# Gravitational Collapse for Polytropic Gaseous Stars: Self-similar Solutions

Yan Guo, Mahir Hadžić, Juhi Jang, and Matthew Schrecker Division of Applied
Mathematics, Brown University, Providence, RI 02912, USA, Email:
<EMAIL_ADDRESS>of Mathematics, University College London, London
WC1E 6XA, UK. Email<EMAIL_ADDRESS>of Mathematics, University
of Southern California, Los Angeles, CA 90089, USA, and Korea Institute for
Advanced Study, Seoul, Korea. Email<EMAIL_ADDRESS>of
Mathematics, University College London, London WC1E 6XA, UK. Email:
<EMAIL_ADDRESS>
###### Abstract

In the supercritical range of the polytropic indices
$\gamma\in(1,\frac{4}{3})$ we show the existence of smooth radially symmetric
self-similar solutions to the gravitational Euler-Poisson system. These
solutions exhibit gravitational collapse in the sense that the density blows-
up in finite time. Some of these solutions were numerically found by Yahil in
1983 and they can be thought of as polytropic analogues of the Larson-Penston
collapsing solutions in the isothermal case $\gamma=1$. They each contain a
sonic point, which leads to numerous mathematical difficulties in the
existence proof.

###### Contents

  1. 1 Introduction and the main result
    1. 1.1 Methodology
  2. 2 The sonic point
    1. 2.1 The formal Taylor expansion
    2. 2.2 Branch selection
    3. 2.3 Larson-Penston-Hunter- (LPH-) type solutions
    4. 2.4 The induction argument and the series convergence
  3. 3 Solution to the right of the sonic point
  4. 4 Solution to the left of the sonic point
    1. 4.1 Continuity properties
    2. 4.2 Invariant structures
    3. 4.3 Properties of the fundamental set $Y$
    4. 4.4 Asymptotics at the scaling origin, $y=0$
  5. 5 Proof of the main theorem
  6. A Well-posedness away from singular points
  7. B Combinatorial bootstrap - convergence of the series at the sonic point
  8. C Interval arithmetic
    1. C.1 Proofs of $s(\omega_{0})>0$ and Proposition 2.9
    2. C.2 Proof of (2.89)–(2.91)
    3. C.3 Proof of Lemma 3.4
  9. D Proof of Proposition 4.7

##  1 Introduction and the main result

The rigorous description of stellar collapse in the context of Newtonian
gravity is a fundamental mathematical problem. It is believed, at least for
some classes of initial data, that on approach to singularity a self-
gravitating gaseous star will enter an approximately self-similar regime [19,
25, 27, 29, 14], which will intertwine the spatial and the time scales in a
universal manner dictated by the scaling symmetries of the problem. The
purpose of this paper is to construct radially symmetric examples of exactly
self-similar imploding solutions for the full range of the supercritical
polytropic pressure laws.

A self-gravitating Newtonian star is described using the gravitational Euler-
Poisson equations, coupling the isentropic compressible Euler equations to a
gravitational potential. In three spatial dimensions, under the assumption of
radial symmetry, these equations take the form

$\displaystyle\partial_{t}\rho+\partial_{r}(\rho u)+\frac{2}{r}\rho u$
$\displaystyle=0,$ (1.1)
$\displaystyle\rho\big{(}\partial_{t}u+u\partial_{r}u\big{)}+\partial_{r}p+\frac{1}{r^{2}}\rho
m$ $\displaystyle=0,$ (1.2)

where the principal unknowns $\rho(t,r)$ and $u(t,r)$ are the density and
radial velocity of the star, respectively, and depend only on time $t$ and the
radial coordinate $r=|x|$. Equation (1.1) gives the conservation of mass and
(1.2) is the conservation of momentum with the given pressure law and
gravitational force. We will assume throughout that the pressure $p=P(\rho)$
satisfies the polytropic equation of state

$P(\rho)=\kappa\rho^{\gamma},\quad\gamma\in(1,\frac{4}{3}),\>\kappa>0,$ (1.3)

and the mass function $m(t,r)$ is defined by

$m(t,r)=4\pi\int_{0}^{r}\sigma^{2}\rho(t,\sigma)\,\operatorname{d}\\!\sigma.$
(1.4)

Notice that the term $\frac{m}{r^{2}}$ appearing in the momentum equation
(1.2) corresponds to the radial component of the gravitational force field
$\nabla\phi$ generated by the gravitational potential $\phi$, which by
definition solves the Poisson equation

$\Delta\phi=4\pi\rho,\quad\lim_{|x|\to\infty}\phi(t,x)=0.$

This is easily checked under the assumptions of radial symmetry.

A natural criticality scale is introduced in the problem by varying the
polytropic index $\gamma$ in the pressure law (1.3). It is easily checked that
the nonlinear flow associated with (1.1)–(1.3) is invariant under the scaling
transformation

$\displaystyle\rho(t,r)\mapsto\lambda^{-\frac{2}{2-\gamma}}\rho\Big{(}\frac{t}{\lambda^{\frac{1}{2-\gamma}}},\frac{r}{\lambda}\Big{)},$
(1.5) $\displaystyle
u(t,r)\mapsto\lambda^{-\frac{\gamma-1}{2-\gamma}}u\Big{(}\frac{t}{\lambda^{\frac{1}{2-\gamma}}},\frac{r}{\lambda}\Big{)}.$

This scaling is in fact the only invariant scaling for the compressible Euler-
Poisson system, by contrast to the compressible Euler equations which allow
for a 2-parameter family of invariant scalings, see for example [22].

When $\gamma>\frac{4}{3}$ the problem is mass-subcritical with respect to the
scaling (1.5), see [13]. In this case, under the assumption of finite total
mass and energy, it is known that no collapsing solutions can exist, see [5].
In the mass-critical case, there is a well-known finite-dimensional family of
collapsing stars discovered by Goldreich and Weber [8], see also [21, 7, 6].
The goal of this paper is to prove the existence of self-similar solutions
describing gravitational collapse in the mass-supercritical regime
$\gamma\in(1,\frac{4}{3})$.

Motivated by (1.5), we define the self-similar variable

$y=\frac{r}{\sqrt{\kappa}(-t)^{2-\gamma}}$ (1.6)

and formally look for solutions to (1.1)–(1.3) of the form

$\displaystyle\rho(t,r)=(-t)^{-2}\tilde{\rho}(y),$ (1.7) $\displaystyle
u(t,r)=\sqrt{\kappa}(-t)^{1-\gamma}\tilde{u}(y).$

Substituting this ansatz into the continuity equation (1.1) and dropping the
tilde-s, we derive

$\displaystyle\rho^{\prime}\big{(}u+(2-\gamma)y)+\rho
u^{\prime}+2\rho+\frac{2}{y}\rho u=0.$ (1.8)

Multiplying through by $y^{2}$, we simplify to find

$(4-3\gamma)y^{2}\rho=\big{(}y^{2}\rho(u+(2-\gamma)y)\big{)}^{\prime}$

which we integrate to get a representation for the self-similar local mass as

$\int_{0}^{y}z^{2}\rho(z)\,\operatorname{d}\\!z=\frac{y^{2}\rho(u+(2-\gamma)y)}{4-3\gamma}.$
(1.9)

Thus we derive from the momentum equation (1.2) the second self-similar
equation

$\rho(u+(2-\gamma)y)u^{\prime}+\gamma\rho^{\gamma-1}\rho^{\prime}+(\gamma-1)\rho
u+\frac{4\pi}{4-3\gamma}\rho^{2}(u+(2-\gamma)y)=0.$ (1.10)

It will be convenient in what follows to work with the re-scaled relative
velocity, rather than working directly with the velocity $u$. The new relative
velocity is defined as

$\omega=\frac{u+(2-\gamma)y}{y}.$ (1.11)

We therefore arrive at the self-similar ODE system

$\displaystyle y\omega\rho^{\prime}+y\rho\omega^{\prime}=$
$\displaystyle\,(4-3\gamma)\rho-3\rho\omega,$ (1.12) $\displaystyle
y\rho\omega\big{(}y\omega^{\prime}+\omega-(2-\gamma))+\gamma\rho^{\gamma-1}\rho^{\prime}=$
$\displaystyle\,-(\gamma-1)y\rho\omega-(\gamma-1)(\gamma-2)y\rho-\frac{4\pi}{4-3\gamma}\rho^{2}y\omega.$

Equivalently we may rewrite the system in the form

$\displaystyle\rho^{\prime}=$
$\displaystyle\frac{y\rho\left(2\omega^{2}+(\gamma-1)\omega-\frac{4\pi\rho\omega}{4-3\gamma}+(\gamma-1)(2-\gamma)\right)}{\gamma\rho^{\gamma-1}-y^{2}\omega^{2}},$
(1.13) $\displaystyle\omega^{\prime}=$
$\displaystyle\,\frac{4-3\gamma-3\omega}{y}-\frac{y\omega\left(2\omega^{2}+(\gamma-1)\omega-\frac{4\pi\rho\omega}{4-3\gamma}+(\gamma-1)(2-\gamma)\right)}{\gamma\rho^{\gamma-1}-y^{2}\omega^{2}}.$

We refer to (1.13) as the self-similar Euler-Poisson system. Clearly, this
system has a singularity at $y=0$. However, there is a further singularity
which occurs whenever $\gamma\rho^{\gamma-1}-y^{2}\omega^{2}=0$. This is of
fundamental importance and the presence of such singularities, as we shall
explain below, is unavoidable in the study of smooth self-similar solutions to
(1.13) satisfying physically reasonable boundary conditions. This motivates
the following definition.

######  Definition 1.1 (Sonic point).

Let $(\rho(\cdot),\omega(\cdot))$ be a $C^{1}$-solution to the self-similar
Euler-Poisson system (1.13) on the interval $(0,\infty)$. A point
$y_{*}\in(0,\infty)$ such that

$\gamma\rho^{\gamma-1}(y_{*})-y_{*}^{2}\omega^{2}(y_{*})=0$

is called a sonic point.

If $y_{\ast}$ is a sonic point, then the hypersurface defined by the relation
$r(t)=\sqrt{\kappa}y_{\ast}(-t)^{2-\gamma}$ corresponds to the backward
acoustic cone emanating from the origin $(0,0)$ ([1, 22]). It satisfies the
relation $\dot{r}(t)=u(t,r(t))-c_{s}(t,r(t))$, where $c_{s}$ is the sound
speed $c_{s}=\frac{dP}{d\rho}=\sqrt{\kappa\gamma\rho^{\gamma-1}}$.

We are looking for smooth solutions which are both regular at the (scaling)
origin $y=0$ and satisfy suitable decay conditions as $y\to\infty$. Taking
Taylor expansions at the origin and in the far-field (as $y\to\infty$), we see
that we require the initial and asymptotic boundary conditions

$\displaystyle\rho(0)>0,\quad$ $\displaystyle\omega(0)=\frac{4-3\gamma}{3},$
(1.14) $\displaystyle\rho(y)\sim y^{-\frac{2}{2-\gamma}}\text{ as
}y\to\infty,\quad$ $\displaystyle\lim_{y\to\infty}\omega(y)=2-\gamma.$ (1.15)

From these conditions, it is clear that any continuous solution of (1.13) and
(1.14)–(1.15) must have at least one sonic point $y_{*}>0$.

In the isothermal case $\gamma=1$, the existence of global solutions
satisfying (1.13) and (1.14)–(1.15) has a long history in the astrophysics
literature, primarily relying on numerical methods. In their seminal works,
Larson [19] and Penston [25] independently showed (numerically) the existence
of a globally defined solution satisfying (1.14)–(1.15) and with only a single
sonic point present. Subsequently, Hunter [15] numerically constructed a full
sequence of further solutions, commonly referred to as Hunter-type solutions,
each of which also has a single sonic point (see also the work of Shu [27] and
the later work of Whitworth and Summers [28]). Despite the physically
simplifying assumption $\gamma=1$, these families of solutions attracted a lot
of attention in the physics literature as possible prototype models for the
behaviour of the core in late stages of gravitational collapse. In fact, the
Larson-Penston (henceforth, LP) solutions were judged to be the more stable
solutions under subsequent numerical analysis [1, 20, 24]. They also play an
important role in the Newtonian criticality theory and the resolution of the
so-called self-similarity hypothesis, see [14].

However, the assumption that the flow is isothermal received criticism on
physical grounds, for example by Yahil [29], who pointed out that the physical
condition of finite energy is violated unless $\gamma>\frac{6}{5}$. The value
$\gamma=\frac{6}{5}$ plays the role of the energy-critical exponent with
respect to the scaling (1.5), see [13]. More importantly, different values of
$\gamma>1$ allow us to encode stars with different thermodynamic properties
and it is therefore important to understand the space of self-similar flows in
the range $\gamma\in(1,\frac{4}{3})$. In the range
$\gamma\in(\frac{6}{5},\frac{4}{3})$, Yahil [29] constructed a family of
numerical self-similar solutions to (1.13) with finite energy. These solutions
share certain characteristics with the isothermal LP solution. For example,
the physical radial velocity remains strictly negative (except at the origin,
where it vanishes) up to the collapse time in both the Yahil solutions and the
LP solutions. This property does not hold for Hunter solutions and has been
tied to the possible dynamic instabilities of such solutions by Maeda–Harada
[20]. This leads us to the following definition.

######  Definition 1.2 (Yahil-type solution).

Let $\gamma\in(1,\frac{4}{3})$. A pair of $C^{1}$ functions $(\rho,\omega)$
defined on a connected interval $I\subset[0,\infty)$ satisfying the self-
similar Euler-Poisson system (1.13) is said to be of Yahil-type if

  * (i)

There exists a unique sonic point $y_{*}\in I$;

  * (ii)

For all $y\in I$, $\rho(y)>0$ and for all $y\in I\setminus\\{0\\}$, $u(y)<0$.

Recently, the first three authors were able to construct LP solutions in the
case $\gamma=1$ in [11]. The main result of this paper is to show that Yahil
solutions exist for the full physical range $\gamma\in(1,\frac{4}{3})$,
including the finite energy range ($\gamma>\frac{6}{5}$).

######  Theorem 1.3.

For each $\gamma\in(1,\frac{4}{3})$, there exists a global, real-analytic,
Yahil-type solution $(\rho,\omega)$ of (1.13), (1.14)–(1.15) with a single
sonic point $y_{*}$ and satisfying the natural, physical conditions

$\displaystyle\rho(y)>0\text{ for all
}y\in[0,\infty),\quad-\frac{2}{3}y<u(y)<0\text{ for all }y\in(0,\infty).$
(1.16)

In addition, both $\rho$ and $\omega$ are strictly monotone on their domain of
definition:

$\displaystyle\rho^{\prime}(y)<0\text{ for all
}y\in(0,\infty),\quad\omega^{\prime}(y)>0\text{ for all }y\in(0,\infty).$
(1.17)

The proof of this theorem is a consequence of a delicate analysis of the
nonautonomous dynamical system (1.13) in the regions separated by the sonic
point $y_{\ast}$, presented in Sections 3 and 4. The combination of results
derived in these two sections gives Theorem 1.3 and the short argument is
given in Section 5.

The most famous class of special solutions to the radially symmetric Euler-
Poisson system are the Lane-Emden steady stars [3], known to be of finite mass
and energy if $\gamma\in[\frac{6}{5},2)$. Their dynamic stability is a
classical subject, and in the case $\gamma>\frac{4}{3}$ they are known to be
linearly stable and conditionally nonlinearly stable [26]. By contrast, when
$\gamma\in[\frac{6}{5},\frac{4}{3})$ the Lane-Emden stars are unstable [16,
17]. In the critical case $\gamma=\frac{4}{3}$, the Lane-Emden stars are
spectrally stable, but nonlinearly unstable. The latter statement follows by
observing that the above mentioned Goldreich-Weber (henceforth GW) collapsing
stars can be chosen initially to be arbitrarily close to the corresponding
steady Lane-Emden stars. In fact, due to the mass-critical nature of the
problem, the GW collapse is a consequence of an effective separation of
variables in the problem, where the solution corresponds to a time-modulated
spatial profile, which satisfies a Lane-Emden-like equation. By time-reversal,
there also exist global-in-time expanding GW-solutions, whose nonlinear
stability was shown in [13].

The solutions constructed in Theorem 1.3 ($1<\gamma<\frac{4}{3}$) are very
different from the GW solutions ($\gamma=\frac{4}{3}$), and owe their
existence to a subtle balancing of the three dominant forces in the problem:
inertia, pressure, and gravity. A completely different portion of the phase-
space is populated by the so-called dust-like collapsing stars, which have
been shown to exist in [12]. The solutions constructed in [12] do not honour
the scaling invariance implied by (1.5), but are instead to a leading order
approximated by the so-called dust solutions, which solve (1.1)–(1.2) without
the pressure term $p$.

As explained above, the most exciting physical feature of the self-similar
solutions that we construct is their behaviour in the inner core region, as a
possible model of typical stellar collapse scenario. Nevertheless, for
completeness we also discuss some global properties of the solution, in
particular the size of the total mass and total energy. The solutions
constructed in Theorem 1.3 have infinite total mass

$M[\rho]=4\pi\int_{0}^{\infty}\rho(t,r)r^{2}\operatorname{d}\\!r,$

as can easily be seen from the asymptotic behaviour (1.15). A short
calculation shows that for any fixed $t<0$, asymptotically as $r\to\infty$

$\displaystyle\rho(t,r)\sim_{r\to\infty}r^{-\frac{2}{2-\gamma}},\ \
m(t,r)\sim_{r\to\infty}r^{\frac{4-3\gamma}{2-\gamma}},\ \
\phi(t,r)\sim_{r\to\infty}r^{\frac{2(1-\gamma)}{2-\gamma}},$ (1.18)

where $m(t,r):=4\pi\int_{0}^{r}\rho(t,s)s^{2}\operatorname{d}\\!s$ is the mass
contained in a ball of radius $r$. On the other hand, the total energy

$E[\rho,u]=4\pi\int_{0}^{\infty}\left(\frac{1}{2}\rho
u^{2}+\frac{1}{\gamma-1}\rho^{\gamma}-\frac{1}{8\pi}|\partial_{r}\phi|^{2}\right)r^{2}\operatorname{d}\\!r$

of the solutions constructed in Theorem 1.3 is finite when
$\gamma\in(\frac{6}{5},\frac{4}{3})$ and infinite for
$\gamma\in(1,\frac{6}{5}]$. This can be easily seen from (1.18) and the
asymptotic behaviour $u(t,r)_{r\to\infty}\sim r^{\frac{1-\gamma}{2-\gamma}}$
for any fixed $t<0$, which is established later in Lemma 3.9.

A further surprising outcome of our work is the provision of a new context
within which to consider the above mentioned distinction between the LP- and
Hunter-type solution. In the context of the isothermal problem ($\gamma=1$),
the demand that the solution be regular produces two possible algebraic
“branches” for the Taylor expansion coefficients at the sonic point. The LP-
solution constructed in [11] belongs to one of them, all the Hunter solutions
to the other, and the branches intersect at exactly one point. When
$\gamma>1$, we will show that there are two analogous branches. Remarkably, in
the formal $\gamma\to 1$ limit one of them converges to two portions of the
two isothermal branches that together form a continuous curve containing both
the LP- and Hunter solutions in the isothermal case. We thus term the
solutions coming off this $\gamma>1$-branch the Larson-Penston-Hunter- (LPH-)
type solutions. We comment on this further in Section 1.1, while the detailed
analysis can be found in Section 2.

###  1.1 Methodology

Due to the importance that the sonic condition will play throughout all of the
subsequent analysis, we define here a function

$G(y;\rho,\omega)=\gamma\rho^{\gamma-1}-y^{2}\omega^{2}.$ (1.19)

######  Definition 1.4 (Sonic, supersonic, and subsonic).

We say that the flow is subsonic whenever $G(y;\rho,\omega)>0$, supersonic
when $G(y;\rho,\omega)<0$, and sonic when $G(y;\rho,\omega)=0$.

For convenience, we denote by $h(\rho,\omega)$ the function

$h(\rho,\omega)=2\omega^{2}+(\gamma-1)\omega-\frac{4\pi\rho\omega}{4-3\gamma}+(\gamma-1)(2-\gamma).$
(1.20)

The system (1.13) may then be written concisely as

$\displaystyle\rho^{\prime}=$ $\displaystyle\,\frac{y\rho
h(\rho,\omega)}{G(y;\rho,\omega)},$ (1.21) $\displaystyle\omega^{\prime}=$
$\displaystyle\,\frac{4-3\gamma-3\omega}{y}-\frac{y\omega
h(\rho,\omega)}{G(y;\rho,\omega)}.$

There are two known, explicit solutions to the system (1.21), the Friedman
solution

$\omega_{F}=\frac{4-3\gamma}{3},\quad\rho_{F}=\frac{1}{6\pi}$ (1.22)

which satisfies the initial condition (1.14) at the origin, but fails the
asymptotic boundary condition (1.15), and the far-field solution,

$\omega_{f}=2-\gamma,\quad\rho_{f}=ky^{-\frac{2}{2-\gamma}},\text{ where
}k=\Big{(}\frac{\gamma(4-3\gamma)}{2\pi(2-\gamma)^{2}}\Big{)}^{\frac{1}{2-\gamma}},$
(1.23)

which satisfies the asymptotic boundary condition (1.15) but fails the initial
condition (1.14) . Note that the constant $k>0$ is well-defined due to
$\gamma<4/3$.

The Friedman and far-field solutions have sonic points at $y_{F}(\gamma)$,
$y_{f}(\gamma)$, respectively, with

$\displaystyle
y_{F}(\gamma)=\frac{3}{4-3\gamma}\sqrt{\frac{\gamma}{(6\pi)^{(\gamma-1)}}},\quad
y_{f}(\gamma)=\frac{\sqrt{\gamma}}{2-\gamma}\Big{(}\frac{4-3\gamma}{2\pi}\Big{)}^{\frac{\gamma-1}{2}}.$
(1.24)

For all $\gamma\in(1,\frac{4}{3})$, we see that
$0<y_{f}(\gamma)<y_{F}(\gamma)<\infty$. Henceforth, we will drop the explicit
dependence on $\gamma$ for $y_{f}$ and $y_{F}$, emphasising that for each
$\gamma\in(1,\frac{4}{3})$, $[y_{f},y_{F}]$ is a compact interval.

The system of ODE (1.13) is challenging since, as explained above, the flow
must pass through a sonic point. The requirement of smoothness at such a point
then leads to a number of mathematical difficulties. Generically, if we
stipulate that some $y_{\ast}\in(0,\infty)$ be a sonic point, then the flow
around that point will not be global. It is only for special values of
$y_{\ast}$ where the corresponding solution is in fact globally defined on
$[0,\infty)$. In a recent pioneering study of self-similar solutions for the
compressible Euler system with the equation of state $P(\rho)=\rho^{\gamma}$
($\gamma>1$), Merle, Raphaël, Rodnianski, and Szeftel [22] systematically
developed the existence theory for $C^{\infty}$-self-similar solutions of the
Euler flow; the underlying $2\times 2$ system of ODE is in this case
autonomous (in contrast to (1.13)). The smoothness of the self-similar
solutions across the sonic point is in fact a crucial ingredient in the proof
of their (finite codimension) nonlinear stability [23].

We will seek a solution with sonic point at some $y_{*}\in(y_{f},y_{F})$.
Making the formal Taylor expansion around the sonic point $y_{*}$, we set

$\rho(y)=\sum_{N=0}^{\infty}\rho_{N}(y-y_{*})^{N},\quad\omega(y)=\sum_{N=0}^{\infty}\omega_{N}(y-y_{*})^{N}.$
(1.25)

In order to have a smooth solution through $y_{*}$, we require that the values
$\rho_{0}=\rho(y_{*})$ and $\omega_{0}=\omega(y_{*})$ are constrained by the
two identities

$G(y_{*};\rho_{0},\omega_{0})=0,\quad h(\rho_{0},\omega_{0})=0.$ (1.26)

For all $y_{*}\in[y_{f},y_{F}]$, we will show below that there is a unique
pair $(\rho_{0},\omega_{0})$ satisfying these two conditions. When we come to
solve for the first order coefficients $(\rho_{1},\omega_{1})$, however, we
see that the picture becomes more complicated. In fact, there are again two
possible branches from which the coefficients may be chosen. However, as we
next explain, it is natural to view the $\gamma=1$-case as a degenerate case.
Namely, the possible pairs lie on graphs as shown in Figure 1, parametrised by
$\omega_{0}$ (equivalently, by $y_{*}$). In this case, the LP solution
constructed in [11] lies in the region of the LP branch for which
$\omega_{0}<\frac{1}{2}$ (equivalently $y_{*}>2$) while the numerically
constructed Hunter solutions all lie in the region $\omega_{0}>\frac{1}{2}$
(equivalently $y_{*}<2$), compare also [15, Fig. 2].

Figure 1: Plot of possible values $R=y_{*}\rho_{1}/\rho_{0}$ as a function of
$\omega_{0}\in[\frac{1}{3},1]$ in case $\gamma=1$.

However, as soon as we increase $\gamma>1$, a bifurcation occurs. The picture
then looks like one of the cases in Figure 2. The Hunter and LP solutions in
the case $\gamma=1$ actually live on the same branch of the solutions, a
feature that is concealed in the isothermal case by the degeneracy that makes
the branches coalesce at this value. For $\gamma>1$, the analogue of the
isothermal LP solution is the global solution with a unique sonic point
$y_{*}$ such that the first order coefficient $\rho_{1}$ lies on the joint LP-
Hunter (henceforth LPH) branch, and with maximal $y_{*}$ (equivalently minimal
$\omega_{0}$) - this is the lower (blue) branch in Figure 2. Such a solution
which will be shown to correspond to the Yahil solution that we are looking to
construct, see Definition 1.2.

Figure 2: Plot of possible values $R=y_{*}\rho_{1}/\rho_{0}$ as functions of
$\omega_{0}\in[\frac{4-3\gamma}{3},2-\gamma]$ for $\gamma=1.01$ and $1.3$

Once we have correctly identified the branch of solutions on which the LPH-
type solution should lie, we seek the globally defined Yahil-type solution
whose Taylor expansion at the sonic point is of LPH-type (see Definition 2.12
for the precise meaning). We then proceed in four key steps, as in the earlier
work of the first three authors, [11].

Step 1: The first step is to complete the Taylor expansion at each potential
sonic point $y_{*}\in[y_{f},y_{F}]$ to obtain a local, analytic solution
around $y_{*}$, denoted by

$(\rho(y;y_{\ast}),\omega(y;y_{\ast})).$

When clear from the context, we shall occasionally drop the dependence on
$y_{\ast}$ in the notation above. In comparison to [11], the convergence of
the Taylor series is significantly complicated by the presence of the term
$\rho^{\gamma-1}$ with its non-integer power. Various technical tricks are
employed, using the Faà di Bruno formula, to control the size of the
coefficients arising in the expansion, while interval arithmetic is employed
to control rigorously the sign of three key quantities (see (2.89)–(2.91) and
Appendix C.2 below).

Step 2: Second, we show that the local solution arising from each
$y_{*}\in[y_{f},y_{F}]$ may be extended globally to the right, remains
supersonic, and satisfies the correct asymptotic boundary condition (1.15).
This is based on the identification of several invariant regions to the right.
Compared with the isothermal case, the key property to show is that the flow
remains supersonic, a fact that is no longer trivially true. The asymptotics
follow easily from the global existence and bounds obtained.

Step 3: The third, and key, step, is to show that there exists a critical
value $\bar{y}_{*}$ for which the local analytic solution extends smoothly up
to the singular point at the origin with limit
$\omega(y)\to\frac{4-3\gamma}{3}$ as $y\to 0$. Similarly to [11], this
$\bar{y}_{*}$ is found as the infimum of a fundamental set

$Y=\Big{\\{}y_{*}\in(y_{f},y_{F})\,|\,\text{ there exists }y\text{ such that
}\omega(y;\tilde{y}_{*})=\frac{4-3\gamma}{3}\text{ for all
}\tilde{y}_{*}\in[y_{*},y_{F})\Big{\\}}.$

It is here that many of the additional complications arising from the choice
$P(\rho)=\rho^{\gamma}$, $\gamma>1$ make themselves felt. Many of the
invariances that were easily available in the case $\gamma=1$ are either
significantly harder to prove or fail altogether. For example, we no longer
have that the region $\\{\omega>\frac{1}{2}\\}$ is invariant as $y$ decreases.
These losses are due to the non-linear structure of the quantities
$h(\rho,\omega)$ and $G(y;\rho,\omega)$. Whereas, for $\gamma=1$, the sets in
phase space in which $h$ or $G$ have a constant sign are simply half-spaces
(parametrised by $y$ due to the non-autonomous nature of the system), for
$\gamma>1$, they have a much more complicated structure, with a change in the
geometry of the set $\\{h(\rho,\omega)=0\\}$ especially at
$\gamma=\frac{10}{9}$, see Lemma 2.1. This makes itself felt at a number of
levels. For example, the sets $\\{\omega^{\prime}(y)=0\\}$ and
$\\{\omega^{\prime\prime}(y)=0\\}$ in the $(\rho,\omega)$ have an intersection
in the region $\\{h<0,G>0\\}$, something which cannot happen for $\gamma=1$,
while there are no obvious invariant regions for $G$ either.

To resolve the difficulties caused by these features, we prove a new and
stronger property for the relative velocity $\omega(y;y_{*})$ for all
$y_{*}\in Y$: monotonicity with respect to $y$. By a careful analysis of the
phase plane and a continuity argument, we are able to show that for all
$y_{*}\in Y$, the function $\omega(y;y_{*})$ is strictly monotone as long as
it remains above the Friedman solution. This property, which is proved in the
key Proposition 4.14 below, allows us to propagate a lower bound for the
quantity $G$ to the left, preventing the formation of additional sonic points
and allowing us to extend the solution as far as the origin, $y=0$.

Step 4: The final step in the scheme is to show that the solution
$(\rho(y;\bar{y}_{*}),\omega(y;\bar{y}_{*}))$ connects smoothly to the origin.
More precisely, we show that the solution is analytic on $(0,\infty)$ and
$C^{1}$ at the scaling origin $y=0$. With a little extra work, one can show
that the solution is in fact smooth at $y=0$, but the argument is not included
in this paper, as it is not central to our proof of the existence of Yahil-
type solutions. This is achieved by exploiting again the monotonicity proved
for $\omega$ to demonstrate that $\omega(\cdot;\bar{y}_{*})$ attains the
boundary condition $\omega(0;\bar{y}_{*})=\frac{4-3\gamma}{3}$ and that the
density remains both bounded and monotone. This greatly simplifies the proof
of the equivalent step in [11] and removes the need for a topological upper-
and lower solution argument of the kind used in [11].

At three points throughout the proof (twice in the Taylor expansion at the
sonic point in Propositions 2.9 and 2.10 and then once more in extending the
solution to the right in the technical Lemma 3.4), we require an understanding
of the sign of key quantities depending polynomially on $\omega$ and $\gamma$.
As the quantities are significantly too complicated to control by hand, we
employ rigorous interval arithmetic, a means of computer-assisted proof that
has been used several times recently to resolve open questions in the theory
of PDE, see for example [2, 4, 9]. A useful overview of the method and its
applications, along with a wealth of references to recent applications, can be
found in [10]. Unlike in these works, our use of interval arithmetic is
elementary, as we perform most of the analysis directly, only employing
interval arithmetic to find bounds for the maxima and minima of certain
explicit polynomial quantities.

The paper is organised as follows. Details of the sonic point expansion, the
definition of the LPH-type solutions, and the local existence of real-analytic
solutions in the vicinity of the sonic point are presented in Section 2. In
Section 3 we show that for any $y_{\ast}\in[y_{f},y_{F}]$ there exists an LPH-
type solution on $[y_{\ast},\infty)$ and provide a detailed asymptotic
description of the solution as $y\to\infty$. Section 4 is devoted to the
existence problem to the left of the sonic point, and contains some of the key
conceptual insights of the paper. In particular, Proposition 4.17 shows that
there exists a $\bar{y}_{\ast}\in[y_{f},y_{F}]$ such that the associated local
LPH-type solution extends to the whole interval $(0,\bar{y}_{\ast}]$. The main
theorem is then easily obtained by gluing together the constructed left- and
right solutions, and the proof is presented in Section 5.

Several technical lemmas are stated and proved in Appendices A and B. Appendix
A contains the standard existence and uniqueness argument away from the sonic
points, while Appendix B contains the details of an involved combinatorial
argument used to prove the existence of real-analytic solution in a
neighbourhood of a sonic point. Several of our arguments in Sections 3 and 4
involve complicated multinomial expressions depending on $\gamma$,
$\omega_{0}$, and $y_{\ast}$. Their signs play a crucial role in the proofs
and we resort to rigorous, computer-assisted proofs by way of interval
arithmetic to check the relevant signs. Appendix C contains all the details of
such arguments including the associated interval arithmetic code. Finally,
Appendix D contains a detailed proof of some of the key continuity properties
of the LPH-type solutions, used heavily in Section 4. Such a proof is not
standard in the literature, but is quite similar to a related proof in [11],
and the details are therefore moved to an appendix.

Acknowledgments. Y. Guo’s research is supported in part by NSF DMS-grant
2106650. M. Hadžić’s and M. Schrecker’s research is supported by the EPSRC
Early Career Fellowship EP/S02218X/1. J. Jang’s research is supported by the
NSF DMS-grant 2009458 and the Simons Fellowship (grant number 616364).

##  2 The sonic point

As discussed in the introduction, our strategy for constructing a solution to
the system (1.13) is to begin from a sonic point $y_{*}$, obtain a solution
locally around this point, and then to extend to both the left and to the
right. The purpose of this section is to provide the solution locally around
the sonic point. This is a difficult endeavour, as it requires us to first
clarify how the condition of smoothness (in fact analyticity) at the sonic
point affects our definition of the solution we seek after. This will lead us
to the notion of the Larson-Penston-Hunter (LPH) branch. The next step
involves a combinatorial argument that shows that locally around the sonic
point there indeed exist analytic solutions of the LPH-type.

###  2.1 The formal Taylor expansion

Any smooth solution to the flow (1.13) must satisfy that, at any sonic point,
$y_{*}$, the values $\rho_{0}=\rho(y_{*})$, $\omega_{0}=\omega(y_{*})$ satisfy
the constraint

$h(\rho_{0},\omega_{0})=2\omega_{0}^{2}+\big{(}\gamma-1-\frac{4\pi\rho_{0}}{4-3\gamma}\big{)}\omega_{0}+(\gamma-1)(2-\gamma)=0.$
(2.27)

For notational reasons, we define

$f_{1}(\omega)=\frac{4-3\gamma}{4\pi\omega}\big{(}2\omega^{2}+(\gamma-1)\omega+(\gamma-1)(2-\gamma)\big{)},$
(2.28)

so that $h(\rho,\omega)=0$ corresponds to $\rho=f_{1}(\omega)$. The structure
of the level set $h(\rho,\omega)=0$, equivalently $\rho=f_{1}(\omega)$, will
play an important role, both in solving for the Taylor coefficients at the
sonic point (see Lemma 2.2 below), but also in demonstrating certain crucial
invariances along the flow in Section 4.2.

######  Lemma 2.1.

Let $\gamma\in(1,\frac{4}{3})$ and consider the function $f_{1}(\omega)$ on
the domain $\omega\in(0,2-\gamma)$. On this domain, $f_{1}$ is uniformly
convex with a global minimum at

$\omega_{*}=\sqrt{\frac{(\gamma-1)(2-\gamma)}{2}}.$ (2.29)

For $\gamma\in(1,\frac{10}{9})$, the inequality
$\omega_{*}<\frac{4-3\gamma}{3}$ holds while the inequality is reversed if
$\gamma\in(\frac{10}{9},\frac{4}{3})$ and equality holds at
$\gamma=\frac{10}{9}$.

In particular, $f_{1}^{\prime}(\omega_{*})=0$ and, if
$\gamma\in(1,\frac{10}{9}]$, we have $f_{1}^{\prime}(\omega)\geq 0$ for all
$\omega\in[\frac{4-3\gamma}{3},2-\gamma]$ (with strict inequality if at least
one of $\gamma<\frac{10}{9}$ or $\omega>\frac{4-3\gamma}{3}$ holds). If
$\gamma\in(\frac{10}{9},\frac{4}{3})$, then for
$\omega\in[\frac{4-3\gamma}{3},\omega_{*})$, $f_{1}^{\prime}(\omega)<0$ and
for $\omega\in(\omega_{*},2-\gamma]$, $f_{1}^{\prime}(\omega)>0$.

The proof is by a simple, direct calculation, and so we omit it.

Figure 3: $\\{h(\rho,\omega)=0\\}$ level sets for $\gamma=1.08$,
$\frac{10}{9}$, and $1.2$

Figure 3 plots the level set $h(\rho,\omega)=0$ in the $(\rho,\omega)$ plane
for the cases $\gamma=1.08,\frac{10}{9},1.2$ and
$\omega\in\big{[}\frac{4-3\gamma}{3},2-\gamma\big{]}$. The minimum for
$\gamma\geq\frac{10}{9}$ occurs at
$\omega=\sqrt{\frac{(\gamma-1)(2-\gamma)}{2}}\geq\frac{4-3\gamma}{3}$ as
stated in Lemma 2.1.

######  Lemma 2.2.

For any $y_{*}\in[y_{f},y_{F}]$, where $y_{f}$, $y_{F}$ are defined as in
(1.24), there exists a unique pair $(\rho_{0}(y_{*}),\omega_{0}(y_{*}))$ with
$\rho_{0}(y_{*})\geq\underline{\rho}>0$, where $\bar{\rho}$ depends only on
$\gamma$, satisfying

$G(y_{*},\rho_{0},\omega_{0})=0,\quad h(\rho_{0},\omega_{0})=0.$ (2.30)

Moreover, the mapping $y_{*}\mapsto\omega_{0}(y_{*})$ is a strictly decreasing
function for $y_{*}\in[y_{f},y_{F}]$ with

$\omega_{0}(y_{f})=2-\gamma,\quad\omega_{0}(y_{F})=\frac{4-3\gamma}{3}.$

###### Proof.

We begin by recalling the definition of the function $f_{1}(\omega)$ from
(2.28) and define also a function $f_{2}(\omega;y_{*})$ so that

$\displaystyle f_{1}(\omega)=$
$\displaystyle\,\frac{4-3\gamma}{4\pi\omega}\big{(}2\omega^{2}+(\gamma-1)\omega+(\gamma-1)(2-\gamma)\big{)},$
(2.31) $\displaystyle f_{2}(\omega;y_{*})=$
$\displaystyle\,\Big{(}\frac{y_{*}^{2}\omega^{2}}{\gamma}\Big{)}^{\frac{1}{\gamma-1}}.$

As discussed above, the constraint $h(\rho_{0},\omega_{0})=0$ is equivalent to
$\rho_{0}=f_{1}(\omega_{0})$, while we see easily that
$G(y_{*},\rho_{0},\omega_{0})=0$ if and only if
$\rho_{0}=f_{2}(\omega_{0};y_{*})$. So we seek $\omega_{0}(y_{*})$ such that
$f_{1}(\omega_{0}(y_{*}))=f_{2}(\omega_{0}(y_{*});y_{*})$. This value is then
defined to be $\rho_{0}(y_{*})$. We easily check that

$f_{1}(\frac{4-3\gamma}{3})=\frac{1}{6\pi},\quad
f_{1}(2-\gamma)=\frac{4-3\gamma}{2\pi}.$

Moreover, as $y_{f}$, $y_{F}$ are the sonic points corresponding to the far-
field and Friedman solutions, respectively, we also know that

$f_{2}(\frac{4-3\gamma}{3};y_{F})=f_{1}(\frac{4-3\gamma}{3})=\frac{1}{6\pi},\quad
f_{2}(2-\gamma;y_{f})=f_{1}(2-\gamma)=\frac{4-3\gamma}{2\pi}.$

Noting then that

$\partial_{\omega}f_{2}(\omega;y_{*}),\partial_{y_{*}}f_{2}(\omega;y_{*})>0,$
(2.32)

we see that for any $y_{*}\in(y_{f},y_{F})$, we have

$\displaystyle
f_{2}(\frac{4-3\gamma}{3};y_{*})<f_{2}(\frac{4-3\gamma}{3};y_{F})=\frac{1}{6\pi},\quad
f_{2}(2-\gamma;y_{*})>f_{2}(2-\gamma;y_{f})=\frac{4-3\gamma}{2\pi},$

and so

$\displaystyle(f_{2}(\cdot;y_{*})-f_{1})(\frac{4-3\gamma}{3})<0<(f_{2}(\cdot;y_{*})-f_{1})(2-\gamma).$

Thus, by the intermediate value theorem, we see that $\omega_{0}(y_{*})$
exists as required, and hence so does $\rho_{0}(y_{*})$. The uniqueness
follows from the following observations:

$\displaystyle f_{1}^{\prime}(\omega)=$
$\displaystyle\,-\frac{f_{1}(\omega)}{\omega}+\frac{4-3\gamma}{4\pi\omega}\big{(}4\omega+(\gamma-1)\big{)},$
$\displaystyle\partial_{\omega}f_{2}(\omega;y_{*})=$
$\displaystyle\,\frac{2\omega
y_{*}^{2}}{(\gamma-1)\gamma}\Big{(}\frac{y_{*}^{2}\omega^{2}}{\gamma}\Big{)}^{\frac{1}{\gamma-1}-1}=\frac{2f_{2}(\omega;y_{*})}{(\gamma-1)\omega}.$

Thus at any point $\omega>0$ such that $f_{2}(\omega;y_{*})\geq
f_{1}(\omega)$, we have

$\displaystyle\partial_{\omega}\big{(}f_{2}(\omega;y_{*})-f_{1}(\omega)\big{)}=$
$\displaystyle\,\frac{2f_{2}(\omega;y_{*})}{(\gamma-1)\omega}+\frac{f_{1}(\omega)}{\omega}-\frac{4-3\gamma}{4\pi\omega}\big{(}4\omega+(\gamma-1)\big{)}$
$\displaystyle\geq$
$\displaystyle\,\frac{1}{\omega}\Big{(}\frac{\gamma+1}{\gamma-1}f_{1}(\omega)-\frac{4-3\gamma}{4\pi}\big{(}4\omega+(\gamma-1)\big{)}\Big{)}$
$\displaystyle\geq$
$\displaystyle\,\frac{4-3\gamma}{4\pi\omega}\Big{(}\frac{\gamma+1}{\gamma-1}\big{(}2\omega+(\gamma-1)+\frac{(\gamma-1)(2-\gamma)}{\omega}\big{)}-\big{(}4\omega+(\gamma-1)\big{)}\Big{)}$
$\displaystyle\geq$
$\displaystyle\,\frac{4-3\gamma}{4\pi\omega}\Big{(}2\omega\frac{3-\gamma}{\gamma-1}+2+\frac{(\gamma+1)(2-\gamma)}{\omega}\Big{)}>0$

and so the uniqueness follows easily. The monotonicity properties of
$y_{*}\mapsto\omega_{0}(y_{*})$ then follow directly from (2.32) and
$f_{2,\omega}(\omega_{0}(y_{*}),y_{*})-f_{1,\omega}(\omega_{0}(y_{*}))>0$ as

$\displaystyle 0=$
$\displaystyle\,\frac{\operatorname{d}\\!}{\operatorname{d}\\!{y_{*}}}(f_{2}(\omega_{0}(y_{*});y_{*})-f_{1}(\omega_{0}(y_{*}))$
$\displaystyle=$
$\displaystyle\,\big{(}f_{2,\omega}(\omega_{0}(y_{*}),y_{*})-f_{1,\omega}(\omega_{0}(y_{*}))\big{)}\omega_{0}^{\prime}(y_{*})+f_{2,y_{*}}(\omega_{0}(y_{*}),y_{*}).$

To find the uniform lower bound $\rho_{0}\geq\underline{\rho}>0$, we note that
$\rho_{0}=f_{1}(\omega_{0})$. As
$\omega_{0}\in[\frac{4-3\gamma}{3},2-\gamma]$, we easily obtain
$f_{1}(\omega_{0})\geq\frac{(4-3\gamma)(\gamma-1)}{4\pi}>0$ as required. ∎

We seek a local solution around a sonic point $y_{*}\in[y_{f},y_{F}]$ via a
Taylor expansion. To that end, we now derive the necessary relations for the
coefficients of the expansion. Suppose we have an analytic solution of system
(1.13). Then, after rearranging, we have

$\displaystyle\big{(}\gamma\rho^{\gamma-1}-y^{2}\omega^{2}\big{)}\rho^{\prime}=$
$\displaystyle\,{(\gamma-1)y\rho(\omega+2-\gamma)}-{2y\rho\omega}\big{(}\frac{2\pi\rho}{4-3\gamma}-\omega\big{)},$
(2.33)
$\displaystyle\big{(}\gamma\rho^{\gamma-1}-y^{2}\omega^{2}\big{)}\omega^{\prime}=$
$\displaystyle\,\frac{4-3\gamma-3\omega}{y}\big{(}\gamma\rho^{\gamma-1}-y^{2}\omega^{2}\big{)}-{(\gamma-1)y\omega(\omega+2-\gamma)}+{2y\omega^{2}}\big{(}\frac{2\pi\rho}{4-3\gamma}-\omega\big{)}.$
(2.34)

We write the formal power series

$\rho(y)=\sum_{N=0}^{\infty}\rho_{N}(y-y_{*})^{N},\quad\omega(y)=\sum_{N=0}^{\infty}\omega_{N}(y-y_{*})^{N}.$
(2.35)

By Lemma 2.2, we see that a choice of $y_{*}\in[y_{f},y_{F}]$ defines a unique
pair of values $(\rho_{0},\omega_{0})$ for the Taylor series. We define the
obvious notation

$\displaystyle(\omega^{2})_{N}=\sum_{k+j=N}\omega_{k}\omega_{j},$
$\displaystyle\quad(\rho\omega)_{N}=\sum_{k+j=N}\rho_{k}\omega_{j},$ (2.36)
$\displaystyle(\rho^{2}\omega)_{N}=\sum_{k+j+l=N}\rho_{k}\rho_{j}\omega_{l},$
$\displaystyle\quad(\rho\omega^{2})_{N}=\sum_{k+j+l=N}\rho_{k}\omega_{j}\omega_{l}.$

We recall the Faà di Bruno formula for the $N$-th derivative of a composition,

$\frac{\operatorname{d}\\!^{N}}{\operatorname{d}\\!y^{N}}\big{(}f(g(y))\big{)}=\sum_{(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}}\frac{N!}{\lambda_{1}!\cdots\lambda_{N}!}f^{(\lambda_{1}+\cdots+\lambda_{N})}(g(y))\prod_{j=1}^{N}\Big{(}\frac{g^{(j)}(y)}{j!}\Big{)}^{\lambda_{j}},$
(2.37)

where

$M_{N}=\\{(\lambda_{1},\ldots,\lambda_{N})\in(\mathbb{Z}_{\geq
0})^{N}\,|\,\sum_{j=1}^{N}j\lambda_{j}=N\\}.$

Taking $f(x)=x^{\gamma-1}$, $g(y)=\rho(y)$ in this formula, we therefore
obtain

$\displaystyle\frac{\frac{\operatorname{d}\\!^{N}}{\operatorname{d}\\!y^{N}}\big{(}\rho^{\gamma-1}(y)\big{)}\big{|}_{y=y_{*}}}{N!}$
(2.38) $\displaystyle=\sum_{(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}}\frac{(\gamma-1)\cdots(\gamma-(\lambda_{1}+\cdots+\lambda_{N}))\rho_{0}^{\gamma-(\lambda_{1}+\cdots+\lambda_{N})-1}}{\lambda_{1}!\cdots\lambda_{N}!}\prod_{j=1}^{N}\rho_{j}^{\lambda_{j}}=:P_{N}$

and thus we have the power series

$\displaystyle\rho^{\gamma-1}=$
$\displaystyle\,\sum_{N=0}^{\infty}\frac{\frac{\operatorname{d}\\!^{N}}{\operatorname{d}\\!y^{N}}\big{(}\rho^{\gamma-1}(y)\big{)}\big{|}_{y=y_{*}}}{N!}(y-y_{*})^{N}=\sum_{N=0}^{\infty}P_{N}(y-y_{*})^{N}.$
(2.39)

Throughout this section, for $N<0$, we set coefficients $\rho_{N},\omega_{N}$
etc to be zero.

######  Lemma 2.3.

For each $N\geq 1$, the power series coefficients satisfy the relations

$\displaystyle\sum_{k+j=N}$ $\displaystyle(k+1)\rho_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle=$
$\displaystyle(\gamma-1)(2-\gamma)\big{(}y_{*}\rho_{N}+\rho_{N-1}\big{)}+(\gamma-1)\big{(}y_{*}(\rho\omega)_{N}+(\rho\omega)_{N-1}\big{)}$
(2.40)
$\displaystyle-\frac{4\pi}{4-3\gamma}\big{(}y_{*}(\rho^{2}\omega)_{N}+(\rho^{2}\omega)_{N-1}\big{)}+2\big{(}y_{*}(\rho\omega^{2})_{N}+(\rho\omega^{2})_{N-1}\big{)},$
$\displaystyle\sum_{k+j=N}$ $\displaystyle(k+1)\omega_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle=$
$\displaystyle\frac{4-3\gamma}{y_{*}}\sum_{k+j=N}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
(2.41)
$\displaystyle-\frac{3}{y_{*}}\sum_{k+j+l=N}\omega_{l}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle-(\gamma-1)(2-\gamma)\big{(}y_{*}\omega_{N}+\omega_{N-1}\big{)}-(\gamma-1)\big{(}y_{*}(\omega^{2})_{N}+(\omega^{2})_{N-1}\big{)}$
$\displaystyle+\frac{4\pi}{4-3\gamma}\big{(}y_{*}(\rho\omega^{2})_{N}+(\rho\omega^{2})_{N-1}\big{)}-2\big{(}y_{*}(\omega^{3})_{N}+(\omega^{3})_{N-1}\big{)}.$

###### Proof.

We begin the proof by noting the identities, for a general power series,

$\displaystyle y\sum_{N=0}^{\infty}b_{N}(y-y_{*})^{N}=$
$\displaystyle\,\sum_{N=0}^{\infty}\big{(}y_{*}b_{N}+b_{N-1}\big{)}(y-y_{*})^{N},$
$\displaystyle y^{2}\sum_{N=0}^{\infty}b_{N}(y-y_{*})^{N}=$
$\displaystyle\,\sum_{N=0}^{\infty}\big{(}y_{*}^{2}b_{N}+2y_{*}b_{N-1}+b_{N-2}\big{)}(y-y_{*})^{N},$

where we define $b_{N}=0$ for any $N<0$.

Step 1: Derive (2.40).
We begin by substituting the power series into (2.33). The left hand side of
this equation then becomes

$\displaystyle\Big{(}\sum_{N=0}^{\infty}\big{(}\gamma
P_{N}-y_{*}^{2}(\omega^{2})_{N}-2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}(y-y_{*})^{N}\Big{)}\Big{(}\sum_{N=0}^{\infty}(N+1)\rho_{N+1}(y-y_{*})^{N}\Big{)}$
(2.42)
$\displaystyle=\sum_{N=0}^{\infty}\Big{(}\sum_{k+j=N}(k+1)\rho_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}\Big{)}(y-y_{*})^{N}.$

The right hand side of (2.33) becomes

$\displaystyle(\gamma-1)\sum_{N=0}^{\infty}\Big{(}(2-\gamma)\big{(}y_{*}\rho_{N}+\rho_{N-1}\big{)}+y_{*}(\rho\omega)_{N}+(\rho\omega)_{N-1}\Big{)}(y-y_{*})^{N}$
(2.43)
$\displaystyle-2\sum_{N=0}^{\infty}\Big{(}\frac{2\pi}{4-3\gamma}\big{(}y_{*}(\rho^{2}\omega)_{N}+(\rho^{2}\omega)_{N-1}\big{)}-\big{(}y_{*}(\rho\omega^{2})_{N}+(\rho\omega^{2})_{N-1}\big{)}\Big{)}(y-y_{*})^{N}.$

Equating the $N$-th order terms of (LABEL:eq:powerseries1) and
(LABEL:eq:powerseries2), we have the claimed relation (2.40), that is, for all
$N\in\mathbb{N}\cup\\{0\\}$,

$\displaystyle\sum_{k+j=N}$ $\displaystyle(k+1)\rho_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
(2.44) $\displaystyle=$
$\displaystyle(\gamma-1)(2-\gamma)\big{(}y_{*}\rho_{N}+\rho_{N-1}\big{)}+(\gamma-1)\big{(}y_{*}(\rho\omega)_{N}+(\rho\omega)_{N-1}\big{)}$
$\displaystyle-\frac{4\pi}{4-3\gamma}\big{(}y_{*}(\rho^{2}\omega)_{N}+(\rho^{2}\omega)_{N-1}\big{)}+2\big{(}y_{*}(\rho\omega^{2})_{N}+(\rho\omega^{2})_{N-1}\big{)}.$

Step 2: Derive (2.41).
To prove from (2.34), we begin by expanding the term
$\frac{4-3\gamma-3\omega}{y}\big{(}\gamma\rho^{\gamma-1}-y^{2}\omega^{2}\big{)}$
by noting first that

$\frac{1}{y}=\frac{1}{y_{*}}\sum_{N=0}^{\infty}\frac{(-1)^{N}}{y_{*}^{N}}(y-y_{*})^{N}.$

Then we find

$\displaystyle\frac{4-3\gamma-3\omega}{y}$
$\displaystyle\big{(}\gamma\rho^{\gamma-1}-y^{2}\omega^{2}\big{)}$
$\displaystyle=$
$\displaystyle\Big{(}\frac{1}{y_{*}}\sum_{N=0}^{\infty}\frac{(-1)^{N}}{y_{*}^{N}}(y-y_{*})^{N}\Big{)}\Big{(}4-3\gamma-3\sum_{N=0}^{\infty}\omega_{N}(y-y_{*})^{N}\Big{)}$
$\displaystyle\times\Big{(}\sum_{N=0}^{\infty}\big{(}\gamma
P_{N}-y_{*}^{2}(\omega^{2})_{N}-2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}(y-y_{*})^{N}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{4-3\gamma}{y_{*}}\sum_{N=0}^{\infty}\Big{(}\sum_{k+j=N}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}\Big{)}(y-y_{*})^{N}$
$\displaystyle-\frac{3}{y_{*}}\sum_{N=0}^{\infty}\Big{(}\sum_{k+j+l=N}\omega_{l}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}\Big{)}(y-y_{*})^{N}.$

Thus, expanding (2.34) and equating terms of the same order, we find

$\displaystyle\sum_{k+j=N}$ $\displaystyle(k+1)\omega_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
(2.45) $\displaystyle=$
$\displaystyle\frac{4-3\gamma}{y_{*}}\sum_{k+j=N}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle-\frac{3}{y_{*}}\sum_{k+j+l=N}\omega_{l}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle-(\gamma-1)(2-\gamma)\big{(}y_{*}\omega_{N}+\omega_{N-1}\big{)}-(\gamma-1)\big{(}y_{*}(\omega^{2})_{N}+(\omega^{2})_{N-1}\big{)}$
$\displaystyle+\frac{4\pi}{4-3\gamma}\big{(}y_{*}(\rho\omega^{2})_{N}+(\rho\omega^{2})_{N-1}\big{)}-2\big{(}y_{*}(\omega^{3})_{N}+(\omega^{3})_{N-1}\big{)}.$

This is (2.41). ∎

Before studying the solvability of this system for the higher order
coefficients, we first collect a pair of identities satisfied by the first
order coefficients, $(\rho_{1},\omega_{1}$).

######  Lemma 2.4 (First order Taylor coefficients).

Let $\gamma\in(1,\frac{4}{3})$ and consider the formal Taylor expansion
(2.35). Let

$R:=\frac{y_{*}\rho_{1}}{\rho_{0}}\text{ and }\ W:=y_{*}\omega_{1}.$ (2.46)

Then the pair $(R,W)$ satisfies the following system of algebraic equations:

$\displaystyle(\gamma-1)\omega_{0}^{2}R^{2}-2\omega_{0}WR+(\gamma-1)(\omega_{0}+2-\gamma)R-2\omega_{0}W+(\gamma-1)(2-\gamma)\frac{W}{\omega_{0}}=0,$
(2.47) $\displaystyle
2\omega_{0}W^{2}-(\gamma-1)\omega_{0}^{2}RW+W\big{(}-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}$
$\displaystyle+\big{(}(5-3\gamma)\omega_{0}^{2}+(5-3\gamma)(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\omega_{0}R-2(4-3\gamma-3\omega_{0})\omega_{0}^{2}=0,$
(2.48)

with the additional constraint

$R\omega_{0}+W=4-3\gamma-3\omega_{0}.$ (2.49)

###### Proof.

In the case $N=1$, we note that $P_{1}=(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}$,
$(\omega^{2})_{1}=2\omega_{1}\omega_{0}$ etc to find from (2.40)

$\displaystyle\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}^{2}-2y_{*}^{2}\rho_{1}\omega_{1}\omega_{0}-2y_{*}\rho_{1}\omega_{0}^{2}$
(2.50)
$\displaystyle=(\gamma-1)(2-\gamma)(y_{*}\rho_{1}+\rho_{0})+(\gamma-1)(y_{*}\rho_{0}\omega_{1}+y_{*}\rho_{1}\omega_{0}+\rho_{0}\omega_{0})$
$\displaystyle-\frac{4\pi}{4-3\gamma}(y_{*}\rho_{0}^{2}\omega_{1}+2y_{*}\rho_{0}\omega_{0}\rho_{1}+\rho_{0}^{2}\omega_{0})+2(y_{*}\rho_{1}\omega_{0}^{2}+2y_{*}\rho_{0}\omega_{0}\omega_{1}+\rho_{0}\omega_{0}^{2})$
$\displaystyle=(\gamma-1)y_{*}\rho_{0}\omega_{1}-\frac{4\pi}{4-3\gamma}(y_{*}\rho_{0}^{2}\omega_{1}+y_{*}\rho_{0}\omega_{0}\rho_{1})+4y_{*}\rho_{0}\omega_{0}\omega_{1},$

where we have used (2.27) twice.

From (2.41) we get

$\displaystyle\omega_{1}$
$\displaystyle\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}^{2}\omega_{0}-2y_{*}\omega_{1}\omega_{0}^{2}$
(2.51) $\displaystyle=$
$\displaystyle\,\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{(}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}-(\gamma-1)(2-\gamma)(y_{*}\omega_{1}+\omega_{0})$
$\displaystyle-(\gamma-1)(2y_{*}\omega_{0}\omega_{1}+\omega_{0}^{2})+\frac{4\pi}{4-3\gamma}(y_{*}\omega_{0}^{2}\rho_{1}+2y_{*}\rho_{0}\omega_{0}\omega_{1}+\rho_{0}\omega_{0}^{2})-2(3y_{*}\omega_{0}^{2}\omega_{1}+\omega_{0}^{3})$
$\displaystyle=$
$\displaystyle\,\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{(}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}+\frac{4\pi}{4-3\gamma}y_{*}\rho_{1}\omega_{0}^{2}-2y_{*}\omega_{0}^{2}\omega_{1}$
$\displaystyle+(\gamma-1)(2-\gamma)y_{*}\omega_{1},$

where we have used (2.27) again. Rearranging (LABEL:eq:rho1quad1), we can use
(2.27) further to write

$\displaystyle 0=$
$\displaystyle\,\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}^{2}-2y_{*}^{2}\omega_{0}\omega_{1}\rho_{1}-2y_{*}\omega_{0}^{2}\rho_{1}-(\gamma-1)y_{*}\rho_{0}\omega_{1}$
$\displaystyle+\frac{4\pi}{4-3\gamma}y_{*}\rho_{0}^{2}\omega_{1}+\frac{4\pi}{4-3\gamma}y_{*}\rho_{0}\omega_{0}\rho_{1}-4y_{*}\rho_{0}\omega_{0}\omega_{1}$
$\displaystyle=$
$\displaystyle\,\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}^{2}-2y_{*}^{2}\omega_{0}\omega_{1}\rho_{1}+(\gamma-1)(\omega_{0}+2-\gamma)y_{*}\rho_{1}-2y_{*}\rho_{0}\omega_{0}\omega_{1}$
$\displaystyle+(\gamma-1)(2-\gamma)y_{*}\frac{\rho_{0}}{\omega_{0}}\omega_{1}.$

Thus, using also the sonic condition to replace
$\gamma\rho_{0}^{\gamma-2}=\frac{y_{*}^{2}\omega_{0}^{2}}{\rho_{0}}$ and
dividing through by $\rho_{0}$, we recall the definitions of $R$, $W$ and
arrive at

$(\gamma-1)\omega_{0}^{2}R^{2}-2\omega_{0}RW+(\gamma-1)(\omega_{0}+2-\gamma)R-2\omega_{0}W+(\gamma-1)(2-\gamma)\frac{W}{\omega_{0}}=0,$
(2.52)

that is, we have (2.47).

Working now from (2.51), we rearrange to find

$\displaystyle 0=$
$\displaystyle\,2\omega_{0}W^{2}-(\gamma-1)\omega_{0}^{2}RW+2\omega_{0}^{2}W+(4-3\gamma-3\omega_{0})\big{(}(\gamma-1)\omega_{0}^{2}R-2\omega_{0}W-2\omega_{0}^{2}\big{)}$
$\displaystyle+\frac{4\pi}{4-3\gamma}\rho_{0}\omega_{0}^{2}R-2\omega_{0}^{2}W+(\gamma-1)(2-\gamma)W$
$\displaystyle=$
$\displaystyle\,2\omega_{0}W^{2}+W\big{(}-(\gamma-1)\omega_{0}^{2}R-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}$
$\displaystyle+(4-3\gamma-3\omega_{0})(\gamma-1)\omega_{0}^{2}R+\omega_{0}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}R$
$\displaystyle-2(4-3\gamma-3\omega_{0})\omega_{0}^{2},$

which is exactly (2.48).

To show (2.49), we work from (1.21). Multiplying the first equation by $\rho$,
the second by $\omega$ and summing, we obtain

$(\rho\omega)^{\prime}=\frac{4-3\gamma-3\omega}{y}\rho.$

Substituting in the formal Taylor expansion and grouping the terms at order
zero, we find

$\rho_{1}\omega_{0}+\rho_{0}\omega_{1}=\frac{4-3\gamma-3\omega_{0}}{y_{*}}\rho_{0}.$

Multiplying through by $\frac{y_{*}}{\rho_{0}}$ and recalling (2.46) we arrive
at (2.49). ∎

######  Remark 2.5.

The coefficients of the quadratics in (2.47)–(2.48) depend only on $\gamma$
and on $\omega_{0}$ (hence also on $y_{*}$).

Our next lemma establishes the key recursive relation that will allow us to
compute the $N$-th order Taylor coefficients in terms of
$(\rho_{k},\omega_{k})$, $0\leq k\leq N-1$.

######  Lemma 2.6.

Let $N\geq 2$ and define the matrix $\mathcal{A}_{N}$ by

$\mathcal{A}_{N}=\begin{pmatrix}\mathcal{A}^{N}_{11}&\mathcal{A}^{N}_{12}\\\
\mathcal{A}^{N}_{21}&\mathcal{A}^{N}_{22},\end{pmatrix},$ (2.53)

where the matrix coefficients $\mathcal{A}^{N}_{ij}$, $i,j\in\\{1,2\\}$ depend
on $N$, $\gamma$, $\omega_{0}$, $\rho_{1}$ and $\omega_{1}$ and are given
explicitly by (2.67)–(2.70) below. Then the coefficients
$(\rho_{N},\omega_{N})$ in the formal series expansion (2.35) satisfy the
algebraic equation

$\mathcal{A}_{N}\begin{pmatrix}\rho_{N}\\\
\omega_{N}\end{pmatrix}=\begin{pmatrix}\mathcal{F}_{N}\\\
\mathcal{G}_{N}\end{pmatrix},$ (2.54)

where the polynomials $\mathcal{F}_{N}$ and $\mathcal{G}_{N}$ are given by
(2.58) and (2.65) below.

###### Proof.

We begin from (2.40) and group the terms on the left hand side as follows.

$\displaystyle N\rho_{N}\big{(}\gamma
P_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}+\rho_{1}\big{(}\gamma
P_{N}-2y_{*}^{2}\omega_{N}\omega_{0}\big{)}-\rho_{1}y_{*}^{2}\sum_{\begin{subarray}{c}j+k=N,\\\
j,k\neq N\end{subarray}}\omega_{j}\omega_{k}$ (2.55)
$\displaystyle-\rho_{1}\big{(}2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}+\sum_{\begin{subarray}{c}k+j=N\\\
j\neq 0,1,N\end{subarray}}(k+1)\rho_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle=\rho_{N}\Big{(}N\big{(}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}+\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}\Big{)}-2y_{*}^{2}\rho_{1}\omega_{0}\omega_{N}$
$\displaystyle\quad+\mathcal{F}_{N}^{I},$

where

$\displaystyle\mathcal{F}_{N}^{I}=$
$\displaystyle-\rho_{1}y_{*}^{2}\sum_{\begin{subarray}{c}j+k=N,\\\ j,k\neq
N\end{subarray}}\omega_{j}\omega_{k}-\rho_{1}\big{(}2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}$
$\displaystyle+\sum_{\begin{subarray}{c}k+j=N\\\ j\neq
0,1,N\end{subarray}}(k+1)\rho_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle+\gamma\rho_{1}\sum_{\begin{subarray}{c}(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}\\\
\lambda_{N}=0\end{subarray}}\frac{(\gamma-1)\cdots(\gamma-(\lambda_{1}+\cdots+\lambda_{N}))\rho_{0}^{\gamma-1-(\lambda_{1}+\cdots+\lambda_{N})}}{\lambda_{1}!\cdots\lambda_{N}!}\prod_{j=1}^{N}\rho_{j}^{\lambda_{j}},$

and we have applied the definition of $P_{j}$ to isolate the term with a
$\rho_{N}$ contribution as

$\displaystyle P_{N}=$
$\displaystyle\,(\gamma-1)\rho_{0}^{\gamma-2}\rho_{N}+\sum_{\begin{subarray}{c}(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}\\\
\lambda_{N}=0\end{subarray}}\frac{(\gamma-1)\cdots(\gamma-(\lambda_{1}+\cdots+\lambda_{N}))\rho_{0}^{\gamma-1-(\lambda_{1}+\cdots+\lambda_{N})}}{\lambda_{1}!\cdots\lambda_{N}!}\prod_{j=1}^{N}\rho_{j}^{\lambda_{j}}$

and also recalled

$P_{1}=(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}.$

Studying the right hand side of (2.40), we find expand to isolate terms at
order $N$ and then apply (2.27) to eliminate terms with factors of
$\frac{4\pi}{4-3\gamma}$ as follows:

$\displaystyle(\gamma-1)(2-\gamma)y_{*}\rho_{N}+(\gamma-1)y_{*}\rho_{N}\omega_{0}+(\gamma-1)y_{*}\rho_{0}\omega_{N}-2\frac{4\pi}{4-3\gamma}y_{*}\rho_{N}\omega_{0}\rho_{0}-\frac{4\pi}{4-3\gamma}y_{*}\rho_{0}^{2}\omega_{N}$
(2.56)
$\displaystyle+2y_{*}\rho_{N}\omega_{0}^{2}+4y_{*}\rho_{0}\omega_{0}\omega_{N}+(\gamma-1)(2-\gamma)\rho_{N-1}+(\gamma-1)\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j=N\\\
k\neq 0,N\end{subarray}}\rho_{k}\omega_{j}+(\rho\omega)_{N-1}\Big{)}$
$\displaystyle-\frac{4\pi}{4-3\gamma}\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j+l=N\\\
k,j,l\neq
N\end{subarray}}\rho_{k}\rho_{j}\omega_{l}+(\rho^{2}\omega)_{N-1}\Big{)}+2\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j+l=N\\\
k,j,l\neq
N\end{subarray}}\rho_{k}\omega_{j}\omega_{l}+(\rho\omega^{2})_{N-1}\Big{)}$
$\displaystyle=-\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}y_{*}\rho_{N}+2y_{*}\rho_{0}\omega_{0}\omega_{N}-(\gamma-1)(2-\gamma)y_{*}\frac{\rho_{0}}{\omega_{0}}\omega_{N}+\mathcal{F}_{N}^{II},$

where

$\displaystyle\mathcal{F}_{N}^{II}=$
$\displaystyle\,(\gamma-1)(2-\gamma)\rho_{N-1}+(\gamma-1)\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j=N\\\
k\neq 0,N\end{subarray}}\rho_{k}\omega_{j}+(\rho\omega)_{N-1}\Big{)}$
$\displaystyle-\frac{4\pi}{4-3\gamma}\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j+l=N\\\
k,j,l\neq
N\end{subarray}}\rho_{k}\rho_{j}\omega_{l}+(\rho^{2}\omega)_{N-1}\Big{)}+2\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j+l=N\\\
k,j,l\neq
N\end{subarray}}\rho_{k}\omega_{j}\omega_{l}+(\rho\omega^{2})_{N-1}\Big{)},$

where we have applied (2.27). Thus, as (LABEL:eq:rhoNeq1) is equal to
(LABEL:eq:rhoNeq2), we rearrange to arrive at

$\displaystyle\rho_{N}\Big{(}N\big{(}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}$
$\displaystyle\qquad+\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}+y_{*}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+\omega_{N}\Big{(}-2y_{*}^{2}\rho_{1}\omega_{0}-2y_{*}\rho_{0}\omega_{0}+(\gamma-1)(2-\gamma)y_{*}\frac{\rho_{0}}{\omega_{0}}\Big{)}$
$\displaystyle=\mathcal{F}_{N}^{II}-\mathcal{F}_{N}^{I}$ (2.57)
$\displaystyle=:\mathcal{F}_{N}(\rho_{0},\ldots,\rho_{N-1},\omega_{0},\ldots,\omega_{N-1})$
(2.58)

Thus we have found

$\displaystyle\rho_{N}\Big{(}(N+1)(\gamma-1)\frac{y_{*}^{2}\omega_{0}^{2}}{\rho_{0}}\rho_{1}-2Ny_{*}^{2}\omega_{1}\omega_{0}-2(N-1)y_{*}\omega_{0}^{2}+y_{*}\big{(}(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
(2.59)
$\displaystyle+\omega_{N}\Big{(}-2y_{*}^{2}\rho_{1}\omega_{0}-2y_{*}\rho_{0}\omega_{0}+(\gamma-1)(2-\gamma)y_{*}\frac{\rho_{0}}{\omega_{0}}\Big{)}$
$\displaystyle=\mathcal{F}_{N}.$

Considering now (2.41), we expand the left hand side as above as

$\displaystyle N\omega_{N}\big{(}\gamma
P_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}+\omega_{1}\big{(}\gamma
P_{N}-2y_{*}^{2}\omega_{N}\omega_{0}\big{)}-\omega_{1}y_{*}^{2}\sum_{\begin{subarray}{c}j+k=N,\\\
j,k\neq N\end{subarray}}\omega_{j}\omega_{k}$ (2.60)
$\displaystyle-\omega_{1}\big{(}2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}+\sum_{\begin{subarray}{c}k+j=N\\\
j\neq 0,1,N\end{subarray}}(k+1)\omega_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle=\omega_{N}\Big{(}N\big{(}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}-2y_{*}^{2}\omega_{1}\omega_{0}\Big{)}+\rho_{N}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\omega_{1}$
$\displaystyle\quad+\mathcal{G}_{N}^{I},$

where

$\displaystyle\mathcal{G}_{N}^{I}=-\omega_{1}\big{(}2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}-\omega_{1}y_{*}^{2}\sum_{\begin{subarray}{c}j+k=N,\\\
j,k\neq N\end{subarray}}\omega_{j}\omega_{k}$ (2.61)
$\displaystyle+\sum_{\begin{subarray}{c}k+j=N\\\ j\neq
0,1,N\end{subarray}}(k+1)\omega_{k+1}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle+\gamma\omega_{1}\sum_{\begin{subarray}{c}(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}\\\
\lambda_{N}=0\end{subarray}}\frac{(\gamma-1)\cdots(\gamma-(\lambda_{1}+\cdots+\lambda_{N}))\rho_{0}^{\gamma-1-(\lambda_{1}+\cdots+\lambda_{N})}}{\lambda_{1}!\cdots\lambda_{N}!}\prod_{j=1}^{N}\rho_{j}^{\lambda_{j}},$

where we have applied the definition of $P_{j}$ to isolate the term with a
$\rho_{N}$ contribution.

Working with the right hand side of (2.41), we have

$\displaystyle\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{(}\gamma
P_{N}-y_{*}^{2}(\omega^{2})_{N}\big{)}-(\gamma-1)(2-\gamma)y_{*}\omega_{N}-2(\gamma-1)y_{*}\omega_{N}\omega_{0}$
(2.62)
$\displaystyle+\frac{4\pi}{4-3\gamma}y_{*}\rho_{N}\omega_{0}^{2}+2\frac{4\pi}{4-3\gamma}y_{*}\rho_{0}\omega_{0}\omega_{N}-6y_{*}\omega_{0}^{2}\omega_{N}$
$\displaystyle+\widetilde{\mathcal{G}}_{N}^{II},$

where

$\displaystyle\widetilde{\mathcal{G}}_{N}^{II}=\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{(}-2y_{*}(\omega^{2})_{N-1}-(\omega^{2})_{N-2}\big{)}$
(2.63) $\displaystyle+\frac{4-3\gamma}{y_{*}}\sum_{\begin{subarray}{c}k+j=N\\\
j\neq N\end{subarray}}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}$
$\displaystyle-\frac{3}{y_{*}}\sum_{\begin{subarray}{c}k+j+l=N\\\ j\neq
N\end{subarray}}\omega_{l}\frac{(-1)^{k}}{y_{*}^{k}}\big{(}\gamma
P_{j}-y_{*}^{2}(\omega^{2})_{j}-2y_{*}(\omega^{2})_{j-1}-(\omega^{2})_{j-2}\big{)}-(\gamma-1)(2-\gamma)\omega_{N-1}$
$\displaystyle-(\gamma-1)\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j=N\\\ k\neq
0,N\end{subarray}}\omega_{k}\omega_{j}+(\omega^{2})_{N-1}\Big{)}+\frac{4\pi}{4-3\gamma}\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j+l=N\\\
k,j,l\neq
N\end{subarray}}(\rho_{k}\omega_{j}\omega_{l})+(\rho\omega^{2})_{N-1}\Big{)}$
$\displaystyle-2\Big{(}y_{*}\sum_{\begin{subarray}{c}k+j+l=N\\\ k,j,l\neq
N\end{subarray}}(\omega_{k}\omega_{j}\omega_{l})+(\omega^{3})_{N-1}\Big{)}.$

Grouping the terms on the first two lines here, we again expand $P_{N}$ to
find the contribution

$\displaystyle\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{(}\rho_{N}\gamma(\gamma-1)\rho_{0}^{\gamma-2}-2y_{*}^{2}\omega_{0}\omega_{N}\big{)}+(\gamma-1)(2-\gamma)y_{*}\omega_{N}-2y_{*}\omega_{0}^{2}\omega_{N}$
(2.64)
$\displaystyle+\frac{\omega_{0}}{\rho_{0}}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}y_{*}\rho_{N}-\frac{4-3\gamma-3\omega_{0}}{y_{*}}y_{*}^{2}\sum_{\begin{subarray}{c}j+k=N\\\
j,k\neq N\end{subarray}}\omega_{j}\omega_{k}$
$\displaystyle+\frac{4-3\gamma-3\omega_{0}}{y_{*}}\gamma\sum_{\begin{subarray}{c}(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}\\\
\lambda_{N}=0\end{subarray}}\frac{(\gamma-1)\cdots(\gamma-(\lambda_{1}+\cdots+\lambda_{N}))\rho_{0}^{\gamma-1-(\lambda_{1}+\cdots+\lambda_{N})}}{\lambda_{1}!\cdots\lambda_{N}!}\prod_{j=1}^{N}\rho_{j}^{\lambda_{j}},$

where we have again applied (2.27). Setting

$\displaystyle\mathcal{G}_{N}^{II}=$
$\displaystyle\,\widetilde{\mathcal{G}}_{N}^{II}-\frac{4-3\gamma-3\omega_{0}}{y_{*}}y_{*}^{2}\sum_{\begin{subarray}{c}j+k=N\\\
j,k\neq N\end{subarray}}\omega_{j}\omega_{k}$
$\displaystyle+\frac{4-3\gamma-3\omega_{0}}{y_{*}}\gamma\sum_{\begin{subarray}{c}(\lambda_{1},\ldots,\lambda_{N})\in
M_{N}\\\
\lambda_{N}=0\end{subarray}}\frac{(\gamma-1)\cdots(\gamma-(\lambda_{1}+\cdots+\lambda_{N}))\rho_{0}^{\gamma-1-(\lambda_{1}+\cdots+\lambda_{N})}}{\lambda_{1}!\cdots\lambda_{N}!}\prod_{j=1}^{N}\rho_{j}^{\lambda_{j}},$

we substitute (LABEL:eq:omNeq3) back into (LABEL:eq:omNeq2) and equate with
(LABEL:eq:omNeq1) to arrive at

$\displaystyle\omega_{N}\Big{(}N\big{(}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{*}^{2}\omega_{1}\omega_{0}-2y_{*}\omega_{0}^{2}\big{)}-2y_{*}^{2}\omega_{1}\omega_{0}\Big{)}+\rho_{N}\gamma(\gamma-1)\rho_{0}^{\gamma-2}\omega_{1}$
$\displaystyle-\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{(}\rho_{N}\gamma(\gamma-1)\rho_{0}^{\gamma-2}-2y_{*}^{2}\omega_{0}\omega_{N}\big{)}-(\gamma-1)(2-\gamma)y_{*}\omega_{N}+2y_{*}\omega_{0}^{2}\omega_{N}$
$\displaystyle-\frac{\omega_{0}}{\rho_{0}}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}y_{*}\rho_{N}$
$\displaystyle=\mathcal{G}_{N}^{II}-\mathcal{G}_{N}^{I}$
$\displaystyle=:\mathcal{G}_{N}.$ (2.65)

Thus we have

$\displaystyle\omega_{N}\Big{(}N(\gamma-1)\frac{y_{*}^{2}\omega_{0}^{2}}{\rho_{0}}\rho_{1}-2(N+1)y_{*}^{2}\omega_{1}\omega_{0}-2(N+2)y_{*}\omega_{0}^{2}+2(4-3\gamma)y_{*}\omega_{0}-(\gamma-1)(2-\gamma)y_{*}\Big{)}$
(2.66)
$\displaystyle+\rho_{N}\Big{(}\frac{y_{*}^{2}\omega_{0}^{2}}{\rho_{0}}(\gamma-1)\big{(}\omega_{1}-\frac{4-3\gamma-3\omega_{0}}{y_{*}}\big{)}-y_{*}\frac{\omega_{0}}{\rho_{0}}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle=\mathcal{G}_{N}.$

So we have found the claimed identity with

$\displaystyle\mathcal{A}_{11}=y_{*}\Big{(}(N+1)(\gamma-1)\omega_{0}^{2}\frac{y_{*}\rho_{1}}{\rho_{0}}-2Ny_{*}\omega_{1}\omega_{0}-2(N-1)\omega_{0}^{2}+\big{(}(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)},$
(2.67)
$\displaystyle\mathcal{A}_{12}=y_{*}\rho_{0}\Big{(}-2\frac{y_{*}\rho_{1}}{\rho_{0}}\omega_{0}-2\omega_{0}+\frac{(\gamma-1)(2-\gamma)}{\omega_{0}}\Big{)},$
(2.68)
$\displaystyle\mathcal{A}_{21}=\frac{y_{*}}{\rho_{0}}\Big{(}\omega_{0}^{2}(\gamma-1)\big{(}y_{*}\omega_{1}-(4-3\gamma-3\omega_{0})\big{)}-\omega_{0}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)},$
(2.69)
$\displaystyle\mathcal{A}_{22}=y_{*}\Big{(}N(\gamma-1)\omega_{0}^{2}\frac{y_{*}\rho_{1}}{\rho_{0}}-2(N+1)y_{*}\omega_{1}\omega_{0}-2(N+2)\omega_{0}^{2}+2(4-3\gamma)\omega_{0}-(\gamma-1)(2-\gamma)\Big{)}.$
(2.70)

∎

######  Lemma 2.7.

Consider the formal series expansion (2.35) and recall the definitions

$R=\frac{y_{*}\rho_{1}}{\rho_{0}}\text{ and }W=y_{*}\omega_{1}.$ (2.71)

Then the map $N\mapsto\det(\mathcal{A}_{N})$ is a quadratic polynomial of the
form

$\det\mathcal{A}_{N}=\sum_{j=0}^{2}A_{j}N^{j},$ (2.72)

where $A_{0}$, $A_{1}$, and $A_{2}$ are $(\gamma,\omega_{0},R,W)$-dependent
functions given by the formulas:

$\displaystyle A_{2}=$
$\displaystyle\,\big{(}-2(3-\gamma)\omega_{0}^{2}+\omega_{0}(\gamma-1)(5\gamma-9)-(\gamma-1)(2-\gamma)(\gamma+1)\big{)}\omega_{0}^{2}R$
$\displaystyle+8\omega_{0}^{3}W+4\omega_{0}^{4},$ (2.73) $\displaystyle
A_{1}=$
$\displaystyle\,-\big{(}2(3-\gamma)\omega_{0}^{2}+2\omega_{0}(\gamma-1)+(\gamma-1)(2-\gamma)(\gamma+1)\big{)}\omega_{0}^{2}R$
$\displaystyle+\big{(}8\omega_{0}^{2}-4(4-3\gamma)\omega_{0}-2(\gamma-1)\omega_{0}\big{)}\omega_{0}W+\big{(}4\omega_{0}^{4}-14\omega_{0}^{3}+10\gamma\omega_{0}^{3}\big{)},$
(2.74) $\displaystyle A_{0}=$
$\displaystyle\,2\big{(}\omega_{0}^{2}(\gamma-1)-\omega_{0}(\gamma-1)-\gamma(\gamma-1)(2-\gamma)\big{)}\omega_{0}^{2}R$
$\displaystyle+\big{(}-16\omega_{0}^{2}+4(\gamma-1)\omega_{0}^{2}+4(4-3\gamma)\omega_{0}-2(\gamma-1)\omega_{0}-2(\gamma-1)(2-\gamma)(\gamma+1)\big{)}\omega_{0}W$
$\displaystyle+(6\gamma-30)\omega_{0}^{4}+(6\gamma^{2}-44\gamma+46)\omega_{0}^{3}+(3\gamma^{3}-12\gamma^{2}+11\gamma-2)\omega_{0}^{2}$
$\displaystyle+(3\gamma^{4}-10\gamma^{3}+5\gamma^{2}+10\gamma-8)\omega_{0}$
(2.75)

###### Proof.

We begin with the following identity. Multiplying (2.47) by
$(\gamma-1)\omega_{0}^{2}$ and (2.48) by $2\omega_{0}$ and summing, we get

$\displaystyle(\gamma-1)^{2}\omega_{0}^{4}R^{2}-4(\gamma-1)\omega_{0}^{3}RW+4\omega_{0}^{2}W^{2}$
(2.76)
$\displaystyle=-(\gamma-1)^{2}\omega_{0}^{2}(\omega_{0}+2-\gamma)R+2(\gamma-1)\omega_{0}^{3}W-(\gamma-1)^{2}(2-\gamma)W\omega_{0}$
$\displaystyle-2\omega_{0}W\big{(}-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}$
$\displaystyle-2\big{(}(5-3\gamma)\omega_{0}^{2}+(5-3\gamma)(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\omega_{0}^{2}R+4(4-3\gamma-3\omega_{0})\omega_{0}^{3}$
$\displaystyle=\omega_{0}^{2}R\Big{(}-(\gamma-1)^{2}(\omega_{0}+2-\gamma)-2\big{(}(5-3\gamma)\omega_{0}^{2}+(5-3\gamma)(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+\omega_{0}W\Big{(}2(\gamma-1)\omega_{0}^{2}-(\gamma-1)^{2}(2-\gamma)-2\big{(}-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+4(4-3\gamma-3\omega_{0})\omega_{0}^{3}.$

Now we expand the determinant as

$\displaystyle y_{*}^{-2}$ $\displaystyle\det\mathcal{A}$ $\displaystyle=$
$\displaystyle\,\Big{(}(N+1)(\gamma-1)\omega_{0}^{2}R-2NW\omega_{0}-2(N-1)\omega_{0}^{2}+\big{(}(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle\>\times\Big{(}N(\gamma-1)\omega_{0}^{2}R-2(N+1)W\omega_{0}-2(N+2)\omega_{0}^{2}+2(4-3\gamma)\omega_{0}-(\gamma-1)(2-\gamma)\Big{)}$
$\displaystyle-\Big{(}-2R\omega_{0}-2\omega_{0}+\frac{(\gamma-1)(2-\gamma)}{\omega_{0}}\Big{)}$
$\displaystyle\>\times\Big{(}\omega_{0}^{2}(\gamma-1)\big{(}W-(4-3\gamma-3\omega_{0})\big{)}-\omega_{0}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\,N(N+1)(\gamma-1)^{2}\omega_{0}^{4}R^{2}-\big{(}2N^{2}+2(N+1)^{2}\big{)}(\gamma-1)\omega_{0}^{3}RW+4N(N+1)\omega_{0}^{2}W^{2}$
$\displaystyle+R\Big{(}(N+1)(\gamma-1)\omega_{0}^{2}\big{(}-2(N+2)\omega_{0}^{2}+2(4-3\gamma)\omega_{0}-(\gamma-1)(2-\gamma)\big{)}$
$\displaystyle\hskip
25.60747pt+N(\gamma-1)\omega_{0}^{2}\big{(}-2(N-1)\omega_{0}^{2}+\big{(}(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\big{)}\Big{)}$
$\displaystyle+W\Big{(}-2N\omega_{0}\big{(}-2(N+2)\omega_{0}^{2}+2(4-3\gamma)\omega_{0}-(\gamma-1)(2-\gamma)\big{)}$
$\displaystyle\hskip
25.60747pt-2(N+1)\omega_{0}\big{(}-2(N-1)\omega_{0}^{2}+\big{(}(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\big{)}\Big{)}$
(2.77)
$\displaystyle+\Big{(}-2(N-1)\omega_{0}^{2}+\big{(}(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle\quad\times\Big{(}-2(N+2)\omega_{0}^{2}+2(4-3\gamma)\omega_{0}-(\gamma-1)(2-\gamma)\Big{)}$
$\displaystyle+2\omega_{0}^{3}(\gamma-1)RW$
$\displaystyle+R\Big{(}-2\omega_{0}^{3}(\gamma-1)(4-3\gamma-3\omega_{0})-\omega_{0}^{2}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+W\Big{(}-\omega_{0}^{2}(\gamma-1)\big{(}-2\omega_{0}+(\gamma-1)(2-\gamma)\frac{1}{\omega_{0}}\big{)}\Big{)}$
$\displaystyle+\big{(}2\omega_{0}-(\gamma-1)(2-\gamma)\frac{1}{\omega_{0}}\big{)}$
$\displaystyle\quad\times\Big{(}-\omega_{0}^{2}(\gamma-1)(4-3\gamma-3\omega_{0})-\omega_{0}\big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}.$

We first re-group the quadratic terms in $(R,W)$ and substitute
(LABEL:eq:quadsubstitute) to get

$\displaystyle N($ $\displaystyle
N+1)(\gamma-1)^{2}\omega_{0}^{4}R^{2}-\big{(}2N^{2}+2(N+1)^{2}-2\big{)}(\gamma-1)\omega_{0}^{3}RW+4N(N+1)\omega_{0}^{2}W^{2}$
$\displaystyle=$ $\displaystyle
N(N+1)\big{(}(\gamma-1)^{2}\omega_{0}^{4}R^{2}-4(\gamma-1)\omega_{0}^{3}RW+4\omega_{0}^{2}W^{2}\big{)}$
$\displaystyle=$ $\displaystyle N(N+1)$
$\displaystyle\times\Big{(}\omega_{0}^{2}R\Big{(}-(\gamma-1)^{2}(\omega_{0}+2-\gamma)-2\big{(}(5-3\gamma)\omega_{0}^{2}+(5-3\gamma)(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle\quad+\omega_{0}W\Big{(}2(\gamma-1)\omega_{0}^{2}-(\gamma-1)^{2}(2-\gamma)-2\big{(}-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle\quad+4(4-3\gamma-3\omega_{0})\omega_{0}^{3}\Big{)}.$

Substituting this into (2.77), we group the terms by order in $N$ as

$\displaystyle\det$
$\displaystyle\mathcal{A}_{N}=y_{*}^{2}\Big{(}A_{2}N^{2}+A_{1}N+A_{0}\Big{)},$
(2.78)

where

$\displaystyle A_{2}=$
$\displaystyle\,\omega_{0}^{2}R\Big{(}-(\gamma-1)^{2}(\omega_{0}+2-\gamma)-2\big{(}(5-3\gamma)\omega_{0}^{2}+(5-3\gamma)(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+\omega_{0}W\Big{(}2(\gamma-1)\omega_{0}^{2}-(\gamma-1)^{2}(2-\gamma)-2\big{(}-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+4(4-3\gamma-3\omega_{0})\omega_{0}^{3}-4(\gamma-1)\omega_{0}^{4}R+8\omega_{0}^{3}W+4\omega_{0}^{4},$
$\displaystyle A_{1}=$
$\displaystyle\,\omega_{0}^{2}R\Big{(}-(\gamma-1)^{2}(\omega_{0}+2-\gamma)-2\big{(}(5-3\gamma)\omega_{0}^{2}+(5-3\gamma)(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+\omega_{0}W\Big{(}2(\gamma-1)\omega_{0}^{2}-(\gamma-1)^{2}(2-\gamma)-2\big{(}-2(4-3\gamma-3\omega_{0})\omega_{0}+(\gamma-1)(2-\gamma)\big{)}\Big{)}$
$\displaystyle+4(4-3\gamma-3\omega_{0})\omega_{0}^{3}-4(\gamma-1)\omega_{0}^{4}R+(\gamma-1)\omega_{0}^{2}R\big{(}2(4-3\gamma)\omega_{0}+(\gamma-1)\omega_{0}\big{)}$
$\displaystyle-8\omega_{0}^{3}W-2\omega_{0}W\big{(}2(4-3\gamma)\omega_{0}+(\gamma-1)\omega_{0}\big{)}$
$\displaystyle+2\omega_{0}^{2}\Big{(}4\omega_{0}^{2}-2(4-3\gamma)\omega_{0}+(\gamma-1)(2-\gamma)\Big{)}-2\omega_{0}^{2}\Big{(}2\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\Big{)}$

and $A_{0}$ is the remainder. Simplifying these expressions and that for
$A_{0}$ results in (2.73)–(2.75) to conclude the proof. ∎

###  2.2 Branch selection

To find solutions that are smooth through the sonic point, we must first
calculate the first order Taylor coefficients $(\rho_{1},\omega_{1})$ as
functions of the parameters $\gamma$ and $y_{\ast}$.

######  Lemma 2.8 (The two solution branches).

Let $\gamma\in(1,\frac{4}{3})$ be given and let
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$. There exist exactly two pairs
$(R_{i},W_{i})$, $i=1,2$ solving the system of algebraic equations (2.47),
(2.48), (2.49). The functions $R_{i}$ are given by

$\displaystyle
R_{1}=\frac{(9-7\gamma)\omega_{0}^{2}-8\omega_{0}^{3}-\sqrt{\omega_{0}^{3}s(\omega_{0})}}{2\omega_{0}^{3}(\gamma+1)},$
(2.79) $\displaystyle
R_{2}=\frac{(9-7\gamma)\omega_{0}^{2}-8\omega_{0}^{3}+\sqrt{\omega_{0}^{3}s(\omega_{0})}}{2\omega_{0}^{3}(\gamma+1)},$
(2.80)

where

$\displaystyle s(\omega_{0})=$
$\displaystyle\,-4(4-3\gamma)(\gamma+1)(\gamma-1)(2-\gamma)+\big{(}(\gamma-1)(\gamma^{2}-5\gamma+5)+\gamma^{2}+6\gamma-3\big{)}\omega_{0}$
(2.81)
$\displaystyle-8(3\gamma^{2}-15\gamma+14)\omega_{0}^{2}+8(5-3\gamma)\omega_{0}^{3}$

is strictly positive for all $\omega_{0}\in[\frac{4-3\gamma}{3},2-\gamma]$,
$\gamma\in(1,\frac{4}{3})$.
For any $i=1,2$, $W_{i}$ is determined by $R_{i}$ through the formula

$W_{i}=4-3\gamma-3\omega_{0}-\omega_{0}R_{i}.$ (2.82)

###### Proof.

By rearranging (2.47), we see

$\Big{(}2\omega_{0}(R+1)-\frac{(\gamma-1)(2-\gamma)}{\omega_{0}}\Big{)}W=(\gamma-1)\omega_{0}^{2}R^{2}+(\gamma-1)(\omega_{0}+2-\gamma)R.$
(2.83)

Rearranging (2.49) to solve for $W$ as

$W=4-3\gamma-3\omega_{0}-\omega_{0}R,$ (2.84)

we obtain the claimed relation (2.82). We then substitute this into (2.83) to
obtain the following quadratic for $R$:

$\displaystyle\Big{(}2\omega_{0}(R+1)-\frac{(\gamma-1)(2-\gamma)}{\omega_{0}}\Big{)}\Big{(}4-3\gamma-3\omega_{0}-\omega_{0}R\Big{)}=(\gamma-1)\omega_{0}^{2}R^{2}+(\gamma-1)(\omega_{0}+2-\gamma)R$
(2.85)

with roots $R_{1}$, $R_{2}$ as claimed in (2.79)–(2.80) from the quadratic
formula. We postpone the verification that $s(\omega_{0})>0$ to Appendix C.1.

One can check that equation (2.48) is also satisfied by these roots by
similarly substituting (2.82) into (2.48) then simplifying. This again yields
a quadratic in $R$ which, on inspection, turns out to be exactly (2.85) up to
a factor of $\omega_{0}$, and hence has the same roots. ∎

We will see in the following Subsection 2.3 that the physically relevant
solution branch is that given by $(R_{1},W_{1})$. We therefore collect some
useful estimates on the coefficients derived from this branch.

######  Proposition 2.9.

Let $\gamma\in(1,\frac{4}{3})$ be given and let
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$ and consider the branch
$(R_{1},W_{1})$ defined in Lemma 2.8. Then

$-\frac{4}{(4-3\gamma)(2-\gamma)}<R_{1}<-\frac{1}{2-\gamma}.$ (2.86)

Moreover, if $\gamma\in[\frac{10}{9},\frac{4}{3})$, then the upper bound on
$R_{1}$ may be taken to satisfy

$R_{1}\leq-\frac{2\gamma}{(2-\gamma)(\gamma+1)},$ (2.87)

where the inequality is strict provided either $\gamma>\frac{10}{9}$ or
$y_{*}<y_{F}$.

Finally,

$W_{1}>0\text{ for }y_{*}>y_{f}\text{ and }W_{1}|_{y_{*}=y_{f}}=0.$ (2.88)

###### Proof.

The proof relies in part on interval arithmetic and it is presented in detail
in Appendix C.1. ∎

######  Proposition 2.10 (Positivity of $\det\mathcal{A}_{N}$).

Let $\gamma\in(1,\frac{4}{3})$ be given and let
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$. Let $A_{0},A_{1},A_{2}$ be
functions of $\rho_{0},\omega_{0},R,W$ given by (2.73)–(2.75) and assume that
$R=R_{1}$ and $W=W_{1}$, where the branch $(R_{1},W_{1})$ is defined in Lemma
2.8.

  1. (i)

The following inequalities hold:

$\displaystyle A_{2}$ $\displaystyle>0,$ (2.89) $\displaystyle 4A_{2}+A_{1}$
$\displaystyle>0,$ (2.90) $\displaystyle 4A_{2}+2A_{1}+A_{0}$
$\displaystyle>0$ (2.91)

  2. (ii)

There exist constants $c_{1},c_{2}>0$, depending only on $\gamma$, so that

$c_{1}N^{2}\leq\det\mathcal{A}_{N}\leq c_{2}N^{2},\ \ N\geq 2.$ (2.92)

In particular, the matrix $\mathcal{A}_{N}$ is invertible for all $N\geq 2$
and the formal Taylor coefficients $(\rho_{N},\omega_{N})$ are well-defined
through the formula

$\begin{pmatrix}\rho_{N}\\\
\omega_{N}\end{pmatrix}=\mathcal{A}_{N}^{-1}\begin{pmatrix}\mathcal{F}_{N}\\\
\mathcal{G}_{N}\end{pmatrix},\ \ N\geq 2,$ (2.93)

where the source terms $\mathcal{F}_{N}$, $\mathcal{G}_{N}$ are defined in
Lemma 2.6.

  3. (iii)

There exists a constant $\beta_{0}=\beta_{0}(y_{\ast},\gamma)>0$ such that

$\displaystyle|\rho_{N}|\leq\frac{\beta_{0}}{N}\left(|\mathcal{F}_{N}|+\frac{1}{N}|\mathcal{G}_{N}|\right)$
(2.94)
$\displaystyle|\omega_{N}|\leq\frac{\beta_{0}}{N}\left(|\mathcal{G}_{N}|+\frac{1}{N}|\mathcal{F}_{N}|\right).$
(2.95)

###### Proof.

Proof of part (i). The proof of (2.89)–(2.91) relies on interval arithmetic
and it is presented in detail in Appendix C.2.

Proof of part (ii). Since
$\det\mathcal{A}_{2}=y_{*}^{2}\big{(}4A_{2}+2A_{1}+A_{0}\big{)}>0$ by (2.91)
and, for $N\geq 2$,
$\frac{\operatorname{d}\\!}{\operatorname{d}\\!N}\det\mathcal{A}_{N}=y_{*}^{2}\big{(}2NA_{2}+A_{1}\big{)}\geq
y_{*}^{2}\big{(}4A_{2}+A_{1}\big{)}$, it follows from (2.90)–(2.91) that

$\displaystyle\det\mathcal{A}_{2}=y_{*}^{2}\big{(}4A_{2}+2A_{1}+A_{0}\big{)}>0,$
(2.96)
$\displaystyle\frac{\operatorname{d}\\!}{\operatorname{d}\\!N}\det\mathcal{A}_{N}=y_{*}^{2}\big{(}2NA_{2}+A_{1}\big{)}\geq
y_{*}^{2}\big{(}4A_{2}+A_{1}\big{)}>0.$ (2.97)

These estimates then easily imply (2.92). Claim (2.93) is an obvious
consequence of the invertibility of $\mathcal{A}_{N}$ and Lemma 2.6.

Proof of part (iii). From (2.93) it follows that

$\displaystyle\rho_{N}=\frac{\mathcal{A}_{22}}{\det\mathcal{A}_{N}}\mathcal{F}_{N}-\frac{\mathcal{A}_{12}}{\det\mathcal{A}_{N}}\mathcal{G}_{N},$
(2.98)
$\displaystyle\omega_{N}=\frac{\mathcal{A}_{11}}{\det\mathcal{A}_{N}}\mathcal{G}_{N}-\frac{\mathcal{A}_{21}}{\det\mathcal{A}_{N}}\mathcal{F}_{N},$
(2.99)

and thus (2.94)–(2.95) follow directly from (2.92) and (2.67)–(2.70).

∎

###  2.3 Larson-Penston-Hunter- (LPH-) type solutions

In order to distinguish the relevant solution branch for the first order
Taylor coefficients, we compare directly to the situation in the case
$\gamma=1$.

######  Lemma 2.11.

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in[y_{f}(\gamma),y_{F}(\gamma)]$, and
consider the functions $R_{i}$, $i=1,2$ as in Lemma 2.8 as functions of both
$\omega_{0}(y_{*})$ and $\gamma$. As $\gamma\to 1$, these coefficients satisfy
the limits

$\displaystyle R_{1}(\omega_{0})\to$
$\displaystyle\,\frac{1-4\omega_{0}-|1-2\omega_{0}|}{2\omega_{0}},$ (2.100)
$\displaystyle R_{2}(\omega_{0})\to$
$\displaystyle\,\frac{1-4\omega_{0}+|1-2\omega_{0}|}{2\omega_{0}}.$

###### Proof.

The identities for the limit as $\gamma\to 1^{+}$ for $R_{i}$ follow directly
from the identities (2.79)–(2.80). ∎

Thus, to maintain compatibility with the LP solution in the case $\gamma=1$,
we note that, in that case, the sonic point lies in the interval $(2,3)$ with
$\frac{\rho^{\prime}(y_{*})y_{*}}{\rho(y_{*})}=-1$ (compare [11]), and hence
the LP-type branch, for $\gamma>1$, should be chosen to be the $1$-branch. In
this case, we find that the limit of $W_{1}(\omega_{0})$ as $\gamma\to 1$ is
$1-2\omega_{0}$, again in compatibility with the $\gamma=1$ case resolved in
[11].

This motivates the following definition.

######  Definition 2.12 (Larson-Penston-Hunter (LPH) type solutions).

Let $\gamma\in(1,\frac{4}{3})$ be given and let
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$. We say that a sequence
$(\rho_{N},\omega_{N})$, $N\in\mathbb{N}$ associated with a formal power
series expansion

$\displaystyle\rho(y)=\sum_{N=0}^{\infty}\rho_{N}(y-y_{*})^{N},\quad\omega(y)=\sum_{N=0}^{\infty}\omega_{N}(y-y_{*})^{N},$
(2.101)

is of Larson-Penston-Hunter (LPH) type if the following properties are
satisfied

  1. (i)

$G(y_{*},\rho_{0},\omega_{0})=0,\quad h(\rho_{0},\omega_{0})=0.$ (2.102)

  2. (ii)

$\rho_{1}:=\frac{\rho_{0}R_{1}}{y_{\ast}},\ \ \ \
\omega_{1}:=\frac{W_{1}}{y_{*}},$ (2.103)

where the pair $(R_{1},W_{1})$ corresponds to the branch defined by (2.79) and
(2.82) from Lemma 2.8.

  3. (iii)

For any $N\geq 2$, the coefficients $(\rho_{N},\omega_{N})$ satisfy the
recursive relation (2.93).

If the series (2.101) converge, we say that the functions $\rho$ and $\omega$
are of LPH-type.

######  Remark 2.13.

As shown in Proposition 2.10, the matrix $\mathcal{A}_{N}$ defined in Lemma
2.6 is indeed invertible for all $N\geq 2$ and therefore for any LPH-type
sequence the coefficients $(\rho_{N},\omega_{N})$, $N\geq 2$ are therefore
uniquely determined as functions of $\rho_{0},\omega_{0},\rho_{1},\omega_{1}$.

###  2.4 The induction argument and the series convergence

In order to prove the convergence of the formal power series (2.35) we prove
the crucial lemma, which establishes favourable growth bounds for the
coefficients $(\rho_{N},\omega_{N})$. The proof is based on involved
combinatorial arguments that are presented in Appendix B, culminating in Lemma
B.6.

######  Lemma 2.14.

Let $\gamma\in(1,\frac{4}{3})$ and $\alpha\in(1,2)$ be given. Let
$(\rho_{N},\omega_{N})$, $N\in\mathbb{N}$ be the coefficients in the formal
Taylor expansion of $\rho,\omega$ about $y=y_{\ast}$ given by Proposition
2.10. Then there exists a constant $C>1$ such that for any
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$ the bounds

$\displaystyle\left|\rho_{N}\right|\leq\frac{C^{N-\alpha}}{N^{3}},$ (2.104)
$\displaystyle\left|\omega_{N}\right|\leq\frac{C^{N-\alpha}}{N^{3}},$ (2.105)

hold for all $N\geq 2$.

###### Proof.

We use mathematical induction to prove the lemma. When $N=2$ clearly there
exists a constant $\bar{C}=\bar{C}(y_{\ast},\alpha)>0$ such that the claimed
bounds hold true as the recursive relation (2.93) defining
$(\rho_{2},\omega_{2})$ involves only products of continuous functions
composed with $(y_{*},\rho_{0},\omega_{0},\rho_{1},\omega_{1})$, all of which
are bounded.

Suppose now that for some $N\geq 3$, (2.104)–(2.105) hold for all $2\leq m\leq
N-1$. This implies that the assumptions (B.227)–(B.228) hold true and thus by
Lemma B.6 we conclude that (B.229)–(B.230) hold. Therefore, from Proposition
2.10 and (B.229)–(B.230) we obtain

$\displaystyle\left|\rho_{N}\right|\leq\frac{c\beta_{0}\beta
C^{N-\alpha}}{N^{3}}\left(\frac{1}{C^{\alpha-1}}+\frac{1}{C^{2-\alpha}}+\frac{1}{CN}\right),$
(2.106)

for some universal constant $c>0$. Similarly,

$\displaystyle\left|\omega_{N}\right|\leq\frac{c\beta_{0}\beta
C^{N-\alpha}}{N^{3}}\left(\frac{1}{C^{\alpha-1}}+\frac{1}{C^{2-\alpha}}+\frac{1}{CN}\right).$
(2.107)

It is now clear that we can choose $C=C(\gamma,y_{\ast})$ sufficiently large
so that the claimed estimates (2.104)–(2.105) hold at $N$. Since $y_{\ast}$
ranges over a compact interval and all the constants involved vary
continuously in $y_{\ast}$, we may choose the constant $C$ above uniformly in
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$. We conclude by induction on $N$. ∎

######  Theorem 2.15.

Let $\gamma\in(1,\frac{4}{3})$ be given and for any
$y_{\ast}\in[y_{f}(\gamma),y_{F}(\gamma)]$ consider the sequence
$(\rho_{N},\omega_{N})$, $N\in\mathbb{N}$ which corresponds to the formal
Taylor coefficients associated with an LPH-type solution. Then there exists a
$\nu>0$ independent of $y_{\ast}$ such that the series

$\rho(y;y_{\ast}):=\sum_{N=0}^{\infty}\rho_{N}(y-y_{\ast})^{N},\ \
\omega(y;y_{\ast}):=\sum_{N=0}^{\infty}\omega_{N}(y-y_{\ast})^{N}$

converge absolutely and the functions
$(\rho(\cdot;y_{*}),\omega(\cdot;y_{*}))$ are real analytic solutions to
(1.13) on the interval $(y_{*}-\nu,y_{*}+\nu)$. Moreover, $y_{*}$ is a sonic
point for the flow, there are no other sonic points on the interval, and the
solutions are continuous with respect to $y_{*}\in[y_{f},y_{F}]$.

###### Proof.

Let $\alpha\in(1,2)$ be fixed. By Lemma 2.14 there exists a constant
$C=C(\gamma,\alpha)$ such that

$\displaystyle\left|\sum_{N=2}^{\infty}\rho_{N}(y-y_{\ast})^{N}\right|\leq\sum_{N=2}^{\infty}|\rho_{N}||y-y_{\ast}|^{N}\leq\sum_{N=2}^{\infty}\frac{|C(y-y_{\ast})|^{N}}{C^{\alpha}N^{3}}<\infty,$
(2.108)

and therefore the formal power series
$\sum_{N=0}^{\infty}\rho_{N}(y-y_{\ast})^{N}$ converges absolutely as long as
$|y-y_{\ast}|<\nu$, for any $0<\nu<\frac{1}{C}$. Similarly, the power series
$\sum_{N=0}^{\infty}\omega_{N}(y-y_{\ast})^{N}$ also converges absolutely as
long as $|y-y_{\ast}|<\nu$. The real analyticity is clear. Recalling (1.19) we
have

$\displaystyle G(y;\rho,\omega)$
$\displaystyle=\gamma\rho(y)^{\gamma-1}-y^{2}\omega(y)^{2}$
$\displaystyle=\gamma\left(\sum_{N=0}^{\infty}\rho_{N}(y-y_{\ast})^{N}\right)^{\gamma-1}-(y_{\ast}+(y-y_{\ast}))^{2}\left(\sum_{N=0}^{\infty}\omega_{N}(y-y_{\ast})^{N}\right)^{2}$
$\displaystyle=\left(\gamma(\gamma-1)\rho_{0}^{\gamma-2}\rho_{1}-2y_{\ast}\omega_{0}(1+y_{\ast}\omega_{1})\right)(y-y_{\ast})+O(|y-y_{\ast}|^{2})$
$\displaystyle=\left((\gamma-1)y_{\ast}^{2}\omega_{0}^{2}\frac{\rho_{1}}{\rho_{0}}-2y_{\ast}\omega_{0}(1+y_{\ast}\omega_{1})\right)(y-y_{\ast})+O(|y-y_{\ast}|^{2})$
$\displaystyle=y_{\ast}\omega_{0}\left((\gamma-1)y_{\ast}\omega_{0}\frac{\rho_{1}}{\rho_{0}}-2-2y_{\ast}\omega_{1}\right)(y-y_{\ast})+O(|y-y_{\ast}|^{2})$
$\displaystyle=y_{\ast}\omega_{0}\left((\gamma-1)\omega_{0}R_{1}-2-2W_{1}\right)(y-y_{\ast})+O(|y-y_{\ast}|^{2}),$
(2.109)

where we have used the sonic condition $G(y_{\ast},\rho,\omega)=0$ in the
second and the third line, and the notation $(R_{1},W_{1})$, see Lemma 2.8.
Now observe that $\omega_{0}>0$ by Lemma 2.2, and $R_{1}<0$, $W_{1}\geq 0$ by
Proposition 2.9. Therefore $(\gamma-1)\omega_{0}R_{1}-2-2W_{1}<0$ and
therefore, upon possibly choosing a smaller $\nu>0$, it follows that
$G(y;\rho,\omega)$ is strictly positive for $y\in(y_{\ast}-\nu,y_{\ast})$ and
strictly negative for $y\in(y_{\ast},y_{\ast}+\nu)$. In particular, the right-
hand side of (1.13) is well-defined and it is straightforward to verify that
$(\rho,\omega)$ is a solution to (1.13). ∎

In the final proposition of this section, we collect some remaining facts
concerning the LPH Taylor expansions.

######  Proposition 2.16.

Let $\gamma\in(1,\frac{4}{3})$. For $y_{*}\in[y_{f}(\gamma),y_{F}(\gamma)]$,
the following properties hold at the sonic point:

  * (i)

The branch $(R_{1},W_{1})$ that we take for the re-scaled first derivatives at
the sonic point $y_{*}$ satisfies
$(R_{1},W_{1})(y_{f})=(-\frac{2}{2-\gamma},0)$, $W_{1}(y_{*})>0$ for all
$y_{*}\in(y_{f},y_{F}]$.

  * (ii)

The local LPH-type solution obtained by Theorem 2.15 with $y_{*}=y_{f}$ is
exactly the far-field solution

$(\rho(y;y_{f}),\omega(y;y_{f}))\equiv(\rho_{f}(y),\omega_{f}(y))=(ky^{-\frac{2}{2-\gamma}},2-\gamma).$

  * (iii)

The local LPH-type solution obtained by Theorem 2.15 with $y_{*}=y_{F}$ is not
the Friedman solution:
$(\rho(\cdot;y_{F}),\omega(\cdot;y_{F}))\neq(\rho_{F},\omega_{F})$.

###### Proof.

(i) By Lemma 2.2, we know
$\omega_{0}(y_{*})\in[\frac{4-3\gamma}{3},2-\gamma]$. Then, by Proposition
2.9, we have $W_{1}(\omega_{0})\geq 0$ for all
$\omega_{0}\in[\frac{4-3\gamma}{3},2-\gamma]$ with equality if and only if
$\omega_{0}=2-\gamma$. In addition, $R_{1}(2-\gamma)=-\frac{2}{2-\gamma}$ by
direct computation from (2.79).
(ii) To see that the solution obtained at $y_{f}$ is the far-field solution,
it is enough to note that $\rho_{0}$ is uniquely determined by $y_{*}$ also
through the relation $\rho_{0}=f_{1}(\omega_{0}(y_{*}))$, and hence we have
that $\omega_{0}(y_{f})=2-\gamma=\omega_{f}(y_{f})$ and
$\rho_{0}(y_{f})=\rho_{f}({y}_{f})$. Thus the solution locally around the
sonic point is determined entirely by the choice of the branch $(R_{1},W_{1})$
for the first order terms in the Taylor expansion. As $W_{1}=0$,
$R_{1}=-\frac{2}{2-\gamma}$ are equal to the corresponding values for the far-
field solution, the Taylor expansions of the solution derived from the choice
$y_{*}=y_{f}$ and the far-field solution are equal. Thus the solutions are
locally equal (as both are analytic functions) and, by uniqueness theory for
the ODE system away from the sonic point and $y=0$, therefore globally equal
on all of $(0,\infty)$.
(iii) As in item (i), we know that $W_{1}(y_{F})>0$ by Proposition 2.9, hence
$\omega_{1}(y_{F})>0$ also. As the Friedman solution satisfies
$\omega_{F}^{\prime}(y)\equiv 0$ for all $y$, the two solutions are not equal.
∎

##  3 Solution to the right of the sonic point

Now that we have established the existence of a local solution to (1.13)
around each choice of sonic point $y_{*}\in[y_{f},y_{F}]$, we show in this
section that the local solution can be extended to the right on the whole
interval $(y_{*},\infty)$ while remaining strictly supersonic and satisfying
suitable asymptotics. For $y_{*}=y_{f}$, we know from Proposition 2.16 the
obtained solution is simply the far-field solution $(\rho_{f},\omega_{f})$
which is globally defined and supersonic for all $y>y_{f}$. We will therefore
restrict in the sequel to the case $y_{*}\in(y_{f},y_{F}]$.

The strategy of the section is to identify certain inequalities that propagate
along the flow to the right and provide qualitative control on the solutions.
Because the system (1.13) is non-autonomous, we cannot argue simply from a
fixed phase plane analysis, but instead we make use of dynamical arguments
that prevent the crossing of certain critical values by particular quantities
fundamental to the flow. After a number of technical lemmas, we prove the key
continuation estimates in Proposition 3.5. We then demonstrate that the flow
remains strictly supersonic to the right and so deduce that it exists globally
on $(y_{*},\infty)$ in Lemma 3.8. Finally, in Lemmas 3.9 and 3.10, we study
the asymptotics and monotonicity of the solution.

For each $y_{*}\in[y_{f}(\gamma),y_{F}(\gamma)]$, let
$(\rho,\omega)=(\rho(\cdot;y_{*}),\omega(\cdot;y_{*}))$ be the local LPH-type
solution of Theorem 2.15. We define the maximal extension time to the right as

$y_{\max}(y_{*}):=\sup\\{y>y_{*}\,|\,(\rho,\omega)\text{ extends as a strictly
supersonic solution of \eqref{eq:EPSS} on }(y_{*},y)\\},$ (3.110)

where we recall the definition of supersonicity from Definition 1.4.

The first lemma in this section states and proves the basic estimates that we
will use to propagate the solution and verifies that they hold in a small
neighbourhood of the sonic point.

######  Lemma 3.1 (Initial inequalities).

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in(y_{f},y_{F}]$ (recall that we
suppress the dependence of $y_{f}$, $y_{F}$ on $\gamma$ where clear) and let
$(\rho,\omega)$ be the unique LPH-type solution to (1.13) to the right of
$y_{*}$ given by Theorem 2.15. Then there exists $\bar{\nu}>0$ (depending on
$y_{*}$) such that for $y\in(y_{*},y_{*}+\bar{\nu})$, the strictly supersonic
flow satisfies also the inequalities

$\frac{4-3\gamma}{3}<\omega(y)<2-\gamma,\quad\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}>0,\quad-\frac{4}{(4-3\gamma)(2-\gamma)}<\frac{\rho^{\prime}y}{\rho}<-\frac{1}{2-\gamma}.$
(3.111)

###### Proof.

By Theorem 2.15, the existence of $\nu>0$ such that the solution remains
supersonic on $(y_{*},y_{*}+\nu)$ is clear. Moreover, by Lemma 2.2, we know
that if $y_{*}\in(y_{f},y_{F})$, we have
$\frac{4-3\gamma}{3}<\omega(y_{*})<2-\gamma$, and hence, as $\omega$ is
continuous on $[y_{*},y_{*}+\nu]$, there exists $\bar{\nu}\in(0,\nu)$ such
that

$\frac{4-3\gamma}{3}<\omega(y)<2-\gamma\text{ for
}y\in(y_{*},y_{*}+\bar{\nu}).$

On the other hand, if $y_{*}=y_{F}$, then $\omega(y_{*})=\frac{4-3\gamma}{3}$
and, by Proposition 2.9, $\omega^{\prime}(y_{*})>0$, hence by possibly
shrinking $\bar{\nu}$, we again have the claimed estimate.

Similarly, by Proposition 2.9 and smoothness of the flow, by possibly
shrinking $\bar{\nu}$, we retain the final inequality of (3.111)

$-\frac{4}{(4-3\gamma)(2-\gamma)}<\frac{\rho^{\prime}y}{\rho}<-\frac{1}{2-\gamma}.$

Finally, we check the second condition in (3.111) through the following
observation:

$\displaystyle\frac{4\pi
y_{*}^{2}\rho_{0}\omega_{0}}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho_{0}^{\gamma-1}=$
$\displaystyle\,\frac{4\pi
y_{*}^{2}\rho_{0}\omega_{0}}{4-3\gamma}-\frac{2}{2-\gamma}y_{*}^{2}\omega_{0}^{2}$
(3.112) $\displaystyle=$
$\displaystyle\,y_{*}^{2}\Big{(}\frac{2-2\gamma}{2-\gamma}\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)\Big{)}>0$

for $\omega_{0}\in[\frac{4-3\gamma}{3},2-\gamma)$, where we have used
$\rho_{0}=f_{1}(\omega_{0})$ (compare (2.28)) in the second line to eliminate
$\rho_{0}$, and observe that the quadratic function of $\omega_{0}$ in the
parentheses factorises as

$\frac{2-2\gamma}{2-\gamma}\omega_{0}^{2}+(\gamma-1)\omega_{0}+(\gamma-1)(2-\gamma)=-(\gamma-1)(2-\gamma-\omega)\big{(}\frac{2\omega}{2-\gamma}+1\big{)}$

to deduce the sign. By again exploiting continuity of the flow and possibly
shrinking $\bar{\nu}$, we conclude. ∎

We will also need the following two lemmas.

######  Lemma 3.2.

Let $\gamma\in(1,\frac{4}{3})$. For any $C^{1}$ solution $(\rho,\omega)$ of
(1.13), the following identities hold along the flow at any point $y>0$ such
that $y$ is not a sonic point:

$\displaystyle\omega^{\prime}$
$\displaystyle=\frac{4-3\gamma-3\omega}{y}-\frac{\omega}{\rho}\rho^{\prime},$
(3.113) $\displaystyle\big{(}\rho\omega
y^{\frac{2}{2-\gamma}}\big{)}^{\prime}=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\frac{\rho}{y}(4-3\gamma)\big{(}1-\frac{\omega}{2-\gamma}\big{)},$
(3.114) $\displaystyle\big{(}\omega y^{\frac{2}{2-\gamma}}\big{)}^{\prime}=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\Big{(}\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{\omega}{\rho}\rho^{\prime}\Big{)},$
(3.115)
$\displaystyle\big{(}\omega^{2}y^{\frac{2}{2-\gamma}}\big{)}^{\prime}=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\Big{(}2\omega\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{2}{2-\gamma}\frac{\omega^{2}}{y}-2\frac{\omega^{2}}{\rho}\rho^{\prime}\Big{)}.$
(3.116)

###### Proof.

Identity (3.113) is a trivial consequence of (1.21). Identity (3.114) follows
from using (3.113) in the following:

$\displaystyle\big{(}\rho\omega y^{\frac{2}{2-\gamma}}\big{)}^{\prime}=$
$\displaystyle\,(\rho\omega)^{\prime}y^{\frac{2}{2-\gamma}}+\frac{2}{2-\gamma}y^{\frac{2}{2-\gamma}}\frac{\rho}{y}\omega$
$\displaystyle=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\frac{4-3\gamma-3\omega}{y}\rho+\frac{2}{2-\gamma}y^{\frac{2}{2-\gamma}}\frac{\rho}{y}\omega$

and grouping the $\omega$ terms.

To obtain (3.115), we again apply (3.113) to find

$\displaystyle\big{(}\omega y^{\frac{2}{2-\gamma}}\big{)}^{\prime}=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\big{(}\frac{4-3\gamma-3\omega}{y}-\frac{\omega}{\rho}\rho^{\prime}+\frac{2}{2-\gamma}\frac{\omega}{y}\big{)}$

and group terms. The proof of (3.116) is similar. ∎

######  Lemma 3.3.

Let $\gamma\in(1,\frac{4}{3})$, let $(\rho,\omega)$ be a $C^{1}$ solution of
(1.13) and suppose that $y>0$ is not a sonic point of the flow.

  * (i)

For any $m\geq 0$, the derivative of $\rho$ may be expressed through the
following relation:

$\frac{\rho^{\prime}y}{\rho}+\frac{m}{2-\gamma}=y^{-\frac{2(\gamma-1)}{2-\gamma}}\frac{P_{m}(y,\rho,\omega)}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}},$
(3.117)

where

$\displaystyle P_{m}(y,\rho,\omega)=$
$\displaystyle-\frac{4-m-2\gamma}{2-\gamma}y^{\frac{2}{2-\gamma}}\omega^{2}-(\gamma-1)y^{\frac{2}{2-\gamma}}\big{(}\omega+(2-\gamma)\big{)}$
(3.118)
$\displaystyle-\frac{m}{2-\gamma}\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-1}+\frac{4\pi
y^{\frac{2}{2-\gamma}}\omega\rho}{4-3\gamma}.$

We usually suppress the explicit dependence of $P_{m}$ on $(\rho,\omega)$,
writing instead $P_{m}(y)=P_{m}(y,\rho(y),\omega(y))$ where clear.

  * (ii)

At any point $y_{1}$ at which the flow is smooth and not sonic and where
$P_{m}(y_{1})=0$, the derivative of $P_{m}$ satisfies the identity

$\displaystyle{P}_{m}^{\prime}(y_{1})=\frac{y_{1}^{\frac{2}{2-\gamma}}}{y_{1}\omega(y_{1})}Q_{m}\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)},$
(3.119)

where

$\displaystyle Q_{m}(\omega,\mathcal{R})=\bigg{(}$
$\displaystyle\big{(}1-\frac{\omega}{2-\gamma}\big{)}\Big{(}-\frac{4-m-2\gamma}{2-\gamma}(4-3\gamma)\omega^{2}+(\gamma-1)(4-3\gamma)(2-\gamma)\Big{)}$
(3.120)
$\displaystyle-\frac{2(4-m-2\gamma)(m-1)\omega^{3}}{(2-\gamma)^{2}}-\frac{m(\gamma-1)}{2-\gamma}\omega^{2}-2(\gamma-1)\omega$
$\displaystyle+\mathcal{R}\frac{m}{(2-\gamma)^{2}}\Big{(}(4-3\gamma)(2-\gamma)-\omega\big{(}4-3\gamma+(\gamma-1)(2-m)\big{)}\Big{)}\bigg{)}\Big{|}_{y_{1}}.$

###### Proof.

(i) To show (3.117), we let $m\geq 0$. Then, rearranging the first equation of
(1.13), we find

$\displaystyle\frac{\rho^{\prime}y}{\rho}=$
$\displaystyle\,\frac{-2y^{2}\omega^{2}-(\gamma-1)y^{2}(\omega+2-\gamma)+\frac{4\pi
y^{2}\omega\rho}{4-3\gamma}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}$ (3.121)
$\displaystyle=$
$\displaystyle\,\frac{-\frac{m}{2-\gamma}\big{(}y^{2}\omega^{2}-\gamma\rho^{\gamma-1}\big{)}+\big{(}\frac{m}{2-\gamma}-2\big{)}y^{2}\omega^{2}-\frac{m}{2-\gamma}\gamma\rho^{\gamma-1}-(\gamma-1)y^{2}(\omega+2-\gamma)+\frac{4\pi
y^{2}\omega\rho}{4-3\gamma}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}$
$\displaystyle=$
$\displaystyle\,-\frac{m}{2-\gamma}+\frac{-\frac{4-m-2\gamma}{2-\gamma}y^{2}\omega^{2}-(\gamma-1)y^{2}\big{(}\omega+(2-\gamma)\big{)}-\frac{m}{2-\gamma}\gamma\rho^{\gamma-1}+\frac{4\pi
y^{2}\omega\rho}{4-3\gamma}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}},$

and pulling out a factor of $y^{-\frac{2(\gamma-1)}{2-\gamma}}$ leaves us with
the claimed identity.

(ii) By (3.114)–(3.116), as the flow is smooth at $y_{1}$,

$\displaystyle{P}_{m}^{\prime}(y)=y^{\frac{2}{2-\gamma}}\bigg{(}$
$\displaystyle-\frac{4-m-2\gamma}{2-\gamma}\Big{(}2\omega\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{2}{2-\gamma}\frac{\omega^{2}}{y}-\frac{2\omega^{2}}{\rho}\rho^{\prime}\Big{)}$
(3.122)
$\displaystyle-(\gamma-1)\Big{(}\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{\omega}{\rho}\rho^{\prime}\Big{)}-\frac{2(\gamma-1)}{y}$
$\displaystyle+4\pi\frac{\rho}{y}\big{(}1-\frac{\omega}{2-\gamma})-\frac{m(\gamma-1)}{2-\gamma}\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-2}\big{(}\rho^{\prime}+\frac{2}{2-\gamma}\frac{\rho}{y}\big{)}\bigg{)}.$

From the identity $P_{m}(y_{1})=0$, we rearrange to find

$4\pi\rho=\frac{(4-m-2\gamma)(4-3\gamma)}{2-\gamma}\omega+(\gamma-1)(4-3\gamma)\big{(}1+\frac{2-\gamma}{\omega}\big{)}+\frac{m(4-3\gamma)}{2-\gamma}\gamma\frac{\rho^{\gamma-1}}{\omega
y_{1}^{2}},$ (3.123)

where all functions are evaluated at $y_{1}$. In addition, by (3.117), as
$P_{m}(y_{1})=0$ we also have
$\frac{\rho^{\prime}y}{\rho}=-\frac{m}{2-\gamma}$.

Substituting (3.123) and $\frac{\rho^{\prime}y}{\rho}=-\frac{m}{2-\gamma}$
into (3.122), we have

$\displaystyle{P}_{m}^{\prime}(y_{1})=y^{\frac{2}{2-\gamma}}\bigg{(}$
$\displaystyle\frac{1-\frac{\omega}{2-\gamma}}{y}\Big{(}-\frac{4-m-2\gamma}{2-\gamma}(4-3\gamma)\omega+(\gamma-1)(4-3\gamma)\frac{2-\gamma}{\omega}\Big{)}$
$\displaystyle-\frac{2(4-m-2\gamma)(m-1)\omega^{2}}{(2-\gamma)^{2}y}-\frac{m(\gamma-1)}{2-\gamma}\frac{\omega}{y}-\frac{2(\gamma-1)}{y}$
$\displaystyle+\frac{m(4-3\gamma)}{2-\gamma}\big{(}1-\frac{\omega}{2-\gamma}\big{)}\frac{\gamma\rho^{\gamma-1}}{\omega
y^{3}}-\frac{m(\gamma-1)(2-m)}{(2-\gamma)^{2}}\gamma\rho^{\gamma-1}\frac{1}{y^{3}}\bigg{)}\Big{|}_{y_{1}}$
$\displaystyle=y^{\frac{2}{2-\gamma}-1}\bigg{(}$
$\displaystyle\big{(}1-\frac{\omega}{2-\gamma}\big{)}\Big{(}-\frac{4-m-2\gamma}{2-\gamma}(4-3\gamma)\omega+(\gamma-1)(4-3\gamma)\frac{2-\gamma}{\omega}\Big{)}$
$\displaystyle-\frac{2(4-m-2\gamma)(m-1)\omega^{2}}{(2-\gamma)^{2}}-\frac{m(\gamma-1)}{2-\gamma}{\omega}-2(\gamma-1)$
$\displaystyle+\frac{\gamma\rho^{\gamma-1}}{y^{2}}\frac{m}{(2-\gamma)^{2}\omega}\Big{(}(4-3\gamma)(2-\gamma)-\omega\big{(}4-3\gamma+(\gamma-1)(2-m)\big{)}\Big{)}\bigg{)}\Big{|}_{y_{1}},$

which yields the required inequality after factoring out $\omega^{-1}$. ∎

With these identities, we will show that as long as the flow remains strictly
supersonic, the inequalities of Lemma 3.1 above also hold strictly. For the
proof, we will require also the following technical lemma containing
properties of the functions $Q_{m}$.

######  Lemma 3.4.

Define the functions

$Q_{m}^{+}(\omega)=Q_{m}(\omega,0),\quad
Q_{m}^{-}(\omega)=Q_{m}(\omega,\omega^{2}),$ (3.124)

where we recall the definition of $Q_{m}$ from Lemma 3.3. Then, for any
$\gamma\in(1,\frac{4}{3})$, there exists $\delta_{0}>0$ such that for all
$\omega\in[\frac{4-3\gamma}{3},2-\gamma]$, we have

$\displaystyle Q_{m}^{\pm}(\omega)<0$ $\displaystyle\text{ for all
}m\in\Big{[}1,\frac{2\gamma}{\gamma+1}+\delta_{0}\Big{]},$ (3.125)
$\displaystyle Q_{\frac{4}{4-3\gamma}}^{\pm}(\omega)>0.$ (3.126)

The proof is deferred to Appendix C.3.

We are now able to state and prove the continuation estimates for the
extension of the LPH-type solutions on their maximal supersonic interval of
existence, $(y_{*},y_{\max}(y_{*}))$.

######  Proposition 3.5.

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in[y_{f},y_{F}]$, and let
$(\rho,\omega)$ be the extension of the unique LPH-type solution obtained from
Theorem 2.15 to $(y_{*},y_{\max}(y_{*}))$. Then the following strict
inequalities hold on the whole interval $(y_{*},y_{\max}(y_{*}))$:

$\frac{4-3\gamma}{3}<\omega<2-\gamma,\quad\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}>0,\quad-\frac{4}{(4-3\gamma)(2-\gamma)}\frac{\rho}{y}<\rho^{\prime}<-\frac{1}{2-\gamma}\frac{\rho}{y}.$
(3.127)

Moreover, on this interval, we retain $\rho>0$.

###### Proof.

We begin the proof by observing that the upper and lower bounds on
$\rho^{\prime}$ of (3.127) guarantee that as long as the flow lives to the
right of $y_{*}$ and satisfies the weak forms of these inequalities, we always
retain $|(\log\rho)^{\prime}|\leq C$, and hence $\rho>0$. Thus we assume this
throughout the following.

By (3.111), we know that all of the inequalities (3.127) hold on the interval
$(y_{*},y_{*}+\bar{\nu})$. By the smoothness and extendability of the flow
guaranteed by Proposition A.1 and Theorem 2.15, the set

$\mathfrak{Y}:=\big{\\{}y_{1}\in(y_{*},y_{\max})\,|\,\text{\eqref{ineq:rightinvariance}
holds on }(y_{*},y_{1}]\big{\\}}$

is clearly relatively open in $(y_{*},y_{\max})$.

We therefore work to show that $\mathfrak{Y}$ is also relatively closed. We
therefore suppose $(y_{*},y_{1})\subset\mathfrak{Y}$, i.e., we assume that
(3.127) holds on the interval $(y_{*},y_{1})$ with $y_{1}<y_{\max}$. Showing
that (3.127) holds strictly at $y_{1}$ also is then sufficient to conclude the
proof. Clearly the weak versions of (3.127) hold on $(y_{*},y_{1}]$ and the
flow is strictly supersonic on this whole interval. As we have guaranteed
already that $\rho_{0}>\rho(y_{1})>0$ and $\omega(y_{1})$ is bounded, we may
apply again the local existence theorem, Proposition A.1, to deduce that the
flow can be smoothly extended past $y_{1}$, and hence is smooth at $y_{1}$
itself.

From (3.113), we see that as $\rho^{\prime}\leq 0$ and
$\omega\geq\frac{4-3\gamma}{3}>0$, then

$\omega^{\prime}\geq\frac{4-3\gamma-3\omega}{y},$

and hence

$(y^{3}\omega)^{\prime}\geq(4-3\gamma)y^{2},$

leading to

$y^{3}\omega(y)\geq\frac{4-3\gamma}{3}y^{3}+y_{*}^{3}\big{(}\omega(y_{*}+\frac{\bar{\nu}}{2})-\frac{4-3\gamma}{3}\big{)}>\frac{4-3\gamma}{3}y^{3},$

for all $y\in[y_{*},y_{1}]$. Clearly then $\omega(y_{1})>\frac{4-3\gamma}{3}$
also, as required.

To close the upper bound on $\omega$, we first rearrange the first equation of
(1.13) as

$\displaystyle\frac{\rho^{\prime}y}{\rho}=$
$\displaystyle\,\frac{-2y^{2}\omega^{2}-(\gamma-1)y^{2}(\omega+2-\gamma)+\frac{4\pi
y^{2}\omega\rho}{4-3\gamma}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}$ (3.128)
$\displaystyle=$
$\displaystyle\,-\frac{2}{2-\gamma}+\frac{(\gamma-1)y^{2}\big{(}\frac{2\omega^{2}}{2-\gamma}-\omega-(2-\gamma)\big{)}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}+\frac{4\pi
y^{2}\omega\rho}{4-3\gamma}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}.$

Note that, by assumption, on $[y_{*}+\bar{\nu},y_{1}]$,
$y^{2}\omega^{2}-\gamma\rho^{\gamma-1}>0$. We apply also (3.113) to calculate

$\displaystyle\big{(}2$
$\displaystyle-\gamma-\omega\big{)}^{\prime}=-\frac{\frac{4-3\gamma}{2-\gamma}\big{(}2-\gamma-\omega\big{)}}{y}+\frac{\omega}{y}\Big{(}\frac{\rho^{\prime}y}{\rho}+\frac{2}{2-\gamma}\Big{)}$
(3.129) $\displaystyle=$
$\displaystyle\,-\frac{\frac{4-3\gamma}{2-\gamma}\big{(}2-\gamma-\omega\big{)}}{y}-\frac{\omega}{y}\frac{(\gamma-1)y^{2}\big{(}\frac{2\omega}{2-\gamma}+1\big{)}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}(2-\gamma-\omega)+\frac{\omega}{y}\frac{\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}.$

Defining

$W(y)=\exp\Big{(}\int^{y}_{y_{*}+\bar{\nu}}\frac{1}{\tilde{y}}\Big{(}\frac{4-3\gamma}{2-\gamma}+\omega\frac{(\gamma-1)\tilde{y}^{2}\big{(}\frac{2\omega}{2-\gamma}+1\big{)}}{\tilde{y}^{2}\omega^{2}-\gamma\rho^{\gamma-1}}\Big{)}\operatorname{d}\\!\tilde{y}\Big{)},$

we have

$\big{(}W(2-\gamma-\omega)\big{)}^{\prime}=W\frac{\omega}{y}\frac{\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}.$

As $\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}\geq 0$ on
$[y_{*}+\bar{\nu},y_{1}]$, we have

$W(2-\gamma-\omega)\geq W(2-\gamma-\omega)\big{|}_{y_{*}+\bar{\nu}}>0,$

and hence

$\omega<2-\gamma\text{ on }[y_{*},y_{1}].$ (3.130)

Turning now to $\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}$, we
suppose for a contradiction that

$\frac{4\pi
y_{1}^{2}\rho(y_{1})\omega(y_{1})}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho(y_{1})^{\gamma-1}=0.$

From (3.128), at $y_{1}$, we therefore have

$\frac{\rho^{\prime}(y_{1})y_{1}}{\rho(y_{1})}+\frac{2}{2-\gamma}=\frac{(\gamma-1)y_{1}^{2}\big{(}\frac{2\omega(y_{1})}{2-\gamma}+1\big{)}\big{(}\omega(y_{1})-(2-\gamma)\big{)}}{y_{1}^{2}\omega(y_{1})^{2}-\gamma\rho(y_{1})^{\gamma-1}}<0$
(3.131)

due to $\omega<2-\gamma$. Note now the simple scaled identity

$\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}=y^{-\frac{2(\gamma-1)}{2-\gamma}}\Big{(}\frac{4\pi
y^{\frac{2}{2-\gamma}}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-1}\Big{)}.$

Differentiating the term in the bracket, we use (3.114) to see

$\displaystyle\Big{(}$ $\displaystyle\frac{4\pi
y^{\frac{2}{2-\gamma}}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-1}\Big{)}^{\prime}\Big{|}_{y=y_{1}}$
$\displaystyle=4\pi
y^{\frac{2}{2-\gamma}}\frac{\rho}{y}\big{(}1-\frac{\omega}{2-\gamma}\big{)}-\frac{2\gamma(\gamma-1)}{2-\gamma}\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-2}y^{\frac{2}{2-\gamma}}\big{(}\rho^{\prime}+\frac{2}{2-\gamma}\frac{\rho}{y}\big{)}\Big{|}_{y=y_{1}}$
$\displaystyle\geq 4\pi
y^{\frac{2}{2-\gamma}}\frac{\rho}{y}\big{(}1-\frac{\omega}{2-\gamma}\big{)}\Big{|}_{y=y_{1}}>0,$

where we have used (3.131) in the first inequality on the last line and
$\omega(y_{1})<2-\gamma$ in the second. But this contradicts the assumption
that $y_{1}$ is the first point at which $\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}=0$, hence
the derivative must be non-positive. So

$\Big{(}\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}\Big{)}\Big{|}_{y_{1}}>0.$

Next, we consider the quantity

$\frac{\rho^{\prime}y}{\rho}+\frac{1}{2-\gamma}.$

Applying (3.117) in the case $m=1$, we get

$\displaystyle\frac{\rho^{\prime}y}{\rho}+\frac{1}{2-\gamma}=y^{-\frac{2(\gamma-1)}{2-\gamma}}\frac{P_{1}(y,\rho,\omega)}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}},$
(3.132)

where we recall from (3.118) that

$P_{1}=-\frac{(3-2\gamma)}{2-\gamma}y^{\frac{2}{2-\gamma}}\omega^{2}-(\gamma-1)y^{\frac{2}{2-\gamma}}\big{(}\omega+(2-\gamma)\big{)}-\frac{1}{2-\gamma}\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-1}+\frac{4\pi
y^{\frac{2}{2-\gamma}}\omega\rho}{4-3\gamma}.$

By assumption, we have $P_{1}(y)<0$ for all $y\in(y_{*},y_{1})$. By
Proposition 3.5 and Proposition A.1, as the flow is assumed supersonic, the
flow may be extended smoothly to the right of $y_{1}$, and hence is smooth at
$y_{1}$. Suppose now that at $y_{1}$,
$\frac{\rho^{\prime}y}{\rho}=-\frac{1}{2-\gamma}$ for the first time
(otherwise we are done). Then we must also have that
$P_{1}^{\prime}(y_{1})\geq 0$, $P_{1}(y_{1})=0$, and hence, at $y_{1}$, by
(3.119),

$\displaystyle
P_{1}^{\prime}(y_{1})=\frac{y^{\frac{2}{2-\gamma}}}{y\omega}Q_{1}$
$\displaystyle\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)}.$

Note that $Q_{m}(\omega,\mathcal{R})$ is linear in $\mathcal{R}$ and that, as
the flow is supersonic, we have always
$0\leq\frac{\gamma\rho^{\gamma-1}}{y^{2}}\leq\omega^{2}$. Thus,

$Q_{1}\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)}\leq\max\big{\\{}Q_{1}(\omega(y_{1}),0),Q_{1}(\omega(y_{1}),\omega(y_{1})^{2})\big{\\}}<0$

by Lemma 3.4. Thus $P^{\prime}(y_{1})<0$, contradicting $P^{\prime}(y_{1})\geq
0$. So we obtain

$\frac{\rho^{\prime}y}{\rho}+\frac{1}{2-\gamma}<0,\text{ for
}y\in[y_{*},y_{1}].$

To conclude the final inequality, the lower bound for
$\frac{\rho^{\prime}y}{\rho}$, we let $m=\frac{4}{4-3\gamma}$ and apply again
(3.117) to find

$\displaystyle\frac{\rho^{\prime}y}{\rho}+\frac{4}{(4-3\gamma)(2-\gamma)}=y^{-\frac{2(\gamma-1)}{2-\gamma}}\frac{P_{m}(y,\rho,\omega)}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}.$
(3.133)

If $y_{1}$ is the first point where
$\frac{\rho^{\prime}y}{\rho}=-\frac{4}{(4-3\gamma)(2-\gamma)}$, then
$P_{m}(y_{1})=0$, $P_{m}^{\prime}(y_{1})\leq 0$ and so, at $y_{1}$, by
(3.119), we have

$\displaystyle
P_{m}^{\prime}(y_{1})=\frac{y^{\frac{2}{2-\gamma}}}{y\omega}Q_{m}\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)}.$

Again, as $Q_{m}(\omega,\mathcal{R})$ is linear in $\mathcal{R}$ and
$0\leq\frac{\gamma\rho^{\gamma-1}}{y^{2}}\leq\omega^{2}$, we have

$Q_{m}\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)}\geq\min\big{\\{}Q_{m}(\omega(y_{1}),0),Q_{m}(\omega(y_{1}),\omega(y_{1})^{2})\big{\\}}<0$

by Lemma 3.4. This contradicts the assumption $P_{m}(y_{1})=0$, and hence we
have

$\frac{\rho^{\prime}y}{\rho}+\frac{4}{(4-3\gamma)(2-\gamma)}>0.$ (3.134)

∎

To show that the flow remains supersonic to the right, and hence the global
existence to the right, we need a slightly sharper upper bound on the
derivative of the density, provided by the following lemma.

######  Lemma 3.6.

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in[y_{f},y_{F}]$ and define
$R_{1}=\frac{\rho_{1}y_{*}}{\rho_{0}}$ as in Proposition 2.9. Let
$(\rho,\omega)$ be the extension of the unique LPH-type solution obtained from
Theorem 2.15 to $(y_{*},y_{\max}(y_{*}))$. Then there exists $\delta>0$ such
that, for any
$R>\max\\{R_{1},-\frac{2\gamma}{(2-\gamma)(\gamma+1)}-\delta\\}$, we retain
the inequality $\frac{\rho^{\prime}y}{\rho}<R$ on the whole of
$(y_{*},y_{\max})$.

######  Remark 3.7.

In effect, this says that if
$\frac{\rho_{1}y_{*}}{\rho_{0}}<-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$, then
we retain $\frac{\rho^{\prime}y}{\rho}<-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$
as long as the flow stays supersonic. If, on the other hand, we only have
$\frac{\rho_{1}y_{*}}{\rho_{0}}\geq-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$,
then we will at least keep
$\frac{\rho^{\prime}y}{\rho}\leq\frac{\rho_{1}y_{*}}{\rho_{0}}$ as long as the
flow stays supersonic.

###### Proof.

Choose $\delta>0$ such that $\delta(2-\gamma)<\delta_{0}$ with $\delta_{0}$
the constant defined in Lemma 3.4 and let
$m\in(1,\frac{2\gamma}{\gamma+1}+\delta(2-\gamma))$ be such that
$R_{1}<-\frac{m}{2-\gamma}$. Applying again (3.117), we find

$\displaystyle\frac{\rho^{\prime}y}{\rho}+\frac{m}{2-\gamma}=y^{-\frac{2(\gamma-1)}{2-\gamma}}\frac{{P_{m}}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}},$
(3.135)

where

$P_{m}=-\frac{(4-m-2\gamma)}{2-\gamma}y^{\frac{2}{2-\gamma}}\omega^{2}-(\gamma-1)y^{\frac{2}{2-\gamma}}\big{(}\omega+(2-\gamma)\big{)}-\frac{m}{2-\gamma}\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-1}+\frac{4\pi
y^{\frac{2}{2-\gamma}}\omega\rho}{4-3\gamma}.$

Suppose now that at $y_{1}$, $\frac{\rho^{\prime}y}{\rho}=-\frac{m}{2-\gamma}$
for the first time, so that ${P_{m}}(y)<0$ for all $y\in(y_{*},y_{1})$. By
Proposition 3.5 and Proposition A.1, as the flow is assumed supersonic, the
flow may be extended smoothly to the right of $y_{1}$, and hence is smooth at
$y_{1}$. Suppose now that at $y_{1}$,
$\frac{\rho^{\prime}y}{\rho}=-\frac{m}{2-\gamma}$ for the first time
(otherwise we are done). Then we must also have that
$P_{m}^{\prime}(y_{1})\geq 0$, $P_{m}(y_{1})=0$, and hence, at $y_{1}$, by
(3.119),

$\displaystyle
P_{m}^{\prime}(y_{1})=\frac{y^{\frac{2}{2-\gamma}}}{y\omega}Q_{m}$
$\displaystyle\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)}.$

Note that $Q_{m}(\omega,\mathcal{R})$ is linear in $\mathcal{R}$ and that, as
the flow is supersonic, we have always
$0\leq\frac{\gamma\rho^{\gamma-1}}{y^{2}}\leq\omega^{2}$. Then,

$Q_{m}\Big{(}\omega(y_{1}),\frac{\gamma\rho(y_{1})^{\gamma-1}}{y_{1}^{2}}\Big{)}\leq\max\big{\\{}Q_{m}(\omega(y_{1}),0),Q_{m}(\omega(y_{1}),\omega(y_{1})^{2})\big{\\}}.$

Applying Lemma 3.4, for $m\in[1,\frac{2\gamma}{\gamma+1}+\delta(2-\gamma)]$,
$\omega\in[\frac{4-3\gamma}{3},2-\gamma]$ this is strictly negative, leading
to the desired contradiction. ∎

With this, we may prove that the flow remains supersonic to the right for all
$y>y_{*}$, concluding the proof of existence to the right.

######  Lemma 3.8.

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in[y_{f},y_{F}]$. Then
$y_{\max}(y_{*})=\infty$, i.e. the unique LPH-type solution $(\rho,\omega)$ to
the right of $y_{*}$ obtained from Theorem 2.15 extends smoothly as a strictly
supersonic solution of (1.13) to the whole of $(y_{*},\infty)$.

###### Proof.

Let now

$S=y^{\frac{2}{2-\gamma}}\omega^{2}-\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-1}.$

By Theorem 2.15, there exists $\delta>0$ such that $S>0$ on
$(y_{*},y_{*}+\delta]$.

By Proposition 3.5 and the local existence and uniqueness Proposition A.1, the
only obstruction to continuing the solution to the right is if strict
supersonicity fails.

Suppose for a contradiction that $y_{\max}(y_{*})<\infty$. Then there exists
$y_{0}\in(y_{*},y_{\max}]$ such that $\liminf_{y\to y_{0}^{-}}S(y)=0$ where
$S(y)>0$ on $(y_{*},y_{0})$. The flow is then smooth on $(y_{*},y_{0})$, but
may not extend smoothly up to $y_{0}$.

A simple calculation using (3.114)–(3.116) shows that, for all
$y\in(y_{*},y_{0})$,

$\displaystyle S^{\prime}(y)=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\Big{(}2\omega\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{2}{2-\gamma}\frac{\omega^{2}}{y}-2\frac{\omega^{2}\rho^{\prime}}{\rho}\Big{)}$
$\displaystyle-(\gamma-1)\gamma\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\gamma-2}y^{\frac{2}{2-\gamma}}\big{(}\rho^{\prime}+\frac{2}{2-\gamma}\frac{\rho}{y}\big{)}$
$\displaystyle=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\Big{(}2\omega\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{2}{2-\gamma}\frac{\omega^{2}}{y}-2\frac{\omega^{2}\rho^{\prime}}{\rho}\Big{)}$
$\displaystyle+(\gamma-1)(S-y^{\frac{2}{2-\gamma}}\omega^{2})\big{(}\frac{\rho^{\prime}}{\rho}+\frac{2}{2-\gamma}\frac{1}{y}\big{)}.$

Rearranging this identity, we obtain

$\displaystyle S^{\prime}(y)=$
$\displaystyle\,y^{\frac{2}{2-\gamma}}\Big{(}2\omega\frac{(4-3\gamma)(1-\frac{\omega}{2-\gamma})}{y}-\frac{2}{2-\gamma}\frac{\omega^{2}}{y}-2\omega^{2}\frac{\rho^{\prime}}{\rho}-(\gamma-1)\omega^{2}\frac{\rho^{\prime}}{\rho}-2\frac{(\gamma-1)}{2-\gamma}\frac{\omega^{2}}{y}\Big{)}$
(3.136)
$\displaystyle+S(y)(\gamma-1)\big{(}\frac{\rho^{\prime}}{\rho}+\frac{2}{2-\gamma}\frac{1}{y}\big{)}$
$\displaystyle=$ $\displaystyle\,y^{\frac{2}{2-\gamma}-1}\omega
F(\omega,\frac{\rho^{\prime}y}{\rho})+S(y)(\gamma-1)\big{(}\frac{\rho^{\prime}}{\rho}+\frac{2}{2-\gamma}\frac{1}{y}\big{)},$

where

$F(\omega,R):=2(4-3\gamma)\big{(}1-\frac{\omega}{2-\gamma}\big{)}-\omega(\gamma+1)\big{(}R+\frac{2\gamma}{(2-\gamma)(\gamma+1)}\big{)}.$

As the flow is smooth (analytic) through $y_{*}$ by construction, then this
identity also holds at $y_{*}$, where $S(y_{*})=0$. In particular, this gives
us the inequality

$F(\omega_{0},R_{1})=\delta^{*}>0,$ (3.137)

where we have defined, as usual, $R_{1}=\frac{\rho_{1}y_{*}}{\rho_{0}}$. We
distinguish now two cases: $R_{1}<-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$ and
$R_{1}\geq-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$.

Case 1: Suppose that $R_{1}<-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$. Then, by
Lemma 3.6, there exists $\delta>0$ such that

$\frac{\rho^{\prime}y}{\rho}\leq-\frac{2\gamma}{(2-\gamma)(\gamma+1)}-\delta\text{
for all }y\in(y_{*},y_{0}).$

Thus as we have also $\frac{4-3\gamma}{3}<\omega<2-\gamma$, we obtain

$F(\omega,\frac{\rho^{\prime}y}{\rho})\geq\omega(\gamma+1)\delta\geq\delta\frac{(\gamma+1)(4-3\gamma)}{3}=:\tilde{\delta}>0.$

By the estimates of Proposition 3.5, there exists $M>0$, depending only on
$y_{*}$, $y_{0}$ and $\gamma$, such that, for all $y\in(y_{*},y_{0})$,

$y^{1-\frac{2}{2-\gamma}}\omega^{-1}\Big{|}(\gamma-1)\big{(}\frac{\rho^{\prime}}{\rho}+\frac{2}{2-\gamma}\frac{1}{y}\big{)}\Big{|}\leq
M.$

Thus, if $S(y)\leq\frac{\tilde{\delta}}{2M}$, we obtain from (3.136)
$S^{\prime}(y)>0$, contradicting $\liminf_{y\to y_{0}}S(y)=0$.

Case 2: Suppose now that $R_{1}\geq-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$. By
Proposition 2.9, this forces $\gamma\leq\frac{10}{9}$. As $\rho^{\prime}<0$ by
Proposition 3.5, we know that on $(y_{*}+\bar{\nu},y_{0})$ ($\bar{\nu}$ taken
as in Lemma 3.1), we have $\rho<\rho_{0}-\delta$ for some small $\delta>0$. By
Lemma 2.1 (compare also Figure 3), there exists $\bar{\varepsilon}>0$,
depending on $\delta$, $\rho_{0}$ and $\gamma\leq\frac{10}{9}$, such that if
$0\leq h(\rho,\omega)<\bar{\varepsilon}$, $\omega>\frac{4-3\gamma}{3}$ and
$\rho<\rho_{0}-\delta$, then $\omega<\omega_{0}$. Here $h(\rho,\omega)$ is as
defined above in (1.20).

By Proposition 3.5, we have a bound $M>0$, depending only on $y_{*}$, $y_{0}$
and $\gamma$, such that

$\big{|}\frac{\rho^{\prime}}{y\rho}\big{|}+y^{1-\frac{2}{2-\gamma}}\omega^{-1}\Big{|}(\gamma-1)\big{(}\frac{\rho^{\prime}}{\rho}+\frac{2}{2-\gamma}\frac{1}{y}\big{)}\Big{|}\leq
M\text{ on }(y_{*},y_{0}).$ (3.138)

Let $\varepsilon>0$ be such that $\varepsilon
M<\min\\{\delta^{*},\bar{\varepsilon}\\}$. As $S$ is differentiable on
$(y_{*},y_{0})$, there exists $y_{1}\in(y_{*},y_{0})$ such that

$S^{\prime}(y_{1})\leq 0\text{ and }S(y_{1})=\varepsilon.$

From the first equation of (1.21), we obtain

$\big{|}h(\rho(y_{1}),\omega(y_{1}))\big{|}=\Big{|}-S(y_{1})\frac{\rho^{\prime}(y_{1})}{y_{1}\rho(y_{1})}\Big{|}\leq\varepsilon
M<\bar{\varepsilon}.$

Thus, by construction of $\bar{\varepsilon}$, we also obtain
$\omega(y_{1})<\omega_{0}$.

We use Lemma 3.6 to see that $\frac{\rho^{\prime}y}{\rho}\leq R_{1}$ on
$(y_{*},y_{0})$ and so, noting that $\frac{\partial}{\partial
R}F(\omega,R)<0$, we have
$F(\omega(y_{1}),\frac{\rho^{\prime}(y_{1})y_{1}}{\rho(y_{1})})\geq
F(\omega(y_{1}),R_{1})$. Now as
$R_{1}\geq-\frac{2\gamma}{(2-\gamma)(\gamma+1)}$, it is clear from the
definition of $F$ that $\frac{\partial}{\partial\omega}F(\omega,R_{1})<0$, and
so, as $\omega(y_{1})<\omega_{0}$, we obtain

$F(\omega(y_{1}),\frac{\rho^{\prime}(y_{1})y_{1}}{\rho(y_{1})})\geq
F(\omega(y_{1}),R_{1})>F(\omega_{0},R_{1})=\delta^{*}>0,$

and so, using (3.138) and $\varepsilon M<\delta^{*}$ in (3.136), we find
$S^{\prime}(y_{1})>0$, a contradiction to the definition of $y_{1}$. ∎

######  Lemma 3.9 (Asymptotics).

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in[y_{f},y_{F}]$. Then the local LPH-
type solution $(\rho,\omega)$ obtained from Theorem 2.15 may be extended to
the right as a Yahil-type solution of (1.13) on the whole interval
$[y_{*},\infty)$.

Moreover, as $y\to\infty$, the asymptotics of $(\rho,\omega)$ are as follows.
There exist constants $\bar{k}_{1}>0$ and $\bar{k}_{2}>0$ such that

$y^{\frac{1}{2-\gamma}}(2-\gamma-\omega(y))\to\bar{k}_{1},\quad
y^{\frac{2}{2-\gamma}}\rho(y)\to\bar{k}_{2}\quad\text{ as }y\to\infty.$

###### Proof.

The global existence to the right follows from Proposition 3.5 and Lemma 3.8,
while the negativity of $u(y)$ follows directly from the bounds
$\frac{4-3\gamma}{3}<\omega(y)<2-\gamma$.

We begin by showing the asymptotics for $\omega$. Recall from (3.129) the
identity

$\displaystyle\big{(}2$ $\displaystyle-\gamma-\omega\big{)}^{\prime}=$ (3.139)
$\displaystyle-\frac{\frac{4-3\gamma}{2-\gamma}\big{(}2-\gamma-\omega\big{)}}{y}-\frac{\omega}{y}\frac{(\gamma-1)y^{2}\big{(}\frac{2\omega}{2-\gamma}+1\big{)}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}(2-\gamma-\omega)+\frac{\omega}{y}\frac{\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}.$

From Lemma 3.6 and the initial estimate
$\frac{y_{*}\rho_{1}}{\rho_{0}}<-\frac{1}{2-\gamma}$, we see that there exists
$\epsilon>0$ such that
$\rho^{\prime}\leq\big{(}-\frac{1}{2-\gamma}-\epsilon\big{)}\frac{\rho}{y}$.
As also $\rho>0$, we easily see that

$0<\rho(y)\leq Cy^{-\frac{1}{2-\gamma}-\epsilon},$

and so, for $y$ large, we may estimate

$\Big{|}\frac{\omega}{y}\frac{\frac{4\pi
y^{2}\rho\omega}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}\Big{|}\leq
Cy^{-1-\frac{1}{2-\gamma}-\epsilon}.$

We re-write the middle term of (3.139) as

$\displaystyle-\frac{\omega}{y}$
$\displaystyle\frac{(\gamma-1)y^{2}\big{(}\frac{2\omega}{2-\gamma}+1\big{)}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}(2-\gamma-\omega)$
$\displaystyle=$
$\displaystyle\,-\frac{1}{y}(\gamma-1)\big{(}\frac{2}{2-\gamma}+\frac{1}{\omega}\big{)}(2-\gamma-\omega)+\frac{\omega}{y}\frac{\gamma\rho^{\gamma-1}(\gamma-1)y^{2}\big{(}\frac{2\omega}{2-\gamma}+1\big{)}}{y^{2}\omega^{2}(y^{2}\omega^{2}-\gamma\rho^{\gamma-1})}(2-\gamma-\omega)$
$\displaystyle=$
$\displaystyle\,-\frac{\frac{3(\gamma-1)}{2-\gamma}(2-\gamma-\omega)}{y}-\frac{(\gamma-1)(2-\gamma-\omega)^{2}}{y\omega(2-\gamma)}+O\big{(}y^{-3-\frac{1}{2-\gamma}}\big{)}.$

Thus, we find

$\displaystyle\big{(}2$
$\displaystyle-\gamma-\omega\big{)}^{\prime}\leq-\frac{\frac{1}{2-\gamma}(2-\gamma-\omega)}{y}+Cy^{-1-\frac{1}{2-\gamma}-\epsilon},$

leading to the desired estimate

$0<2-\gamma-\omega(y)\leq Cy^{-\frac{1}{2-\gamma}},$

as claimed. With this quantitative decay established, it is easier to see that
this decay is also sharp by using this estimate to treat the quadratic term in
$(2-\gamma-\omega)$ as higher order and so obtain a lower bound of the same
form: $2-\gamma-\omega\geq cy^{-\frac{1}{2-\gamma}}$. Indeed, we easily see
that the quantity
$\big{(}y^{\frac{1}{2-\gamma}}(2-\gamma-\omega)\big{)}^{\prime}$ is integrable
as $y\to\infty$, giving the existence of $\bar{k}_{1}$ as claimed.

Treating now $\rho$, we see from (3.128) that

$\displaystyle\big{(}y^{\frac{2}{2-\gamma}}\rho\big{)}^{\prime}=$
$\displaystyle\,\frac{y^{\frac{2}{2-\gamma}}\rho}{y}\big{(}\frac{\rho^{\prime}y}{\rho}+\frac{2}{2-\gamma}\big{)}$
$\displaystyle=$
$\displaystyle\,\frac{y^{\frac{2}{2-\gamma}}\rho}{y}\Big{(}\frac{-(\gamma-1)y^{2}\big{(}\frac{2\omega}{2-\gamma}+1\big{)}(2-\gamma-\omega)+\frac{4\pi
y^{2}\omega\rho}{4-3\gamma}-\frac{2}{2-\gamma}\gamma\rho^{\gamma-1}}{y^{2}\omega^{2}-\gamma\rho^{\gamma-1}}\Big{)}$

and the asymptotics just obtained for $2-\gamma-\omega$ and $\rho$ immediately
yield that $y^{\frac{2}{2-\gamma}}\rho$ remains bounded as $y\to\infty$. In
particular, the right hand side of this identity is integrable as
$y\to\infty$, giving the claimed convergence of $y^{\frac{2}{2-\gamma}}\rho$.
∎

######  Lemma 3.10.

Let $\gamma\in(1,\frac{4}{3})$, $y_{*}\in[y_{f},y_{F}]$, and let
$(\rho,\omega)$ be the global Yahil-type solution to the right of (1.13)
obtained as the extension of the LPH-type solution from Theorem 2.15. Then the
solution remains monotone (strictly monotone for $y_{*}>y_{f}$) in both $\rho$
and $\omega$.

###### Proof.

In the case $y_{*}=y_{f}$, we know that the solution to the right is exactly
the far-field solution
$(\rho_{f},\omega_{f})=(ky^{-\frac{2}{2-\gamma}},2-\gamma)$. We therefore need
only to consider the case $y_{*}>y_{f}$ for which $\omega^{\prime}(y_{*})>0$.
Moreover, by the estimate $\rho^{\prime}<-\frac{1}{2-\gamma}\frac{y}{\rho}$ of
Proposition 3.5 above, we have $\rho^{\prime}<0$ for all $y>y_{*}$. It remains
only to show that we retain also $\omega^{\prime}(y)>0$.

Suppose now that there exists a point $y_{0}>y_{*}$ such that
$\omega^{\prime}(y_{0})=0$. Then, from (1.13), we have

$\frac{4-3\gamma-3\omega(y_{0})}{y_{0}}=\frac{y_{0}\omega(y_{0})h(y_{0})}{G(y_{0})}.$
(3.140)

Differentiating $h(\rho,\omega)$, we obtain

$\displaystyle\frac{\operatorname{d}\\!}{\operatorname{d}\\!y}h(\rho,\omega)=$
$\displaystyle\,4\omega\omega^{\prime}+(\gamma-1)\omega^{\prime}-\frac{4\pi}{4-3\gamma}\omega\rho^{\prime}-\frac{4\pi}{4-3\gamma}\rho\omega^{\prime}$
(3.141) $\displaystyle=$
$\displaystyle\,2\omega\omega^{\prime}-\frac{4\pi}{4-3\gamma}\omega\rho^{\prime}+\frac{h(\rho,\omega)-(\gamma-1)(2-\gamma)}{\omega}\omega^{\prime}$
$\displaystyle=$
$\displaystyle\,\big{(}2\omega^{2}-(\gamma-1)(2-\gamma)+h(\rho,\omega)\big{)}\frac{4-3\gamma-3\omega}{y\omega}-y\frac{h(4\omega^{2}+(\gamma-1)\omega)}{G(\rho,\omega,y)}.$

Thus, at $y_{0}$,

$\displaystyle h^{\prime}(y_{0})=$
$\displaystyle\,-\frac{4\pi}{4-3\gamma}\omega\rho^{\prime}=-\frac{4\pi}{4-3\gamma}\frac{y\rho\omega
h}{G}.$ (3.142)

Arguing directly, we differentiate $G$ to obtain

$\displaystyle G^{\prime}=$
$\displaystyle\,(\gamma-1)\gamma\rho^{\gamma-2}\rho^{\prime}-2y\omega^{2}-2y^{2}\omega\omega^{\prime}$
$\displaystyle=$
$\displaystyle\,(\gamma-1)\gamma\rho^{\gamma-1}\frac{yh}{G}-2y\omega^{2}-2y^{2}\omega\omega^{\prime}.$

Thus, at $y_{0}$,

$\displaystyle G^{\prime}(y_{0})=$
$\displaystyle\,(\gamma-1)\gamma\rho^{\gamma-1}\frac{4-3\gamma-3\omega}{y\omega}-2y\omega^{2}.$
(3.143)

We now further differentiate the second equation of (1.21) to obtain

$\displaystyle\omega^{\prime\prime}=$
$\displaystyle\,-\frac{3\omega^{\prime}}{y}-\frac{4-3\gamma-3\omega}{y^{2}}-\frac{\omega
h}{G}-\frac{y\omega^{\prime}h}{G}-\frac{y\omega h^{\prime}}{G}+\frac{y\omega
hG^{\prime}}{G^{2}}.$

Hence, at $y_{0}$, we find

$\displaystyle\omega^{\prime\prime}(y_{0})=-\frac{4-3\gamma-3\omega}{y^{2}}-\frac{\omega
h}{G}-\frac{y\omega h^{\prime}}{G}+\frac{y\omega
hG^{\prime}}{G^{2}}=-2\frac{\omega h}{G}-\frac{y\omega
h^{\prime}}{G}+\frac{y\omega hG^{\prime}}{G^{2}},$

where we have used (3.140) in the second equality. Substituting (3.142) into
the second term and (3.143) into the third term, we get

$\displaystyle\omega^{\prime\prime}(y_{0})=$ $\displaystyle\,-\frac{2\omega
hG}{G^{2}}+\frac{\frac{4\pi}{4-3\gamma}y^{2}\omega^{2}\rho
h}{G^{2}}+\frac{y\omega
h\big{(}(\gamma-1)\gamma\rho^{\gamma-1}\frac{4-3\gamma-3\omega}{y\omega}-2y\omega^{2}\big{)}}{G^{2}}$
$\displaystyle\geq$ $\displaystyle\,\frac{\omega
h}{G^{2}}\Big{(}2(y^{2}\omega^{2}-\gamma\rho^{\gamma-1})+y^{2}\big{(}2\omega^{2}+(\gamma-1)\omega+(\gamma-1)(2-\gamma)\big{)}$
$\displaystyle\quad+(\gamma-1)\gamma\rho^{\gamma-1}\frac{4-3\gamma-3\omega}{\omega}-2y^{2}\omega^{2}\Big{)},$

where we have used that $h>0$ (from $\rho^{\prime}<0$) to obtain
$\frac{4\pi}{4-3\gamma}\rho\omega>2\omega^{2}+(\gamma-1)\omega+(\gamma-1)(2-\gamma)$.
Grouping terms, we then find

$\displaystyle\omega^{\prime\prime}(y_{0})\geq$ $\displaystyle\,\frac{\omega