,1],H^{k}(\mathbb {T}{}^3,\mathbb {R}{}^4)\bigr ) \end{aligned}$$

to IVP (1.48)–(1.49) where $T_*\in [0,1)$ is the maximal time of existence. From the computations carried out in Step 1 of the proof, this solution determines via (1.58) and (1.60) a solution

$$\begin{aligned} \mathscr {W}{}= (\bar{W}{},\bar{W}{}\!_J) \in C^0\bigl ((T_*,1], H^{k}(\mathbb {T}{}^3,\mathbb {R}{}^{16})\bigr )\cap C^1\bigl ((T_*,1],H^{k-1}(\mathbb {T}{}^3,\mathbb {R}{}^{16})\bigr )\nonumber \\ \end{aligned}$$

(2.38)

of the IVP

$$\begin{aligned} \mathscr {A}{}^0\partial _{t}\mathscr {W}{}+ \mathscr {A}{}^I \partial _{I}\mathscr {W}{}&= \frac{\mu }{t}\mathscr {A}{}^0\mathbb {P}{}\mathscr {W}{}+ \mathscr {F}{}\quad \text {in} (T_*,1]\times \mathbb {T}{}^3, \end{aligned}$$

(2.39)

$$\begin{aligned} \mathscr {W}{}&= \mathscr {W}{}_0 := (W_0,\partial _{J}W_0) \quad \text {in} \{1\}\times \mathbb {T}{}^3, \end{aligned}$$

(2.40)

where we observe that

$$\begin{aligned} \Vert \mathscr {W}{}_0\Vert _{H^k} \lesssim \Vert W_0\Vert _{H^{k+1}} \le \delta . \end{aligned}$$

(2.41)

On the other hand, by Step 2 we can apply¹⁴ Theorem 3.8. from [3] to the time transformed version of (2.11) as described in [3, Section 3.4] to deduce, for $\delta >0$ chosen small enough and the initial data satisfying (2.41), the existence of a unique solution

$$\begin{aligned}{} & {} \mathscr {W}{}^* \in C^0\bigl ((0,1],H^k(\mathbb {T}{}^3,\mathbb {R}{}^{16})\bigr )\cap L^\infty \bigl ((0,1],H^k(\mathbb {T}{}^3,\mathbb {R}{}^{16}))\bigr )\\{} & {} \quad \cap C^1\bigl ((0,1],H^{k-1}(\mathbb {T}{}^3,\mathbb {R}{}^{16})\bigr ) \end{aligned}$$

to the IVP (2.39)–(2.40) with $T_*=0$ that satisfies the following properties:

• (1) The limit of $\mathbb {P}{}^\perp \mathscr {W}{}^*$ as $t\searrow 0$, denoted $\mathbb {P}{}^\perp \mathscr {W}{}^*(0)$, exists in $H^{k-1}(\mathbb {T}{}^3,\mathbb {R}{}^{16})$.

• (2) The solution $\mathscr {W}{}^*$ is bounded by the energy estimate

  $$\begin{aligned} \Vert \mathscr {W}{}^*(t)\Vert _{H^k}^2 + \int _{t}^1 \frac{1}{\tau } \Vert \mathbb {P}{}\mathscr {W}{}^*(\tau )\Vert _{H^k}^2\, d\tau \lesssim \Vert \mathscr {W}{}_0\Vert _{H^k}^2 \end{aligned}$$

  (2.42)

  for all $t\in (0,1]$, where the implied constant depends on $\delta$.

• (3) For any given $\sigma >0$, the solution $\mathscr {W}{}^*$ satisfies the decay estimate

  $$\begin{aligned} \Vert \mathbb {P}{}\mathscr {W}{}^*(t)\Vert _{H^{k-1}} \lesssim t^{\mu -\sigma } {\quad \text {and}\quad }\Vert \mathbb {P}{}^\perp \mathscr {W}{}^*(t) - \mathbb {P}{}^\perp \mathscr {W}{}^*(0)\Vert _{H^{k-1}} \lesssim t^{\mu -\sigma } \end{aligned}$$

  (2.43)

  for all $t\in (0,1]$, where the implied constants depend on $\delta$ and $\sigma$.

By uniqueness, the two solutions $\mathscr {W}{}$ and $\mathscr {W}{}^*$ to the IVP (2.39)–(2.40) must coincide on their common domain of definition, and consequently, $\mathscr {W}{}(t)=\mathscr {W}{}^*(t)$ for all $t\in (T_*,1]$. But this implies by (2.38), the energy estimate (2.42), and Sobolev’s inequality [21, Thm. 6.2.1] that

$$\begin{aligned} \Vert \bar{W}{}(t)\Vert _{W^{1,\infty }} \lesssim \Vert \bar{W}{}(t)\Vert _{H^k} \le \Vert \mathscr {W}{}(t)\Vert _{H^{k-1}} \lesssim \Vert \mathscr {W}{}_0\Vert , \quad T^*<t\le 1. \end{aligned}$$

By shrinking $\delta$ if necessary, we can, by (2.41), make $\Vert \mathscr {W}{}_0\Vert _{H^k}$ as small as we like, which in turn, implies via the above estimate that we can bound $\bar{W}{}$ by $\Vert \bar{W}{}(t)\Vert _{W^{1,\infty }} \le \frac{R}{2}$ for all $t\in (T^*,1]$, where $R>0$ is as determined in Step 2 of the proof. This bound is sufficient to guarantee that the matrices $A^i$ and the source term $A^0\mathscr {G}{}$ from the symmetric hyperbolic system (1.45) remain well defined and that the matrix $A^0$ continues to be positive definite. By the continuation principle and the maximality of $T_*$, we deduce that $T_*=0$, and hence that $\mathscr {W}{}(t)=\mathscr {W}{}^*(t)$ for all $t\in (0,1]$. From this and the energy estimate (2.42), it then follows with the help of the definitions (1.41)–(1.42), (1.58), (1.60), (2.15) and (2.38) that

$$\begin{aligned} \mathcal {E}{}(t) + \int _t^1 \tau ^{2\mu -1}\bigl (\Vert D\tilde{\zeta }{}(\tau )\Vert _{H^k}^2+\Vert Dw_1(\tau )\Vert _{H^k}^2\bigr )\,d\tau \lesssim \Vert W_0\Vert _{H^k}^2, \quad 0<t\le 1, \end{aligned}$$

where

$$\begin{aligned}{} & {} \mathcal {E}{}(t)=\Vert \tilde{\zeta }{}(t)\Vert _{H^k}^2+\Vert w_1(t)\Vert _{H^k}^2+t^{2\mu }\Bigl (\Vert D\tilde{\zeta }{}(t)\Vert _{H^k}^2\\{} & {} \quad +\Vert Dw_1(t)\Vert _{H^k}^2+\Vert w_2(t)\Vert _{H^{k+1}}^2+\Vert w_3(t)\Vert _{H^{k+1}}^2\Bigr ). \end{aligned}$$

We further obtain from the decay estimate (2.43) and (2.33) the existence of functions $\tilde{\zeta }{}_*, w_1^* \in H^{k-1}(\mathbb {T}{}^3)$ and $\bar{w}{}_2^*,\bar{w}{}_3^* \in H^{k}(\mathbb {T}{}^3)$ such that

$$\begin{aligned} \bar{\mathcal {E}{}}(t) \lesssim t^{\mu -\sigma } \end{aligned}$$

(2.44)

for all $t\in (0,1]$ where

$$\begin{aligned}{} & {} \bar{\mathcal {E}{}}(t)=\Vert \tilde{\zeta }{}(t) - \tilde{\zeta }{}_*\Vert _{H^{k-1}}\\{} & {} +\Vert w_1(t) - w_1^*\Vert _{H^{k-1}} +\Vert t^\mu w_2(t) - \bar{w}{}_2^*\Vert _{H^{k}}+\Vert t^\mu w_3(t) - \bar{w}{}_3^*\Vert _{H^{k}}. \end{aligned}$$

We also note by (1.4), (1.14), (1.17) and (1.25)–(1.28) that u and $W=(\tilde{\zeta }{},w_J)^{\textrm{tr}}$ determine a solution of the relativistic Euler equations (1.1) on the spacetime region $M=(0,1]\times \mathbb {T}{}^3$ via the formulas (1.50)–(1.54).

To complete the proof, we find from differentiating (1.50) that the density gradient can be expressed as

$$\begin{aligned} \frac{\partial _{I}\rho }{\rho } = \frac{(1+K)(t^{2\mu }+ e^{2(u+w_1)})^{\frac{1+K}{2}}\partial _{I}\tilde{\zeta }{}-(1+K)(t^{2\mu }+ e^{2(u+w_1)})^{\frac{K-1}{2}}e^{2(u+w_1)}\partial _{I}w_1 }{(t^{2\mu }+ e^{2(u+w_1)})^{\frac{1+K}{2}}}.\nonumber \\ \end{aligned}$$

(2.45)

Since $\mu >0$, we can choose $\sigma >0$ small enough so that $\mu -\sigma >0$. Doing so then implies by (2.44) that $\tilde{\zeta }{}$ and $w_1$ converge in $H^{k-1}(\mathbb {T}{}^3)$ to $\tilde{\zeta }{}_*$ and $w_1^*$ as $t\searrow 0$. Since u(t) converges as well by Proposition 1.1, it is then not difficult to verify from (2.45) and the Sobolev and Moser inequalities [21, Thms. 6.2.1 & 6.4.1] that

$$\begin{aligned} \lim _{t\searrow 0} \Bigl \Vert \frac{\partial _{I}\rho }{\rho } - (1+K)\partial _{I}(\tilde{\zeta }{}_*-w_1^*) \Bigr \Vert _{H^{k-2}} = 0, \end{aligned}$$

which completes the proof.

3 Numerical solutions

3.1 Numerical setup

In the numerical setup that we use to solve the system (1.64)–(1.65), the computational domain is $[0,2\pi ]$ with periodic boundary condition, the variables $\texttt {z}$ and $\texttt {w}$ are discretised in space using 2nd order central finite differences, and time integration is performed using a standard 2nd order Runge–Kutta method (Heun’s Method). As a consequence, our code is second order accurate.¹⁵

3.1.1 Convergence tests

We have verified the second order accuracy of our code with convergence tests involving perturbations of both types of homogeneous solutions (1.5) and (1.9). In our convergence tests, we have evolved the system (1.64)–(1.65) staring from the the two initial data sets

$$\begin{aligned} (\texttt{z}{}_{0},\texttt{w}{}_{0})&= (0,0.1\sin (x))\end{aligned}$$

(3.1)

$$\begin{aligned} \text {and} (\texttt{z}{}_{0},\texttt{w}{}_{0})&= (0,0.1\sin (x)+0.15) \end{aligned}$$

(3.2)

using resolutions of $N =$ 200, 400, 800, 1600, 3200, and 6400 grid points. The initial data (3.1) and (3.2) satisfy the conditions (1.7) and (1.8), respectively, and the solutions generated from this initial data represent perturbations of the homogeneous solutions (1.5) and (1.9), respectively.

To estimate the error, we took the base 2 log of the absolute value of the difference between each simulation and the highest resolution run. The results for are shown in Figs. 1, 2, 3 and, 4 from which the second order convergence is clear.

Fig. 1

Convergence plots of w at various times. $K=0.5,(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x)+0.15)$

Fig. 2

Convergence plots of $\texttt{z}{}$ at various times. $K=0.5,(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x)+0.15)$

Fig. 3

Convergence plots of w at various times. $K=0.5,(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x))$

Fig. 4

Convergence plots of $\texttt{z}{}$ at various times. $K=0.5,(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x))$

3.1.2 Code validation

A simple way to test the validity of our code is to verify that numerical solutions to (1.64)–(1.65) that are generated from initial data $(\texttt{z}{}_0,\texttt{w}{}_0)$ with $\texttt{w}{}_0>0$ satisfy the decay rates of Proposition 1.1

$$\begin{aligned} |u(t)-u(0)| \lesssim t^{2\mu } {\quad \text {and}\quad }|u'\!(t)| \lesssim t^{2\mu -1}, \end{aligned}$$

(3.3)

and Theorem 1.2

$$\begin{aligned} \Vert \tilde{\zeta }{}(t) - \tilde{\zeta }{}_*\Vert _{H^{k-1}}+\Vert w_1(t) - w_1^*\Vert _{H^{k-1}}+\Vert t^\mu w_2(t) - \bar{w}{}_2^*\Vert _{H^{k}}\nonumber \\ +\Vert t^\mu w_3(t) - \bar{w}{}_3^*\Vert _{H^{k}} \lesssim t^{\mu -\sigma }, \;\; \sigma >0. \end{aligned}$$

(3.4)

We first note that, by equating (1.26) and (1.63) and recalling that $W = 0$ for homogeneous solutions, u(t) can be expressed in terms of a homogeneous solution $\texttt{w}{}_{H}(t)$ of (1.64)–(1.65) as $u(t) = \ln (\texttt{w}{}_{H}(t))$. The decay rates for the homogeneous solution (3.3) can then be re-written in terms of $\texttt{w}{}_{H}$ as

$$\begin{aligned} |\ln (\texttt{w}{}_{H}(t))-\ln (\texttt{w}{}_{H}(0))|&\lesssim t^{2\mu }, \end{aligned}$$

(3.5)

$$\begin{aligned} \Bigl |\frac{\texttt{w}{}^{\prime }_{H}(t)}{\texttt{w}{}_{H}(t)}\Bigr |&\lesssim t^{2\mu -1}. \end{aligned}$$

(3.6)

Similarly, for non-homogeneous solutions, we can express $w_{1}$ in terms of $\texttt{w}{}$ by setting $w_{2}=w_{3}=0$ and equating (1.26) and (1.63) to get $w_{1} = \ln (\texttt{w}{}(t,x)) - \ln (\texttt{w}{}_{H}(t))$. The decay rate (3.4), in the $H^{1}$ norm, is then

$$\begin{aligned} \Vert \texttt{z}{}(t,x) -\texttt{z}{}(0,x)\Vert _{H^{1}}+\Vert [\ln (\texttt{w}{}(t,x))-\ln (\texttt{w}{}_{H}(t))]\nonumber \\ -[\ln (\texttt{w}{}(0,x))-\ln (\texttt{w}{}_{H}(0)]\Vert _{H^{1}} \lesssim t^{\mu -\sigma }. \end{aligned}$$

(3.7)

We have estimated $\texttt{z}{}|_{t=0},\texttt{w}{}|_{t=0}, \texttt{w}{}_{H}|_{t=0}$ by taking the values of the functions at a time-step close to $t=0$ and calculated $\texttt{w}{}^{\prime }_{H}(t)$ using second order central finite differences. As shown in Fig. 5, the numerical solutions clearly replicate the above decay rates suggesting the code is correctly implemented.

Fig. 5

Log–log decay plots of numerical solutions (Blue) against the corresponding bound (Orange) and the bound multiplied by a constant c (Yellow). $K = 0.5, N = 1000$. Initial data for the homogeneous solution is $(\texttt{z}{}(0,x),\texttt{w}{}_{H}(0,x)) = (0,1)$. Initial data for the non-homogeneous solution is $(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x)+1)$

3.2 Numerical behaviour

Beyond the convergence tests, we have generated numerical solutions to the system (1.64)–(1.65) from a variety of initial data sets $(\texttt{z}{}_0,\texttt{w}{}_0)$ for which $\texttt{w}{}_0$ satisfies the conditions (1.7) and (1.8). We employed resolutions ranging from 1000 to 160,000 grid points in our simulations. For initial data satisfying (1.7), we chose functions $\texttt{w}{}_0$ that cross the x-axis at least twice,¹⁶ while for initial data satisfying (1.8), $\texttt{w}{}_0$ does not cross the x-axis at all.

All of the solutions in this article displayed in the figures are generated from initial data of the form

$$\begin{aligned} (\texttt{z}{}_0,\texttt{w}{}_{0}) = (0, a\sin (x+\theta )+c) \end{aligned}$$

for some particular choice of the constants $a,c, \theta \in {\mathbb {R}}$. From our numerical solutions, we observe, for the full parameter range $1/3<K<1$ and all choices of the initial data with a sufficiently small, that $\texttt{z}{}$ and $\texttt{w}{}$ remain bounded and converge pointwise as $t\searrow 0$; see Figs. 6 and 7.

3.2.1 Derivative blow-up at $t=0$

While $\texttt{z}{}$ and $\texttt{w}{}$ remain bounded, our numerical simulations reveal that derivatives of the solutions of sufficiently high order blow-up at $t=0$ for the parameter values $1/3<K<1$ and initial data satisfying (1.7). In Table 1, we list, for a selection of K values, the corresponding minimum value of $\ell$ for which $\sup _{x\in \mathbb {T}{}^1}\bigl (|\partial _{x}^{\ell } \texttt{z}{}(t,x)|+|\partial _{x}^{\ell }\texttt{w}{}(t,x)|\bigr ) \nearrow \infty$ as $t\searrow 0$. From these values, it appears that $\ell$ is a monotonically decreasing function of K.

Table 1 Observed value of $\ell$ for various K

Fig. 6

Plots of $\texttt {w}$ at various times. $K=0.6,\;\; N = 1000,(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x))$

Fig. 7

Plots of $\texttt{z}{}$ at various times. $K=0.6,\;\; N = 1000,(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x))$

3.2.2 Asymptotic behaviour and approximations

For the full range of parameter $1/3<K<1$ and all choices of initial data, we observe that our numerical solutions display ODE-like behaviour near $t=0$. In particular, these solutions can be approximated by solutions of the asymptotic system (1.66)–(1.67) at late times using the following procedure:

• (i) Generate a numerical solution $(\texttt{z}{},\texttt{w}{})$ of (1.64)–(1.65) from initial data $(\texttt{z}{}_{0},\texttt{w}{}_{0})$ specified at time $t_{0}>0$.

• (ii) Fix a time $\tilde{t}_{0} \in (0,t_{0})$ when the numerical solution $(\texttt{z}{},\texttt{w}{})$ first appears to be dominated by ODE behaviour.

• (iii) Fix initial data for the asymptotic system (1.66)–(1.67) at $t=\tilde{t}_{0}$ by setting

  $$\begin{aligned} (\tilde{\texttt{z}{}}{}_{0},\tilde{\texttt{w}{}}{}_{0}) = (\texttt{z}{},\texttt{w}{})|_{t=\tilde{t}{}_0}. \end{aligned}$$

• (iv) Solve the asymptotic system (1.66)–(1.67) with initial data as chosen above in (iii) to obtain the asymptotic solution $(\tilde{\texttt{z}{}}{},\tilde{\texttt{w}{}}{})$ where

  $$\begin{aligned} \tilde{\texttt{z}{}}{}= \tilde{\texttt{z}{}}{}_0, \end{aligned}$$

  (3.8)

  and $\tilde{\texttt{w}{}}{}$ is defined implicitly by

  $$\begin{aligned} \frac{(3 K-\mu -1) \ln \left( (3 K-1) t^{2 \mu }-(K-1) \mu \tilde{\texttt{w}{}}{}^2\right) }{2 (3 K-1) \mu }-\frac{\ln (|\tilde{\texttt{w}{}}{}|(1-3K)}{1-3 K}=c \end{aligned}$$

  (3.9)

  and

  $$\begin{aligned} c = \frac{(3 K-\mu -1) \ln \left( (3 K-1) \tilde{t}_{0}^{2 \mu }-(K-1) \mu \tilde{\texttt{w}{}}{}_{0}^2\right) }{2 (3 K-1) \mu }-\frac{\ln (|\tilde{\texttt{w}{}}{}_{0}|(1-3K))}{1-3 K}. \end{aligned}$$

• (v) Compare the numerical solution $(\texttt{z}{},\texttt{w}{})$ to the asymptotic solution $(\tilde{\texttt{z}{}}{},\tilde{\texttt{w}{}}{})$ on the region $(0,\tilde{t}{}_0)\times \mathbb {T}{}^1$.

Using this procedure, we find that numerical solutions $(\texttt{z}{},\texttt{w}{})$ of the system (1.66)–(1.67) can be remarkably well-approximated by solutions $(\tilde{\texttt{z}{}}{},\tilde{\texttt{w}{}}{})$ of the asymptotic system. In particular, by setting $t=0$ in (3.9) and noting that we can solve for $\tilde{\texttt{w}{}}{}|_{t=0}$ to get

$$\begin{aligned} \tilde{\texttt{w}{}}{}_{f}:= \tilde{\texttt{w}{}}{}|_{t=0}=\frac{\text {sgn}(\tilde{\texttt{w}{}}{}_{0})|\tilde{\texttt{w}{}}{}_0|^{\frac{1}{1-K}}}{(\tilde{t}{}_{0}^{2\mu }+\tilde{\texttt{w}{}}{}_{0}^{2})^{\frac{K}{2(1-K)}}} \end{aligned}$$

(3.10)

where $\text {sgn}(x)$ is the sign function, we have, with the help of (3.8), that

$$\begin{aligned} (\texttt{z}{},\texttt{w}{})|_{t=0} \approx (\tilde{\texttt{z}{}}{}_0, \tilde{\texttt{w}{}}{}_f). \end{aligned}$$

(3.11)

It is worth noting that this ODE-like asymptotic behaviour of solutions generated from initial data satisfying (1.8) is expected by Theorem 1.2. What is interesting is that this behaviour of solutions persists for initial data that violates (1.8).

To illustrate how well solutions $(\texttt{z}{},\texttt{w}{})$ of (1.64)–(1.65) can be approximated by solutions $(\tilde{\texttt{z}{}}{},\tilde{\texttt{w}{}}{})$ of the asymptotic system (1.66)–(1.67) near $t=0$, we compare in Fig. 8 the plot of $\tilde{\texttt{w}{}}{}_f=\tilde{\texttt{w}{}}{}|_{t=0}$, for a fixed choice of $\tilde{t}{}_0$ and $\tilde{\texttt{w}{}}{}_0$ (see (3.10)), with that of $\texttt{w}{}(t)$ at times close to zero. From the figure, it is clear that the agreement is almost perfect for times close enough to zero.

Fig. 8

Comparison of numerical solution $\texttt {w}$ (Blue) and $\tilde{\texttt{w}{}}{}_{f}$ (Orange). $K=0.6,\;\;N = 1000,\;\;(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\cos (x))$, $(\tilde{t}_{0},\tilde{\texttt{w}{}}{}_{0}) = (9.93\times 10^{-4},\texttt{w}{}|_{t=9.93\times 10^{-4}})$

3.2.3 Behaviour of the density gradient

By (1.14), (1.17), (1.25) and (1.62)–(1.63), the density can be written in terms of $\texttt {z}$ and $\texttt {w}$ as $\rho = (\texttt {w}^{2}+t^{2\mu })^{-\frac{K+1}{2}}\rho _{c}t^\frac{2(K+1)}{1-K}e^{(1+K)\texttt {z}}$ where $\rho _c \in (0,\infty )$. Differentiating this expression, we find after a short calculation that the density gradient is given by

$$\begin{aligned} \frac{\partial _{x}\rho }{\rho }= (1+K)\biggl (\partial _{x}\texttt{z}{}- \frac{\texttt{w}{}}{(t^{2\mu }+\texttt{w}{}^2)}\partial _{x}\texttt{w}{}\biggr ). \end{aligned}$$

(3.12)

Using this formula to compute the density gradient for numerical solutions of (1.64)–(1.65), we observe from our numerical solutions that density gradient displays markedly different behaviour depending on whether or not it is generated from initial data satisfying (3.2). For solutions generated from initial data satisfying (3.2), we find that the density gradient remains bounded and converges as $t\searrow 0$ to a fixed function, which is expected by Theorem 1.2. An example of this behaviour is provided in Fig. 9. On the other hand, the density gradient of solutions generated from initial data violating (3.2) develop steep gradients and blows-up at $t=0$ at isolated spatial points; see Fig. 10 for an example of this behaviour.

As in Sect. 3.2.2, we can compare the density gradient of the full numerical solutions with the density gradient computed from a solutions of the asymptotic equation. We do this by evaluating (3.12) at $t=0$ and using (3.11) to approximate the density gradient at $t=0$ by

$$\begin{aligned} \frac{\partial _{x}\rho }{\rho }\biggl |_{t=0}\approx (1+K)\biggl (\partial _{x}\tilde{\texttt{z}{}}{}_0 - \frac{(\tilde{t}{}_0^{2\mu }+(1-K)\tilde{\texttt{w}{}}{}_0^2)}{(1-K)(\tilde{t}{}_0^{2\mu }+\tilde{\texttt{w}{}}{}_0^2)\tilde{\texttt{w}{}}{}_0}\partial _{x}\tilde{\texttt{w}{}}{}_0\biggr ). \end{aligned}$$

This formula identifies, at least heuristically, that the blow-up at $t=0$ in the density gradient is due the vanishing of $\texttt{w}{}$. Once again the agreement between the numerical and asymptotic plots is close enough that the two are practically indistinguishable as can be seen from Fig. 11.

Fig. 9

Plots of density gradient, $\frac{\partial _{x}\rho }{\rho }$, at various times. K = 0.6, N = 1000, $(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x)+0.15)$

Fig. 10

Plots of density gradient, $\frac{\partial _{x}\rho }{\rho }$, at various times. K = 0.6, N = 1000, $(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x))$

Fig. 11

Plots of density gradient, $\frac{\partial _{x}\rho }{\rho }$, calculated from numerical results (Blue) and the asymptotic map (Green). K=0.45, N = 160,000, $(\texttt{z}{}_{0},\texttt{w}{}_{0}) = (0,0.1\sin (x))$. Points near $\texttt{w}{}_{0} = 0$ in the asymptotic map have been removed to emphasise agreement of the plots away from the singularities