# The Stability of Irrotational Shocks and the Landau Law of Decay

Daniel Ginsberg and Igor Rodnianski

(August 28, 2024)

###### Abstract

We consider the long-time behavior of irrotational solutions of the three-
dimensional compressible Euler equations with shocks, hypersurfaces of
discontinuity across which the Rankine-Hugoniot conditions for irrotational
flow hold. Our analysis is motivated by Landau’s analysis of spherically-
symmetric shock waves, who predicted that at large times, not just one, but
two shocks emerge. These shocks are logarithmically-separated from the
Minkowskian light cone and the fluid velocity decays at the non-time-
integrable rate $1/(t(\log t)^{1/2})$. We show that for initial data, which
need not be spherically-symmetric, with two shocks in it and which is
sufficiently close, in appropriately weighted Sobolev norms, to an $N$-wave
profile, the solution to the shock-front initial value problem can be
continued for all time and does not develop any further singularities. In
particular this is the first proof of global existence for solutions (which
are necessarily singular) of a quasilinear wave equation in three space
dimensions which does not verify the null condition. The proof requires
carefully-constructed multiplier estimates and analysis of the geometry of the
shock surfaces.

###### Contents

  1. 1 Introduction
    1. 1.1 Strategy of the proof
      1. 1.1.1 The determinism conditions and formulation as an initial-boundary value problem
      2. 1.1.2 The energy estimates and the basic energy identity
      3. 1.1.3 The bootstrap argument, the decay estimates, and the choice of multipliers
    2. 1.2 Modulated profiles and location of the shocks
    3. 1.3 The full compressible Euler equations and the restricted shock front problem
    4. 1.4 Further background on the problem and related results
    5. 1.5 Further developments
    6. 1.6 Acknowledgements
  2. 2 Notation and definitions
    1. 2.1 The multiplier and commutator fields
    2. 2.2 Basic assumptions about the positions of the shocks
    3. 2.3 The basic structure of the equations
    4. 2.4 The wave equation for the perturbations
    5. 2.5 The commutator fields in each region
    6. 2.6 Volumes and areas
  3. 3 Multiplier identities
    1. 3.1 The energy-momentum tensor
    2. 3.2 Modified multiplier identites
      1. 3.2.1 Assumptions on perturbative quantities
    3. 3.3 Estimates for the modified scalar and energy currents in the exterior regions
  4. 4 Formulas for the energy-momentum tensor and scalar currents
    1. 4.1 The energy-momentum tensor on the constant time slices
    2. 4.2 The energy-momentum tensor along the shocks
      1. 4.2.1 The energy-momentum tensor on the spacelike side of the shock
      2. 4.2.2 The energy-momentum tensor on the timelike side of the shock
    3. 4.3 The scalar currents
  5. 5 The energy estimates
    1. 5.1 The energy estimates to the right of the right shock
    2. 5.2 The energy estimates in the region between the shocks
    3. 5.3 The energy estimates in the region to the left of the left shock
    4. 5.4 The Morawetz estimate
  6. 6 The nonlinear equations and the main theorem
    1. 6.1 The equations for the perturbations in each region
    2. 6.2 The boundary conditions along the timelike sides of the shock
      1. 6.2.1 The higher-order wave equations
    3. 6.3 The definitions of the energies
    4. 6.4 The quantities that control the geometry of the shocks
    5. 6.5 Assumptions about the initial data
    6. 6.6 The statement of the main theorem
    7. 6.7 The proof of the bootstrap proposition
  7. 7 Basic consequences of the bootstrap assumptions
    1. 7.1 Pointwise estimates
    2. 7.2 Time-integrated estimates for the potentials
    3. 7.3 Estimates for quantities along the shocks
  8. 8 Estimates for the scalar currents
    1. 8.1 Control of the metric perturbation $\gamma$, the currents $P$ and the inhomogeneous terms
    2. 8.2  Control of the scalar currents
  9. 9 The higher-order boundary conditions
    1. 9.1 Estimates for derivatives in terms of tangential derivatives
      1. 9.1.1 Estimates for $Z^{I}-Z_{T}^{I}$ in the Minkowskian case
      2. 9.1.2 Estimates for $Z^{I}-Z_{T}^{I}$ when $\mathcal{Z}=\mathcal{Z}_{m_{B}}$
    2. 9.2 Proof of Proposition 9.1
      1. 9.2.1 Supporting lemmas for the proof of Proposition 9.1
      2. 9.2.2 Control of $\Upsilon_{I,L}^{+}$
    3. 9.3 Proof of Proposition 9.2
      1. 9.3.1 Supporting lemmas for the proof of Proposition 9.2
  10. 10 The transport equation for the boundary-defining function
    1. 10.1 The asymptotic behavior of the shocks
  11. A Basic properties of the vector fields in ${\mathcal{Z}_{m}}$ and $\mathcal{Z}_{m_{B}}$
    1. A.1 Commutators with $\mathcal{Z}$
    2. A.2 Commutators with $\mathcal{Z}_{m_{B}}$
  12. B Derivation of the wave equation for the potential
    1. B.1 The continuity equation
  13. C The equation for $r\Phi$ and the higher-order continuity equation
    1. C.1 The higher-order equation in the exterior regions
    2. C.2 The higher-order equations in the central region
  14. D The Rankine-Hugoniot conditions
  15. E Stokes’ theorem
  16. F Hardy and Poincaré-type inequalities
  17. G Global Sobolev inequalities
  18. H The modified energy and scalar currents
    1. H.1 The proof of Proposition H.1
    2. H.2 The proof of Proposition H.2
    3. H.3 Estimates for a linear term verifying the null condition

##  1 Introduction

We consider the isentropic compressible Euler equations describing an ideal
gas in $\mathbb{R}^{3}$,

$\displaystyle\partial_{t}\rho+\partial_{i}(\rho v^{i})$ $\displaystyle=0,$
(1.1) $\displaystyle\partial_{t}(\rho v_{i})+\partial_{j}(\rho
v^{j}v_{i})+\partial_{i}p$ $\displaystyle=0,\qquad i=1,2,3.$ (1.2)

Here, $v=(v_{1},v_{2},v_{3})$ denotes the fluid velocity, $\rho\geq 0$ denotes
the mass density, $p$ denotes the pressure, and we are summing over repeated
upper and lower indices. The pressure $p$ is determined from the density
$p=P(\rho)$ for a given equation of state $P$, which is assumed to be smooth,
monotone and convex. The local well-posedness theory for the system
(1.1)-(1.2) with initial data lying in appropriate function spaces is
classical [22]. On the other hand, it is well-known that regular solutions to
(1.1)-(1.2) can develop singularities in finite time [46, 8, 48, 40]. In
particular, they may develop _shocks_ , surfaces across which the velocity $v$
and density $\rho$ are bounded but not differentiable. It was shown by Majda
[36, 38] that given initial data for (1.1)-(1.2) which already has a shock in
it, the solution and the shock can be continued for a short time. Given this,
it is natural to ask what happens at large times, after the formation of a
shock.

This question was first addressed by Landau [26] (whose conclusions were
rediscovered in a somewhat sharper form by Whitham [51]), who considered the
long-time behavior of _irrotational_ and _spherically symmetric_ solutions to
(1.1)-(1.2). Using a combination of geometric and approximation arguments,
Landau argued that far away from a spherically symmetric body, where sound
waves decay like $1/r$, not just one but _two_ shocks eventually emerge; these
shocks are approximately located at $\\{r=t\pm(\log t)^{1/2}\\}$, and the
velocity along the shocks decays at the non-integrable rate
$|v|\sim\frac{1}{t(\log t)^{1/2}}$.

To translate Landau’s picture into precise mathematical language, we first
observe that the _irrotational_ isentropic Euler equations reduce to a
quasilinear wave equation for the potential $\Phi$ such that $v=\nabla\Phi$,

$\Box\Phi+\partial_{\alpha}({\gamma}^{\alpha\beta}(\partial\Phi)\partial_{\beta}\Phi)=0,$
(1.3)

where $\Box$ denotes the Minkowskian wave operator and ${\gamma}(0)=0$. We
note that under the conditions $P^{\prime}(1)>0,P^{\prime\prime}(1)\neq 0$,
(1.3) does not satisfy the classical null condition and as a consequence
solutions may develop singularities in finite time even for initial data that
are small, smooth and well-localized. In some situations these singularities
are shocks in which case one can attempt to extend the local classical
solution to a global weak solution containing shocks.

Landau’s result can be interpreted as consisting of two statements:

  * •

At least in the small data regime, or alternatively, far out, the final state
of any solution contains two spherical shocks,

  * •

At large times $t$, the shocks are located at $\\{r-t\sim\pm(\log
t)^{1/2}\\}$, and the velocity along the shocks decays with the rate
$\sim\frac{1}{t(\log t)^{1/2}}$.

These types of statements, but with the shock separation $\sim t^{1/2}$ and
the shock strength decay $\sim t^{-1/2}$, are known for $1+1$-dimensional
system of conservation laws [28], [32], and even for large data in the case of
scalar conservation laws. They are however completely out of reach for higher
dimensional problems, and even more so outside of spherical symmetry.

The result of Landau can be motivated by the following heuristic, very
different from his original arguments. Introduce null coordinates $u=r-t$ and
$v=r+t$ and define $\Psi=r\Phi$. Restricting to the wave zone $r\sim t$ and
dropping nonlinear terms which verify the null condition, and which should
play no role in the long-time behavior (see below however), in spherical
symmetry the equation (1.3) takes the form

$-4\partial_{v}\partial_{u}\Psi+\frac{2}{v}\partial_{u}(\partial_{u}\Psi)^{2}=0,$
(1.4)

(see (B.39) and (C.27)) and introducing $s=\log v$ and $B=\partial_{u}\Psi$,
we find that $B$ satisfies Burgers’ equation

$\partial_{s}B+\frac{1}{2}\partial_{u}B^{2}=0.$ (1.5)

It was shown by Hopf [18, 28], that at large times, the solution of (1.5)
converges to

$\Sigma=\begin{cases}\frac{u}{s},\qquad\text{ when }|u|\leq s^{1/2},\\\
0,\qquad\text{ otherwise },\end{cases}$ (1.6)

which is a classical solution of (1.5) away from

$\Gamma^{L}_{\Sigma}=\\{u=s^{1/2}\\},\qquad\Gamma^{R}_{\Sigma}=\\{u=-s^{1/2}\\},$
(1.7)

across which the classical Rankine-Hugoniot conditions for Burgers’ equation
hold; that is, the above is a solution of (1.5) with (Burgers’) shocks.
Unwinding definitions, the solution (1.3)-(1.7) has velocity of size
$|v|\sim|B|/r\sim\frac{1}{t(\log t)^{1/2}}$ at the shocks $u=\pm s^{1/2}$.
Thus, Landau’s result could be possibly understood as the statement that at
large times, the solution to (1.3) should (in appropriate variables) approach
the profile (1.6)-(1.7).

It is worth pointing out that in the above heuristic replacing the original
equation by the effective Burgers equation required removing the nonlinear
terms satisfying the null condition. We argued that one can do so since such
terms do not influence the long term behavior. This assertion however is well
established only for smooth solutions of quasilinear wave equations. A priori
there is no reason as to why the same logic should apply to solutions with
shocks. In fact, it is not difficult to see that the dropped terms, evaluated
on the profile (1.6), will contribute $\delta$-functions to the equation. To
properly account for this one needs to observe that in fact the profile
$\Sigma$ can be upgraded to a 2-dimensional family of profiles

$\Sigma_{\xi,\eta}=\begin{cases}\frac{u}{s},\qquad\text{ when
}-\eta\,s^{1/2}\leq u\leq\xi\,s^{1/2},\\\ 0,\qquad\text{ otherwise
},\end{cases}$ (1.8)

with arbitrary constants $\xi,\eta\geq 0$. Then, first, the correct statement
about the solutions of the Burgers equation is that they converge to one of
the profiles $\Sigma_{\xi,\eta}$. This, of course, is already in [18]. And,
second, is that it is precisely the freedom of choice of $\xi,\eta$ that could
allow one to modulate, that is to make $\eta$ and $\xi$ $s$-dependent, to have
any hope to account for the terms neglected in the original equation. None of
this has been implemented even in spherical symmetry.

The goal of this paper is to partially justify Landau’s description of the
late-time behavior of irrotational solutions to (1.1)-(1.2). We do not address
the formation of a second shock (or the first one, for that matter), nor do we
show that arbitrary solutions with two shocks must behave as in Landau’s
prediction. What we do show is that initial data, _not necessarily spherically
symmetric_ , which is sufficiently close to the model shock profile
(1.6)-(1.8) (which has two shocks already in the initial data) leads to a
solution to the shock front problem which remains close to the modulated model
shock for all times. What that means is that the solution can be decomposed
into the sum of the profile $\Sigma_{\eta(s,\omega),\xi(s,\omega)}$ with
functions $\eta(s,\omega),\xi(s,\omega)$ depending on $s=\log(t+r)$ and
$\omega\in\mathbb{S}^{2}$ (the shock surfaces are no longer spherically
symmetric) which converge to bounded limits $\eta(\omega),\xi(\omega)$ as
$s\to+\infty$, and sound waves, which are smooth away from the shock surfaces
(note that $u=t-r$ and thus the right shock lies in the region $u<0$)

$\Gamma^{L}_{\Sigma}=\\{u=\xi(s,\omega)s^{1/2}\\},\qquad\Gamma^{R}_{\Sigma}=\\{u=-\eta(s,\omega)s^{1/2}\\},$

and which decay faster (this statement applies only to the region between the
shocks where the profile is nontrivial) than the profile itself. The functions
$\eta(\omega),\xi(\omega)$ encode the asymptotic behavior of the shocks and,
together with the asymptotic behavior of the profile $\Sigma$, provide the
precise statement of the Landau law of decay for weak compressible shocks. We
note that the $N$-wave shape of the profile $\Sigma$ in (1.6), which we assume
our initial data is close to, is precisely the shape that Landau claims should
emerge at late times.

The statement about asymptotic behavior of such solutions contains their
global existence as weak solutions containing two shocks. In particular we
show that such solutions do not develop any further singularities, either away
from the shocks or on their surfaces. The latter is particularly interesting
in view of the fact that in the absence of spherical symmetry shock surfaces
may be unstable to corrugation [27].

The question of existence of higher dimensional global solutions containing
shocks had been raised by Majda in his work on local well-posedness of shock
solutions. This paper in particular resolves open problem 4.6.2 from [35] in
the irrotational setting.

From the point of view of theory of general quasilinear wave equations (1.3),
our result is the first proof of global well-posedness (for solutions with
initial data given in a small neighborhood, in a weighted Sobolev norm, of the
two-shock profile) for such an equation that does not verify the null
condition; we emphasize that such solutions are not (and cannot be expected to
be) smooth, but instead are smooth away from two hypersurfaces across which
natural jump conditions hold. While the question of global well-posedness
(with small initial data in appropriately weighted Sobolev spaces) for
quasilinear wave equations (even systems) of the type

$\Box\Phi^{i}+\partial_{\alpha}(h^{\alpha\beta}(\partial\Phi)\partial_{\beta}\Phi^{i})+q^{\alpha\beta}_{ijk}(\partial\Phi)\partial_{\alpha}\Phi^{j}\partial_{\beta}\Phi^{k}=0,$

satisfying the null condition, [25],

$\partial_{\ell^{\gamma}}h^{\alpha\beta}(0)\ell_{\gamma}\ell_{\alpha}\ell_{\beta}=0,\qquad
q^{\alpha\beta}_{ijk}(0)\ell_{\alpha}\ell_{\beta}=0,\quad{\text{for
all}}\,\,i,j,k\,\,{\text{and all null}}\,\,\ell:\,m^{-1}(\ell,\ell)=0$ (1.9)

with $m$ – the Minkowski metric, is always answered in the affirmative and has
been very well understood, going back to [9, 24] and can, in some cases, be
even extended with the same answer to systems satisfying the weak null
condition [31] and nonlinearities depending on $\Phi$ instead of
$\partial\Phi$, see e.g. [30], [23], in the absence of the null or the weak
null conditions, the question has been completely open. In those cases, the
analysis stopped at the statement of singularity formation, going back to [20]
and [47] and, in the specific context of the compressible Euler equations,
followed by the more recent results referred to earlier, or the statement of
almost global existence, going back to [21], asserting that a classical
solution will exist on the time interval exponential in the inverse size of
initial data.

To our knowledge, no examples of global solutions, classical or weak, are
known for either the wave equation (1.3) without the null condition or the
compressible Euler equations on $\mathbb{R}^{3}$ in the regime of small data
or near the equilibrium state $v=0,\rho=1$, respectively. Even in other
regimes, we are not aware of any results on the wave equation, and for the
compressible Euler equations, the only exceptions are the results in [17, 45,
42] (and related works) where global classical solutions (in [45, 42]
considered as a free boundary problem with physical vacuum) had been
constructed for initial data with velocity satisfying an expansion condition
with the density $\rho$ vanishing outside of compact set. Such problems and
the corresponding solutions, of course, lie far away from the problem on the
whole $\mathbb{R}^{3}$ near the equilibrium state $v=0,\rho=1$ studied here.

We now formulate the equation and jump conditions as well as the notion of
shock front initial data more precisely. A rough version of our main theorem,
Theorem 6.1 can be found in Theorem 1.1.

In terms of the enthalpy $w$,

$w(\rho)=\int_{1}^{\rho}\frac{P^{\prime}(\lambda)}{\lambda}d\lambda,$ (1.10)

the equations (1.2) read

$\partial_{t}v_{i}+\partial_{j}(v^{j}v_{i})+\partial_{i}w=0.$ (1.11)

It follows from this equation in the usual way that if
$\omega=\textrm{\,curl\,}v$ vanishes initially and the solution remains
smooth, then $\omega=0$ at later times as well. It is therefore sensible to
look for solutions of the form $v=\nabla\Phi$, and inserting this into (1.11)
we find

$\partial_{t}\Phi+\frac{1}{2}|\nabla_{x}\Phi|^{2}=-w(\rho).$ (1.12)

If $P^{\prime}>0$, we can solve (1.10) for $\rho=\rho(w)$ and we can then
solve (1.12) for $\rho=\varrho(\partial\Phi)$. The dynamics are then
completely determined by the continuity equation (1.1), which is the following
quasilinear wave equation,

$\partial_{\mu}H^{\mu}(\partial\Phi)=0,\qquad\text{ where
}H^{0}(\partial\Phi)=\varrho(\partial\Phi)\text{ and
}H^{i}(\partial\Phi)=\varrho(\partial\Phi)\nabla^{i}\Phi.$ (1.13)

Here, and in what follows, Greek indices $\mu,\nu,...$ run over 0,1,2,3 and
Latin indices $i,j,...$ run over spatial indices 1,2,3. This is precisely the
wave equation (1.3). Under the convexity assumption on the equation of state
$P^{\prime}(1)>0$ and $P^{\prime\prime}(1)\neq 0$, see Appendix B, the
coefficients $\gamma^{\alpha\beta}$ do not satisfy the null condition (1.9):

$\partial_{\ell^{\delta}}\gamma^{\alpha\beta}(0)\ell_{\alpha}\ell_{\beta}\ell_{\delta}\neq
0,\quad\forall\ell:\,m^{-1}(\ell,\ell)=0$ (1.14)

In fact, if we parametrize all null vectors $\ell=\lambda(-1,\omega)$ with
$\lambda\in\mathbb{R}$ and $\omega\in\mathbb{S}^{2}$, then the right hand side
of the above is simply $c\lambda^{3}$ for some $c\neq 0$. By rescaling $\Phi$
one can actually assume

$\partial_{\ell^{\delta}}\gamma^{\alpha\beta}(0)\ell_{\alpha}\ell_{\beta}\ell_{\delta}=-\lambda^{3}$
(1.15)

Now, let $\Gamma\subset\mathbb{R}^{1+3}$ be a $C^{2}$ hypersurface. We say
that $\Phi$ has a _shock_ along $\Gamma$ if $\Phi$ is a classical solution to
(1.13) away from $\Gamma$, and along $\Gamma$ the Rankine-Hugoniot conditions
hold,

$\displaystyle\zeta_{\mu}[H^{\mu}(\partial\Phi)]$ $\displaystyle=0,$ (1.16)
$\displaystyle[\Phi]$ $\displaystyle=0.$ (1.17)

Here, $\zeta$ is a space-time one-form whose null space at each point $(t,x)$
is the tangent space $T_{(t,x)}\Gamma$ to $\Gamma$, and $[q]$ denotes the jump
in the quantity $q$ across $\Gamma$: if $D^{\pm}$ denote the regions to either
side of $\Gamma$ and $q_{\pm}$ denote the limits of $q$ at $\Gamma$ taken from
the regions $D^{\pm}$, then $[q]=q_{+}-q_{-}$.

We discuss the nature of the conditions (1.16)-(1.17) in Section 1.1.1 and
their relation to the compressible Euler equations (1.1)-(1.2) in Section 1.3.
For now, just note that (1.16) ensures that $\Phi$ is a weak solution to
(1.13) and (1.17) ensures that $v=\nabla_{x}\Phi$ is a weak solution to
$\textrm{\,curl\,}v=0$. The surface $\Gamma$ needs to be determined along with
$\Phi$ so that (1.16)-(1.17) hold. We then come to the following initial value
problem.

######  Definition 1 (The (restricted) shock front initial value problem).

Let $\Gamma_{0}\subset\mathbb{R}^{3}$ be a $C^{2}$ surface and let
$(\Phi_{0}^{-},\Phi_{1}^{-})$ and $(\Phi_{0}^{+},\Phi_{1}^{+})$ be initial
data posed at $t=t_{0}$ for the wave equation (1.13) defined on either side of
$\Gamma^{0}$. We say that $(\Gamma,\Phi^{-},\Phi^{+})$ is a solution to the
(restricted) shock front problem if the hypersurface
$\Gamma\subset\mathbb{R}^{1+3}$ satisfies
$\Gamma\cap\\{t=t_{0}\\}=\Gamma_{0}$, and if the $\Phi^{\pm}$ are classical
solutions of (1.13) on either side of the surface $\Gamma$ with initial data
$(\Phi_{0}^{\pm},\Phi_{1}^{\pm})$ so that at $\Gamma$, the jump conditions
(1.16)-(1.17) hold.

The above definition is extended to the case of more than one shock in the
natural way. It was shown in [38] that the above initial-value problem has a
unique local-in-time solution for _shock front initial data_ , initial data
$(\Gamma_{0},\Phi^{\pm}_{0},\Phi^{\pm}_{1})$ satisfying certain compatibility
and determinism conditions, discussed in Section 1.1.1.

We can now give the rough statement of our main theorem, Theorem 6.1.

######  Theorem 1.1.

Fix shock front initial data, posed at a large initial time, which is
sufficiently close, in appropriate weighted Sobolev norms, to the model shock
profile (1.8). That is, we assume for some sufficiently large time $t_{0}$,
the data for the potential $\Phi$ is close to the profile

$\Phi=\begin{cases}\frac{(t_{0}-r)^{2}}{r\log(t_{0}+r)},&\qquad\text{ when
}-r^{R}(t_{0},\omega)\,\log t_{0}^{1/2}\leq t_{0}-r\leq
r^{L}(t_{0},\omega)\,\log t_{0}^{1/2},\\\ 0,&\qquad\text{ otherwise
},\end{cases}$

with the functions $r^{L}(t_{0},\omega),r^{R}(t_{0},\omega)$ sufficiently
close to constants $\xi,\eta>0$, bounded away from $0$. Then the shock front
initial value problem from Definition 1 for the equation (1.3), satisfying the
condition (1.15), has a unique global-in-time solution
$(\Gamma^{L},\Gamma^{R},\Phi^{L},\Phi^{C},\Phi^{R})$ with two shocks
$\Gamma^{L},\Gamma^{R}$, where $\Gamma^{R}$ lies to the exterior of
$\Gamma^{L}$, and where $\Phi^{L}$ is defined in the region $D^{L}$ to the
left of the left shock $\Gamma^{L}$, $\Phi^{C}$ is defined in the region
$D^{C}$ between the shocks $\Gamma^{L},\Gamma^{R},$ and $\Phi^{R}$ is defined
in the region $D^{R}$ to the right of the right shock $\Gamma^{R}$. See Figure
1.

The solution has the following asymptotic behavior.

  * •

The time slices $\Gamma^{A}_{t^{\prime}}=\Gamma^{A}\cap\\{t=t^{\prime}\\}$ are
described by

$\Gamma^{L}_{t}=\\{x\in\mathbb{R}^{3}:r=t-(\log
t)^{1/2}r^{L}(t,\omega)\\},\quad\Gamma^{R}_{t}=\\{x\in\mathbb{R}^{3}:r=t+(\log
t)^{1/2}r^{R}(t,\omega)\\},$ (1.18)

where $r=|x|$ and $\omega=x/r$, for sufficiently smooth functions
$r^{L},r^{R}$ with bounded limits as $t\to\infty$.

  * •

The potentials $\Phi^{L},\Phi^{C},\Phi^{R}$ enjoy the following pointwise
decay estimates along $D^{A}_{t^{\prime}}=D^{A}\cap\\{t=t^{\prime}\\}$,

$\lim_{t\to\infty}(1+t)(1+\log
t)^{1/2}\left(\|\partial\Phi^{L}(t)\|_{L^{\infty}(D^{L}_{t})}+\|\partial(\Phi^{C}-\tfrac{(t-r)^{2}}{2r\log(t+r)})\|_{L^{\infty}(D^{C}_{t})}+\|\Phi^{R}(t)\|_{L^{\infty}(D^{R}_{t})}\right)=0.$
(1.19)

That is, the solution remains close to an appropriately modulated version of
the model shock profile (1.6)-(1.7) for all time and no further singularities
emerge.

The function $(t-r)^{2}/(2r\log(t+r))=u^{2}/(2rs)$ is just the profile (1.6)
expressed at the level of $\Phi$ instead of $B=\partial_{u}(r\Phi)$. Note that
$\partial(u^{2}/(2rs))\sim\frac{1}{(1+t)(1+\log t)^{1/2}}$ when $u\sim\pm\log
t^{1/2}$. This is precisely the rate given by the Landau law for the decay of
the shock strength. Of course, we prove and will require more detailed
information than (1.19).

Note that at the level of the fluid variables $\rho,v$ the data at time
$t_{0}$ is assumed to be close to

$v=\begin{cases}\frac{(t_{0}-r)}{r\log(t_{0}+r)}\frac{x}{r},&\qquad\text{ when
}-r^{R}(t_{0},\omega)\,\log t_{0}^{1/2}\leq t_{0}-r\leq
r^{L}(t_{0},\omega)\,\log t_{0}^{1/2},\\\ 0,&\qquad\text{ otherwise
},\end{cases}$

while $\rho$ can be found from the Bernoulli equation (1.12). For the initial
profile, $v$ vanishes identically both in front of the right shock and also
behind the left shock. As a consequence, in those regions $\rho=1$. Moreover,
globally, the value of $v$ is bounded by $1/(t_{0}\sqrt{l}ogt_{0})$, so that
in $L^{\infty}$ norm $v$ is globally close to $0$. By the same token, provided
that the equation of state $p=P(\rho)$ is convex in the neighborhood of
$\rho=1$, the density $\rho$ is uniformly globally close to $1$.

Figure 1: The surfaces in Theorem 1.1. The initial data is posed along the
time slices $D_{t_{0}}^{L},D_{t_{0}}^{C},D_{t_{0}}^{R}$. The shocks are
logarithmically-separated from the Minkowskian light cone and satisfy
$\Gamma^{L}\sim\\{(t,x):t-|x|=\xi(\log t)^{1/2}\\}$ and
$\Gamma^{R}\sim\\{(t,x):t-|x|=-\eta(\log t)^{1/2}\\}$ for positive constants
$p,q$. Each shock is spacelike with respect to the wave equation (1.21) in the
region to the exterior of the shock but timelike with respect to the wave
equation (1.21) in the region to the interior of the shock.

###  1.1 Strategy of the proof

We now describe the nature of the problem (1.13) with jump conditions
(1.16)-(1.17) and the strategy we use to prove our main theorem. In section
1.1.1 we reformulate the system as an initial-boundary value problem for the
potentials $\psi_{A}$ and the positions of the shocks. In section 1.1.2, we
describe the construction of the energy norms that we will use, and in
sections 1.1.3, we describe the main issues that come up in the course of the
proof of the energy estimates and their resolutions.

In the following sections we will continue using the null variables

$u=t-r,\quad v=t+r,\quad s=\log v.$

####  1.1.1 The determinism conditions and formulation as an initial-boundary
value problem

Given the regions $D^{L},D^{C},D^{R}$ as in the above theorem, we let $\sigma$
denote the following approximate solution (the “model shock profile”) to
(1.13)

$\sigma(t,x)=\begin{cases}\frac{u^{2}}{2rs},\qquad&\text{ in }D^{C}\\\
0,\qquad&\text{ in }D^{L},D^{R},\end{cases}$ (1.20)

whose definition is motivated by (1.6). If we let $\phi_{A}=\Phi_{A}-\sigma$,
where index $A$ refers to the regions $L,C$ and $R$, the perturbations
$\phi_{A}$ satisfy quasilinear wave equations of the form

$\partial_{\alpha}(h_{A}^{\alpha\beta}(\partial\phi_{A})\partial_{\beta}\phi_{A})=\partial_{\alpha}(g_{A}^{\alpha\beta}\partial_{\beta}\phi^{A})+\partial_{\alpha}\widetilde{j}^{\alpha}(\partial\phi_{A})=0,\qquad\text{
in }D^{A},$ (1.21)

where $\widetilde{j}$ is a quadratic nonliearity, and where the linearized
metrics $g_{A}$ are given by $g_{L}=g_{R}=m$ and $g_{C}=m_{B}$, where $m$
denotes the Minkowski metric and $m_{B}$ is the “Burgers’ metric ”,

$m=-dt^{2}+dx^{2},\qquad m_{B}=m+\frac{u}{vs}dv^{2}.$ (1.22)

In (1.21), we are suppressing various small and rapidly-decaying error terms
that appear in the central region, which arise from the fact that $\sigma$ is
not an exact solution to (1.13).

We then need to solve the equations (1.21) in the regions $D^{L},D^{C},D^{R}$.
These regions are separated by the shocks $\Gamma^{L},\Gamma^{R}$ (see Figure
1), which are assumed close to the model shocks
$\Gamma^{L}_{\Sigma}=\\{u=s^{1/2}\\}$, $\Gamma^{R}_{\Sigma}=\\{u=-s^{1/2}\\}$.
A calculation (see Lemma 2.1) reveals that the shocks and the metrics
$m,m_{B}$ satisfy the following _determinism conditions_ : the right shock
$\Gamma^{R}$ is spacelike with respect to the Minkowski metric (and thus with
respect to small perturbations of the Minkowski metric), but _timelike_ with
respect to the metric $m_{B}$. As a consequence, the solution to (1.23) in the
rightmost region $D^{R}$, and in particular along the right side of the right
shock $\Gamma^{R}$, is entirely determined for $t\geq t_{0}$ by initial data
posed in $D^{R}_{t_{0}}$. On the other hand, the solution of the equation
(1.24) in the central region $D^{C}$ is determined both by initial data in
$D^{C}_{t_{0}}$ and boundary data along $\Gamma^{R}$, which needs to be chosen
so that the Rankine-Hugoniot conditions (1.16)-(1.17) hold. Similarly, the
left shock $\Gamma^{L}$ is spacelike with respect to the metric $m_{B}$ but
timelike with respect to $m$, and so in the leftmost region we need to
prescribe boundary data along the left side of $\Gamma^{L}$.

In what follows, it will be more natural to work in terms of the variable
$\psi_{A}=r\phi_{A}$, in which case, in $D^{L}$ and $D^{R}$, (1.21) takes the
form

$-4\partial_{v}\partial_{u}\psi_{A}+\not{\Delta}\psi_{A}+\partial_{\alpha}j^{\alpha}(\partial\psi_{A})=0,\qquad\text{
for }A\in\\{L,R\\},$ (1.23)

for a nonlinearity $j$, and in $D^{C}$, it takes the form

$-4\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\partial_{u}\psi_{C}+\not{\Delta}\psi_{C}+\partial_{\alpha}j^{\alpha}(\partial\psi_{C})=0.$
(1.24)

In Section D, we show that at the right shock, the Rankine-Hugoniot conditions
(1.16)-(1.17) imply a nonlinear boundary condition for $\psi_{C}$ of the form

$\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\psi_{C}+N(\partial\psi_{C})=\partial_{v}\psi_{R}+N(\partial\psi_{R})\qquad\text{
at }\Gamma^{R},$ (1.25)

for a quadratic nonlinearity $N$, which determines (at least in the linearized
sense when $N$ can be ignored) $\psi_{C}$ along $\Gamma^{R}$ in terms of
$\psi_{R}$. At the left shock, we instead have a nonlinear boundary condition
for $\psi_{L}$ of the form

$\partial_{v}\psi_{L}+N(\partial\psi_{L})=\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\psi_{C}+N(\partial\psi_{C})\qquad\text{
at }\Gamma^{L},$ (1.26)

which, together with (1.23), determines $\psi_{L}$ along $\Gamma^{L}$ in terms
of $\psi_{C}$. The above expressions are motivated by the fact that the fields
$\partial_{v},\partial_{v}+\frac{u}{vs}\partial_{u}$ are null vectors for the
metrics $m,m_{B}$ respectively.

Since $\psi_{R}$ is determined entirely from initial data, once the position
of the right shock is known, the condition (1.25) gives boundary data for
$\psi_{C}$ along $\Gamma^{R}$ in terms of the data coming from $\psi_{R}$.
This data and the equation (1.24) determine $\psi_{C}$ uniquely in the region
$D^{C}$ if the position of the left shock $\Gamma^{L}$ is known. The condition
(1.26) then plays the same role at the left shock and determines $\psi_{L}$
along $\Gamma^{L}$ in terms of the data coming from $\psi_{C}$. In the above
discussion, we have assumed that the shocks were fixed but in reality we need
to determine them at the same time as we determine the $\psi_{A}$. In Section
D, we show that the Rankine-Hugoniot conditions (1.16)-(1.17) give evolution
equations for the positions of the shocks. We parametrize the shocks by
$\Gamma^{A}=\\{(t,x)\in\mathbb{R}^{1+3}:u=\beta_{s}^{A}(\omega)\\}$ with
$\omega=x/|x|\in\mathbb{S}^{2}$, for functions
$\beta_{s}^{A}:\mathbb{S}^{2}\to\mathbb{R}$, and (1.16)-(1.17) lead to the
following evolution equation,

$\frac{d}{ds}\beta^{A}_{s}(\omega)-\frac{1}{2s}\beta_{s}^{A}(\omega)=\left(\frac{1}{2}(\partial_{u}\psi_{A}+\partial_{u}\psi_{C})+N(\partial\psi_{A},\partial\psi_{C})\right)\bigg{|}_{u=\beta_{s}(\omega)},$
(1.27)

where $N$ collects nonlinear error terms. Note that if the right-hand side is
negligible this gives $\beta_{s}^{A}\sim\beta_{s_{0}}^{A}s^{1/2}$ for initial
data $\beta_{s_{0}}^{A}$ and thus we can recover the assumption that the
shocks are close to the model shocks if this holds initially.

We have arrived at the following initial-boundary value problem. Given
functions $\beta_{0}^{A}$ for $A\in\\{L,R\\}$ which describe the positions of
the initial shocks and so that the initial shocks are close to the initial
model shock surfaces (1.7), and given small initial data for the wave
equations (1.23),(1.24) on the initial time slices
$D^{L}_{t_{0}},D^{C}_{t_{0}},D^{R}_{t_{0}}$ (defined in the natural way in
terms of the data $\beta^{A}_{0}$), solve the wave equations

$\displaystyle-4\partial_{v}\partial_{u}\psi_{R}+\not{\Delta}\psi_{R}+\partial_{\alpha}\gamma(\partial\psi_{R})$
$\displaystyle=0,$ $\displaystyle\qquad\text{ in }D^{R},$ (1.28)
$\displaystyle-4\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\partial_{u}\psi_{C}+\not{\Delta}\psi_{C}+\partial_{\alpha}\gamma(\partial\psi_{C})$
$\displaystyle=0,$ $\displaystyle\qquad\text{ in }D^{C},$ (1.29)
$\displaystyle-4\partial_{v}\partial_{u}\psi_{L}+\not{\Delta}\psi_{L}+\partial_{\alpha}\gamma(\partial\psi_{L})$
$\displaystyle=0,$ $\displaystyle\qquad\text{ in }D^{L},$ (1.30)

subject to the boundary conditions

$\displaystyle\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\psi_{C}+N(\partial\psi_{C})$
$\displaystyle=\partial_{v}\psi_{R}+N(\partial\psi_{R})$
$\displaystyle\qquad\text{ along }\Gamma^{R},$ (1.31)
$\displaystyle\partial_{v}\psi_{L}+N(\partial\psi_{L})$
$\displaystyle=\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\psi_{C}+N(\partial\psi_{C})$
$\displaystyle\qquad\text{ along }\Gamma^{L},$ (1.32)

and where the surfaces $\Gamma^{A}$ are given by
$\Gamma^{A}=\\{(t,x):u=\beta_{s}^{A}(\omega)\\}$ for $\beta_{s}^{A}$ solving

$\displaystyle\frac{d}{ds}\beta_{s}^{A}(\omega)-\frac{1}{2s}\beta_{s}^{A}(\omega)$
$\displaystyle=\left(\frac{1}{2}(\partial_{u}\psi_{C}+\partial_{u}\psi_{A})+N(\partial\psi_{A},\partial\psi_{C})\right)\bigg{|}_{u=\beta_{s}^{A}(\omega)},$
(1.33)

By the local existence theory from [38] and the above-mentioned determinism
conditions, we are guaranteed a local-in-time unique (in the class of 2-shock
solutions) solution to the above problem. Our goal is to continue this local-
in-time solution for all time.

####  1.1.2 The energy estimates and the basic energy identity

Our proof of global existence uses a carefully constructed hierarchy of
weighted high-order energy estimates whose weights are designed to capture the
expected decay rate of solutions in each of the three regions
$D^{L},D^{C},D^{R}$. These energy estimates are obtained by commuting the
equations (1.21) with families of vector fields (the “commutator fields”) that
commute well with the linearized wave operators and then multiplying the
resulting equation by by $X\psi_{A}^{I}$ for well-chosen vector fields $X$
(the “multiplier fields”) and integrating over the region bounded between two
time slices $D^{A}_{t}$ and the shocks, where $\psi_{A}^{I}=Z^{I}\psi_{A}$
denotes a collection of vector fields $Z$ applied to $\psi_{A}$.

In the exterior regions $D^{L},D^{R}$, we use the standard Minkowskian vector
fields as commutator fields and in the central region $D^{C}$ we use the
commutator fields
$\mathcal{Z}_{m_{B}}=\\{s\partial_{u},v\partial_{v},x_{i}\partial_{j}-x_{j}\partial_{i}\\}$.
The multiplier fields we use are described below, and all of the fields we use
are recorded in sections 2.1 and 2.5. The multiplier fields need to be chosen
large enough that bounds for the resulting energies are strong enough to imply
good pointwise decay estimates, but small enough that the nonlinear error
terms we encounter in the course of proving the energy estimates can be
handled.

The basic calculation that leads to energy estimates is as follows. If the
time slices $D^{A}_{t}$ are bounded between a spacelike (with respect to the
linearized metric $g_{A}$) shock $\Gamma^{S}$ and a timelike shock
$\Gamma^{T}$ (either of which can be empty), integrating with respect to the
measure $r^{-2}dxdt$, we arrive at the identity

$\int_{D^{A}_{t_{1}}}Q_{h_{A}}(X,N_{h_{A}}^{D_{t}^{A}})=\int_{D^{A}_{t_{0}}}Q_{h_{A}}(X,N_{h_{A}}^{D^{A}_{t_{0}}})+\int_{\Gamma^{S}_{t_{0},t_{1}}}Q_{h_{A}}(X,N_{h_{A}}^{\Gamma_{S}})-\int_{\Gamma^{T}_{t_{0},t_{1}}}Q_{h_{A}}(X,N_{h_{A}}^{\Gamma_{T}})+\int_{t_{0}}^{t_{1}}\int_{D^{A}_{t}}K_{X,h_{A}}.$
(1.34)

On the spacelike surfaces $S\in\\{D^{A}_{t},\Gamma^{S}\\}$ above,
$N^{\Gamma^{S}}_{h_{A}}$ denotes the future-directed normal vector field to
$S$ defined with respect to the metric $h_{A}$, and on the timelike surface
$\Gamma^{T}$, $N^{\Gamma^{T}}_{h_{A}}$ denotes the outward-pointing normal
vector field. We are also abbreviating
$\Gamma_{t_{0},t_{1}}=\Gamma\cap\\{t_{0}\leq t\leq t_{1}\\}$, and all surface
integrals are taken with respect to the measure induced by $r^{-2}dxdt$.

The quantity $Q_{h_{A}}$ is the energy-momentum tensor associated to the
metric $h_{A}$ and $\psi_{A}^{I}$,

$Q_{h_{A}}(X,Y)=X\psi_{A}^{I}Y\psi_{A}^{I}-\frac{1}{2}h_{A}(X,Y)h_{A}^{-1}(\partial\psi_{A}^{I},\partial\psi_{A}^{I}),$
(1.35)

and the scalar current $K_{X,h_{A}}$ associated to $X$ and $h_{A}$ takes the
form $K_{X,h_{A}}=K_{X,g_{A}}+K_{X,\text{nonlinear}}$, where
$K_{X,\text{nonlinear}}$ collects the nonlinear terms and $K_{X,g_{A}}$ is the
scalar current associated to the linearized metric $g_{A}$. For the moment,
the exact expressions for these quantities are not important.

We now work out how we expect the above quantities to behave if the shocks are
close to the model shocks (1.7) and the potentials $\psi_{A}$ are sufficiently
small. First, the vector field $n=\partial_{u}$ is a null vector for both of
the linearized metrics $g_{A}\in\\{m,m_{B}\\}$, and these metrics each admit
an additional null field $\ell^{g_{A}}$ with
${\ell^{m}}=\partial_{v},\ell^{m_{B}}=\partial_{v}+\frac{u}{vs}\partial_{u}$.
If the multiplier field $X$ takes the form
$X=X^{n}_{g_{A}}n+X^{\ell}_{g_{A}}\ell^{g_{A}}$, and if $\psi_{A}$ is small
enough that $h_{A}(\partial\psi_{A})\sim g_{A}$, then the quantity on the time
slices is

$Q_{h_{A}}(X,N^{D_{t}^{A}}_{h_{A}})\sim
X_{g_{A}}^{n}(n\psi_{A}^{I})^{2}+X^{\ell}_{g_{A}}\left((\ell^{g_{A}}\psi^{I}_{A})^{2}+|\not{\nabla}\psi^{I}_{A}|^{2}\right),$
(1.36)

which is coercive (positive definite) if $X$ is future-directed and timelike
with respect to $g_{A}$, $g_{A}(X,X)<0$.

Along the spacelike surface $\Gamma^{S}$, provided $\Gamma^{S}$ is
sufficiently close to the appropriate model shock (1.8) and $\psi_{A}$ is
sufficiently small, we instead have (see Section 4.2)

$Q_{h_{A}}(X,N^{\Gamma_{S}}_{h_{A}})\sim\lambda(v)X^{n}_{g_{A}}(n\psi_{A}^{I})^{2}+X^{\ell}_{g_{A}}(\ell^{g_{A}}\psi_{A}^{I})^{2}+\left(\lambda(v)X^{\ell}_{g_{A}}+X^{n}_{g_{A}}\right)|\not{\nabla}\psi_{I}^{A}|^{2},$
(1.37)

where the weight $\lambda$ is given by
$\lambda(v)={\eta^{A}}(1+v)^{-1}(1+s)^{-1/2}$, with $\eta^{L}=\xi$ and
$\eta^{R}=\eta$, the positive constants appearing in (1.8). The expression in
(1.37) is positive-definite if $X$ is timelike and future-directed.

On the other hand, even if $X$ is timelike and future-directed, the energy-
momentum tensor along the timelike surface $\Gamma_{T}$ is not coercive and we
instead have

$-Q_{h_{A}}(X,N^{\Gamma_{T}}_{h_{A}})\sim\lambda(v)X^{n}_{g_{A}}(n\psi_{A}^{I})^{2}-X^{\ell}_{g_{A}}(\ell^{g_{A}}\psi_{A}^{I})^{2}+\left(\lambda(v)X^{\ell}_{g_{A}}-X^{n}_{g_{A}}\right)|\not{\nabla}\psi_{A}^{I}|^{2}$
(1.38)

Note that the coefficient of $|\not{\nabla}\psi_{A}^{I}|^{2}$ need not be
positive. Combining the above, for spacelike and future-directed multiplier
fields $X$ we arrive at an energy identity of the form

$E_{X}(t_{1})+S_{X}(t_{1})+B^{+}_{X}(t_{1})+\not{B}_{X}(t_{1})\lesssim
E_{X}(t_{0})+B^{-}_{X}(t_{1})+\int_{t_{0}}^{t_{1}}\int_{D^{A}_{t}}|K_{X,\text{nonlinear}}|,$
(1.39)

where the energies on the time slices are

$E_{X}(t)=\int_{D^{A}_{t}}X^{n}_{g_{A}}(n\psi_{A}^{I})^{2}+X^{\ell}_{g_{A}}\left((\ell^{g_{A}}\psi_{A}^{I})^{2}+|\not{\nabla}\psi_{A}^{I}|^{2}\right),$
(1.40)

the space-time integrated quantity $S_{X}(t_{1})$ is contributed by the linear
part of the scalar current,

$S_{X}(t_{1})=\int_{t_{0}}^{t_{1}}\int_{D^{A}_{t}}-K_{X,g_{A}},$ (1.41)

and the boundary terms $B^{\pm}_{X},\not{B}_{X}$ are

$\displaystyle B^{+}_{X}(t_{1})$
$\displaystyle=\int_{\Gamma^{S}_{t_{0},t_{1}}}\lambda(v)X^{n}_{g_{A}}(n\psi_{A}^{I})^{2}+X^{\ell}_{g_{A}}(\ell^{g_{A}}\psi_{A}^{I})^{2}+(\lambda(v)X^{\ell}_{g_{A}}+X^{n}_{g_{A}})|\not{\nabla}\psi|^{2}$
(1.42)
$\displaystyle\qquad\qquad+\int_{\Gamma^{T}_{t_{0},t_{1}}}\lambda(v)X^{n}_{g_{A}}(n\psi_{A}^{I})^{2}$
(1.43) $\displaystyle B^{-}_{X}(t_{1})$
$\displaystyle=\int_{\Gamma^{T}_{t_{0},t_{1}}}X_{g_{A}}^{\ell}(\ell^{g_{A}}\psi)^{2},\qquad\not{B}_{X}(t_{1})=\int_{\Gamma^{T}_{t_{0},t_{1}}}\left(\lambda(v)X^{\ell}_{g_{A}}-X^{n}_{g_{A}}\right)|\not{\nabla}\psi_{A}^{I}|^{2}.$
(1.44)

To illustrate the methodology of these energy estimates consider the exact
Minkowski wave equation $\Box\phi=0$ which, relative to $\psi=r\phi$, takes
the form

$-4\partial_{v}\partial_{u}\psi+\not{\Delta}\psi=0,$

in the right region $D^{R}$ where $u\leq-\eta s^{1/2}$, i.e.
$t-r\leq\eta\log^{1/2}(t+r)$. Take $X$ to be the Killing field
$X=\partial_{t}=2(\partial_{u}+\partial_{v})$. Then $K_{X}=0,$ the surface
$\Gamma^{R}=\\{u=-\eta s^{1/2}\\}$ is spacelike, and our energy identity takes
the form

$\int_{D^{R}_{t_{1}}}\left(|\partial_{u}\psi|^{2}+|\partial_{v}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)+\int_{\Gamma^{R}_{t_{0},t_{1}}}\left(\frac{|\partial_{u}\psi|^{2}}{(1+v)(1+s)^{1/2}}+|\partial_{v}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)=\int_{D^{R}_{t_{0}}}\left(|\partial_{u}\psi|^{2}+|\partial_{v}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)$

The small weight $\lambda(v)=\eta(1+v)^{-1}(1+s)^{1/2}$ appears in the above
estimate due to the fact that the surface $\Gamma^{R}$ is very close (within
$\sim\log^{1/2}v$) to the null cone $u=0$. If we extended this estimate all
the way to the null cone $\\{u=0\\}$, the corresponding energy flux would not
contain the term $|\partial_{u}\psi|^{2}$ at all.

On the other hand, the corresponding energy estimate in the left region
$D^{L}=\\{u\geq\xi s^{1/2}\\}$, where the surface $\Gamma^{L}=\\{u=\xi
s^{1/2}\\}$ is timelike, is

$\displaystyle\int_{D^{L}_{t_{1}}}$
$\displaystyle\left(|\partial_{u}\psi|^{2}+|\partial_{v}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)+\int_{\Gamma^{L}_{t_{0},t_{1}}}\frac{|\partial_{u}\psi|^{2}}{(1+v)(1+s)^{1/2}}$
$\displaystyle=\int_{D^{L}_{t_{0}}}\left(|\partial_{u}\psi|^{2}+|\partial_{v}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)+\int_{\Gamma^{L}_{t_{0},t_{1}}}\left(|\partial_{v}\psi|^{2}+\left(1-\frac{1}{(1+v)(1+s)^{1/2}}\right)|\not{\nabla}\psi|^{2}\right)$

Unlike the previous case, the future energy at time $t_{1}$ requires not just
the control of the energy at $t_{0}$ but also part of the energy flux along
$\Gamma^{L}$. Note that the boundary condition (1.32) would allow control of
$|\partial_{v}\psi|^{2}$ along $\Gamma^{L}$ but not of the term involving
$|\not{\nabla}\psi|^{2}$. This indicates that even for local existence theory,
standard energy estimates with $X=\partial_{t}$ would not be sufficient.

In general, as in the above example, the estimate (1.39) only gives very weak
control over $n\psi_{A}^{I}$ along the spacelike and timelike sides of the
shocks, but strong control over $\ell\psi_{A}^{I}$ along the spacelike sides
of the shocks. On the other hand, in the regions $D^{L},D^{C}$, we need to
treat $X^{\ell}_{g_{A}}(\ell\psi_{A}^{I})^{2}$ as an error term along the
timelike sides of the shocks. Also, the term $\not{B}_{X}$ need not have a
sign and if $\lambda(v)X^{\ell}_{g_{A}}-X^{n}_{g_{A}}<0$ we also need to be
able to bound this term.

In reality, the argument we use to establish our energy estimates is more
delicate than the calculation described above. In particular, the fact that
our nonlinearities do not satisfy the null condition means that we need to
treat the nonlinear error terms carefully; this is described in more detail in
Section 3.2. In Section 4, we collect various estimates involving the energy
momentum tensors which are used to justify estimates of the form (1.37) and
(1.38), and we also derive expressions for the linear scalar currents
$K_{X,g_{A}}$. Finally, the basic energy estimates (which are analogous to
(1.39)) we rely on are carried out in section 5.

####  1.1.3 The bootstrap argument, the decay estimates, and the choice of
multipliers

Our proof of global existence rests on a bootstrap argument, which requires
propagating a bound of the form $E_{X}(t)+B_{X}^{+}(t)\lesssim\epsilon^{2}$
for a small parameter $\epsilon$ from $t=t_{0}$ to $t=t_{1}$. All of the
multipliers we will consider will have the property that $S_{X}\geq 0$, and in
light of the identity (1.39), propagating this bound requires getting control
over (a) the nonlinear part of the scalar current $K_{X,\text{nonlinear}}$,
(b) the boundary integral $B_{X}^{-}$, which does not come with a favorable
sign and must be treated as another error term, and (c) $\not{B}_{X}$, if the
multiplier $X$ is such that $\lambda(v)X^{\ell}_{g_{A}}-X^{n}_{g_{A}}<0$.
These issues are not independent and must be resolved in tandem with one
another. We discuss these issues below.

#### Issue (a): Controlling the nonlinear scalar current

As is the case with every supercritical nonlinear equation, the mechanism
behind any global existence and stability statement is decay. For quasilinear
wave equations (1.3) on $\mathbb{R}^{3+1}$ this is a well known subtle issue
in view of the slow decay rate of linear waves, i.e., solutions of the linear
wave equation $\Box\phi=0$ on Minkowski space. Using the methods of energy
estimates with appropriate commutators and multipliers which can then be
adapted to the study of the nonlinear problems, such waves can be shown, [24],
to satisfy the bounds

$|\partial^{k}_{u}\partial_{v}^{j}\not{\nabla}^{i}\phi|\leq\frac{C_{ijk}}{(1+t)^{i+j}(1+|t-r|)^{1/2+k}}.$
(1.45)

In view of the fact that for the nonlinear wave equation (1.3) with quadratic
nonlinearities dependent on $\partial\phi$, the statement of global existence
for classical small data solutions requires time integrability of the
pointwise norm of the second derivatives of $\phi$, that is

$\int_{t_{0}}^{\infty}\|\partial^{2}\phi\|_{L^{\infty}}dt<\epsilon,$ (1.46)

for a generic equation of the type (1.3) such a statement will not hold true,
since the linear waves already violate the required integrability criterion.
It is precisely this phenomenon that led to the notion of the null condition,
imposing structure on the form of the quadratic terms, which guarantees that
for equations satisfying the null condition (1.46) is not necessary and (1.45)
is sufficient, and also to the result that for (scalar) equations that do not
satisfy the null condition small data solutions develop singularities in
finite time.

For our solutions, which are no longer classical and contain shocks, the
mechanism behind their global existence and stability statements is still
decay. As before, to control quadratic terms (which do not satisfy the null
condition) requires the time integrability of the pointwise norm of the second
derivatives. The alert reader will notice that second derivatives for shock
solutions contain $\delta$-functions of the shock surfaces and that even away
from the shocks, such an estimate does not hold for either the model shock
solution, for which in the central region $\partial^{2}\Phi\sim 1/(t(\log
t)^{1/2})$, or the linear waves (still).

The first issue is resolved by observing that the integrability statement
should hold in each region $D^{L},D^{R},D^{C}$ separately. Of course, since
the integrability/decay properties are derived from the energy estimates, both
the latter and the derivation of the former from the latter now have to be
properly localized.

The second merely suggests that we should rewrite our equation (1.3) for the
perturbation $\phi_{A}=\Phi_{A}-\sigma$ as is done in (1.21) and hope that
$\phi_{A}$ (and the source terms, omitted in (1.21), coming from the profile
$\sigma$) decays faster than the model shock profile. One of the challenges
here is that the improvement of the rate of decay of $\phi_{A}$ over the one
for the shock profile is truly marginal. In fact, pointwise, we can only
establish that

$|\partial^{2}\phi_{L}|\lesssim\frac{1}{t\log t(\log\log t)^{1/2}}$ (1.47)

which is still non-integrable. One of the novelties in this work is that in
the absence of an integrable pointwise estimate, (1.46) is established
directly.

Finally, to overcome slow decay of the linear waves we must take advantage of
the geometries of the regions $D^{L},D^{R},D^{C}$. We begin with the region
$D^{R}$ which is bounded from above by a spacelike (relative to the Minkowski
metric) hypersurface which is close to the model shock $t=r-\eta(\log
r)^{1/2}$. The solution of (1.21) in such a region is determined completely
from its initial data. The region is located (logarithmically) below the light
cone $t=r$. This indicates that the uniform bound on free waves
$|\partial^{2}\phi_{R}|\lesssim 1/t$ is not sharp. In fact, (1.45) already
suggests that using the fact that in such a region $|u|=|t-r|\geq(\log
t)^{1/2}$ we could have the bound

$|\partial^{2}\phi|\lesssim\frac{1}{t(\log t)^{5/4}}$

which is integrable. In this region, using the multiplier
$X_{R}=(1+|u|)^{\mu}\partial_{t}+r(\log r)^{\nu}\partial_{v}$, with
sufficiently large $\mu,\nu$ we can derive even stronger estimates. The
analysis of both the linear and the full nonlinear problem is straightforward.
This particular choice of the multiplier is motivated by the weighted
estimates from [30] and the $r^{p}$ method from [14]. We note that the
existence problem in regions which lie strictly below the light cone is
connected with the so-called "boost problem" considered in [12], see also the
recent work [50].

In the region $D^{C}$ the profile $\sigma$ is non-trivial and, as a result,
the linearized problem (1.21) contains the wave equation with respect to the
"Burgers metric"

$m_{B}=m+\frac{t-r}{(t+r)\log(t+r)}dv^{2}.$

Even though the deviation from the Minkowski metric is of order $1/(t(\log
t)^{1/2})$ (since in this region $|t-r|\lesssim(\log(t+r))^{1/2}$) and decays,
its influence on the behavior of linear waves is nontrivial and that behavior
is very different from that of free waves on Minkowski space. The outgoing
(radial) characteristics of the metric $m_{B}$ can be parametrized as

$u=K\log v$

(compare with the outgoing characteristics in Minkowski space given by $u=K$.)
As with the 1-dimensional rarefaction waves, the characteristics are
spreading. The quantitative effect of spreading on the behavior of linear
waves on such background is additional decay. To capture it we use the
multiplier $X_{C}=\log v\partial_{u}+v\partial_{v}$. In fact, both the
multipliers and the commutators, employed in the energy estimates in this
region, should be adapted to the metric $m_{B}$ and its properties. The result
is that in this region

$|\partial^{2}\phi|\lesssim\frac{1}{t(\log t)^{3/2}}$

The most difficult region is $D^{L}$. It is bounded on the right by a timelike
(relative to the Minkowski metric) hypersurface close to the model shock
$t=r+\xi(\log r)^{1/2}$. We are faced with the quasilinear wave equation
(1.21) supplemented with the boundary condition (1.32) along the timelike
hypersurface. The behavior of free waves in Minkowski space given by (1.45)
indicates that they decay faster in the interior of the light cone $t=r$. In
particular, in the region $D_{L}$ (1.45) would suggest the bound

$|\partial^{2}\phi_{L}|\lesssim\frac{1}{t(\log t)^{5/4}}.$ (1.48)

We are however no longer dealing with the free waves on Minkowski space but
rather with the solutions of the Minkowski wave equation on a bounded domain
with a boundary condition, and as such there is no reason to expect (1.48) to
hold. For an obvious example, consider such an equation in the cylindrical
domain $r\leq 1$ with Dirichlet or Neumann boundary conditions along $r=1$.
Linear waves for such an equation do not decay at all! The behavior of linear
(and nonlinear) waves in $D_{L}$ is entirely determined by the domain itself
and the boundary condition. To take advantage of both we employ two different
multipliers. The first is a logarithmically amplified version of the scaling
vector field

$X_{L}=uf(u)\partial_{u}+vf(v)\partial_{v},\qquad f(z)=z\log z(\log\log
z)^{\alpha},\qquad\text{ where }1<\alpha<3/2$ (1.49)

and the second, a logarithmically enhanced version of the Morawetz multiplier

$X_{M}=(\log(1+r)^{1/2}f(\log(1+r))+1)\partial_{r}.$ (1.50)

The latter is critical to establishing the integrability estimate (1.46). The
logic behind our choice of multipliers will be explained momentarily.

To summarize: to control the nonlinear scalar current, the crucial point is to
show that a bound for the energy $E_{X}\lesssim\epsilon^{2}$ implies the
following time-integrated estimate (recall that $\phi=r\psi$)

$\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\|\partial^{2}\psi_{A}\|_{L^{\infty}(D^{A}_{t})}\,dt\lesssim\epsilon.$
(1.51)

Taking into account the definition of the energies $E_{X}$, by the Klainerman-
Sobolev inequality, simple properties of our commutator fields, and the fact
that by assumption $|u|\gtrsim s^{1/2}\sim(1+\log t)^{1/2}$ in the exterior
regions $D^{L},D^{R}$, we find the pointwise bounds

$|\partial^{2}\psi_{A}|\lesssim\frac{1}{(1+\log
t)^{3/4}}\frac{1}{|X^{n}_{g^{A}}|^{1/2}}\epsilon,\quad\text{ in
}D^{L},D^{R},\quad\text{and}\qquad|\partial^{2}\psi_{C}|\lesssim\frac{1}{1+\log
t}\frac{1}{|X^{n}_{g^{C}}|^{1/2}}\epsilon.$ (1.52)

We therefore want to pick $X$ so that the coefficients $X^{n}_{g^{A}}$ are
large enough that the right-hand sides here are time-integrable; in either
case, we are “just” missing a few factors of $\log t$. However, we are not
free to choose arbitrary multiplier fields $X$; for one thing, we need to
guarantee that $-K_{X,g_{A}}\leq 0$. Moreover, the lack of null structure in
this problem and the need to be able to control various nonlinear error terms
places a limit on the size of the multipliers we consider. We will see that
this is relatively straightforward to handle in the rightmost and central
regions, but it presents serious difficulties in the leftmost region,
discussed in more detail in the next section.

#### Issue (b): Controlling the error terms along the timelike shocks

We now consider issue (b), which is only relevant in the central region and
the region to the left of the left shock. To control the boundary term
$B_{X}^{-}$, we use the boundary conditions (1.31)-(1.32); note that these
identities, at the linear level, relate $\ell^{g_{A}}\psi_{A}$ along the
timelike side of $\Gamma_{A}$ to $\ell^{g_{A}^{+}}\psi_{A}^{+}$ with
$g_{L}^{+}=g_{C},g_{C}^{+}=g_{R}$ denoting the linearized metric on the
spacelike side of $\Gamma_{A}$ and $\psi_{A}^{+}$ the corresponding potential.
Since our commutator fields are not tangent to the shocks, getting control of
$\ell^{g_{A}}\psi_{A}^{I}$ requires first decomposing the commutator fields
into components which are tangent to the shock and components which are
transverse to the shock. This in turn requires getting bounds for high-order
derivatives of the function $\beta$, which defines the shock, and for which we
will need to differentiate the evolution equation (1.33). This decomposition
is performed in Section 9 and control of high-order derivatives of $\beta$ is
established in Section 10.

Handling the above is somewhat involved, but the main difficulties in handling
the nonlinear boundary terms can be understood already when $|I|=0$. If we
directly use (1.31)-(1.32) as appropriate, we find

$\int_{\Gamma^{T}_{t_{0},t_{1}}}X^{\ell}_{g_{A}}(\ell^{g_{A}}\psi_{A})^{2}\lesssim\int_{\Gamma^{T}_{t_{0},t_{1}}}\frac{X^{\ell}_{g_{A}}}{(1+v)^{2}}|\partial\psi_{A}|^{4}+\int_{\Gamma^{T}_{t_{0},t_{1}}}X^{\ell}_{g}(\ell^{g_{A}^{+}}\psi_{A}^{+})^{2}+\frac{X^{\ell}_{g_{A}}}{(1+v)^{2}}|\partial\psi_{A}^{+}|^{4}.$
(1.53)

The last two terms will not cause any serious difficulties: we will always
have better estimates available for $\psi_{A}^{+}$ than for $\psi_{A}$. We
therefore focus on the first term. In order to handle this term, it turns out
that the main difficulty lies in establishing the estimate

$\int_{\Gamma^{T}_{t_{0},t_{1}}}\frac{X^{\ell}_{g_{A}}}{(1+v)^{2}}(n\psi_{A})^{4}\lesssim\epsilon^{2}\int_{\Gamma^{T}_{t_{0},t_{1}}}\lambda(v)X^{n}_{g_{A}}(n\psi_{A})^{2},$
(1.54)

where the quantity on the right-hand side is essentially the only control we
get over the solution along the timelike side of the shock from (1.39). We
remark that the fact that we need to handle a term of this form ultimately
derives from the fact that our nonlinearities do not satisfy the null
condition. Recalling $\lambda(v)=\xi(1+v)^{-1}(1+s)^{-1/2}$, this bound
requires that

$\frac{X^{\ell}_{g_{A}}}{1+v}(n\psi_{A})^{2}\lesssim\frac{X^{n}_{g_{A}}}{(1+s)^{1/2}}.$
(1.55)

This places a limitation on the size of the multipliers we can afford to use,
because the Klainerman-Sobolev inequality and the bounds for our energies give
us

$(n\psi_{A})^{2}\lesssim\frac{1}{1+|u|}\frac{1}{X^{n}_{g^{A}}}E_{X}(t)\lesssim\frac{1}{(1+s)^{1/2}}\frac{1}{X^{n}_{g^{A}}}\epsilon$
(1.56)

along the shocks. Inserting this into the above we find that if we want to
close estimates for the nonlinear boundary terms, we must choose the
multipliers so that the following condition holds true,

$\frac{X^{\ell}_{g^{A}}}{1+v}\lesssim|X^{n}_{g^{A}}|^{3}.$ (1.57)

This same restriction also appears even in deriving the energy identity (1.39)
and is needed to guarantee that the statements (1.37)-(1.38) hold; see in
particular Section 3.2.1 and Lemmas 4.4-4.6 where we prove bounds for the
nonlinear energy currents along the shocks.

We now come to the main difficulty: we want to choose our multiplier field
large enough that the pointwise bounds (1.52) imply the time-integrated bound
(1.51), but not so large that the condition (1.57) fails, and at the same time
we must ensure that the linear scalar current satisfies $K_{X,g_{A}}\leq 0$.
The above difficulty at the shocks is of course not present in $D^{R}$ since
there is no timelike shock to contend with, and there, as mentioned above, we
can afford to use the multiplier $X_{R}=(1+|u|)^{\mu}\partial_{t}+r(\log
r)^{\nu}\partial_{v}$, for large $\mu,\nu$.

In the central region, it turns out that the above issues are not difficult to
resolve and we can afford to use the multiplier $X=\log
v\partial_{u}+v\partial_{v}$, which is more than large enough for our
purposes. This strategy however raises issues in dealing with point (c) above,
see the next subsection.

In the leftmost region, on the other hand, this issue is nontrivial to
resolve, and the above considerations lead to the fact that we cannot afford
to use a larger multiplier than (1.49).

The estimate (1.39) coming from $X_{L}$ is still useful, since it gives
control over the quantity $uf(u)(n\psi_{A}^{I})^{2}\sim(\log t)^{1/2}f((\log
t)^{1/2})(n\psi_{A}^{I})^{2}$ near the shock (which is the most dangerous
region from the point of view of our estimates). This allows us to prove a
Morawetz inequality, obtained using the spacelike multiplier field (1.50). If
we use this in (1.34), the resulting integrals over the time slices are not
positive-definite, but the field $X_{M}$ has been chosen so that these
integrals can be bounded in terms of the energies $E_{X_{L}}$, which leads to
a bound of the form $S_{X_{M}}(t_{1})\lesssim\epsilon^{2}$. The advantage of
using this multiplier is that the scalar current $-K_{X_{M},m}$ comes with a
favorable sign and it turns out that the above bound for $S_{X_{M}}$ directly
implies the bound (1.51), which ultimately allows us to close our estimates.
This argument is carried out in section 5.4.

#### Issue (c): Controlling the angular error term $\not{B}_{X}$

We now consider issue (c) above. Again, since there is no timelike boundary to
contend with in the rightmost region, this only plays a role in the leftmost
and central regions.

In the leftmost region $D^{L}$, one interesting and important aspect of the
choice of the multiplier $X_{L}$ (1.49) is that the angular flux
$\not{B}_{X_{L}}$ along $\Gamma^{L}$ is actually positive. We have discussed
earlier that if $X_{L}=\partial_{t}$ that term is negative and, unlike the
term involving $|\partial_{v}\psi|^{2}$ in ${B}_{X_{L}}^{-}$ , it could not be
controlled from the boundary condition.

In the central region, however, the multiplier
$X_{C}={\left(s+\frac{u}{s}\right)}\partial_{u}+v\partial_{v}$ which we use to
establish our pointwise decay estimates does not satisfy this condition. As a
result, $\not{B}_{X}$ needs to be treated as an error term and we need to find
a way to control it.

For this, we couple the estimate obtained from $X_{C}$ with an estimate
obtained by using the much weaker multiplier
$X_{T}=v\ell^{m_{B}}+\left(\frac{u}{s}+\frac{\eta}{4s^{1/2}}\right)n$ (the
“top-order multiplier”). The resulting estimate is too weak to give useful
decay estimates, but this multiplier has been chosen so that
$\not{B}_{X_{T}}\geq 0$. To get the needed decay estimate, the idea is to
prove the multiplier estimate with $X_{C}$, but after commuting fewer vector
fields than we commute with in the estimate for $X_{T}$. It turns out that one
can control the resulting angular boundary term $\not{B}_{X_{C}}$ by
integrating along the shock, after bounding
$|\not{\nabla}\psi_{A}|\lesssim(1+v)^{-1}|\Omega\psi_{A}|$. This relies on the
Hardy estimate from Lemma F.4 and is carried out in Lemma 7.7; see in
particular the bound (7.51).

###  1.2 Modulated profiles and location of the shocks

Recall that the shocks

$\Gamma^{A}=\\{(t,x)\in\mathbb{R}^{1+3}:t-r=\beta_{s}^{A}(\omega)\\}$

with $A=L,R$ and $\omega=x/|x|\in\mathbb{S}^{2}$ are parametrized by the
functions $\beta_{s}^{A}:\mathbb{S}^{2}\to\mathbb{R}$ with $s=\log(t+r)$ which
satisfy the following evolution equation

$\frac{d}{ds}\beta_{s}^{A}(\omega)-\frac{1}{2s}\beta_{s}^{A}(\omega)=\left(\frac{1}{2}(\partial_{u}\psi_{C}+\partial_{u}\psi_{A})+N(\partial\psi_{A},\partial\psi_{C})\right)\bigg{|}_{u=\beta_{s}^{A}(\omega)}.$
(1.58)

These functions appear as modulation parameters of our shock profile

$\sigma(t,x)=\begin{cases}\frac{u^{2}}{2rs},\qquad&\text{ in }\beta^{L}\leq
u\leq\beta^{R}\\\ 0,\qquad&\text{ otherwise}.\end{cases}$

When $\beta^{A}=C_{A}s^{1/2}$ with constant $C_{A}$, $\sigma$ is a 2-shock
solution of the Burgers equation

$\partial_{s}\sigma+\frac{1}{2}\partial_{u}(\sigma^{2})=0.$

Modulating the profile $\sigma$ by making $\beta^{A}/s^{1/2}$ to depend
nontrivially on $s$ and $\omega$ allows to adapt the profile to fit the
equation (1.3) and, in particular, account for the correct location of the
shocks. The Rankine-Hugoniot conditions then lead to the evolution equations
for $\beta^{A}$ and connect $\beta^{A}$ to the solutions of (1.21) or (1.23).
Due to the dependence of $\beta$ on $\omega$ and from the point of view of
(1.58), the space of modulation parameters is infinite dimensional.

As discussed in the previous section dealing with the higher derivatives of
$\psi$ requires decomposing them along the shocks into its transversal and
tangential parts. This is done in order to take advantage of the (higher
order) boundary conditions. Let $Z$ be an arbitrary vector field. Along the
shock, it can be decomposed in the form

$Z=Z_{T}+Z(\beta-u)\partial_{u}$

where $Z_{T}$ is tangent to the shock. Applying this repeatedly we see that
the decomposition for $\partial^{n}(\partial\psi)$ will involve $(n+1)$
derivatives (with respect to $s$ or $\omega$) of $\beta$. Going back to (1.58)
then shows that to control those would require either the control of $(n+1)$
derivatives of $\psi$, if one the derivatives is the $s$-derivative, or even
$(n+2)$ derivatives of $\psi$ otherwise. Even in the best case scenario,
$\beta$ and $\psi$ couple to each other linearly at the highest order. This
was already a major issue in the local existence theory of Majda [36, 37, 38].
In his work, the general approach is different as the shock is straightened at
the expense of making the linearized equations for $\psi$ more complicated.
For a global problem like the one considered here shock straightening can be
costly and is avoided. To avoid the loss of derivatives which can arise when
one commutes (1.58) and the boundary conditions for $\psi$ with $(n+1)$
$\omega$-derivatives, we observe that $\beta$ also satisfies another equation

$\not{\nabla}\beta=-\frac{s}{u}[\not{\nabla}\psi]+N(\partial\psi)$

see Appendix D and Remark 12.

Nonetheless, the conclusion of this discussion is that, unlike other problems
where the modulation space is finite dimensional and the modulation parameters
couple quadratically to the unknown fields, in this problem the coupling is
linear and at the same order of differentiability.

The linear coupling also has a major effect on the asymptotic behavior of the
shocks. The logic of the proof requires that the shocks are close to the
surfaces $u=\pm s^{1/2}$ (taking $\eta=\xi=1$). Quantitatively, at the very
least, we need that the functions $\beta^{A}/s^{1/2}$ are uniformly bounded.
From (1.58),

$\left|\frac{\beta^{A}_{s_{1}}}{s_{1}^{1/2}}\right|\lesssim\left|\frac{\beta^{A}_{s_{0}}}{s_{0}^{1/2}}\right|+\int_{s_{0}}^{s_{1}}s^{-1/2}|\partial_{u}\psi|ds$

The energy estimate (1.39) contains the boundary term

$\int_{\Gamma_{t_{0},t_{1}}}\frac{1}{(1+v)(1+s)^{1/2}}X^{n}_{g_{A}}(n\psi_{A})^{2}$

which, in view of our choices of multipliers $X$, discussed above, can be
replaced by

$\int_{\Gamma_{t_{0},t_{1}}}\frac{\log s(\log\log
s)^{\alpha}}{(1+v)}(\partial_{u}\psi_{A})^{2}$

The same estimate also holds for the angular derivatives of $\psi_{A}$. Taking
that into account (and that $ds=1/vdv$),

$\int_{s_{0}}^{s_{1}}s^{-1/2}|\partial_{u}\psi|ds\lesssim\left(\int_{s_{0}}^{s_{1}}s^{-1}(\log
s)^{-1}(\log\log
s)^{-\alpha}\right)^{\frac{1}{2}}\left(\int_{s_{0}}^{s_{1}}\log s(\log\log
s)^{\alpha}|\partial_{u}\psi|^{2}ds\right)^{\frac{1}{2}}\lesssim
E_{X}^{\frac{1}{2}}$

since $\alpha>3/2$. This tells us that not only the functions $\beta/s^{1/2}$
are bounded but that they also have asymptotic limits as $s\to\infty$.

###  1.3 The full compressible Euler equations and the restricted shock front
problem

We now discuss how the problem (1.13) with jump conditions (1.16)-(1.17) is
related to the original problem (1.1)-(1.2). For _smooth_ solutions, it is
well-known that if the initial data for (1.1)-(1.2) is irrotational, then the
solution is irrotational at later times as well. However, this is _not_ true
for solutions with shocks. Indeed, consider (1.1)-(1.2) satisfying the
classical Rankine-Hugoniot conditions

$\displaystyle\zeta_{t}[\rho]+\zeta_{i}[\rho v^{i}]$ $\displaystyle=0,$ (1.59)
$\displaystyle\zeta_{t}[\rho v_{i}]+\zeta_{i}[\rho v^{i}v_{j}]+\zeta_{j}[p]$
$\displaystyle=0,\qquad j=1,2,3,$ (1.60)

across a shock, where here $\zeta=\zeta_{t}dt+\zeta_{i}dx^{i}$ is a one-form
as in (1.16), conormal to the shock. These guarantee that (1.1)-(1.2) hold in
the weak sense across the shock. One can show that if (1.59)-(1.60) hold then
in general $[\omega]\not=0$; in particular, one cannot expect to have a
solution to (1.1)-(1.2) satisfying (1.59)-(1.60) which is irrotational on
_both_ sides of the shock.

To see what is behind the above, recall that (1.1)-(1.2) describe an
isentropic fluid. If we want to take entropy into account, these equations
need to be supplemented with the conservation law for energy

$\partial_{t}(\rho E)+\partial_{i}(\rho v^{i}E+pv^{i})=0,$ (1.61)

where $E=\frac{1}{2}|v|^{2}+e(\rho,p)$, where $e$ is the specific internal
energy. Here, $p$ is no longer determined by the density $\rho$ alone but
instead $p=P(\rho,S)$ where $S$ is the specific entropy, related to the
variables $e$, $p$, $\rho$ and the temperature $T$ by the second law of
thermodynamics $de=TdS-pd(\rho^{-1})$. If $dP/dS\not=0$, irrotationality is
not preserved even for smooth solutions. On the other hand, for classical
solutions, the equation (1.61) together with the other equations is equivalent
to (1.1)-(1.2) supplemented with

$\partial_{t}S+v^{i}\partial_{i}S=0.$ (1.62)

As a result, if $S$ is initially constant and the solution remains smooth, the
motion is determined entirely by (1.1)-(1.2), and the equation (1.13)
completely determines the motion if we additionally assume that the vorticity
is initially zero.

However, if the solution develops a shock we need to supplement the Rankine-
Hugoniot conditions (1.59)-(1.60) with the jump condition

$\zeta_{t}[\rho E]+\zeta_{i}[\rho v^{i}E+pv^{i}]=0.$ (1.63)

In order for the equations (1.1)-(1.2), (1.61) and the jump conditions
(1.59)-(1.60) to be deterministic, it turns out that one needs the entropy to
have a nonzero jump aross the shock. In particular, the solution cannot remain
isentropic on both sides of the shock and as a result it cannot remain
irrotational on both sides either, in light of the fact that
$[\omega]=O([S])$, see equation (1.275) in [7].

The system (1.13) and jump conditions (1.16)-(1.17) can then be understood as
a version of the above non-isentropic problem where we ignore variations due
to entropy, even after the shock has formed. This is precisely the setting of
Christodoulou’s “restricted” shock development program [7]. The main advantage
of working with restricted shocks, beyond the conceptual simplifications of
working with (1.13) instead of (1.1)-(1.2), (1.61), is that one can can ignore
the vorticity, which there is no known way to control at large times.

This problem is of interest in its own right from the point of view of
quasilinear wave equations, and as explained in [7] and [38], it is still
physically relevant despite the above. First, a calculation (see (1.260) in
[7]) shows that the jump in entropy is small if the jump in pressure is small,
$[S]=O([p])^{3}$ and as a result $[\omega]=O([p])^{3}$ is also small, and so
solutions to (1.13) with jump conditions (1.16)-(1.17) are approximate
solutions to the full problem if $[p]$ is small (which is the case in our
setting). In fact, $[p]\sim 1/(t(\log t)^{1/2})$ and, as a result,
$[S],[\omega]=O\left(1/(t^{3}(\log t)^{3/2})\right)$ – negligible from the
point of view of decay.

We also note that (1.16)-(1.17) imply that three of the conditions
(1.59)-(1.60) hold. Indeed, the condition (1.16) is nothing but (1.59), and if
we decompose (1.60) into its components parallel and transverse to $\zeta$, we
find the relations

$\displaystyle[\overline{v}_{i}]$ $\displaystyle=0,$ (1.64)
$\displaystyle\zeta_{t}[\rho v_{\zeta}]+[\rho v_{\zeta}^{2}]+[p]$
$\displaystyle=0,$ (1.65)

with $v_{\zeta}=\zeta_{i}v^{i}$ and with $\overline{v}$ the component of $v$
tangent to the shock. If $v=\nabla\Phi$ where $\Phi$ satisfies the jump
condition (1.17) then (1.64) holds, but there is no guarantee that (1.65)
holds. Our jump conditions (1.16)-(1.17) then ensure that the jump condition
(1.59) associated to the continuity equation holds, and the tangential
components of the conditions (1.60) associated to the momentum equations hold,
but we do not enforce the normal component of (1.60).

###  1.4 Further background on the problem and related results

The mathematical theory of the compressible Euler equations has a long
history, with an enormous amount of literature devoted to it. It would be
impossible for us to survey it, but see for example [8, 13, 29] and the
references therein. We will concentrate on the results more related to the
subject of this paper which can be put into two categories: asymptotic
behavior of solutions for the equations in one space dimension and more recent
work on the problems of breakdown and shock formation and evolution in higher
dimensions.

Breakdown for smooth solutions of (1.1)-(1.2) in one space dimension dates
back to work of Challis [6] and Riemann [43]. The long-time behavior of
solutions of Burgers’ equation, which serves as an important model for the
Euler equations, was studied by Hopf [18] who was able to extract the
asymptotic shape of solutions after shock formation. This was generalized to
other one-dimensional scalar conservation laws by Lax [28, 29]. For a single
convex scalar conservation law the following sharp result, which is
particularly instructive to compare to the main result of this paper, is
proven in [15].

######  Theorem 1.2.

[15] Let $v$ be a BV solution with initial data of compact support of the
equation

$\partial_{t}v+\partial_{x}f(v)=0$

with $f^{\prime\prime}>0$ and $f^{\prime}(0)=0$, $f^{\prime\prime}(0)=1$, and
let $N(t,\eta,\xi)$ denote the $N$-wave (cf. (1.8))

$N(t,p,q)=\begin{cases}\frac{x}{t},\qquad\text{ when }-\eta\,t^{1/2}\leq
x\leq\xi\,t^{1/2},\\\ 0,\qquad\text{ otherwise },\end{cases}$

Then there exist constants $\eta,\xi\geq 0$ depending on the initial data such
that

$\|v(t,\cdot)-N(t,\eta,\xi)\|_{L^{1}}\lesssim t^{-1/2}$

for all sufficiently large $t$.

This result was generalized (for small initial data) to systems of 2
conservation laws in [16] and, finally, to systems of $n$ conservation laws in
[32]. These results should be compared with our Theorem 1.1 which gives the
asymptotic behavior of 2-shock solutions of the equation (1.3) corresponding
to the irrotational compressible Euler equations on $\mathbb{R}^{3}$ or,
generally, the wave equation (1.3) without the null condition, and the
convergence statement in $L^{\infty}$.

The first proof of singularity formation for the compressible Euler equations
in higher dimensions was given by Sideris in [46]. There, the proof is by a
virial argument and does not give any information about the nature of the
singularity. Alinhac’s work [2, 3] on the 2-dimensional version of the
equation (1.3) gave the first constructive proof of the "first time"
singularity formation.

In the monumental work [8] (see also [11]), Christodoulou was able to describe
the maximal classical development for the solutions of the compressible Euler
equations contained in the domain of dependence of the exterior of a sphere of
arbitrary, small, regular initial data which is constant outside of a larger
sphere, and gave a detailed description of the singular boundary. These
results were extended to different regimes, of initial data forming small open
sets of specific profiles for the problem on $\mathbb{R}\times\mathbb{T}^{2}$
in [1] and allowing for nontrivial vorticity at the singular boundary, and
where the authors were able to give a more complete description of the portion
of the maximal development near the crease – first singularity, even in the
absence of strict convexity. For the corresponding results in 2d see also [34]
and [44], where in the latter reference, the authors gave a detailed
description of a maximal development including the portion of a Cauchy horizon
for the problem on $\mathbb{T}^{2}$ for a specific small open set of initial
data. The "first time" singularity formation for the full problem, again for a
specific small open set of initial data, were given in [33], [4]. Shock
formation for a class of quasilinear wave equations in 2d was investigated in
[49] and for a class of large data in 3d in [41].

A different mechanism for blowup for the compressible Euler equations with
smooth data in three dimensions with a very different character was recently
discovered in [40]. The singularities constructed there arise from large
initial data, they are not shocks and instead the density blows up in finite
time.

The problem of local-in-time existence for the multi-dimensional shock front
problem was solved by Majda in the works [36, 37]. There, Majda considered
initial data for a large class of hyperbolic systems, including the
compressible Euler equations, which already has a shock in it and constructed
a local-in-time solution to the shock front problem. In [38], Majda and
Thomann gave a different proof of local existence for the restricted shock
front problem described in Definition 1.

In recent years, starting with the breakthrough results [7] of Christodoulou,
there has also been a great interest in the _shock development problem_ ,
wherein one starts with the singular solution constructed in the process of
the of solving the shock development problem and replaces/extends it with the
weak solution containing a shock. A recent result for the 2d problem with
azimuthal symmetry is in [5]. For earlier results in spherical symmetry see
[10],[52].

###  1.5 Further developments

As we mentioned earlier, our work addresses only part of the picture described
by Landau (in spherical symmetry). In particular, the question of whether
solutions arising from small smooth initial data for large times approach a
2-shock profile remains open (even in spherical symmetry). Already,
constructing an example of the above scenario would be very interesting.

The next obvious step is to address the full problem (1.1)-(1.2), (1.61),
without the irrotational condition and allow for the production of vorticity
and entropy across the shocks. Such a problem in the whole space is completely
intractable for the same reasons as the corresponding 3d problem of shock
formation. Vorticity and entropy waves propagate with the speed of the fluid
and do not decay. As a result, assuming that initially vorticity and entropy
are of compact support, the support will remain compact and, eventually, will
be contained in the interior of the left region $D^{L}$, where the vorticity
waves could undergo vorticity stretching and form singularities of a very
different kind. Nonetheless, it would still be possible and desirable to
consider the problem for the points which lie in the domain of dependence of
the exterior of a sphere. Such a domain would necessarily contain the right
shock $\Gamma^{R}$ in our picture and the vorticity would decay there since it
would be eventually transported away from this domain. As was discussed in
Section 1.3, the vorticity and entropy produced by the shock are proportionate
to the third power of the strength of the shock which is $\sim 1/(t(\log
t)^{1/2})$. As a result, their influence is much weaker than that of the sound
waves and should be easily controlled.

Landau’s paper also discusses the 2-dimensional case. There, two shocks are
supposed to be separated by distance of $\sim t^{1/4}$ and the strength of
shocks should decay with the rate $\sim t^{-3/4}$. This rate is even further
away from integrable than in the 3d case. Additionally, 2d free waves decay
considerably slower than in 3d. Nonetheless, the shocks are further away from
the null cone and both the geometry and preliminary analysis of the problem
indicate that the 2d statements analogous to the ones proven in this paper
likely hold true.

In this paper we considered the global problem involving spherical (but not
spherically symmetric) shocks which are expected to emerge from compactly (or
rapidly decaying) initial data. Of separate interest would be to consider
other geometries and, in particular, investigate the problem of stability of
planar (non-symmetric) shocks.

###  1.6 Acknowledgements

IR acknowledges support through NSF grants DMS-2005464 and a Simons
Investigator Award. DG acknowledges support from the Simons Collaboration on
Hidden Symmetries and a startup grant from Brooklyn College.

##  2 Notation and definitions

Let $t,x^{1},x^{2},x^{3}$ denote the usual rectangular coordinates. We will
work in terms of the Minkowskian null coordinates

$u=t-|x|,\qquad v=t+|x|,\qquad\theta^{1}(x),\qquad\theta^{2}(x),$ (2.1)

where $\theta^{1},\theta^{2}$ are an arbitrary local coordinate system on the
unit sphere $\mathbb{S}^{2}$. We will use $s$ to denote

$s=\log v.$

We will write

$\omega_{i}=\delta_{ij}\omega^{j}=\frac{x_{i}}{|x|},\qquad\not{\Pi}_{i}^{j}=\delta_{i}^{j}-\omega_{i}\omega^{j},\qquad\not{\nabla}_{i}=\not{\Pi}_{i}^{j}\nabla_{j},\quad
i,j=1,2,3,$ (2.2)

where $\nabla$ denotes the covariant derivative defined with respect to the
Minkowski metric,

$m=-dt^{2}+dx^{2}=-dudv+\frac{1}{4}(v-u)^{2}dS(\omega),$ (2.3)

where $dS(\omega)$ denotes the metric on the unit sphere $\mathbb{S}^{2}$.

In the region between the shocks, the perturbation will satisfy a quasilinear
wave equation which is a perturbation of a wave equation with respect to the
“Burgers’ ” metric $m_{B}$,

$m_{B}=-dt^{2}+dx^{2}+\frac{u}{vs}dv^{2}=-dudv+\frac{u}{vs}dv^{2}+\frac{1}{4}(v-u)^{2}dS(\omega)$
(2.4)

We also record that the inverse metrics are given by

$m^{-1}(\xi,\xi)=-4\xi_{u}\xi_{v}+4(v-u)^{-2}|\not{\xi}|^{2}$ (2.5)

and

$m_{B}^{-1}(\xi,\xi)=-4\xi_{u}\xi_{v}-\tfrac{{4}u}{vs}\xi_{u}^{2}+4(v-u)^{-2}|\not{\xi}|^{2}.$
(2.6)

The metrics $g=m,m_{B}$ admit two null vectors $(n,\ell^{g})$ where

$\ell^{m}=\partial_{v},\qquad\ell^{m_{B}}=\partial_{v}+\frac{u}{vs}\partial_{u},\qquad{n=\partial_{u}}$
(2.7)

which satisfy, in either case,

$g(n,\ell^{g})=-\frac{1}{2}.$ (2.8)

For a vector field $X$ we write

$X=X^{\ell}_{g}\ell^{g}+X^{n}_{g}n^{g}+\not{X}$ (2.9)

where

$X^{\ell}_{m}=X^{v},\quad X^{n}_{m}=X^{u},\qquad\qquad
X^{\ell}_{m_{B}}=X^{v},\quad X^{n}_{m_{B}}=X^{u}-\frac{u}{vs}X^{v},$ (2.10)

and where the angular part $\not{X}=\Pi\cdot X$ with $\Pi$ as in (2.2). Note
that from (2.8),

$g(X,Y)=-\frac{1}{2}(X^{n}_{g}Y^{\ell}_{g}+X^{\ell}_{g}Y^{n}_{g})+g(\not{X},\not{Y}).$
(2.11)

###  2.1 The multiplier and commutator fields

For the convenience of the reader, we record here the multiplier fields we use
in each of the three regions $D^{R},D^{C},D^{L}$.

Region | Multipliers | Commutators
---|---|---
$D^{R}$ $(u\lesssim-s^{1/2})$ | $X_{R}=w(u)(\partial_{u}+\partial_{v})+r(\log r)^{\nu}\partial_{v}$ | $\mathcal{Z}=\\{\partial_{\mu},\Omega_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu},S=x^{\alpha}\partial_{\alpha}\\}$
$D^{C}$ $(|u|\lesssim s^{1/2})$ |  $X_{C}={\left(s+\frac{u}{s}\right)}\partial_{u}+v\partial_{v}$, $X_{T}={\big{(}\tfrac{u}{s}+\tfrac{\eta}{{4}s^{1/2}}\big{)}\partial_{u}+v\partial_{v}}$ | $\mathcal{Z}_{m_{B}}=\\{\Omega_{ij},X_{1}=s\partial_{u},X_{2}=v\partial_{v}\\}$
$D^{L}$ $(u\gtrsim s^{1/2})$ |  $X_{L}=uf(u)\partial_{u}+vf(v)\partial_{v}$, $X_{M}=(g(r)+1)(\partial_{v}-\partial_{u})$ | $\mathcal{Z}$

In the above, we use the convention that $x^{i}=x_{i}$ for $i=1,2,3$ and
$x^{0}=-x_{0}=t$. The functions $f,g$ are

$f(z)=\log z(\log\log z)^{\alpha},\qquad g(z)=(\log(1+z))^{1/2}f(\log(1+z))$
(2.12)

and the parameters $\mu,\alpha$ will be chosen subject to (6.20).

The roles of these multipliers are explained in section 1.1.3 and the energies
associated to these multipliers along with the corresponding energy estimates
can be found in section 5.

###  2.2 Basic assumptions about the positions of the shocks

We let $\Gamma^{R},\Gamma^{L}$ denote the right and left shock respectively,
and write $\Gamma^{A}_{t^{\prime}}=\Gamma^{A}\cap\\{t=t^{\prime}\\}$ for
$A=L,R$. We will parametrize the shocks by functions
$\beta_{s}^{L},\beta_{s}^{R}:\mathbb{S}^{2}\to\mathbf{R}$ where the parameter
$s\in[s_{0},s_{1}]$ for some $s_{0},s_{1}>0$, so that the shocks are given by

$\Gamma^{A}=\\{(t,x)|u=\beta^{A}_{\log(t+|x|)}(x/|x|)\\}.$ (2.13)

We will prove energy estimates assuming that the left shock is sufficiently
close to the surface $u=-\xi(\log v)^{1/2}$ and the right shock is
sufficiently close to the surface $u=\eta(\log v)^{1/2}$ for constants
$\xi,\eta>0$. In particular we will assume that the initial positions of the
shocks are parametrized by

$t_{0}-r=\beta^{L}_{0}(\omega),\qquad t_{0}-r=\beta^{R}_{0}(\omega),$ (2.14)

for functions $\beta^{L}_{0},\beta^{R}_{0}$ which are sufficiently close to
the positions of the model shocks,

$\left|\nabla_{\omega}^{j}\left({\xi-}\frac{\beta^{L}_{0}(\omega)}{(1+s^{L}_{0})^{1/2}}\right)\right|+\left|\nabla_{\omega}^{j}\left({\eta+}\frac{\beta^{R}_{0}(\omega)}{(1+s^{R}_{0})^{1/2}}\right)\right|\leq\epsilon,$
(2.15)

for $j=0,1$, and where $s^{L}_{0},s^{R}_{0}$ denote the values of
$s=\log(t+|x|)$ along the shocks $\Gamma^{L},\Gamma^{R}$ at $t=t_{0}$.

We will assume that that for $\epsilon_{1},\epsilon_{2}$ sufficiently small,
we have the bounds

$|\beta^{L}_{s}(\omega)-\beta_{0}^{L}(\omega)s^{1/2}|+(1+s)\left|\frac{d}{ds}\beta^{L}_{s}-\frac{1}{2s}\beta^{L}_{s}\right|\leq\epsilon_{1}(1+s)^{1/2},\qquad|\nabla_{\omega}\beta_{s}^{L}(\omega)|\leq\epsilon_{2}(1+s)^{1/2},$
(2.16)

with the same assumptions at the right shock,

$|\beta^{R}_{s}(\omega)-\beta_{0}^{R}(\omega)s^{1/2}|+(1+s)\left|\frac{d}{ds}\beta^{R}_{s}-\frac{1}{2s}\beta^{R}_{s}\right|\leq\epsilon_{1}(1+s)^{1/2},\qquad|\nabla_{\omega}\beta_{s}^{R}(\omega)|\leq\epsilon_{2}(1+s)^{1/2}.$
(2.17)

To close the estimates along the timelike sides of the shocks, we will also
need to assume control of higher-order norms of the functions $\beta^{A}_{s}$,
see Section 6.4.

It is convenient to introduce the following extension of $\beta^{A}_{s}$ to a
neighborhood of the shocks,

$B^{A}(t,x)=\beta^{A}_{\log(t+|x|)}(x/|x|),$ (2.18)

which satisfies

$\partial_{u}B^{A}(t,x)=0,\qquad\partial_{v}B^{A}(t,x)=\frac{1}{v}\frac{d}{ds}\beta^{A}_{\log(t+|x|)}(x/|x|),\qquad\not{\nabla}B^{A}(t,x)=\frac{1}{r}\left(\nabla_{\omega}\beta^{A}_{\log(t+|x|)}\right)(x/|x|),$
(2.19)

where in the last expression we have identified the abstract sphere
$\mathbb{S}^{2}$ with the subset $\\{|x|=1\\}\subset\mathbf{R}^{4}$. Then the
tangent space to $\Gamma^{A}$ at each point lies in the null space of the one-
form $\zeta^{A}$ given by

$\zeta^{A}=-\frac{1}{2}d(u-B^{A})=-\frac{1}{2}du+\frac{1}{2}\partial_{v}B^{A}dv+\frac{1}{2}\not{\nabla}B^{A}\cdot\not{dx}=-\frac{1}{2}du+\frac{1}{2v}\frac{d}{ds}\beta_{s}^{A}dv+\frac{1}{2r}\nabla_{\omega}\beta_{s}^{A}\cdot\not{dx},$
(2.20)

where $\not{dx}$ denotes the projection of $dx$ to the cotangent space to the
unit spheres and where $s=\log(t+|x|)$.

We will work in terms of a vector field $N_{m}^{\Gamma^{A}}$ which is normal
to $\Gamma^{A}$ with respect to the Minkowski metric, given by raising the
index of (2.20) with the Minkowski metric,

$N^{\Gamma^{A}}_{m}=\partial_{v}-\partial_{v}B^{A}\partial_{u}{+\frac{1}{2}}\not{\nabla}B^{A}\cdot\not{\nabla},$
(2.21)

and similarly we will work in terms of a vector field $N^{\Gamma^{A}}_{m_{B}}$
which is normal to $\Gamma^{A}$ with respect to the Burgers’ metric $m_{B}$,

$N^{\Gamma^{A}}_{m_{B}}=\partial_{v}+\left(2\tfrac{u}{vs}-\partial_{v}B^{A}\right)\partial_{u}{+\frac{1}{2}}\not{\nabla}B^{A}\cdot\not{\nabla}.$
(2.22)

We will often just write $N^{A}_{g}$ in place of $N^{\Gamma^{A}}_{g}$. We have
chosen $N^{A}_{g}$ so that when $\Gamma^{A}$ is spacelike with respect to $g$,
$N^{A}_{g}$ is the future-directed normal to $\Gamma^{A}$, and when
$\Gamma^{A}$ is timelike with respect to $g$, $N^{A}_{g}$ is inward-pointing.
It will be convenient later on to write these formulas in terms of the null
vectors $(n,\ell^{g})$ defined in (2.7). Writing
$N_{g}=N_{g}^{n}n+N_{g}^{\ell}\ell^{g}+\not{N}_{g}$ where $\not{N}_{g}$
denotes the angular part of $N_{g}$, in either case we have

$N^{\ell}_{g}=\frac{g(N_{g},n)}{g(\ell^{g},n)}=-2\zeta(n)=1,\qquad
N^{n}_{g}=\frac{g(N_{g},\ell^{g})}{g(\ell^{g},n)}=-2\zeta(\ell^{g})=-\ell^{g}(B-u).$
(2.23)

We also record the following identities

$g(X,N_{g})=-\frac{1}{2}X(u-B)=-\frac{1}{2}(X^{n}_{g}N_{g}^{\ell}+X^{\ell}_{g}N_{g}^{n})-\frac{1}{2}X\cdot\not{\nabla}B=-\frac{1}{2}(X^{n}_{g}-X^{\ell}_{g}\ell^{g}(B-u))-\frac{1}{2}X\cdot\not{\nabla}B$
(2.24)

where we used (2.11) and (2.23).

A vector field $X$ is called timelike with respect to the metric $g$ when
$g(X,X)<0$ and spacelike if $g(X,X)>0$. We say a surface $\Sigma$ is timelike
(respectively spacelike) with respect to $g$ if the normal field
$N_{g}^{\Sigma}$ to $\Sigma$, associated to the metric $g$ is spacelike
(respectively timelike). If we use (2.24) with $X=N_{g}$, we find

$g(N_{g}^{\Gamma^{A}},N_{g}^{\Gamma^{A}})=\ell^{g}(B^{A}-u)+\frac{1}{4}|\not{\nabla}B^{A}|^{2}.$
(2.25)

When $g=m$ so $\ell^{g}=\partial_{v}$, along $\Gamma^{A}$ where $u=B^{A}$, the
above reads

$\ell^{g}(B^{A}-u)+\frac{1}{4}|\not{\nabla}B^{A}|^{2}=\partial_{v}B^{A}+\frac{1}{4}|\not{\nabla}B^{A}|^{2}=\frac{u}{2vs}+\frac{1}{v}\left(\partial_{s}B^{A}-\frac{B^{A}}{2s}\right)+\frac{1}{4}|\not{\nabla}B^{A}|^{2},$
(2.26)

where the last two terms are negligible, by (2.16)-(2.17) (which are written
at the level of $\beta^{A}=B^{A}|_{\Gamma^{A}}$). When instead $g=m_{B}$, so
$\ell^{g}=\partial_{v}+\frac{u}{vs}\partial_{u}$, we have

$\ell^{g}(B^{A}-u)+\frac{1}{4}|\not{\nabla}B^{A}|^{2}=\partial_{v}B^{A}-\frac{u}{vs}+\frac{1}{4}|\not{\nabla}B^{A}|^{2}=-\frac{u}{2vs}+\frac{1}{v}\left(\partial_{s}B^{A}-\frac{B^{A}}{2s}\right)+\frac{1}{4}|\not{\nabla}B^{A}|^{2}.$
(2.27)

Recalling that at $\Gamma^{R}$, $u\sim-s^{1/2}\eta$ and at $\Gamma^{L}$,
$u\sim s^{1/2}\xi$, where $\eta,\xi$ are the positive constants from (2.15),
if the assumptions (2.16)-(2.17) about the positions of the shocks hold, then
in particular

$\ell^{g}(B^{A}-u)\sim\begin{cases}-\frac{{\eta^{A}}}{2(1+v)(1+s)^{1/2}},\qquad&\text{
when }\Gamma^{A}\text{ is spacelike with respect to }g,\\\
\frac{{\eta^{A}}}{2(1+v)(1+s)^{1/2}},\qquad&\text{ when }\Gamma^{A}\text{ is
timelike with respect to }g,\end{cases}$ (2.28)

with $\eta^{L}=\xi,\eta^{R}=\eta$ positive constants.

Since, by the same assumptions, the angular derivatives of $B^{A}$ are small,
from (2.25) it follows the left shock is timelike with respect to the
Minkowski metric but spacelike with respect to $m_{B}$, while the right shock
is timelike with respect to $m_{B}$ but spacelike with respect to $m$. We
record the result of the above calculation.

######  Lemma 2.1.

For $g=m,m_{B}$, we have
$g(N_{g}^{\Gamma^{A}},N_{g}^{\Gamma^{A}})=\ell^{g}(B^{A}-u)+\frac{1}{4}|\not{\nabla}B^{A}|^{2}$.
Explicitly,

$m(N^{\Gamma^{A}}_{m},N^{\Gamma^{A}}_{m})=\partial_{v}B^{A}+{\frac{1}{4}}|\not{\nabla}B^{A}|^{2},\qquad
m_{B}(N^{\Gamma^{A}}_{m_{B}},N^{\Gamma^{A}}_{m_{B}})=\partial_{v}B^{A}-\frac{u}{vs}+{\frac{1}{4}}|\not{\nabla}B^{A}|^{2}.$
(2.29)

In particular, if the assumptions (2.16),(2.17) about the positions of the
shocks hold, the left shock is timelike with respect to $m$ and spacelike with
respect to $m_{B}$,

$m(N^{\Gamma^{L}}_{m},N^{\Gamma^{L}}_{m})>0,\qquad
m_{B}(N^{\Gamma^{L}}_{m_{B}},N^{\Gamma^{L}}_{m_{B}})<0,$ (2.30)

and the right shock is timelike with respect to $m_{B}$ and spacelike with
respect to $m$,

$m_{B}(N^{\Gamma^{R}}_{m_{B}},N^{\Gamma^{R}}_{m_{B}})>0,\qquad
m(N^{\Gamma^{R}}_{m},N^{\Gamma^{R}}_{m})<0.$ (2.31)

There is a constant $c_{0}$ so that if $h$ is a metric with
$h^{-1}=m^{-1}+\gamma$ where $|\gamma|\leq
c_{0}\frac{1}{1+v}\frac{1}{(1+s)^{1/2}}$ then the same statements (2.30),
(2.31) hold with $m$ replaced by $h$. In the same way, if
$h^{-1}=m_{B}^{-1}+\gamma$ where $|\gamma|\leq
c_{0}\frac{1}{1+v}\frac{1}{(1+s)^{1/2}}$ then the same statements hold with
$m_{B}$ replaced by $h$.

We also record for later use that if (2.16)-(2.17) hold then at the shock
$\Gamma^{A}$

$-\frac{1}{2}g(X,N_{g}^{\Gamma^{A}})\sim\begin{cases}\frac{1}{4}\left(X^{n}_{g}+\frac{{\eta^{A}}}{{2}(1+v)(1+s)^{1/2}}X^{\ell}_{g}\right),&\qquad\text{
when }\Gamma^{A}\text{ is spacelike with respect to }g,\\\
\frac{1}{4}\left(X^{n}_{g}-\frac{{\eta^{A}}}{{2}(1+v)(1+s)^{1/2}}X^{\ell}_{g}\right),&\qquad\text{
when }\Gamma^{A}\text{ is timelike with respect to }g,\\\ \end{cases}$ (2.32)

again with $\eta^{R}=\eta,\eta^{L}=\xi$. This follows from the formula (2.24)
and (2.28). In particular we note that if $X$ is timelike and future-directed,
in the spacelike case, this quantity is positive-definite but in the timelike
case it may take either sign.

###  2.3 The basic structure of the equations

We assume that $\rho$ is given in terms of the density by an equation of state
$p=P(\rho)$. We will assume that the equation of state satisfies
$P^{\prime}(1)>0,P^{\prime\prime}(1)\neq 0$ and $P\in
C^{\infty}(\mathbf{R}\setminus\\{0\\}).$ The enthalpy $w=w(\rho)$ is defined
by

$w(\rho)=\int_{1}^{\rho}\frac{P^{\prime}(\lambda)}{\lambda}d\lambda.$ (2.33)

From Bernoulli’s equation, $w$ is determined from $\partial\Phi$ according to

$w(\partial\Phi)=-\partial_{t}\Phi-\frac{1}{2}|\nabla\Phi|^{2}.$ (2.34)

Since $p^{\prime}>0$ it follows that $\rho\mapsto w(\rho)$ is an invertible
function, which we denote $\rho=\rho(w)$. We then define $\varrho$ by
$\varrho=\varrho(\partial\Phi)=\rho(w(\partial\Phi))$. For the convenience of
the reader we record that for the “polytropic” equation of state
$p(\rho)=\rho^{\gamma}$ with $\gamma>1$, we have

$w(\rho)=\int_{1}^{\rho}\gamma\lambda^{\gamma-2}\,d\lambda=\frac{\gamma}{\gamma-1}\left(\rho^{\gamma-1}-1\right),\qquad\rho(w)=\left(\frac{\gamma-1}{\gamma}w+1\right)^{1/(\gamma-1)}.$
(2.35)

With the above notation, define

$H^{0}(\partial\Phi)=\varrho(\partial\Phi),\qquad
H^{i}(\partial\Phi)=\varrho(\partial\Phi)\nabla^{i}\Phi.$ (2.36)

Then the continuity equation takes the form

$\partial_{\alpha}H^{\alpha}(\partial\Phi)=0,$ (2.37)

with $\partial_{\alpha}=\partial_{x^{\alpha}}$ where $x^{\alpha}$ denote
Cartesian coordinates on $\mathbf{R}^{4}$, and the jump conditions (1.16),
(1.17) take the form

$[H^{\alpha}(\partial\Phi)]\zeta_{\alpha}=0,\qquad[\Phi]=0.$ (2.38)

After an appropriate rescaling of the dependent and independent variables (see
Lemma B.1), the quantities in (2.36) take the form

$H^{\alpha}(\partial\Phi)=m^{\alpha\beta}\partial_{\beta}\Phi+\gamma^{\alpha\beta\delta}\partial_{\beta}\Phi\partial_{\delta}\Phi+G^{\alpha}(\partial\Phi),$
(2.39)

for constants $\gamma^{\alpha\beta\delta}$, where $G^{\alpha}$ is a cubic
nonlinearity, and where the quadratic terms are of the form

$\gamma^{\alpha\beta\delta}\partial_{\beta}\Phi\partial_{\delta}\Phi=-\delta^{\alpha}_{u}(\partial_{u}\Phi)^{2}+\overline{\gamma}^{\alpha\beta\delta}\partial_{\beta}\Phi\partial_{\delta}\Phi.$
(2.40)

Here, we are writing

$\delta^{\alpha}_{u}=\delta^{\alpha\beta}\partial_{\beta}u=\delta^{\alpha}_{0}-\delta^{\alpha}_{i}\omega^{i},\qquad\overline{\gamma}^{\alpha\beta\delta}=\gamma^{\alpha\beta\delta}+\delta_{u}^{\alpha}\delta_{u}^{\beta}\delta_{u}^{\delta}.$
(2.41)

(we have normalized so that
$\gamma^{uuu}=\gamma^{\alpha\beta\delta}\partial_{\alpha}u\partial_{\beta}u\partial_{\delta}u=-1$).
With
$\overline{\gamma}^{u\beta\delta}=\overline{\gamma}^{\alpha\beta\delta}\partial_{\alpha}u$,
the second term satisfies

$|\overline{\gamma}^{u\beta\delta}\partial_{\beta}\Psi_{1}\partial_{\delta}\Psi_{2}|\lesssim|\overline{\partial}\Psi_{1}||\partial\Psi_{2}|+|\partial\Psi_{1}||\overline{\partial}\Psi_{2}|,\qquad\overline{\partial}_{\alpha}:=\partial_{\alpha}-\partial_{\alpha}u\,\partial_{u}$
(2.42)

with
$\overline{\gamma}^{u\beta\delta}=\overline{\gamma}^{0\beta\delta}-\omega_{i}\overline{\gamma}^{i\beta\delta}$,
and the coefficients
$\overline{\gamma}^{\alpha\beta\delta}=\gamma^{\alpha\beta\delta}-\delta^{\alpha\alpha^{\prime}}\gamma^{\alpha^{\prime\prime}\beta\delta}\partial_{\alpha^{\prime}}u\partial_{\alpha^{\prime\prime}}u$
satisfy the bound

$(1+v)^{k}|\partial^{k}\overline{\gamma}^{\alpha\beta\delta}|\lesssim
1,\qquad\text{ when }|u|\leq\min(t/10,1).$ (2.43)

Therefore the continuity equation takes the form

$\partial_{\alpha}(m^{\alpha\beta}\partial_{\beta}\Phi)-\partial_{u}(\partial_{u}\Phi)^{2}+\partial_{\alpha}\left(\overline{\gamma}^{\alpha\beta\delta}\partial_{\beta}\Phi\partial_{\delta}\Phi\right)+\partial_{\alpha}G^{\alpha}(\partial\Phi)=0,$
(2.44)

and with $\zeta$ as in (2.20), the first jump condition in (2.38) reads

$[\partial_{v}\Phi]-[\partial_{u}\Phi]\partial_{v}B+\frac{1}{2}[\not{\nabla}_{i}\Phi]\not{\nabla}^{i}B+[\gamma^{\alpha}(\partial\Phi)]\zeta_{\alpha}=0.$
(2.45)

See Section 9.

###  2.4 The wave equation for the perturbations

Our results are more natural to state in terms of the variable $\Psi=r\Phi$.
The equation (2.44) then takes the form

$-4\partial_{u}\partial_{v}\Psi+\not{\Delta}\Psi+\partial_{\mu}(\gamma^{\mu\nu}(\partial\Phi)\partial_{\nu}\Psi)=F.$
(2.46)

We expand $\Psi=\Sigma+\psi$ where $\Sigma$ is the model shock profile

$\Sigma=\begin{cases}\frac{u^{2}}{2s},\qquad\text{ in }D^{C},\\\
0,\qquad\text{ otherwise. }\end{cases}$ (2.47)

By Lemma C.1, in the exterior regions (where $\Sigma$ vanishes), the
perturbation $\psi$ satisfies the following quasilinear perturbation of the
usual Minkowskian wave equation,

$-4\partial_{u}\partial_{v}\psi+\not{\Delta}\psi+\partial_{\mu}(\gamma^{\mu\nu}(\partial\phi)\partial_{\nu}\psi)+\partial_{\mu}Q^{\mu}=F,$
(2.48)

where $\phi=\psi/r$ and where $Q,F$ are given in Lemma C.1.

In the region between the shocks, the model shock profile contributes a non-
perturbative top-order term and the perturbation $\psi$ instead satisfies an
equation of the form

$-4\partial_{u}\left(\partial_{v}+\frac{u}{vs}\partial_{u}\right)\psi+\partial_{\mu}(m_{B,a}^{\mu\nu}\partial_{\nu}\psi)+\not{\Delta}\psi+\partial_{\mu}(\gamma^{\mu\nu}(\partial\phi)\partial_{\nu}\psi)+\partial_{\mu}Q^{\mu}=F,$
(2.49)

where $P,F$ are given in Lemma C.1 and where
$\partial_{\mu}(m_{B,a}^{\mu\nu}\partial_{\nu}\psi)$ involves linear terms
which can be treated perturbatively. The equation (2.49) is a quasilinear
perturbation of the wave equation with respect to the metric $m_{B}$ from
(2.4).

###  2.5 The commutator fields in each region

In the exterior regions $D^{L},D^{R}$, we will commute the continuity equation
(2.44) with the usual family of Minkowski vector fields,

${\mathcal{Z}_{m}}=\\{\partial_{\alpha},\Omega_{ij},\Omega_{0i},S\\},$ (2.50)

where $\partial_{\alpha}$ denotes differentiation with respect to the usual
rectangular coordinate system on $\mathbf{R}^{4}$ and

$\Omega_{ij}=x_{i}\partial_{j}-x_{j}\partial_{i},\qquad\Omega_{0i}=t\partial_{i}+x_{i}\partial_{t},\qquad
S=x^{\alpha}\partial_{\alpha}.$ (2.51)

It is well-known that these vector fields form an algebra and satisfy the
following commutation properties with the Minkowskian wave operator
$\Box=-\partial_{t}^{2}+\delta^{ij}\partial_{i}\partial_{j}$,

$Z\Box q-\Box Zq=c_{Z}q,\qquad\text{ where }c_{S}=-2,\quad c_{Z}=0\text{
otherwise. }$ (2.52)

In the region between the shocks, we will work with the family

$\mathcal{Z}_{m_{B}}=\\{\Omega_{ij},X_{1}=s\partial_{u},X_{2}=v\partial_{v}\\},$
(2.53)

which spans the tangent space at each point. The field $X_{1}$ satisfies

$X_{1}\ell^{m_{B}}q-\ell^{m_{B}}X_{1}q=0,$ (2.54)

and so it commutes with the spherically-symmetric part of the equation in the
central region,

$X_{1}\partial_{u}\ell^{m_{B}}q-\partial_{u}\ell^{m_{B}}X_{1}q=0.$ (2.55)

The field $X_{2}$ satisfies

$X_{2}\ell^{m_{B}}q-\ell^{m_{B}}X_{2}q=-\ell^{m_{B}}-\frac{u}{vs^{2}}\partial_{u},$
(2.56)

so that in particular,

$X_{2}\partial_{u}\ell^{m_{B}}q-\partial_{u}\ell^{m_{B}}X_{2}q=-\partial_{u}\ell^{m_{B}}q-\partial_{u}\left(\frac{u}{vs^{2}}\partial_{u}q\right).$
(2.57)

###  2.6 Volumes and areas

In what follows, unless mentioned explicitly, all integrals over spacetime
regions are taken with respect to the measure
$d\widehat{\mu}=\frac{1}{r^{2}}dxdt$ as opposed to the standard $d\mu=dxdt$.
We have made this choice because we will be working in terms of the rescaled
variables $\psi=r\phi$ and this simplifies many of the integration-by-parts
identities we will encounter.

As a result, all the surface integrals we encounter are taken with respect to
the surface measure induced by $d\widehat{\mu}$. We will let $dS$ denote the
induced surface measure on the spheres $\Gamma^{A}_{t}$. At each time $t$,
$\Gamma^{A}_{t}$ is the graph over $\mathbb{S}^{2}$ of the function
$r^{A}(t,\omega)$, which is defined by the relation

$t-r^{A}(t,\omega)=\beta^{A}_{s}(\omega),\qquad s=\log(t+r^{A}(t,\omega)).$
(2.58)

Under the assumptions (2.16)-(2.17) on $\beta^{A}_{s}(\omega)$, it follows
that $dS$ is equivalent to $dS(\omega)$, the usual surface measure on the unit
sphere $\mathbb{S}^{2}$,

$dS\sim dS(\omega).$ (2.59)

##  3 Multiplier identities

The goal of this section is to collect the basic identities we will use to
construct energies for the continuity equation (1.13). We consider a linear
wave equation of the form

$\partial_{\mu}(h^{\mu\nu}\partial_{\nu}\psi)+\partial_{\mu}P^{\mu}=F,$ (3.1)

in a region $D$. Here, and for the remainder of this section, the indices
$\mu,\nu$ refer to quantities expressed in the following (Minkowskian) null
coordinate system,

$x^{0}=u=t-r,\quad x^{1}=v=t+r,\quad x^{2}=\theta^{1},\quad x^{3}=\theta^{2},$
(3.2)

where $(\theta^{1},\theta^{2})$ are an arbitrary local coordinate system on
the unit sphere $\mathbb{S}^{2}$. For our applications, in the exterior
regions the metric $h$ will take the form will either take the form
$h^{\mu\nu}=m^{\mu\nu}+\gamma^{\mu\nu}$ where $m^{\mu\nu}$ denote the
components of the reciprocal of the Minkowski metric.

$m=-dudv+\frac{1}{4}(v-u)^{2}d\sigma_{\mathbb{S}^{2}}.$ (3.3)

We note that with our conventions, the Minkowskian wave operator takes the
form

$\partial_{\mu}(m^{\mu\nu}\partial_{\nu}\psi)=-4\partial_{u}\partial_{v}\psi+\not{\Delta}\psi.$
(3.4)

In the region between the shocks, the metric $h$ will take the form
$h^{\mu\nu}=\widetilde{m_{B}}^{\mu\nu}+\gamma^{\mu\nu}=m_{B}^{\mu\nu}+\gamma^{\mu\nu}_{a}+\gamma^{\mu\nu}$,
where $m_{B}^{\mu\nu}$ denote the components of the reciprocal of the metric

$m_{B}=-dudv+\frac{u}{vs}dv^{2}+\frac{1}{4}(v-u)^{2}d\sigma_{\mathbb{S}^{2}}.$
(3.5)

and where $\gamma_{a}^{\mu\nu}=\frac{u}{vs}a^{\mu\nu}$. Here, the
$a^{\mu\nu}=a^{\mu\nu}(u,v,\omega)$ are smooth functions satisfying the symbol
condition $(1+v)^{k}|\partial^{k}a|\lesssim 1$ as well as the null condition

$a^{\mu\nu}\partial_{\mu}u\partial_{\nu}u=0.$ (3.6)

If $\overline{\xi}$ denotes projection of a one-form $\xi$ away from the
cotangent space to $\\{u=const.\\}$,

$\overline{\xi}_{\mu}=\left(\delta_{\mu}^{\nu}-\tfrac{1}{2}\delta^{\nu\nu^{\prime}}\partial_{\nu^{\prime}}u\partial_{\mu}u\right)\xi_{\nu},\qquad|\overline{\xi}|\lesssim|\xi_{v}|+|\not{\xi}|,$
(3.7)

where $\not{\xi}$ denotes the angular part of $\xi$, then for any one-forms
$\xi,\tau$, writing $a(\xi,\tau)=a^{\mu\nu}\xi_{\mu}\tau_{\nu}$, we have
$a(\xi,\tau)=a(\overline{\xi},\tau)+a(\xi,\overline{\tau})$. In particular,
(3.6) implies the bound

$|a(\xi,\tau)|\lesssim|\overline{\xi}||\tau|+|\xi||\overline{\tau}|.$ (3.8)

For any symmetric (2,0)-tensor $g$ and a vectorfield $X$, define the energy
current $J_{X,g}$ by

$J_{X,g}^{\mu}[\psi]=g^{\mu\nu}\partial_{\nu}\psi
X\psi-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi
X^{\mu}$ (3.9)

and the scalar current $K_{X,g}$ by

$K_{X,g}[\psi]=\frac{1}{2}\partial_{\alpha}(g^{\mu\nu}X^{\alpha})\partial_{\mu}\psi\partial_{\nu}\psi-\partial_{\mu}X^{\alpha}g^{\mu\nu}\partial_{\nu}\psi\partial_{\alpha}\psi$
(3.10)

Then we have the basic identity

$\partial_{\mu}(g^{\mu\nu}\partial_{\nu}\psi)X\psi=\partial_{\mu}J^{\mu}_{X,g}+K_{X,g}.$
(3.11)

To keep track of lower-order terms, it is helpful to introduce

$J_{X,g,P}^{\mu}[\psi]=J_{X,g}^{\mu}+P^{\mu}X\psi-X^{\mu}P\psi$ (3.12)

and

$K_{X,g,P}[\psi]=K_{X,g}[\psi]+{(X^{\alpha}\partial_{\alpha}P^{\mu}-P^{\alpha}\partial_{\alpha}X^{\mu})\partial_{\mu}\psi+(\partial_{\alpha}X^{\alpha})P\psi},$
(3.13)

which are defined so that

$\partial_{\mu}(g^{\mu\nu}\partial_{\nu}\psi+P^{\mu})X\psi=\partial_{\mu}J_{X,g,P}^{\mu}+K_{X,g,P}.$
(3.14)

If $\zeta$ is any one-form with $|\zeta|\leq 1$, the energy current
$J_{X,g,P}$ satisfies

$\displaystyle|\zeta(J_{X,g,P})|$
$\displaystyle\lesssim|g||\partial\psi||X\psi|+|\zeta(X)||g(\partial\psi,\partial\psi)|+|P||X\psi|+|\zeta(X)||P||\psi|$
(3.15) $\displaystyle\lesssim|g||X||\partial\psi|^{2}+|P||X||\partial\psi|$
(3.16)

where
$g(\partial\psi,\partial\psi)=g^{\mu\nu}\partial_{\mu}\psi\partial_{\nu}\psi$
and where $\zeta(X)$ denotes the usual action of a one-form on a vector field.
The first bound will be useful along the shocks. The scalar current satisfies
the bound

$|K_{X,g,P}|\lesssim\left(|\partial g||X|+|g||\partial
X|\right)|\partial\psi|^{2}+(|\partial P||X|+|P||\partial X|)|\partial\psi|$
(3.17)

For our applications, we will need to keep better track of the structure of
$K$. It is convenient to work in terms of covariant derivatives
$\nabla_{X}=X^{\mu}\nabla_{\mu}$ defined relative to the Minkowksi metric. We
write

$K_{X,g,P}=\frac{1}{2}(\nabla_{X}g^{\mu\nu})\partial_{\mu}\psi\partial_{\nu}\psi+\frac{1}{2}\partial_{\alpha}X^{\alpha}g^{\mu\nu}\partial_{\mu}\psi\partial_{\nu}\psi-\partial_{\mu}X^{\alpha}g^{\mu\nu}\partial_{\nu}\psi\partial_{\alpha}\psi\\\
+\nabla_{X}P^{\mu}\partial_{\mu}\psi-P^{\alpha}\partial_{\alpha}X^{\mu}\partial_{\mu}\psi+(\partial_{\alpha}X^{\alpha})P\psi\\\
+X^{\alpha}\left(\Gamma_{\mu^{\prime}\alpha}^{\nu}g^{\mu\mu^{\prime}}+\Gamma_{\mu^{\prime}\alpha}^{\mu}g^{\nu\mu^{\prime}}\right)\partial_{\mu}\psi\partial_{\nu}\psi+X^{\alpha}\Gamma^{\mu}_{\nu\alpha}P^{\nu}\partial_{\mu}\psi,$
(3.18)

where the Christoffel symbols (relative to the null coordinate system
$(u,v,\theta^{1},\theta^{2})$) $\Gamma_{\nu\alpha}^{\mu}$ satisfy
$|\Gamma|\lesssim\frac{1}{r}$. In what follows, we just consider the case of a
spherically-symmetric multiplier $X$,

$X=X^{u}(u,v)\partial_{u}+X^{v}(u,v)\partial_{v},$ (3.19)

and in this case we have

$\displaystyle|K_{X,g,P}|$
$\displaystyle\lesssim|(\nabla_{X}g)(\partial\psi,\partial\psi)|+|\partial
X||g(\partial\psi,\partial\psi)|+|g||\partial\psi|\left(|\partial
X^{u}||\partial\psi|+|\partial X^{v}||\partial_{v}\psi|\right)$ (3.20)
$\displaystyle+|(\nabla_{X}P)\cdot\partial\psi|+|\partial
X||P\psi|+|P|\left(|\partial X^{u}||\partial\psi|+|\partial
X^{v}||\partial_{v}\psi|\right)$ (3.21)
$\displaystyle+\frac{1}{r}|X|\left(|g||\partial\psi|^{2}+|P||\partial\psi|\right).$
(3.22)

We also note at this point that it is possible to write the above in terms of
the Lie derivative of $g$. If we return to (3.10) and (3.13) and recall from
the definitions that
$X^{\alpha}\partial_{\alpha}g^{\mu\nu}=\mathcal{L}_{X}g^{\mu\nu}+g^{\mu\alpha}\partial_{\alpha}X^{\nu}+g^{\nu\alpha}\partial_{\alpha}X^{\mu}$
and that
$\mathcal{L}_{X}P^{\mu}=X^{\alpha}\partial_{\alpha}P^{\mu}-P^{\alpha}\partial_{\alpha}X^{\mu}$,
we find instead the bound

$|K_{X,g,P}|\lesssim|(\mathcal{L}_{X}g)(\partial\psi,\partial\psi)|+|\partial
X||g||\partial\psi|^{2}+|\mathcal{L}_{X}P||\partial\psi|,$ (3.23)

which we will use near $r=0$ in place of (3.22) to avoid spurious
singularities at $r=0$.

###  3.1 The energy-momentum tensor

Given a metric $h$, define the energy-momentum tensor

$Q^{h}[\psi](X,Y)=h(J_{X,h}[\psi],Y)=X\psi
Y\psi-\frac{1}{2}h(X,Y)h^{-1}(\partial\psi,\partial\psi).$ (3.24)

We will frequently drop $\psi$ from the notation and just write $Q^{h}(X,Y)$.
For a vector field $P$ we also set

$Q^{h}_{P}(X,Y)=Q^{h}(X,Y)+h(P,Y)X\psi-h(X,Y)P\psi.$ (3.25)

With notation as in (3.12), (3.13), integrating the identity

$\partial_{\mu}\left(h^{\mu\nu}\partial_{\nu}\psi+P^{\mu}\right)X\psi=\partial_{\mu}J_{X,h,P}^{\mu}+K_{X,h,P}$
(3.26)

over a region $D$ bounded by two time slices and a lateral boundary $\Gamma$
and using Lemma E.3, we have the following integral identity which will be
used to get energy estimates.

######  Lemma 3.1.

Fix a metric $h$, vector fields $X,P$ and define $Q^{h}_{P}$ as in
(3.24)-(3.25). Let $D=\cup_{t_{0}\leq t\leq t_{1}}D_{t}$ be a region bounded
by a (possibly empty) timelike boundary $\Gamma_{-}$ and a (possibly empty)
spacelike boundary $\Gamma_{+}$, lying to the future of $D_{t_{0}}$. Suppose
that either $\\{r=0\\}$ is not contained in $D$, or else that $\\{r=0\\}$ is
contained in $D$ and $\lim_{r\to 0}(h^{rr}X^{r})=0$. For a spacelike surface
$\Sigma$, let $N_{h}^{\Sigma}$ denote the future-directed normal vector field
to $\Sigma$ defined relative to the metric $h$, and for a timelike surface
$\Sigma$, let $N_{h}^{\Sigma}$ denote the inward-pointing normal vector field
defined relative to $h$. Suppose that $\lim_{r\to 0}|\psi/r|<\infty$. Then the
following identity holds

$\int_{D_{t_{1}}}Q^{h}_{P}(X,N_{h}^{D_{t_{1}}})+\int_{\Gamma_{+}}Q^{h}_{P}(X,N_{h}^{\Gamma_{+}})-\int_{\Gamma_{-}}Q^{h}_{P}(X,N_{h}^{\Gamma_{-}})-\int_{t_{0}}^{t_{1}}\int_{D_{t}}K_{X,h,P}\\\
=\int_{D_{t_{0}}}Q^{h}_{P}(X,N_{h}^{D_{t_{0}}})+\int_{t_{0}}^{t_{1}}\int_{D_{t}}-\partial_{\mu}(h^{\mu\nu}\partial_{\nu}\psi+P^{\mu})X\psi$
(3.27)

where all integrals over $D$ are taken with respect to the measure
$dudvd\sigma_{\mathbb{S}^{2}}=\tfrac{1}{r^{2}}dxdt$, the integrals over the
boundary terms are taken with respect to the induced surface measure, and
$K_{X,h,P}$ is defined in (3.13).

###  3.2 Modified multiplier identites

For our estimates, we will be considering multipliers
$X=X^{v}\partial_{v}+X^{u}\partial_{u}$ where the coefficient $X^{v}$ is much
larger than $X^{u}$. This causes an issue when closing the nonlinear estimates
because will not be able to control bulk error terms of the form $|\partial
X^{v}||\gamma||\partial\psi|^{2}$, which are present in $K_{X,\gamma}$. For
our applications, $X^{v}\sim v$ (or larger) and thinking of
$|\gamma|\sim\frac{1}{v}|\partial\psi|$, controlling such a term uniformly in
time would require a bound of the form
$\int_{t_{0}}^{t_{1}}\frac{1}{1+t}\|\partial\psi(t)\|_{L^{\infty}}\,dt\lesssim\epsilon^{1/2}$.
We only expect to have such a bound with $\partial\psi$ replaced by
$\partial^{2}\psi$.

These bad terms can be traced back to the terms
$\partial_{u}(\gamma^{uu}\partial_{u}\psi)X^{v}\partial_{v}\psi$ and
$\partial_{v}(\gamma^{vu}\partial_{u}\psi)X^{v}\partial_{v}\psi$ in (3.10). To
handle terms of this type, we need to proceed more carefully and the idea is
to exploit the fact that the combination $\partial_{u}\partial_{v}\psi$ is
expected to be better-behaved than a generic second-order derivative
$\partial^{2}\psi$. To highest order, this combination is already present in
the second term mentioned above and after integrating by parts it is also
present in the first order term. Using the equation for
$\partial_{u}\partial_{v}\psi$, we generate additional terms involving either
$\not{\Delta}\psi$ (which is expected to be better behaved than a generic
second-order derivative), or nonlinear terms.

This leads to a modified version of the identity (3.26) for perturbations
$h^{-1}=g^{-1}+\gamma$ of $g\in\\{m,m_{B}\\}$,

$\left(\partial_{\mu}(h^{\mu\nu}\partial_{\nu}\psi)+\partial_{\mu}P^{\mu}\right)X\psi=\partial_{\mu}J_{X,g,P}^{\mu}+K_{X,g,P}+\partial_{\mu}{\widetilde{J}}_{X,\gamma,P}^{\mu}+\widetilde{K}_{X,\gamma,P},$
(3.28)

where the modified energy current ${\widetilde{J}}$ and scalar current
$\widetilde{K}$ satisfy better bounds than those in (3.16), (3.17).

This calculation is carried out in Section H. The quantities ${\widetilde{J}}$
and $\widetilde{K}$ have rather complicated expressions (see (H.103)-(H.104)
in the Minkowskian case and (H.169)-(H.170) for the version in the central
region) and in this section we will just record the estimates for these
quantities that we will need. These estimates and formulas are proved in
Propositions H.1 and H.2.

For our applications, we will be using (3.28) with $\psi$ replaced with
$Z^{I}\psi$ for a product of vector fields $Z^{I}$ and in that case, $\gamma$
and $P$ will be of the form

$\gamma\sim\frac{1}{1+v}\partial\psi,\qquad P\sim\frac{1}{1+v}\partial
Z^{I_{1}}\psi\cdot\partial Z^{I_{2}}\psi,\qquad|I_{1}|+|I_{2}|\leq|I|-1$
(3.29)

To prove our estimates, we will assume some bounds for the quantities $\gamma$
and $P$ which are designed to capture the expectation that they are of the
form (3.29) and that our bootstrap assumptions hold; see in particular Lemmas
8.1, 8.2 and 8.5.

####  3.2.1 Assumptions on perturbative quantities

We fix a metric $g\in\\{m,m_{B}\\}$ and a multiplier
$X=X^{n}_{g}n+X^{\ell}_{g}\ell^{g}$ with $X^{n}_{g},X^{\ell}_{g}\geq 0$. We
will only consider vector fields $X$ satisfying $X^{n}_{g}\leq X^{\ell}_{g}$.
We define

$|\partial\psi|_{X,g}^{2}=X^{\ell}_{g}\left(|\ell^{g}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)+X^{n}_{g}|n\psi|^{2}.$
(3.30)

We assume that for $\epsilon$ sufficiently small, the perturbation $\gamma$
and our multipliers satisfy the bound

$\displaystyle|\gamma|\leq\epsilon\frac{1}{(1+v)(1+s)^{1/2}},\qquad
X^{\ell}_{g}|\gamma|\leq\epsilon X^{n}_{g}.$ (3.31)

Note that (3.31) implies that

$|X||\gamma||\partial\psi|^{2}\leq\epsilon|\partial\psi|_{X,g}^{2}.$ (3.32)

which will be used to handle many of our error terms. We will also assume that
the initial time $t_{0}$ has been chosen sufficiently large,

$\frac{1}{t_{0}}\leq\epsilon_{0},$ (3.33)

which will be needed to absorb some error terms in the central region.

For most of our multipliers $X$, the first bound in (3.31) will automatically
imply the second bound there. It is only for the estimate in the leftmost
region that the last bound in (3.31) is actually needed, but it makes proving
the estimates more convenient.

###  3.3 Estimates for the modified scalar and energy currents in the exterior
regions

We now collect some estimates for the quantities
${\widetilde{J}},\widetilde{K}$ that will be used in $D^{R}$ and $D^{L}$. In
$D^{R}$ we will need to multiply by the field $X_{R}$ and in $D^{L}$ we will
need to multiply by both $X_{L}$ and $X_{M}$, where

$X_{R}=(1+|u|)^{\mu}\partial_{t}+u\partial_{u}+v\partial_{v},\qquad
X_{L}=uf(u)\partial_{u}+vf(v)\partial_{v},\qquad
X_{M}=((g(r)+1))(\partial_{v}-\partial_{u}),$ (3.34)

where $\mu>0$ and where, with $\alpha$ as in (6.20),

$f(z)=\log z(\log\log z)^{\alpha},\qquad g(z)=(\log(1+z))^{1/2})f(\log(1+z)).$
(3.35)

Note that by contrast with (3.14), which involves only a reference metric $g$,
the right hand side of the identity below is expressed in terms of both $g=m$
and the perturbation $\gamma=h-m$.

######  Proposition 3.1.

Suppose that $\psi$ satisfies (3.1) in either $D^{L}$ or $D^{R}$ and set
$\gamma^{\mu\nu}=h^{\mu\nu}-m^{\mu\nu}$. Let $X$ denote any of
$X_{L},X_{M},X_{R}$ as defined in (3.34), and suppose that $\gamma$ satisfies
the conditions in (3.31) for some $\epsilon>0$. Then

$\left(\partial_{\mu}(h^{\mu\nu}\partial_{\nu}\psi)+\partial_{\mu}P^{\mu}\right)X\psi=\partial_{\mu}J_{X,m}^{\mu}+K_{X,m}+\partial_{\mu}{\widetilde{J}}_{X,\gamma,P}^{\mu}+\widetilde{K}_{X,\gamma,P},$
(3.36)

where the perturbed energy current ${\widetilde{J}}^{\mu}_{X,\gamma,P}$
satisfies the following bounds. With notation as in (3.30), if $\zeta$ is any
one-form with $|\zeta|\leq 1$, if $|u|\leq v/8$ then for any $\delta>0$,

$|\zeta({\widetilde{J}}_{X,\gamma,P})|\lesssim\delta|X^{\ell}_{m}||{\ell^{m}}\psi|^{2}+\left(1+\frac{1}{\delta}\right)|\gamma||\partial\psi|_{X,m}^{2}+|\zeta(X)||\gamma||\partial\psi|^{2}+|\not{\zeta}|^{2}|\partial\psi|_{X,m}^{2}\\\
+\left(1+\frac{1}{\delta}\right)|X||P|^{2}+|X^{n}_{m}|^{1/2}|P||\partial\psi|_{X,m}.$
(3.37)

and if $|u|\geq v/8$, then

$|\zeta({\widetilde{J}}_{X,\gamma,P})|\lesssim|X||\gamma||\partial\psi|^{2}+|X||P||\partial\psi|.$
(3.38)

The modified scalar current $\widetilde{K}$ satisfies the following bounds. If
$|u|\leq v/8$ then

$|\widetilde{K}_{X,\gamma,P}|\lesssim\left(|\nabla\gamma|+\frac{1}{1+|u|}|\gamma|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\nabla_{{\ell^{m}}}\gamma|+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\not{\nabla}\gamma|\right)|\partial\psi|_{X,m}^{2}+|X^{n}_{m}||F||\partial\psi|_{X,m}\\\
+\left(|\nabla
P|+\frac{|P|}{1+|u|}+\frac{|X^{\ell}_{m}|^{1/2}}{|X^{n}_{m}|^{1/2}}|\nabla_{{\ell^{m}}}P|+|X^{\ell}_{m}||\not{\nabla}P|\right)|X^{n}_{m}|^{1/2}|\partial\psi|_{X,m}\\\
+|P||\partial_{u}X^{v}||{\ell^{m}}\psi|+|P||X|\left(|F|+\frac{1}{1+v}|P|\right)$
(3.39)

and in the region $|u|\geq v/8$, we instead have

$|\widetilde{K}_{X,\gamma,P}|\lesssim|\nabla\gamma||X||\partial\psi|^{2}+|\gamma||\partial
X||\partial\psi|^{2}+|\nabla
P||X||\partial\psi|+\left(\frac{1}{r}+\frac{1}{1+v}\right)|X|\left(|\gamma||\partial\psi|^{2}+|P||\partial\psi|\right)$
(3.40)

###### Proof.

It is straightforward to verify that each of the given multipliers satisfy the
condition (H.2), and so the bounds follow from Proposition H.1. ∎

######  Remark 1.

For our applications, $\zeta$ will be the one-form dual to the outward-
pointing normal to a surface $\Sigma$. When $\Sigma=D_{t}$ is a time slice
then it will suffice to bound $|\zeta(X)|\leq|X|$. When $\Sigma=\Gamma$ is one
of our shocks, we will instead have a bound of the form $|\zeta(X)|\lesssim
X^{n}_{g}+(1+v)^{-1}(1+s)^{-1/2}X_{g}^{\ell}$. That is, the “large” component
$X^{\ell}_{g}$ is suppressed. This, and the smallness of $|\gamma|$ expressed
in (3.31), will be needed to close our estimates.

We will need a version of the above when $h$ is instead a perturbation of the
reciprocal of the metric $m_{B}$, up to terms with small coefficients that
verify a null condition. For this we set

$m_{B,a}^{\mu\nu}=m_{B}^{\mu\nu}+\frac{u}{vs}a^{\mu\nu}=m_{B}^{\mu\nu}+\gamma_{a}^{\mu\nu}$
(3.41)

where $a^{\mu\nu}=a^{\mu\nu}(u,v,\omega)$ are smooth functions verifying
$(1+v)^{k}|\partial^{k}a|\lesssim 1$ and the null condition (3.8).

The following is an immediate consequence of Proposition H.2.

######  Proposition 3.2.

Suppose that $\psi$ satisfies (3.1) and set
$\gamma^{\mu\nu}=h^{\mu\nu}-m_{B,a}^{\mu\nu}$ with notation as in (3.41). Fix
a vector field $X=X^{n}_{m_{B}}n+X^{\ell}_{m_{B}}\ell^{m_{B}}$ with
$X^{\ell}_{m_{B}}=v$ and $X^{n}_{m_{B}}\gtrsim(1+s)^{-1/2}$ and $|\partial
X|\lesssim 1$. Suppose that $\gamma$ satisfies the conditions in (3.31) for
some $\epsilon>0$ and that (3.33) holds for sufficiently small $\epsilon_{0}$.
Then

$\left(\partial_{\mu}(h^{\mu\nu}\partial_{\nu}\psi)+\partial_{\mu}P^{\mu}\right)X\psi=\partial_{\mu}J_{X,m_{B,a}}^{\mu}+K_{X,m_{B,a}}+\partial_{\mu}{\widetilde{J}}_{X,\gamma,P}^{\mu}+\widetilde{K}_{X,\gamma,P},$
(3.42)

where the perturbed energy current ${\widetilde{J}}^{\mu}_{X,\gamma,P}$
satisfies the following bound: if $\zeta$ is any one-form with $|\zeta|\leq
1$,

$\displaystyle|\zeta({\widetilde{J}}_{X,\gamma,P})|$
$\displaystyle\lesssim\delta
v|\ell^{m_{B}}\psi|^{2}+\left(1+\frac{1}{\delta}\right)|\gamma||\partial\psi|_{X,m_{B}}^{2}+|\zeta(X)||\gamma||\partial\psi|^{2}+\epsilon|\zeta(J_{X,\gamma_{a}})|$
(3.43)
$\displaystyle+|\not{\zeta}|^{2}|\partial\psi|_{X,m_{B}}^{2}+\left(1+\frac{1}{\delta}\right)v|P|^{2}+\frac{1}{(1+s)^{1/2}}|P||\partial\psi|,$
(3.44)

and the modified scalar current $\widetilde{K}_{X,\gamma,P}$ satisfies

$\displaystyle|\widetilde{K}_{X,\gamma,P}|$
$\displaystyle\lesssim\left(|\nabla\gamma|+\frac{|\gamma|}{1+s}+\frac{|X^{\ell}_{m_{B}}|^{1/2}}{|X^{n}_{m_{B}}|^{1/2}}\left(|\nabla_{{\ell^{m}}}\gamma|+|\not{\nabla}\gamma|\right)\right)|\partial\psi|_{X,m_{B}}^{2}+\frac{1}{(1+v)^{1/4}}|F||\partial\psi|_{X,m_{B}}$
(3.45) $\displaystyle+\left(|\nabla
P^{u}|+\frac{|P^{u}|}{1+s}+|X^{\ell}_{m_{B}}|^{1/2}\left(|\nabla_{\ell^{m_{B}}}P|+|\not{\nabla}P|\right)\right)|X^{n}_{m_{B}}|^{1/2}|\partial\psi|_{X,m_{B}}$
(3.46)
$\displaystyle+\epsilon\left(\frac{1}{(1+v)^{3/2}}|\partial\psi|^{2}+\frac{1}{(1+v)^{1/2}}(|\ell^{m_{B}}\psi|^{2}+|\not{\nabla}\psi|^{2})\right)$
(3.47) $\displaystyle+\frac{1}{(1+s)^{1/2}}\left(|\nabla
P^{u}|+\frac{|P^{u}|}{1+v}\right)|\partial\psi|+v|P|\left(|\nabla
P|+\frac{|P|}{1+v}+|F|\right).$ (3.48)

We now record an analogue of Lemma 3.1. For this we introduce the modified
energy-momentum tensor

${\widetilde{Q}}_{P}^{h}(X,Y)=Q_{P}^{g}(X,Y)+h({\widetilde{J}}_{X,P},Y).$
(3.49)

where $Q_{P}^{g}$ is as in (3.25). By Lemma E.4, we have

######  Lemma 3.2.

Fix a metric $h$, vector fields $X,P$ and define ${\widetilde{Q}}^{h}_{P}$ as
in (3.49). Let $D=\cup_{t_{0}\leq t\leq t_{1}}D_{t}$ be a region bounded by
two time slices, a (possibly empty) timelike boundary $\Gamma_{T}$ and a
(possibly empty) spacelike boundary $\Gamma_{S}$, with $\Gamma_{S}$ lying to
the future of $D_{t_{0}}$. Suppose that either $\\{r=0\\}$ is not contained in
$D$, or else that $\\{r=0\\}$ is contained in $D$ and that with
$\gamma=h^{-1}-g^{-1}$, we have $\lim_{r\to 0}((g^{rr}+\gamma^{rr})X^{r})=0$.
For a spacelike surface $\Sigma$, let $N_{h}^{\Sigma}$ denote the future-
directed normal vector field defined relative to the metric $h$ and if
$\Sigma$ is timelike, let $N_{h}^{\Sigma}$ denote the inward-pointing normal
vector field defined relative to the metric $h$. Suppose that $\lim_{r\to
0}|\psi/r|<\infty$. Then the following identity holds

$\int_{D_{t_{1}}}{\widetilde{Q}}^{h}_{P}(X,N_{h}^{D_{t_{1}}})+\int_{\Gamma_{S}}{\widetilde{Q}}^{h}_{P}(X,N_{h}^{\Gamma_{+}})-\int_{\Gamma_{T}}{\widetilde{Q}}^{h}_{P}(X,N_{h}^{\Gamma_{-}})-\int_{t_{0}}^{t_{1}}\int_{D_{t}}\widetilde{K}_{X,\gamma,P}\\\
=\int_{D_{t_{0}}}{\widetilde{Q}}^{h}_{P}(X,N_{h}^{D_{t_{0}}})+\int_{t_{0}}^{t_{1}}\int_{D_{t}}-\partial_{\mu}(h^{\mu\nu}\partial_{\nu}\psi+P^{\mu})X\psi$
(3.50)

where all integrals over $D_{t}$ are taken with respect to the measure
$dudvd\sigma_{\mathbb{S}^{2}}=\tfrac{1}{r^{2}}dxdt$, the integrals over the
boundary terms are taken with respect to the induced surface measure, and
$\widetilde{K}$ is as in the previous two results.

##  4 Formulas for the energy-momentum tensor and scalar currents

In this section we consider a metric $h$ which is a perturbation of either the
Minkowski metric $m$, or the metric $m_{B}$ defined in (2.4). We collect here
some basic formulas and estimates for the modified energy-momentum tensor
${\widetilde{Q}}^{h}_{P}$ defined in (3.49), the modified energy-momentum
tensor ${\widetilde{Q}}_{P}^{h}$ defined in (3.49), and the linear part of the
scalar current $K_{X,g}$ defined in (3.13).

Each of our metrics $g$ admit spherically-symmetric null vectors
$(n,\ell^{g})$ which we have normalized with $g(\ell^{g},n)=-\frac{1}{2}$.
Since $g(\nabla_{g}\psi,X)=X\psi$ for any vector field $X$, we have

$\nabla_{g}\psi=-2(n\psi)\ell^{g}-2(\ell^{g}\psi)n+\not{\nabla}\psi,\qquad
g(\nabla_{g}\psi,\nabla_{g}\psi)=-4\ell^{g}\psi n\psi+|\not{\nabla}\psi|^{2},$
(4.1)

and so the energy-momentum tensor takes the form

$Q^{g}(X,Y)=X\psi Y\psi+2g(X,Y)\ell^{g}\psi
n\psi-\tfrac{1}{2}g(X,Y)|\not{\nabla}\psi|^{2}.$ (4.2)

If $X=X^{\ell}_{g}\ell^{g}+X^{n}_{g}n$ is spherically symmetric and
$Y=Y^{\ell}_{g}\ell^{g}+Y^{n}_{g}n+\not{Y}$ where $\not{Y}$ is the angular
part of $Y$, we also have

$g(X,Y)=-\tfrac{1}{2}(X^{\ell}_{g}Y^{n}_{g}+X^{n}_{g}Y^{\ell}_{g}),$ (4.3)

and so in this case

$Q^{g}(X,Y)=X^{\ell}_{g}Y^{\ell}_{g}(\ell^{g}\psi)^{2}+X^{n}_{g}Y^{n}_{g}(n\psi)^{2}-\frac{1}{2}g(X,Y)|\not{\nabla}\psi|^{2}+X\psi\not{Y}\psi.$
(4.4)

Before proceeding, we record the following simple result.

######  Lemma 4.1.

Suppose that (2.16)-(2.17) hold. Then $X_{L}$ is timelike and future-directed
with respect to $m$ in $D^{L}$, $X_{R}$ is timelike and future-directed with
respect to $m_{B}$ in $D^{R}$, and the fields $X_{T},X_{C}$ are future-
directed and timelike with respect to $m_{B}$.

###### Proof.

The statements for $X_{L},X_{R}$ are immediate. For $X_{T},X_{C}$, we first
note that expressing the fields $X_{C},X_{T}$ in terms of $n,\ell^{m_{B}}$, we
have

$X_{C}=sn+v\ell^{m_{B}},\qquad X_{T}=\frac{\eta}{{4}s^{1/2}}n+v\ell^{m_{B}},$
(4.5)

and it follows from (4.3) that

$\displaystyle m_{B}(X_{T},X_{T})$
$\displaystyle=-X_{T,m_{B}}^{\ell}X_{T,m_{B}}^{n}=-\frac{\eta}{4}\frac{v}{s^{1/2}}<0,$
(4.6) $\displaystyle m_{B}(X_{C},X_{C})$
$\displaystyle=-X_{C,m_{B}}^{\ell}X_{C,m_{B}}^{n}=-vs<0.$ (4.7)

∎

###  4.1 The energy-momentum tensor on the constant time slices

The normals to the time slices $D^{A}_{t}$ are

$N_{m}^{D_{t}}=\partial_{v}+\partial_{u}=\ell^{m}+n^{m}\qquad
N_{m_{B}}^{D_{t}}=\partial_{v}+\left(1+2\frac{u}{vs}\right)\partial_{u}=\ell^{m_{B}}+\left(1+\frac{u}{vs}\right)n^{m_{B}},$
(4.8)

so using (4.4) and writing
$N_{g}^{D_{t}}=N_{g}^{\ell}\ell^{g}+N_{g}^{n}n^{g}$, we have

$Q^{g}(X,N_{g}^{D_{t}})=N_{g}^{\ell}X^{\ell}_{g}(\ell^{g}\psi)^{2}+N_{g}^{n}X^{n}_{g}(n\psi)^{2}-\frac{1}{2}g(X,N_{g}^{D_{t}})|\not{\nabla}\psi|^{2}.$
(4.9)

As a result,

$\displaystyle Q^{m}(X,N_{m}^{D_{t}})$
$\displaystyle=X_{m}^{\ell}(\ell^{g}\psi)^{2}+X_{m}^{n}(n\psi)^{2}+{\frac{1}{4}}(X^{n}_{m}+X_{m}^{\ell})|\not{\nabla}\psi|^{2},$
(4.10) $\displaystyle Q^{m_{B}}(X,N_{m_{B}}^{D_{t}})$
$\displaystyle=X_{m_{B}}^{\ell}(\ell^{g}\psi)^{2}+X_{m_{B}}^{n}\left(1+\tfrac{u}{vs}\right)(n\psi)^{2}+{\frac{1}{4}}\left(X_{m_{B}}^{n}+X_{m_{B}}^{\ell}\left(1+\tfrac{u}{vs}\right)\right)|\not{\nabla}\psi|^{2}.$
(4.11)

Note that in this setting $X$ is timelike when $X^{\ell}_{g}X^{n}_{g}>0$ and
future-directed when $X^{\ell}_{g}+X^{n}_{g}>0$ when $g=m$ and
$X^{\ell}_{g}+X^{n}_{g}(1+\tfrac{u}{vs})>0$ when $g=m_{B}$ so in particular
these quantities are positive definite when $X$ is timelike and future-
directed. In fact, recalling the definition (3.30) from the previous section,

$|\partial\psi|_{X,g}^{2}=X^{\ell}_{g}\left(|\ell^{g}\psi|^{2}+|\not{\nabla}\psi|^{2}\right)+X^{n}_{g}|n\psi|^{2},$
(4.12)

we have the bound

$Q^{g}(X,N_{g}^{D_{t}})\geq C_{0}|\partial\psi|_{X,g}^{2}$ (4.13)

for a constant $C_{0}>0$, when $X^{\ell}_{g},X^{n}_{g}\geq 0$, for
$g=m,m_{B}$.

By the bounds (3.37) and (3.38), this implies the following bounds for the
perturbed energy-momentum tensor ${\widetilde{Q}}^{h}_{P}(X,N_{h}^{D_{t}})$
when $h$ is a perturbation of one of our metrics $m,m_{B,a}$.

######  Lemma 4.2.

Suppose that either

$\displaystyle g$ $\displaystyle=m$ $\displaystyle\quad\text{ and
}X=X_{L},\text{ or }X_{R},\text{ or }$ (4.14) $\displaystyle g$
$\displaystyle=m_{B}$ $\displaystyle\quad\text{ and }X=X_{T},\text{ or
}X_{C}.$ (4.15)

Fix a metric $h$ and let $\gamma=h^{-1}-m^{-1}$ when $g=m$ and
$\gamma=h^{-1}-m_{B,a}^{-1}$ when $g=m_{B}$, with notation as in (3.41). There
is a constant $\epsilon^{\prime}$ so that if $\gamma$ and $X$ satisfy the
perturbative assumptions (3.31) with $\epsilon<\epsilon^{\prime}$ and if
(3.33) holds with $\epsilon_{0}<\epsilon^{\prime}$, then with
$|\partial\psi|^{2}_{X,g}$ defined as in (4.12) and the modified energy-
momentum tensor ${\widetilde{Q}}$ defined as in (3.49), on the time slices
$D_{t}$ we have

$|\partial\psi|^{2}_{X,g}\lesssim{\widetilde{Q}}_{P}^{h}(X,N^{D_{t}}_{h})+X^{\ell}_{g}|P|^{2}.$
(4.16)

###### Proof.

We first consider the Minkowskian case. We write

${\widetilde{Q}}_{P}^{h}(X,N^{D_{t}}_{h})=\zeta({\widetilde{J}}_{X,h,P})={Q^{m}(X,N^{D_{t}}_{m})}+\zeta({\widetilde{J}}_{X,\gamma,P}),$
(4.17)

where $\zeta=dt$. We just prove the bound in the region $|u|\leq v/8$, the
other region being simpler.

We first use that by the assumption (3.31), $|\gamma|\leq\epsilon$. By the
bounds (3.37) and (3.32) to get

$|\zeta({\widetilde{J}}_{X,\gamma,P})|\lesssim\delta|\partial\psi|^{2}_{X,m}+\delta^{-1}\epsilon|\partial\psi|^{2}_{X,m}+|X||\gamma||\partial\psi|^{2}+\left(\delta^{-1}+1\right)X^{\ell}_{m}|P|^{2}+(X^{n}_{g})^{1/2}|P||\partial\psi|_{X,m}\\\
\lesssim\delta|\partial\psi|^{2}_{X,m}+\left(\delta^{-1}+1\right)\epsilon|\partial\psi|^{2}_{X,m}+\left(\delta^{-1}+1\right)X^{\ell}_{m}|P|^{2},$
(4.18)

where we bounded $X^{n}_{m}\leq X^{\ell}_{m}$. Taking $\delta$ and then
$\epsilon$ sufficiently small, we can arrange for

$|\zeta({\widetilde{J}}_{X,\gamma,P})|\leq\frac{1}{4}C_{0}|\partial\psi|_{X,m}^{2}+CX^{\ell}_{g}|P|^{2},$
(4.19)

with $C_{0}$ as in (4.13), for a constant $C>0$, and the result now follows
from (4.17) and (4.13).

When $g$ is instead a perturbation of $m_{B,a}$, the argument is similar but
we use (3.44) in place of (3.37). We first write

${\widetilde{Q}}_{P}^{h}(X,N^{D_{t}}_{h})={Q^{m_{B}}(X,N^{D_{t}}_{m_{B}})}+\zeta({\widetilde{J}}_{X,\gamma,P})+\zeta(J_{X,\gamma_{a},P}),$
(4.20)

where $\zeta=dt$, where we wrote $h=m_{B}+\gamma+\gamma_{a}$. Using (3.44) to
bound the second term on the right-hand side, we have

$\displaystyle|\zeta({\widetilde{J}}_{X,\gamma,P})|$
$\displaystyle\lesssim\delta
v|\ell^{m_{B}}\psi|^{2}+\delta^{-1}\epsilon|\partial\psi|_{X,m_{B}}^{2}+|X||\gamma||\partial\psi|^{2}+\epsilon|\zeta(J_{X,\gamma_{a}})|$
(4.21)
$\displaystyle\qquad+\left(\delta^{-1}+1\right)v|P|^{2}+\frac{1}{(1+s)^{1/2}}|P||\partial\psi|$
(4.22)