SHOCK FORMATION IN QUASILINEAR WAVE EQUATIONS: AN
OVERVIEW OF THE NEARLY PLANE SYMMETRIC REGIME
JARED SPECK
Abstract. In an influential 1964 article, P. Lax studied genuinely nonlinear 2×2 hyperbolic
PDE systems (in one space dimension). He showed that a large set of smooth initial data lead
to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon
known as wave breaking. In these notes, we study the Cauchy problem for two classes of
quasilinear wave equations in two space dimensions that are closely related to the systems
studied by Lax. When the data have one-dimensional symmetry, Lax’s methods can be
applied to the wave equations to show that a large set of smooth initial data lead to wave
breaking. The main result outlined here is that the Lax-type wave breaking is stable under
small Sobolev-class perturbations of the data that break the symmetry. Moreover, we give a
detailed, constructive description of the asymptotic behavior of the solution all the way up
to the first singularity, which is a shock driven by the intersection of true null (characteristic)
hyperplanes. As a warm-up to the full problem, we first give a complete treatment of the case
in which the initial data have one-dimensional symmetry. To derive our results, we develop
an extension of Christodoulou’s framework for studying shock formation that applies to a
new solution regime in which wave dispersion is not present.
February 25, 2016
Contents
1. Introduction and a Sharp Blow-up Result in Exact Plane Symmetry
1.1. Some background on prior work
1.2. Plane symmetric and nearly plane symmetric solutions
1.3. The classes of wave equations that we treat
1.4. A model wave equation
1.5. Geometric constructions for the model wave equation in plane symmetry
1.6. Geometric constructions related to the method of characteristics
1.7. The size of the initial data
1.8. Smallness assumptions
1.9. Important remarks on “constants”
1.10. The main sharp blow-up result in plane symmetry
2. Beyond Plane Symmetry: The Geometric Setup
2.1. Why is the proof of shock formation so much harder away from plane symmetry?
2.2. An overview of the proof of shock formation away from plane symmetry
2.3. A preview on the vectorfield method tied to an eikonal function u
2.4. Overview of the regime of initial conditions that we treat
2.5. Notational conventions and shorthand notation
2.6. Assumptions on the nonlinearities
2.7. The eikonal function and related constructions
1
2
3
7
8
10
12
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17
18
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31
Stable Shock Formation
2
2.8. Projections
2.9. First and second fundamental forms and covariant differential operators
2.10. Useful expressions for the null second fundamental form
2.11. Expressions for the metrics
2.12. Energy-momentum tensor
2.13. The definition of a deformation tensor
2.14. Norms
2.15. Commutation vectorfields
2.16. Deformation tensor calculations
2.17. Transport equations for µ and Li
2.18. Connection coefficients of the rescaled frame
2.19. Frame decomposition of the wave operator
2.20. The wave equation after one commutation
2.21. Size assumptions on the data
2.22. An overview of the bootstrap assumptions
2.23. Smallness assumptions
3. Generalized energy estimates away from plane symmetry
3.1. The basic strategy for deriving generalized energy estimates
3.2. Energy estimates via the multiplier method
3.3. Details on the behavior of µ
3.4. Details on the top-order energy estimates
3.5. Descending below top order
References
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1. Introduction and a Sharp Blow-up Result in Exact Plane Symmetry
These notes serve as an introduction to the article [12]. The main ideas presented here
are connected to several other important results on the long-time behavior of solutions to
hyperbolic PDEs which we now overview, starting with Peter Lax’s work [20]. In this highly
influential article, Lax showed that genuinely nonlinear 2 × 2 hyperbolic PDE systems1
exhibit finite-time blow-up for a large set of smooth initial data. He showed that the blowup is of wave breaking type, that is, the solution remains bounded but its first derivatives
blow up. Lax’s results are by now considered classic and have been extended in many
directions. Moreover, his robust approach can easily be extended to treat many related
systems in one space dimension that, strictly speaking, have not been addressed in the
literature. In particular, his approach could be used to prove finite-time blow-up for solutions
to various quasilinear wave equations in one space dimension; under suitable assumptions on
the nonlinearities, one could prove blow-up by first writing the wave equation as a first-order
system in the two characteristic derivatives of the solution and then applying Lax’s methods.
In these notes, we study the Cauchy problem for two classes of such wave equations in two
space dimensions, specifically equations (2.6.1a) and (1.3.3a) below. Lax’s methods yield a
proof of blow-up for solutions to these wave equations arising from data with one-dimensional
symmetry, that is, data that depend only on a single real variable x1 ∈ R. Our main goal in
1Such
systems involve two unknowns in one time and one space dimension.
J. Speck
3
these notes is to survey the most important ideas behind the following result: the Lax-type
wave breaking is stable under small perturbations of the data that break the symmetry. To
close the proof, we must derive a sharp description of the blow-up that, even for data with
one-dimensional symmetry, provides more information than does Lax’s approach. It turns
out that in the regime under study, the singularity is a shock. As a preview to the general
case, in Sects. 1.5-1.10, we will restrict our attention to solutions with exact plane symmetry
and give a precise description of the shock formation. We present the main plane symmetric
shock formation result as Theorem 1.1.
Remark 1.1 (Extending the results to the Euler equations). With modest additional
effort, the irrotational2 compressible Euler equations of fluid mechanics, both in the relativistic and non-relativistic cases, can be massaged into the form of the equations treated in these
notes. That is, the results described here can be extended to yield sharp blow-up results for
a class of nearly symmetric solutions to the compressible Euler equations in the absence of
vorticity and strictly away from the trivial vacuum state. The details of the extension are
presented in [12, Appendices A and B] and for brevity, we do not discuss them here.
As we will later see, especially in Sect. 3, the proof away from plane symmetry is much
more challenging due to the necessity of deriving energy estimates. In contrast, in plane
symmetry, one can avoid energy estimates and instead rely exclusively on the method of
characteristics. It is a worthwhile exercise to start with the plane symmetric case because
the right intuition about the blow-up mechanism and the reason for its stability can already
be inferred from the plane symmetric case.
1.1. Some background on prior work. For some evolution equations in more than one
space dimension that enjoy special algebraic structure, short proofs of blow-up by contradiction are known. In contrast, the typical wave equation that we study does not have any
obvious features which suggest a short path to proving blow-up. In particular, the equations do not generally derive from a Lagrangian, admit conserved quantities, or have signed
nonlinearities. They do, however, enjoy a key property: they have special null structures
(which are distinct from the well-known null condition [16] of S. Klainerman). These null
structures manifest in several ways, including the absence of certain terms in the equations
(see, for example, Remark 2.13) as well as the preservation of certain good product structures under suitable commutations and differentiations of the equations (see, for example,
Remark 2.15). The null structures are not visible relative to the standard coordinates. Thus,
to expose them, we construct a dynamic “geometric coordinate system” and a corresponding
vectorfield frame that are adapted to the true characteristics corresponding to the nonlinear
flow. The main ingredient in this construction, which is central to all that follows, is a an
eikonal function, which by definition is a solution u to the eikonal equation
(g −1 )αβ ∂α u∂β u = 0.
(1.1.1)
The eikonal function is in fact one of the coordinate functions of the geometric coordinate
system. In (1.1.1), g denotes the Lorentzian metric3 corresponding to the wave equation.
2This
means that the fluid vorticity vanishes.
to the quasilinear nature of the equations we are considering, g depends on the wave equation
solution variable.
3Due
Stable Shock Formation
4
With the help of u, we are able to exploit the null structures to give a detailed, constructive
description of the singularity, which is a shock in the regime under study. A key feature of
the proof is that the solution remains regular relative to the geometric coordinates at the low
derivative levels. The blow-up occurs in the partial derivatives of the solution relative to the
standard rectangular coordinates and is tied to the degeneration of the change of variables
map between geometric and rectangular coordinates.
The approach that we take in these notes is based on an extension of the remarkable
framework of Christodoulou, who proved [3] detailed shock-formation results for solutions
to the relativistic Euler equations4 in irrotational regions of R1+3 in a very different solution
regime: the small-data dispersive regime. In that regime, relative to a geometric coordinate
system analogous to the one mentioned in the previous paragraph, the solution enjoys time
decay5 at the low derivative levels corresponding to the dispersive nature of waves. The
decay plays an important role in controlling various error terms and showing that they do
not interfere with the shock-formation mechanisms. In contrast, in the regime under study
here, the solutions do not decay. This basic feature is tied to the fact that in one space
dimension, wave equations (and more generally the equations studied by Lax) are essentially
transport equations. For this reason, we must develop a new approach to controlling error
terms and to showing that the solution exists long enough for the shock to form.
As we explain below in more detail, a key ingredient in our analysis is the propagation
of a two-size-parameter ˚
− δ̊ hierarchy all the way up to the shock. Here and throughout,
δ̊ > 0 is a not necessarily small parameter that corresponds to the size of derivatives in a
direction that is transversal to the true characteristics Pu (with Pu denoting a level set of
the eikonal function, see (2.7.2c)) and ˚
≥ 0 is a small parameter that corresponds to the
size of derivatives in directions tangent to the Pu . The fact that we are able to propagate
the hierarchy is deeply tied to the special null structures mentioned above.
1.1.1. Detailed blow-up results in more than one space dimension. Alinhac was the first [1, 2]
to give a sharp description of singularity formation in solutions to quasilinear wave equations in more than one space dimension without symmetry assumptions. He addressed a
compactly supported small-data regime in which dispersive effects are eventually overcome
by sufficiently strong quadratic nonlinearities. For convenience, even though these kinds of
solutions eventually blow-up, we say that they belong to the “small-data dispersive regime.”
Alinhac’s results have been generalized to various equations by several authors; see, for example, [8–10]. In the case of three space dimensions (more precisely, the data are given
on R3 ), Alinhac proved that whenever the nonlinearities in equation (1.3.3a) fail to satisfy
Klainerman’s null condition [16], there exists a set of data of small size ˚
(in a Sobolev norm)
−1
such that the solution decays for a long time at the linear rate t before finally blowing up
at the “almost global existence” time ∼ exp(c˚
−1 ). More precisely, the singularity-forming
˚
quantities6 behave like
, where the O(˚
) term in the denominator
(1 + t) [1 + O(˚
) ln(1 + t)]
4The
results were later extended to apply to the non-relativistic Euler equations in [7].
in our work here, the blow-up in the small-data dispersive regime occurs in the rectangular coordinate
partial derivatives of the solution.
6In Alinhac’s equations of type (1.3.3a), the second rectangular derivatives of the solution blow-up. In
our work on equations of type (2.6.1a), the first rectangular derivatives blow up.
5As
J. Speck
5
depends on the nonlinearities as well as the profile of the data and the blow-up (for some
t > 0) occurs in regions where O(˚
) < 0. Alinhac’s data were posed in an annular region
of R3 , and he assumed that they verified a non-degeneracy condition. His results showed
that the almost global existence lifespan lower bounds, obtained by John and Klainerman
[14, 15, 17] with the help of dispersive estimates that delay7 the singularity formation, are in
fact saturated. Moreover, his results confirmed John’s conjecture [13] regarding the asymptotically correct description of the blow-up time in the limit ˚
↓ 0 for data verifying the
non-degeneracy condition.
Christodoulou’s remarkable work [3] yielded a sharp improvement (described below) of
Alinhac’s results for a similar class of small compactly supported data given on R3 , and he did
not make any non-degeneracy assumption. His main results applied to irrotational regions
of solutions to the special relativistic Euler equations in the small-data dispersive regime.
In such regions, the fluid equations reduce8 to a special case of the wave equation (1.3.3a)
in which additional structure is present. The non-relativistic Euler equations were treated
through the same approach in [7] and feature the same additional structure, including that
the irrotational fluid equations derive from a Lagrangian (and thus can be written in EulerLagrange form) and that solutions possess several conserved quantities associated to various
symmetries of the Lagrangian. These assumptions were used in the proofs, in particular in
exhibiting the good null structure9 enjoyed by the equations. The equations also had some
additional structure due to the assumption that they model a physical fluid. In addition to
assuming that the data are of a small size ˚
in a high Sobolev norm, Christodoulou also made
additional assumptions on the data to ensure that a shock forms. His sufficient conditions
were phrased in terms of certain integrals of the data: shocks form in the solution whenever
the data integrals have the appropriate sign (determined by the nonlinearities) and are not
too small in magnitude relative to ˚
.
Christodoulou’s results were extended [23] to a larger class of equations and data by
Speck (see also the survey article [11], joint with Holzegel, Klainerman, and Wong). In
particular, for data given on R3 , he proved a sharp small-data shock-formation result for
equations (2.6.1a) or (1.3.3a) whenever the null condition fails. That is, he showed that
Christodoulou’s sharp shock-formation results are not tied to the specific structure of the
fluid equations and that the additional structure present in those equations is not needed to
close the proof. Speck also showed that given any sufficiently regular non-trivial compactly
supported initial data, if they are rescaled by a small positive factor, then the solution forms
a shock in finite time. That is, all sufficiently regular data profiles lead to shock formation if
they are suitably rescaled.
Alinhac’s and Christodoulou’s approaches to proving shock formation share many common features. For example, the main idea of Alinhac’s proof was to resolve the singularity
by constructing an eikonal function u, as in (1.1.1). Moreover, near the singularity, he
changed variables to a new “geometric” coordinate system in which u is one of the new
coordinates. Relative to the geometric coordinates, he proved that the solution to (1.3.3a)
7By
“delay,” we mean relative to the case of one space dimension, where the lack of dispersion leads to
blow-up at time O(˚
−1 ).
8Up to simple renormalizations outlined in [12, Appendix B].
9Some of the terms that are generally non-zero happen to vanish in the equations that Christodoulou
studies. The vanishing occurs because he studies equations of the form (1.3.3a) that derive from a Lagrangian.
Stable Shock Formation
6
remains regular all the way up to the point where the characteristics first intersect but that
the change of variables map between the rectangular and geometric coordinates breaks down
there. Changing variables back to rectangular coordinates, he showed that the degeneracy
implies that |∂ 2 Φ| blows up in finite time precisely at the point where the characteristics
intersect. In his proof, Alinhac had to overcome a potential loss of derivatives; we explain
this difficulty in more detail starting in Subsect. 2.3. However, the methods he used did
not immediately eliminate all of the derivative loss and thus differed in a fundamental way
from Christodoulou’s approach. Specifically, to close his energy estimates, Alinhac employed
a Nash-Moser iteration scheme. His scheme featured a free boundary due to the fact that
the blow-up time for each iterate can be slightly different. Although Alinhac gave a sharp
description of the asymptotic behavior of the solution near the singularity, his proof was
not able to reveal information beyond the first blow-up point. Moreover, in order for his
proof to close, the constant-time hypersurface of first blow-up was allowed to contain only one
blow-up point. These fundamental technical limitations were tied to the presence of the free
boundary in his Nash-Moser iteration scheme and they are the reason that he had to make
the non-degeneracy assumption on the data; see [23] for additional discussion regarding his
approach.
We now describe the most important difference between the approaches of Alinhac and
Christodoulou. The main advantage afforded by Christodoulou’s framework, as shown in
[3, 7, 23], is that in the small-data dispersive regime, there is a sharp criterion for blow-up.
Specifically, Christodoulou identified the importance of the inverse foliation density µ, which
is roughly a first derivative of the eikonal function, and showed that the solution blows up
at a given point ⇐⇒ µ vanishes there. In particular, in the small-data dispersive regime,
shocks are the only kinds of singularities that can form. The quantity 1/µ measures the
density of the level sets of the eikonal function and µ = 0 corresponds to infinite density and
the intersection of the characteristics. This quantity plays a central role in the present notes
as well. Since the behavior of µ is local in time and space, the vanishing of µ at one point does
not preclude one from continuing the solution to a neighborhood of other nearby points where
µ > 0. Moreover, {µ = 0} precisely characterizes the singular portion of the boundary of the
maximal development of the data, that is, the portion of the boundary on which the solution
blows up. Thus, Christodoulou’s framework is able to reveal detailed information about the
structure of the maximal development of the data, the shape of the various components of its
boundary, and the behavior of the solution along it. The same information can be extracted
for the solutions that we study here, though we do not present the details. The sharp
description is an essential ingredient in setting up the problem of extending the Euler solution
weakly beyond the first singularity. We note that an essential component of solving this
problem is obtaining information about the shock hypersurface across which discontinuities
occur. The problem was recently solved in spherical symmetry [6], while the non-symmetric
problem remains open and is expected to be of immense difficulty.
1.1.2. Blow-up in a large-data regime featuring a one-parameter scaling of the data. Recently,
Miao and Yu proved [21] a related shock-formation result for the wave equation −∂t2 φ + [1 +
(∂t φ)2 ]∆φ = 0 in three space dimensions with data that are compactly supported in an
annular region of radius ≈ 1 and thin width δ, where δ is a small positive parameter. The
data’s amplitude and their functional dependence on a radial coordinate are rescaled by
J. Speck
7
powers of δ. Consequently, the data and their derivatives verify a hierarchy of estimates
featuring various powers of δ. For example, φ itself has small L∞ size δ3/2 , its rectangular
derivatives ∂α φ have L∞ size δ 1/2 , and a certain derivative of ∂α φ that is transversal to the
true characteristics has large L∞ size δ−1/2 . Due to the largeness, the blow-up of the second
rectangular derivatives of φ happens within one unit of time. The scaling of the data is closely
related to the short-pulse ansatz pioneered by Christodoulou in his aforementioned proof of
the formation of trapped surfaces in solutions to the Einstein-vacuum equations [4]. The
main contribution of [21] was showing how to propagate the δ hierarchy estimates until the
time of first shock formation. In the proof, dispersive effects are not relevant. Instead, the
authors control nonlinear error terms by tracking the powers of δ associated to each factor
in the product. Roughly, the error terms have a product structure, typically of the form
small · large (relative to powers of δ), where the small factor often more than compensates
for the large one. That is, the authors show that the overall powers of δ associated to
the error term products are favorable in the sense that the smallness of δ is sufficient for
controlling them. In this way, a class of large data solutions can be treated using techniques
borrowed from the usual small-data framework.
The results presented here are related to those of [21] but are distinguished by our use of
two size parameters (the parameters ˚
and δ̊ mentioned above), which allows us to treat a set
of initial conditions containing large data and, unlike [21], small data too. As we described
above, a key aspect of our proof is that we can propagate the small size ˚
of the Pu −tangent
derivatives long enough for the shock to form, even though the transversal derivatives can
be of a relatively large size δ̊. To this end, we must exploit the good product/null structure
in the equations in ways that go beyond the δ scaling structures exploited in [21].
1.2. Plane symmetric and nearly plane symmetric solutions. We now describe what
we mean by “plane symmetric” and “nearly plane symmetric.” In our study of nearly plane
symmetric solutions, we will assume that the wave equations are posed on a spacetime with
topology R × Σ, where t ∈ R corresponds to time, (x1 , x2 ) ∈ Σ := R × T corresponds to
space, and the torus10 T := [0, 1) corresponds to the direction that is suppressed in plane
symmetry. We often use the alternate notation x0 = t to denote the time coordinate. We call
{xα }α=0,1,2 the rectangular coordinates because relative to them, the standard Minkowski
metric on R × Σ takes the form mµν = diag(−1, 1, 1); see (1.4.3). We use the notation
∂
∂α :=
to denote the rectangular partial derivative vectorfields, and we often use the
∂xα
alternate notation11 ∂t := ∂0 = ∂x∂ 0 . The vectorfields ∂t , ∂1 , ∂2 are globally defined.
Remark 1.2 (Notation). See Subsect. 2.5 for additional comments on the notation and
our summation conventions.
By a “plane symmetric solution” Ψ, we mean that Ψ happens to depend only on t and
x1 . By “nearly plane symmetric,” we mean that Ψ can depend on x2 , but it must satisfy
smallness assumptions at time 0 that roughly correspond to Ψ having small partial derivatives
10With
the endpoints identified and equipped with the usual smooth orientation and with a corresponding
local rectangular coordinate function x2 .
11Note that ∂ is not the same as the geometric coordinate partial derivative ∂ appearing in equation
t
∂t
(2.7.17) and elsewhere throughout these notes.
Stable Shock Formation
8
with respect to x2 at time 0. Our precise assumptions involve statements about directional
derivatives of Ψ that are tangent and transversal to the true characteristics; see Subsect. 2.23.
Although it is possible to propagate aspects of the smallness all the way to the shock, as we
mentioned at the beginning, one cannot use the rectangular coordinates (t, x1 , x2 ) to identify
the quantities that remain small. One must instead construct geometric coordinates adapted
to the true characteristics.
Remark 1.3. We have make the assumption Σ = R × T mainly for technical convenience;
we expect that suitable wave equations on other manifolds could be treated using techniques
similar to the ones we describe here.
1.3. The classes of wave equations that we treat. The main class of problems that we
study are Cauchy problems for geometric wave equations:12
g(Ψ) Ψ = 0,
(1.3.1a)
(Ψ|Σ0 , ∂t Ψ|Σ0 ) = (Ψ̊, Ψ̊0 ),
(1.3.1b)
where g(Ψ) denotes the covariant wave operator of the Lorentzian metric g(Ψ) and (Ψ̊, Ψ̊0 ) ∈
He19 (Σ0 ) × He18 (Σ0 ) (see Remarks 1.4 and 1.5 just below) are data with support contained in
the compact subset [0, 1] × T of the initial Cauchy hypersurface Σ0 := {t = 0} ' R × T. Here
and throughout,13 g(Ψ) Ψ = (g −1 )αβ (Ψ)Dα Dβ Ψ, where14 D is the Levi-Civita connection of
g(Ψ). We assume that relative to the rectangular coordinates {xα }α=0,1,2 , we have gαβ (Ψ) =
mαβ + O(Ψ), where mαβ = diag(−1, 1, 1) is the standard Minkowski metric and O(Ψ) is
an error term, smooth in Ψ and . |Ψ| in magnitude when |Ψ| is small. We make further
mild assumptions on the nonlinearities ensuring that relative to rectangular coordinates, the
nonlinear terms are effectively quadratic and fail to satisfy Klainerman’s null condition [16];
see Subsect. 2.6 for more details.
Remark 1.4 (Our analysis refers to more than one kind of Sobolev space). Above
and throughout, HeN (Σ0 ) denotes the standard N th order Sobolev space with the correspondnP
o1/2
R
2 2
ing norm kf kHeN (Σ0 ) :=
(∂ f ) d x
, where ∂I~ is a multi-indexed differential
~
|I|≤N
Σ0 I~
operator representing repeated differentiation with respect to the spatial coordinate partial
derivatives and d2 x is the area form of the standard Euclidean metric e on Σ0 , which has the
form e := diag(1, 1) relative to the rectangular coordinates. It is important to distinguish
these L2 −type norms from the more geometric ones that we introduce in Sect 2.14; the two
kinds of norms drastically differ near the shock.
Remark 1.5 (On the number of derivatives). Although our analysis is not optimal
regarding the number of derivatives, we believe that any implementation of our approach
requires significantly more derivatives than does a typical proof of existence of solutions
to a quasilinear wave equation based on energy methods. It is not clear to us whether
this is a limitation of our approach or rather a more fundamental aspect of shock-forming
solutions. Our derivative count is driven by our energy estimate hierarchy, which is based on
√
to arbitrary coordinates, (2.6.1a) is equivalent to ∂α
detg(g −1 )αβ ∂β Ψ = 0
13See Subsect. 2.5 regarding our conventions for indices, and in particular for the different roles played
by Greek and Latin indices.
14Throughout we use Einstein’s summation convention.
12Relative
J. Speck
9
a descent scheme in which the high-order energy estimates are very degenerate, with slight
improvements in the degeneracy at each level in the descent. For our proof to work, we
must obtain at least several orders of non-degenerate energy estimates, which requires many
derivatives. See Sect. 3.2.5 for more details.
Remark 1.6 (Drastically lower regularity data in plane symmetry). In the case
of exact plane symmetry, the regularity of the initial data can be drastically reduced; see
Theorem 1.1. The reason is that the high regularity needed away from plane symmetry is
tied to degenerate energy estimates, which can be avoided in plane symmetry due to the
availability of the method of characteristics.
For convenience, instead of studying the solution in the entire spacetime R × Σ, we study
only the non-trivial future portion of the solution that is completely determined by the
portion of the data lying to the right of the straight line {x1 = 1 − U0 } ∩ Σ0 , where
0 < U0 ≤ 1
(1.3.2)
is a parameter and the data are non-trivial in the region {1 − U0 ≤ x1 ≤ 1} ∩ Σ0 := ΣU0 0
of thickness U0 . See Figure 1 for a picture of the setup in the nearly plane symmetric case,
where the curved null hyperplane portion PUt 0 and the flat null hyperplane portion P0t in
the picture are described in detail in Subsect. 2.7 (see also Subsect. 1.5 for the case of exact
plane symmetry).
PUt 0
P0t
ΣU0 0
x2 ∈ T
U0
non-trivial data
Ψ≡0
trivial data
x1 ∈ R
Figure 1. The spacetime region under study in the nearly plane symmetric case.
The second class of problems to which our results apply is non-geometric wave equations
of the form
(g −1 )αβ (∂Φ)∂α ∂β Φ = 0,
(Φ|Σ0 , ∂t Φ|Σ0 ) = (Φ̊, Φ̊0 ),
(1.3.3a)
(1.3.3b)
Stable Shock Formation
10
where g(∂Φ) is a Lorentzian metric with gαβ (∂Φ) = mαβ + O(∂Φ). We assume that the
data (1.3.3b) are compactly supported as before, but we also assume one extra degree of
differentiability: (Φ̊, Φ̊0 ) ∈ He20 (Σ0 ) × He19 (Σ0 ). It turns out that the study of solutions
to (1.3.3a) can be effectively reduced to the study of equation (2.6.1a). The details of the
reduction are presented in [12, Appendix A], and for brevity, we do not present them here. In
short, by differentiating (1.3.3a) with rectangular coordinate derivatives ∂ν , the question of
the long-time behavior of solutions to (1.3.3a) can be transformed into an equivalent question
of the long-time behavior of solutions to a coupled system of wave equations in the scalar
unknowns Ψν := ∂ν Φ. The main point is that the system comprises scalar equations that
are closely related to equation (2.6.1a). Furthermore, under suitable smallness assumptions,
the system turns out to be rather weakly coupled except in a few key aspects. This structure
was first observed by Christodoulou in [3] for a class of wave equations that derive from a
Lagrangian. In [23], it was show that the structure survives for general wave equations of the
form (1.3.3a). This fact is based on the availability of some good (null) structure present in
certain semilinear terms, which completely vanished15 in Christodoulou’s work [3]. In total,
except for a handful of key aspects, the difference between the coupled system and the scalar
equation (2.6.1a) is small. It is for this reason that we restrict our attention to the scalar
equation in these notes.
1.4. A model wave equation. To illustrate the main ideas behind the formation of shocks,
in the remainder of Sect. 1, we consider the following model spacetime metric on R × Σ:
g = g(Ψ) := −dt ⊗ dt + (1 + Ψ)−1
2
X
dxa ⊗ dxa .
(1.4.1)
a=1
Note that
g −1 := −∂t ⊗ ∂t + (1 + Ψ)
2
X
∂a ⊗ ∂a .
(1.4.2)
a=1
Exercise 1.1 (g −1 is the inverse of g). Show that relative to rectangular coordinates, the
tensorfields g and g −1 from (1.4.1)-(1.4.2) verify gακ (g −1 )κβ = δαβ .
Note also that for |Ψ| small, the metric (1.4.1) can be viewed as a perturbation of the
Minkowski metric m on R × Σ, which takes the following form relative to the rectangular
coordinates:
mµν = diag(−1, 1, 1).
(1.4.3)
(Small)
gµν = gµν (Ψ) := mµν + gµν
(Ψ),
(1.4.4)
That is, we have
(Small)
where gµν
(Ψ) is a given smooth function of Ψ with
(Small)
gµν
(0) = 0.
15More
(1.4.5)
precisely, in [3], Christodoulou rescaled the physical metric by a scalar function multiple in order
to eliminate the semilinear terms.
J. Speck
11
Specifically, in the case of the model metric (1.4.1), we have
(Small)
gµν
(Ψ) = −
2
X
δµa δνa
a=1
Ψ
,
1+Ψ
(1.4.6)
where here and throughout, δαβ is the standard Kronecker delta.
The initial value problem of interest to us is
g(Ψ) Ψ = 0,
(1.4.7)
(Ψ|t=0 , ∂t Ψ|t=0 ) := (Ψ̊, Ψ̊0 ),
(1.4.8)
where g(Ψ) := (g −1 )αβ Dα Dβ denotes the covariant wave operator of the spacetime metric
g, and D denotes the Levi-Civita connection of g.
Remark 1.7 (Basic facts from geometry). We recall the following basic facts from
differential geometry: relative to arbitrary coordinates, D acts on functions f , vectorfields
V , and one-forms ξ as follows:
Dµ f = ∂µ f,
Dµ V =
ν
Dµ ξν =
ν
∂µ V + Γµν α V α ,
∂µ ξν − Γµαν ξα .
(1.4.9a)
(1.4.9b)
(1.4.9c)
Moreover, the action of D extends to arbitrary higher-order tensorfields via the Leibniz rule.
In (1.4.9a)-(1.4.9c), Γµαν := (g −1 )ακ Γµκν is a Christoffel symbol, where
1
{∂α gκβ + ∂β gακ − ∂κ gαβ }
2
are the (fully lowered) rectangular Christoffel symbols.
Γακβ = Γακβ (Ψ, ∂Ψ) :=
(1.4.10)
Exercise 1.2 (The covariant wave equation in rectangular coordinates). Show that
relative to the rectangular coordinates on R × Σ, equation (1.4.7) takes the form
(g −1 )αβ ∂α ∂β Ψ − (g −1 )αβ (g −1 )κλ Γακβ ∂λ Ψ = 0.
(1.4.11)
Remark 1.8. We note that the formula (1.4.11) holds for a general metric g, not just the
model metric (1.4.1).
In deriving geometric energy estimates (see Lemma 3.1), the formula from the next exercise
is useful.
Exercise 1.3 (Useful expression for the covariant divergence of a vectorfield).
Show that relative to arbitrary coordinates, we have the following divergence identity, valid
for all spacetime vectorfields V :
np
o
1
Dα V α = p
∂α
|detg|V α .
(1.4.12)
|detg|
In (1.4.12), g is viewed as 3 × 3 matrix with components expressed relative to the coordinate
system under consideration.
Stable Shock Formation
12
1.5. Geometric constructions for the model wave equation in plane symmetry. We
now restrict our attention to plane symmetric solutions. For convenience, in the remainder
of Sect. 1, we denote x := x1 and ∂x := ∂1 .
Exercise 1.4 (The model wave equation in rectangular coordinates in plane symmetry). Show that in plane symmetry (that is, when Ψ does not depend on the torus
variable x2 ), equation (1.4.7) takes the form
−∂t2 Ψ + (1 + Ψ)∂x2 Ψ = −
1
(∂t Ψ)2 .
1+Ψ
(1.5.1)
1.6. Geometric constructions related to the method of characteristics. To analyze
solutions to (1.5.1), we rely on the method of characteristics, which is tied to the following
frame vectorfields:
Definition 1.1 (Frame vectorfields). We associate the following null vectorfields to the
equation (1.5.1):
√
√
L := ∂t + 1 + Ψ∂x ,
L := ∂t − 1 + Ψ∂x
(1.6.1)
and the Σt −tangent vectorfield16
1
{L − L} .
(1.6.2)
2
Although we later use an analog of X in our analysis of nearly plane symmetric solutions,
we do not use X here.
Note that for |Ψ| small, L is right-pointing and L is left-pointing.
X :=
Remark 1.9 (Null frame). In one space dimension, the pair {L, L} is traditionally known
as a null frame. In n space dimensions, to form a null frame, one would have to supplement
L and L with n − 1 g−orthonormal vectorfields that are g−orthogonal to L and L such that
the entire collection spans the n + 1−dimensional tangent space at each point.
We note the following simple consequence of (1.6.1).
1
{L + L} ,
2
1
∂x = √
{L − L} .
2 1+Ψ
∂t =
(1.6.3a)
(1.6.3b)
Exercise 1.5 (They are null!). Show that L and L are null vectorfields, that is, that17
g(L, L) = g(L, L) = 0. Also show that g(L, L) = −2, g(L, X) = −1, g(X, X) = 1.
Exercise 1.6 (Vectorfield commutators). Compute the following Lie Bracket (that is,
vectorfield commutator) identities:
[L, L] = − √
16Observe
that Xt = 0.
g(X, Y ) := gαβ X α Y β .
17Throughout
∂t Ψ
LΨ + LΨ
∂x =
{L − L} .
4(1 + Ψ)
1+Ψ
(1.6.4)
J. Speck
Exercise 1.7 (Null decomposition of g −1 ).
verifies
1
g −1 = − L ⊗ L −
2
13
Show that the model inverse metric (1.4.2)
1
L ⊗ L + g/−1 ,
(1.6.5)
2
where g/−1 := (1 + Ψ)∂2 ⊗ ∂2 is a symmetric type 20 tensorfield that is g−orthogonal18 to
both L and L. We will later see that a similar formula holds in the general case, away from
plane symmetry (see (2.11.1b)).
Remark 1.10. g/−1 can be viewed as the inverse of the Riemannian metric g/ induced by
the metric (1.4.1) on the one-dimensional tori of constant t and x1 . In plane symmetry, g/
plays no role and can be ignored. Away from plane symmetry, g/ becomes one of the central
objects of study.
As we mentioned at the beginning, the rectangular coordinates are not sufficient for obtaining the kind of sharp estimates that are needed to fully understand the formation of the
shock. To obtain sharp information, we rely on a fundamental object from geometric optics:
an eikonal function.
Definition 1.2 (The eikonal function in plane symmetry). The eikonal function u is
the solution to the initial value problem
u|t=0 := 1 − x.
Lu(t, x) = 0,
(1.6.6)
As we will see, in plane symmetry, we can use the coordinates (t, u) in place of (t, x) in
order to obtain sharp information about the solution, all the way up to the shock.
Definition 1.3 (Geometric coordinates). We refer to (t, u) as the geometric coordinates.
Remark 1.11. Note that the “dynamic” coordinate u depends on the solution itself; this is
a feature of the quasilinear nature of the wave equation (2.6.5).
Away from plane symmetry, we will define u in a different way: as a solution to the eikonal
equation.
Exercise 1.8 (A definition of u that generalizes). Show that the function u from Def. 1.2
is a solution to the eikonal equation
(g −1 )αβ (Ψ)∂α u∂β u = 0,
∂t u > 0,
u|t=0 := 1 − x.
(1.6.7)
We assume for convenience that the data (Ψ̊, Ψ̊0 ) are supported in {0 ≤ x ≤ 1}. This
implies that Ψ ≡ 0 when u ≤ 0. We will focus our attention on the behavior of the solution
spacetime regions of the form
Mt,U0 := {(t0 , x) | 0 ≤ t0 < t and 0 ≤ u(t0 , x) ≤ U0 }.
(1.6.8)
On Mt,U0 , the solution depends only on the data belonging to the interval x ∈ [1 − U0 , 1].
This basic fact will become clear in the proof of Theorem 1.1, which is based on the form of
the evolution equations provided by Prop. 1.3.
18That
is, (g/−1 )αβ Lα = (g/−1 )αβ Lα = 0.
Stable Shock Formation
14
For convenience in notation, we also define
Σt := {(t, x) | x ∈ R},
0
Σut := {(t, x) | 0 ≤ u(t, x) ≤ u0 },
Pu0 := {(t0 , r) | 0 ≤ t0 and u(t0 , x) = u0 }, Put 0 := {(t0 , r) | 0 ≤ t0 ≤ t and u(t0 , x) = u0 }.
(1.6.9a)
Remark 1.12 (On the two coordinate systems). We may think of Ψ as a function defined on the manifolds19 MT,U0 , where the MT,U0 are equipped with two coordinate systems:
the rectangular coordinates (t, x) and the geometric coordinates (t, u). In this way, we may
switch back and forth between thinking of Ψ as function of the rectangular or geometric
coordinates, as we see fit. For T strictly less than the time of first shock formation, the differential structure on MT,U0 corresponding to the rectangular coordinates is C 1 compatible20
with the differential structure corresponding to the geometric coordinates. However, when
the first shock forms, the C 1 compatibility between the two differential structures breaks
down at the shock points; one could say that the coordinate systems degenerate relative to
each other at the shock points.
Remark 1.13 (The characteristics). As is customary, we often refer to the Pu as “the
(right-traveling)21) characteristics”.
From u, we are able to construct an assortment of geometric quantities that can be used
to derive sharp information about the solution. The most important of these in the context
of shock formation is the inverse foliation density.
Definition 1.4 (Inverse foliation density). We define µ > 0 by
√
µ−1 := ∂t u = − 1 + Ψ∂x u,
(1.6.10)
where the last equality in (1.6.10) follows from (1.6.1) and (1.6.6).
The quantity 1/µ measures the density of the level sets of u relative to the constant-time
hypersurfaces Σt . In the solution regime under study, µ will initially be close to 1 and when
it vanishes, the density becomes infinite and the level sets of u (the characteristics) intersect;
see Figure 3 on pg. 35, in which we illustrate (in the nearly plane symmetric case) a scenario
where µ has become small and a shock is about to form. Our arguments will also show that
in the solution regime under study, the rectangular components gαβ remain near those of the
Minkowski metric mαβ = diag(−1, 1, 1) all the way up to the shock. Thus, from (2.7.3), we
infer that the vanishing of µ implies that some rectangular derivative of u blows up. From
experience with model equations in one space dimension such as Burgers’ equation, one might
expect that the intersection of the characteristics is tied to the formation of a singularity
in Ψ. Though it is not obvious, our proof in fact reveals that in the regime under study,
µ = 0 corresponds to blow-up of the first22 rectangular derivatives of Ψ. In particular, on
19Manifolds
20The
with a boundary, that is.
two differentiable structures are in fact C k compatible with k decided by the regularity of the initial
data.
21One could also construct “left-traveling” characteristics corresponding to a second eikonal function u,
defined by replacing equation (1.6.6) with Lu = 0. However, the left-traveling characteristics are not needed
in in these notes.
22For equation (1.3.3a), the blow-up occurs in the second rectangular derivatives of Φ.
J. Speck
15
sufficiently large time intervals, our work affords a sharp description of singularity formation
characterized precisely by the vanishing of µ.
From (1.6.6) and (1.6.10), we see that
1
µt=0 = √
(1.6.11)
1 + Ψt=0
so that |µt=0 − 1| = O(|Ψt=0 |) for |Ψt=0 | small.
Exercise 1.9 (Connection between µ and the change of variables map). Show that
the Jacobian determinant of the change of variables map (t, u) → (t, x) is proportional to µ.
Compare with Exercise 2.9 away from plane symmetry.
Away from plane symmetry, we will not have an explicit formula for L in the spirit of
(1.6.1). Instead, we will have to construct it from the eikonal function and µ.
Exercise 1.10 (A definition of L that generalizes). Show that the vectorfield L from
(1.6.1) can be alternatively defined in the following way:
Lα = −µ(g −1 )αβ ∂β u,
(1.6.12)
where u is defined by (1.6.7) and µ is defined by (1.6.10).
Remark 1.14 (Tangential good, transversal bad). From (1.6.6), we see that L is tangent
to the characteristics Pu . In contrast, L is transversal to the Pu . A basic mantra to keep in
mind is that vectorfields tangent to the characteristics are associated with good behavior and
smallness while vectorfields transversal to the characteristics are associated with dangerous
behavior and largeness. This philosophy carries over to the general case, away from plane
symmetry.
The key idea behind deriving a sharp shock-formation result is to rescale the transversal
vectorfield L in a way that exactly compensates for the blowup (see Remark 1.15). The
rescaling is by µ.
Definition 1.5 (Rescaled left-pointing null vectorfield).
L̆ := µL.
(1.6.13)
We may also define the Σt −tangent vectorfield
X̆ := µX,
(1.6.14)
where X is defined in (1.6.2). Although we later use an analog of X̆ in our analysis of nearly
plane symmetric solutions, we do not use X̆ here.
Remark 1.15 (Reformulating the shock-formation problem as a conventional
long-time existence problem). A fundamental feature of our analysis is that |LΨ| and
|L̆Ψ| will remain uniformly bounded, all the way up to the shock. Therefore, one should keep
in mind that our geometric setup is designed in an effort to reformulate the shock-formation
problem as a conventional long-time existence problem for Ψ, LΨ, and L̆Ψ. Put differently,
we aim to prove long-time-existence-type results relative to the frame {L, L̆}, where L̆ degenerates relative to the rectangular coordinates when µ vanishes. More precisely, L and L
will remain near their flat values L(f lat) := ∂t + ∂x and L(f lat) := ∂t − ∂x , all the way to the
Stable Shock Formation
16
shock and thus the vectorfield L̆ vanishes when µ goes to 0. Thus, one should expect that
|LΨ| will blow-up like µ−1 precisely at the points where µ vanishes (In Theorem 1.1, we will
prove that this is indeed the case).
In the next lemma, we express the vectorfields L and L̆ in terms of the geometric coordinate
partial derivatives.
Lemma 1.1. Relative to the geometric coordinates (t, u), we have23
∂
,
∂t
∂
∂
L̆ = µ + 2 .
∂t
∂u
L=
(1.6.15a)
(1.6.15b)
Proof. From (1.6.1), we find that Lt = 1. Moreover, by (1.6.6), we have Lu = 0. (1.6.15a)
thus follows.
To obtain (1.6.15b), we first use (1.6.1) and (1.6.13) to deduce
L̆t = µ. Next, we use
√
(1.6.1), (1.6.10), and (1.6.13) to compute that L̆u = µ(∂t u − 1 + Ψ∂x u) = 2µ∂t u = 2. The
identity (1.6.15b) thus follows.
Remark 1.16. Using Lemma 1.1, it is easy to check that
[L, L̆] = (Lµ)L.
(1.6.16)
The formula (1.6.16) shows that the commutator [L, L̆] does not have any component in the
Pu −transversal direction L. This is another virtue of including precisely the weight µ in
definition (1.6.13). This fact is very important away from plane symmetry, when to close
the problem, one is forced to commute the wave equation with various vectorfields including
L and to derive energy estimates.
The behavior of µ is decided by the following evolution equation, which plays a central
role in all that follows.
Lemma 1.2 (Evolution equation verified by µ). µ verifies the following evolution equation:
1
1
Lµ = −
µLΨ + L̆Ψ .
(1.6.17)
4 (1 + Ψ)
Exercise 1.11. Derive equation (1.6.17).
Now that we have constructed the appropriate geometric objects, we can express the wave
equation (1.5.1) in a geometric way that is well-suited to the method of characteristics.
Proposition 1.3 (The rescaled system of equations that we use to analyze Ψ).
The wave equation (1.5.1) and the evolution equation (1.6.17) together imply the following
23Throughout
the remainder of this section,
partial differentiation at fixed t.
∂
∂t
denotes partial differentiation at fixed u and
∂
∂u
denotes
J. Speck
17
system of equations:
1
1
(LΨ)L̆Ψ,
2 (1 + Ψ)
n
o
1
1
2
µ (LΨ) + 3(LΨ)L̆Ψ .
L̆LΨ =
4 (1 + Ψ)
LL̆Ψ =
(1.6.18a)
(1.6.18b)
Exercise 1.12. Derive Prop. 1.3.
Exercise 1.13 (Elimination of the Riccati term from equation (1.6.18a)). One of the
key properties of the rescaled vectorfield L̆ is that the evolution equation (1.6.18a) for L̆Ψ
does not involve a Riccati term proportional to (L̆Ψ)2 . Show that in contrast, LΨ verifies an
evolution equation of the form LLΨ = · · · , but that there is a Riccati term proportional to
(LΨ)2 present in · · · , much like the Riccati term µ (LΨ)2 in equation (1.6.18b). The absence
of the Riccati term (L̆Ψ)2 in equation (1.6.18a) is the main reason that we can successfully
reformulate the shock-formation problem as a long-time existence problem relative to the
rescaled frame {L, L̆}, as described in Remark 1.15. We note that if LΨ is initially small
(that is, under a suitable small-data assumption), the Riccati term µ (LΨ)2 on RHS (1.6.18b)
cannot create blow-up because we are restricting our attention to a region of small u−width.
That is, even though its evolution equation involves a Riccati term, LΨ does not have enough
“u−time” to blow up along the integral curves of L̆.
Remark 1.17 (The null structure in equation (1.6.18a)). Note that the nonlinear term
on RHS (1.6.18a) contains only one linear factor that is transversal to the characteristics Pu ,
the factor L̆Ψ. This special “null structure” is closely related24 to Klainerman’s classic null
condition (see Def. 2.1 and Remark 2.2). However, unlike his null condition, it is tied to the
true characteristics rather than the flat Minkowski ones.
We also note that the nonlinearities in equation (1.5.1) (that is, the wave equation in
rectangular coordinates) do not satisfy Klainerman’s null condition. It is only because we
have constructed the rescaled vectorfield L̆ that we are able to rewrite the wave equation in
a form with special null structure. See Remarks 2.12-2.15 for related comments.
1.7. The size of the initial data. We now describe the assumptions on the size of the
data that we use to close our proof of shock formation. In the remainder this section, we
use the following norms, defined on the set of continuous functions f : Ω → R:
kf kC 0 (Ω) := sup |f | .
(1.7.1)
p∈Ω
Definition 1.6 (Two parameters characterizing the size of the data). We characterize
the size of the initial data with the two parameters ˚
and δ̊, which we define as follows:
n
o
max kΨkC 0 (Σ1 ) , kLΨkC 0 (Σ1 ) := ˚
,
(1.7.2a)
0
0
:= δ̊.
(1.7.2b)
L̆Ψ
1
C 0 (Σ0 )
24In
[23], we refer to this structure as the strong null condition.
Stable Shock Formation
18
In addition to ˚
and δ̊, our analysis also refers to the following quantity δ̊∗ , which is
connected to δ̊. For reasons that will become clear, quantity δ̊∗ controls the blow-up time of
solutions to (1.6.18a)-(1.6.18b).
Definition 1.7 (The quantity that controls the blow-up time).
1
δ̊∗ := sup − L̆Ψ ,
4
Σ10
−
(1.7.3)
where [p]− := |p| when p < 0, and [p]− := 0 otherwise.
Remark 1.18. Note that δ̊∗ > 0 whenever the data are non-trivial and compactly supported
in Σ10 .
1.8. Smallness assumptions. For the remainder of these notes, when we say that “A
is small relative to B,” we mean that there exists a continuous increasing function f :
[0, ∞) → (0, ∞) such that A ≤ f (B). In principle, the functions f could always be chosen
to be polynomials with positive coefficients or exponential functions. However, to avoid
lengthening the notes, we typically do not specify the form of f .
To close our proof, we make the following relative smallness assumptions. We continually
adjust the required smallness in order to close our estimates.
• The initial data parameter ˚
(1.7.2a) is small relative to δ̊−1 , where δ̊ is the data-size
parameter from (1.7.2b).
•˚
is small relative to the data-size parameter δ̊∗ from (1.7.3).
The first assumption will allow us to control error terms that, roughly speaking, are of size
˚
δ̊k for some integer k ≥ 0. The second assumption is relevant because below we will show
that the expected blow-up time is25 approximately δ̊−1
∗ , and the assumption will allow us to
show that various error products featuring a small factor ˚
remain small for t < 2δ̊−1
∗ , which
is plenty of time for us to show that a shock forms.
Remark 1.19. δ̊ and δ̊∗ do not have to be small in any absolute sense.
Exercise 1.14 (Data exist that verify the smallness assumptions). Warm up: Construct a sequence of smooth (C ∞ ) data (Ψ̊, Ψ̊0 ) = (Ψt=0 , ∂t Ψt=0 ) that are compactly supported in Σ10 such that ˚
↓ 0 along the sequence while δ̊ and δ̊∗ remain bounded from below
by 1.
Full problem, relevant for the away from plane symmetric case where estimates on higher
derivatives are needed: Fix integers A0 ≥ 1 and B ≥ 0. Construct a sequence of smooth (C ∞ )
data (Ψ̊, Ψ̊0 ) = (Ψt=0 , ∂t Ψt=0 ) that
supported in Σ10 such that for 1 ≤ A ≤ A0
are compactly
↓ 0 along the sequence while δ̊ and δ̊∗ remain
and 0 ≤ B ≤ B0 , we have that LA L̆B Ψ
1
bounded from below by 1.
25Specifically,
C 0 (Σ0 )
for ˚
sufficiently small, the first shock occurs at time (1 + O(˚
))δ̊∗ .
J. Speck
19
1.9. Important remarks on “constants”. Throughout the rest of the notes, we adopt
the following conventions.
• A . B means that there exists C > 0 such that A ≤ CB.
• A = O(B) means that |A| . |B|.
• Constants such as C and c are free to vary from line to line and can be chosen to be
independent of the parameter U0 from (1.3.2). Explicit and implicit constants
are allowed to depend in an increasing, continuous fashion on the datasize parameters δ̊ and δ̊−1
∗ from Sect. 1.7. However, the constants can be
chosen to be independent of ˚
whenever ˚
is sufficiently small (in the sense
−1
of Subsect. 1.8) relative to δ̊ and δ̊∗ .
1.10. The main sharp blow-up result in plane symmetry. In this section, we state
and prove Theorem 1.1, which is our main shock-formation result for solutions to equation
(1.4.7). Our characterization of the shock formation is based on the following quantity, which
characterizes the “worst-case” scenario for µ along Σut .
Definition 1.8.
µ? (t, u) := min{1, min
µ}.
u
Σt
(1.10.1)
Remark 1.20. It is redundant to take the min with 1 in (1.10.1) because µ ≡ 1 along P0
(when the data are supported in Σ10 ); we have done this only to emphasize that µ? (t, u) ≤ 1.
The theorem involves statements about the solution’s classical lifespan in the region of
interest, which is captured in the following definition.
Definition 1.9 (Right-traveling classical lifespan). We define T(Lif espan);U0 , the righttraveling classical lifespan of the solution with parameter U0 , to be the supremum over all
times t > 0 such that Ψ is a C 2 solution (relative to the rectangular coordinates (t, x)) to
equation (1.4.7) in the strip Mt,U0 .
Theorem 1.1 (A sharp description of the plane symmetric solution’s right-traveling classical lifespan). Let (Ψ̊, Ψ̊0 ) ∈ (C 2 × C 1 ) be initial data for equation (1.4.7)
supported in Σ10 and let U0 be the parameter defined in (1.3.2). If ˚
is sufficiently small
−1
relative to δ̊∗ and δ̊ (in the sense of Subsect. 1.9), then the following conclusions hold for
(t, u) ∈ [0, min{T(Lif espan);U0 , 2δ̊−1
∗ }] × [0, U0 ].
Part I. One of the following possibilities (not necessarily disjoint) must occur: either
T(Lif espan);U0 = sup{t > 0 | inf µ > 0},
(1.10.2a)
T(Lif espan);U0 > 2δ̊−1
∗ .
(1.10.2b)
U
Σt 0
or
Stable Shock Formation
20
Part II. There exists a constant C > 0 (see Subsect. 1.9) such that the following estimates
hold for (t, u) ∈ [0, min{T(Lif espan);U0 , 2δ̊−1
∗ }] × [0, 1]:
kΨkC 0 (Σut ) , kLΨkC 0 (Σut ) ≤ C˚
,
,
LL̆Ψ 0 u , L̆LΨ 0 u ≤ C˚
C (Σt )
C (Σt )
L̆Ψ 0 u = L̆Ψ
C (Σt )
(1.10.3a)
(1.10.3b)
C 0 (Σu
0)
+ O(˚
),
kµkC 0 (Σut ) ≤ kµkC 0 (Σu ) + kLµkC 0 (Σu ) t + O(˚
),
0
0
),
kLµkC 0 (Σut ) = kLµkC 0 (Σu ) + O(˚
0
1
µ(t, u) = 1 − L̆Ψ(0, u)t + O(˚
),
4
1
),
µ(t, u) = 1 − L̆Ψ(t, u)t + O(˚
4
µ? (t, 1) = 1 − δ̊∗ t + O(˚
).
(1.10.3c)
(1.10.4a)
(1.10.4b)
(1.10.5a)
(1.10.5b)
(1.10.5c)
Moreover,
1
µ? (t, u) < 1/4 =⇒ Lµ < − δ̊∗ ,
4
µ? (t, u) < 1/4 =⇒ L̆Ψ(t, u) > δ̊∗ .
(1.10.6a)
(1.10.6b)
Part III. T(Lif espan);1 < 2δ̊−1
∗ and in fact, we have
T(Lif espan);1 = {1 + O(˚
)} δ̊−1
∗ .
(1.10.7)
In addition, along ΣT(Lif espan);1 , LΨ blows up precisely on the subset
ΣBlow−up
T(Lif espan);1 := {(T(Lif espan);1 , u) | µ(T(Lif espan);1 , u) = 0}.
(1.10.8)
26
In addition, for each p ∈ ΣBlow−up
Ωp containing p such
T(Lif espan);1 , there exists a past neighborhood
that
δ̊∗
LΨ(q) >
,
q ∈ Ωp .
(1.10.9)
µ(q)
Remark 1.21. The factor of 2 found in the product 2δ̊−1
∗ (see, for example, (1.10.2b)) is
not important and could be replaced with any positive constant larger than 1, but we would
have to further shrink the allowable size of ˚
as the size of the constant increases.
Remark 1.22 (Point of no return for µ). The estimate (1.10.6a) is a quantified version
of the following rough idea: the only way µ can shrink along the integral curves of L is for Lµ
to be significantly negative. An interesting consequence of (1.10.6a) is the following “point
of no return” implication: once µ ≤ 41 , (recall that at t = 0, its value is approximately 1),
µ must continue to shrink along the corresponding integral curves of L until it eventually
26That
is, a neighborhood intersected with the half-space {t ≤ T(Lif espan);1 }.
J. Speck
21
1
vanishes at some point and a shock forms or until t > 2δ̊−1
∗ . The specific value 4 is not
significant: the actual point of no return depends on the smallness of ˚
and 14 is just a
convenient number.
Remark 1.23 (The important role played by (1.10.6a)). Away from plane symmetry,
the estimate (1.10.6a) plays a critical role in allowing us to control certain error integrals
that appear in the energy estimates; see Remark 3.5.
Proof of Theorem 1.1. Throughout we assume that ˚
is small in the sense of Subsect. 1.9
without being explicit about the smallness. To prove P artI and P artII, it suffices to to
derive the estimates (1.10.3a)-(1.10.6b) on MT(Lif espan) ,U0 ∩ M2δ̊−1
. For by (1.6.1), if µ
∗ ,U0
remains uniformly bounded from above and from below away from 0, then the estimates
(1.10.3a)-(1.10.5c) imply that |Ψ|, |∂t Ψ|, and |∂x Ψ| remain uniformly bounded; it is a standard exercise to show that such bounds allow us to extend the solution’s lifespan (in a strip
of u-width U0 ).
Remark 1.24. As we described in Subsect. 1.9, we allow constants C to depend on functions
−1
of δ̊−1
∗ . In particular, since since we are restricting our attention to times t < 2δ̊∗ , we can
incorporate functions of t such as Ct into C. In particular, Ct = O(1) on the time interval
of interest.
The ensuing analysis is based on integrating along characteristics, and we will work relative
to the geometric coordinate system (t, u) of Def. 1.3. To obtain
the desired estimates, we use
a bootstrap continuity argument: let B ⊂ 0, T(Lif espan);U0 be the subset consisting of those
times T such that
√the estimates (1.10.3a)-(1.10.6b) of the theorem hold on MT,U0 , but with
C˚
replaced by ˚
. It is a standard exercise to show that for
sufficiently small, B is a
˚
connected, non-empty,
relatively
closed
subset
of
0,
T
.
To
show that B is relatively
(Lif espan)
open (and thus B = 0, T(Lif espan) ), we improve the bootstrap assumptions with a series of
estimates that we now derive.
Remark 1.25. In plane symmetry, all estimates can be derived with the help of the bootstrap arguments given below and without the need to invoke Gronwall’s inequality, except in
the “short” u−direction. In contrast, away from plane symmetry, we are forced to treat the
most difficult commutator terms in the energy estimates with a rather degenerate Gronwall
estimate in the t direction (see Lemma 3.7 and the proof sketch of inequality (3.4.18)). The
resulting high-order energy estimates are allowed to blow-up like a power of µ−1
? near the
shock, which introduces severe complications into the analysis.
To proceed, we use equation (1.6.18b) and the bootstrap assumptions to obtain
(1.10.10)
L̆LΨ ≤ C |LΨ| .
Using (1.6.15b) we see that if we parametrize the integral curves (flow lines) of L̆ by u,
d
then we can view L̆ = 2 du
along the integral curves. Hence, we may follow the flow lines
backwards until they intersect P0 (along which LΨ ≡ 0) or Σ10 (along which LΨ = O(˚
) by
our small data assumption). In either case, we may apply Gronwall’s inequality and use the
Stable Shock Formation
22
shortness of the interval of u values to obtain27
|LΨ| ≤ C˚
.
We then integrate (1.10.11) along the integral curves of L =
assumption kΨkC 0 (Σ10 ) ≤ ˚
to obtain
(1.10.11)
∂
∂t
and use the small data
|Ψ| ≤ C˚
.
(1.10.12)
We have thus proved (1.10.3a).
Inserting the estimates (1.10.11) and (1.10.12) into the evolution equation (1.6.18a) and
using the bootstrap assumptions, we derive
,
(1.10.13)
LL̆Ψ ≤ C˚
which is half of (1.10.3b). To obtain the other half, we write, with the help of (1.6.16) and
1
1
µLΨ + L̆Ψ LΨ. From this identity, (1.10.11), (1.10.13),
(1.6.17), L̆LΨ = LL̆Ψ+
4 (1 + Ψ)
and the bootstrap assumptions, we obtain the desired bound
.
(1.10.14)
L̆LΨ ≤ C˚
Next, we integrate (1.10.13) along the integral curves of L =
∂
∂t
and obtain
L̆Ψ(t, u) = L̆Ψ(0, u) + O(˚
),
).
L̆Ψ 0 u = L̆Ψ 0 u + O(˚
C (Σt )
(1.10.15a)
(1.10.15b)
C (Σ0 )
In particular, we have derived (1.10.3c).
Inserting the estimates (1.10.11), (1.10.12), and (1.10.15a) into (1.6.17) and using the
bootstrap assumptions, we deduce that
1
Lµ(t, u) = − L̆Ψ(t, u) + O(˚
)
(1.10.16)
4
1
).
= − L̆Ψ(0, u) + O(˚
4
Integrating the second equality in (1.10.16) along the integral curves of L, using (1.6.11),
and using the small data assumption kΨkC 0 (Σ10 ) ≤ ˚
, we deduce
1
µ(t, u) = 1 − L̆Ψ(0, u)t + O(˚
).
4
We have thus proved (1.10.5a). From (1.10.16) and (1.10.17), we deduce
1
µ(t, u) = 1 − L̆Ψ(t, u)t + O(˚
).
4
We have thus proved (1.10.5b).
27Note
(1.10.17)
(1.10.18)
that this argument can be extended to show that in the region MT(Lif espan);1 ,1 , LΨ is identically 0
to the future of the integral curve of L̆ emanating from the point with geometric coordinates (t, u) = (0, 0).
However, we avoid relying on this fact in the proof; we want to use arguments that are connected to some
of the analysis outside of exact plane symmetry, where the statement does not generally hold true.
J. Speck
23
From (1.10.17) and Defs. 1.7 and 1.8, we deduce
µ? (t, 1) = 1 − δ̊∗ t + O(˚
).
(1.10.19)
We have thus obtained (1.10.5c).
(1.10.6b) is an easy consequence of (1.10.5b) and the assumption t < 2δ̊−1
∗ .
(1.10.6a) is an easy consequence of the first equality in (1.10.16) and (1.10.6b).
We have thus obtained P artsI and II.
To derive (1.10.7), we note that by (1.10.2a)-(1.10.2b), we must show that µ? (·, 1) remains
positive until it vanishes at the time {1 + O(˚
)} δ̊−1
∗ . This fact is a simple consequence of
(1.10.5c).
(1.10.8) and (1.10.9) are simple consequences of definition (1.6.13) and (1.10.6b).
We have thus proved the theorem.
2. Beyond Plane Symmetry: The Geometric Setup
In this section, we first overview the proof of shock formation in the nearly plane symmetric
regime. Starting in Subsect. 2.6, we construct the necessary geometric quantities that play a
role in the proof. We also describe our size assumptions on the initial data and overview the
bootstrap assumptions that we use in our analysis. These constructions serve as preliminary
steps in the main new ingredient needed to close the proof away from plane symmetry: energy
estimates. We dedicate all of Sect. 3 to providing an overview of the energy estimates.
2.1. Why is the proof of shock formation so much harder away from plane symmetry? The short answer is simply this: because the plane symmetric problem is effectively
one-dimensional and therefore one can rely exclusively on the method of characteristics,
which is in itself insufficient in two or more space dimensions due to the need to derive
energy estimates in order to propagate the regularity of the initial data.
One might try to attack the problem away from plane symmetry by deriving energy
estimates for the solution and its higher rectangular partial derivatives ∂α and by showing
that the ∂2 derivatives of Ψ are negligible. The problem with such a strategy is that it is
not typically true near the shock ! The reason is that the ∂2 vectorfield, when decomposed
relative to an appropriate frame adapted to the true characteristics, generally contains a
small component that transversal to the characteristics (in contrast to our study of the
model problem in plane symmetry, where ∂2 is tangent to the characteristics). As we have
seen in our study of the plane symmetric case, it is exactly the transversal derivatives of the
solution that blow up at the shock.
Thus, the only hope of closing the estimates away from plane symmetry is
to construct the true characteristics, tied to an eikonal function u, and to try
to show that u serves as a good coordinate, as it did in plane symmetry. As we
explained in Subsect. 1.4, away from plane symmetry, we construct u by solving the the
eikonal equation, that is, the hyperbolic PDE (g −1 )αβ (Ψ)∂α u∂β u = 0 (see (1.6.7)).
2.2. An overview of the proof of shock formation away from plane symmetry.
Stable Shock Formation
24
2.2.1. Features in common with the plane symmetric case. In our study of nearly plane
symmetric solutions, we will treat a regime of initial data that is a small perturbation of
the regime we treated in plane symmetry; see Subsubsect. 2.4 for the details. Under such
initial conditions, we will derive non-degenerate L∞ estimates like the ones (1.10.3a)-(1.10.3c)
obtained in the proof of Theorem 1.1, but supplemented with similar L∞ −smallness-type
estimates for the derivatives of Ψ in a geometric torus direction (representing a slight breaking
of plane symmetry). Unlike in the case of plane symmetry, the derivation these estimates
relies on energy estimates and Sobolev embedding, as we discuss in Subsubsect. 2.2.2.
Once we have obtained the non-degenerate L∞ estimates, the proofs that µ → 0 and
that a shock forms essentially parallel the proofs given in Theorem 1.1 in plane symmetry;
the non-degenerate L∞ estimates imply that the additional terms present away from plane
symmetry make only a negligible contribution to the estimates proved in that theorem. For
this reason, in the nearly plane symmetric case, we do not provide a detailed theorem along
the lines of Theorem 1.1. Instead, we focus on obtaining the energy estimates that lead to
the non-degenerate L∞ estimates.
2.2.2. New difficulties. We now outline some of the new difficulties encountered and the key
ingredients needed to overcome them when extending the proof of shock formation from the
plane symmetric case to the near-plane symmetric one. A lot of the strategy is based on the
framework developed by Christodoulou in [3].
(1) (Dynamic geometric objects, dependent on the solution) As in the proof of
the stability of Minkowski spacetime [5], the proof of shock formation uses a true
outgoing eikonal function u corresponding to the dynamic metric g = g(Ψ) and a
collection of vectorfields dynamically adapted to it.
(2) (Inverse foliation density and shock formation) As in the case of plane symmetry symmetry, shock formation is caused by the degeneracy of u as measured by the
density of its level surfaces relative to the Minkowskian time coordinate t, captured
by the inverse foliation density µ going to 0 in finite time (see Def. 2.4 below).
(3) (A sharp classical lifespan theorem relative to a rescaled frame) At the heart
of the approach lies a sharp classical lifespan result according to which for a sufficiently long time, longer than the expected blow-up time, solutions cannot blow up
unless µ vanishes. That is, one would like to obtain analogs of the relations (1.10.2a)(1.10.6b) derived in plane symmetry. To derive such results, one needs to re-express
the evolution equations as a coupled system between the nonlinear wave equation, expressed relative to a µ-rescaled vectorfield frame, together with a nonlinear transport
equation describing the evolution of the eikonal function u (and hence, by extension,
of µ). In this formulation, the µ-rescaled wave equation does not exhibit any dangerous Riccati-type quadratic term. This is analogous to the absence of the term
(L̆Ψ)2 from RHS (1.6.18a) in the case of plane symmetry (see also Exercise 1.13 and
equation (2.19.3) ). To prove the desired sharp classical lifespan result, we need to
show that the lower-order derivatives of the solution, relative to the rescaled frame,
remain bounded (in L∞ ) all the way up the shock, as they did in Theorem 1.1. To
establish such non-degenerate L∞ estimates, one can no longer rely exclusively on
the method of characteristics. The method of characteristics must be complemented
with appropriate energy estimates for derivatives of the solution with respect to the
J. Speck
25
u-adapted vectorfields. The main technical difficulty one needs to overcome is that
the energy norms of the highest derivatives can degenerate with respect to powers
of µ−1 , as we discuss in point (4). Despite this high-order degeneracy, one needs to
show that the low-order energies remain bounded all the way up to the shock in order
to recover the non-degenerate L∞ estimates via Sobolev embedding.
(4) (Generalized energy estimates) To establish the desired energy-type estimates,
we need to commute the wave equation a large number of times with the adapted
vectorfields,28 a procedure which not only generates a huge number of error terms,
but also seems to lead to a loss of derivatives. To overcome this apparent loss
of derivatives at the top order (which is explained in more detail in Subsect. 2.3), one
uses renormalizations,29 in the spirit30 of [5]. The price one pays for renormalizing is
the introduction of a factor of µ−1 into the top-order energy identities. This leads to
degenerate high-order L2 estimates that are allowed to blow up like a power of µ−1 as
µ → 0; see Prop. 3.6 and Subsubsect. 3.4.3. Establishing these degenerate high-order
L2 estimates and showing that the degeneracy does not propagate down to the lower
levels are the main new advances afforded by Christodoulou’s framework [3].
2.3. A preview on the vectorfield method tied to an eikonal function u. Following
the strategy described in SubSubsect. 2.2, we aim to derive energy estimates that remain
regular at the lower derivative levels, all the way up to the shock. This will allow us to recover
the non-degenerate L∞ estimates that one actually needs to show that the shock forms, as
in our proof of Theorem 1.1. To derive the energy estimates, we use a geometric version
of the energy method known as the vectorfield method. The vectorfield method has two
components, the multiplier method, which we explain in Subsect. 3.2, and the commutator
method, which we now overview.
Put simply, the commutator method is commuting the wave equation with well-chosen
vectorfields in an effort to obtain useful energy estimates for the derivatives of the solution. In
the problem of shock formation, we must rely on vectorfields Z that are adapted to an eikonal
function and that have good commuting properties with the covariant wave operator g(Ψ) .
We cannot expect that the commutators [g(Ψ) , Z] vanish because vanishing commutators
only occur for metrics with Killing fields31; we can only hope that the good properties of the
Z lead to controllable error terms.
To illustrate the main difficulties in obtaining energy estimates, we note the following
general formula32 for the commutator between g and an arbitrary vectorfield Z:
g (ZΨ) = Z(g Ψ) + (Z)π · D 2 Ψ + D (Z)π · DΨ,
(2.3.1)
28 Christodoulou did not give explicit bounds on the number of commutations needed to close the estimates
in [3]. In [12], the authors used 18 commutations. This may be further optimized.
29In n space dimensions with n ≥ 3, one also needs and n − 1−dimensional elliptic estimates.
30
In his work [3], Christodoulou recognizes that similar renormalization procedures can be done in the
context of nonlinear wave equations of type (1.3.3a). A similar observation had previously been used in [19]
in the context of quasilinear wave equations similar to (2.6.5) to derive a low regularity local well-posedness
result.
31A vectorfield V is a Killing field of g if L g = 0.
V
32As we explain in Remark 2.14, to obtain good estimates in the shock-formation problem, it is better to
commute vectorfields through a weighted version of g .
Stable Shock Formation
26
where D denotes the Levi-Civita connection corresponding to the metric g and
the deformation tensor of the vectorfield Z, that is,
π αβ := LZ gαβ = Dα Zβ + Dβ Zα ,
(Z)
(Z)
π denotes
(2.3.2)
where LZ denotes Lie differentiation with respect to Z. The term D (Z)π · DΨ schematically
denotes tensorial products between first covariant derivatives of (Z)π and the first derivatives
of Ψ, and similarly for the term (Z)π · D 2 Ψ.
In SubSubsect. 2.15, we will describe the commutator vectorfields Z needed in the shockformation problem. For illustrative purposes, we discuss here only one of them: the geometric
torus vectorfield Y (see Def. 2.27), which can be viewed as a version of ∂2 that is adapted
to the true characteristics. To construct Y , we will project, using the metric g and the
eikonal function, ∂2 onto the (generally not straight) one-dimensional curves33 `t,u generated
by the intersection of the level sets of u with the constant-time hypersurfaces Σt . The
projection operator can be constructed with the help of the null geodesic gradient vectorfield
L(Geo) := −(g −1 )αβ (Ψ)∂α u∂β corresponding to the eikonal function. Thus, the projection
operator depends on Ψ and the first rectangular derivatives of u. It is then easy to see that
the deformation tensor (Y )π depends on the first derivatives of Ψ and the Hessian H := D 2 u.
Therefore, the term D (Y )π appearing on the right-hand side of the equation
g(Ψ) (Y Ψ) = Y (g(Ψ) Ψ) + (Y )π · D 2 Ψ + D (Y )π · DΨ
(2.3.3)
depends on the second derivatives of Ψ and the third derivatives of u. Hence, to close L2
estimates at a consistent level of derivatives, we need to make sure that we can estimate the
third derivatives of u in terms of two derivatives of Ψ. Note that the eikonal equation (1.6.7)
shows that u depends on Ψ. At first glance of the eikonal equation (1.6.7), one might believe
in the heuristic relationship ∂u ∼ Ψ and hence ∂ 3 u ∼ ∂ 2 Ψ, which is the desired degree of
differentiability. However, as we explain below, only a weakened version, just barely sufficient
for our purposes, of these relationships is true. Furthermore, the weakened version is quite
difficult to prove.
To flesh out the difficulty, we first note that one can derive a Riccati-type matrix evolution
equation for H of the schematic form
L(Geo) H + H 2 = R,
(2.3.4)
where R depends34 on up-to-second-order derivatives of Ψ and up-to-second-order derivatives
of u. Ignoring for now the Riccati-type term H 2 , which actually plays a crucial role in the
blow-up mechanism, we note that the obvious way to estimate H is by integrating the
curvature term R along the integral curves of L(Geo) . The obstacle is that this argument
only allows one to conclude that H has the same degree of differentiability, in directions
transversal to L(Geo) , as R. In particular, using this argument, we can only estimate H in
terms of two derivatives of Ψ, which makes D (Y )π dependent on three derivatives of Ψ. Thus,
the term D (Y )π · DΨ is far from being a lower-order term as one would hope. In fact, it
seems to be an above-top-order term that loses a derivative relative to Ψ and thus obstructs
closure of the estimates. This appears to make equation (2.3.3) useless and casts doubt on
33Actually,
34R
L(Geo) .
the `t,u are diffeomorphic to T.
is in fact the Riemann curvature tensor of the metric g(Ψ) contracted twice with the vectorfield
J. Speck
27
the desired differentiability ∂ 3 u ∼ ∂ 2 Ψ. As we explain in Subsubsect. 3.4.3, the derivative
loss can be overcome by carefully exploiting some special tensorial structures present in the
components of D (Y )π and the components of equation (2.3.4) and, in more than two space
dimensions, by using elliptic estimates. Many of these special structures are closely tied
to the fact that our commutation vectorfields Z are adapted to the eikonal function u; see
Remark 3.4.
2.4. Overview of the regime of initial conditions that we treat. We recall that Pu
denotes a level set of the eikonal function. The main idea of our work is to prove shock
formation for solutions Ψ with initial data that satisfy the following size assumptions: the
pure Pu −transversal derivatives of Ψ are of size ≈ δ̊ > 0, while Ψ and any derivative of Ψ
involving at least one Pu −tangent direction is of small size ˚
. The quantity δ̊ can be either
small or large, but our required smallness of ˚
depends on δ̊; see Sects. 2.21 and 2.23 for
the precise assumptions. This is similar to the size assumptions we made in plane symmetry
in our proof of Theorem 1.1, but we now allow for a small ˚
−size variation of Ψ along
the torus direction T (which in particular allows for a slight breaking of the assumption
of plane symmetry). To avoid lengthening the notes, we generally do not closely track the
dependence of our estimates on δ̊. In particular, we allow the “constants” C appearing in
the estimates to depend on δ̊. There is one crucially important exception: we carefully track
the dependence of a handful of important estimates on a quantity δ̊∗ , which is essentially the
same as the quantity (1.7.3) that we encountered in plane symmetry. In our proof, we show
that we can propagate the ˚
− δ̊ hierarchy (in various norms) all the way up to the time of
first shock formation, which, as in our proof of Theorem 1.1 in plane symmetry, we show
is {1 + O(˚
)} δ̊−1
∗ . As in Theorem 1.1, when proving estimates via a bootstrap argument,
we give ourselves a margin of error by showing that we could propagate the hierarchy for
classical solutions existing up to time 2δ̊−1
∗ , which is plenty of time for the shock to form.
Actually, our results show something stronger: no other singularities besides shocks can form
for times ≤ 2δ̊−1
∗ . The factor of 2 in the previous inequality is not important and could be
replaced with any positive constant larger than 1, but we would have to further shrink the
allowable size of ˚
as the size of the constant increases.
2.5. Notational conventions and shorthand notation. Before launching into some of
the details of the geometric constructions that we need away from plane symmetry, we first
summarize some of our notational conventions; the precise definitions of some of the concepts
referred to here are provided later in the notes.
• Lowercase Greek indices µ, ν, etc. correspond to components with respect to the
rectangular spacetime coordinates x0 , x1 , x2 , and lowercase Latin indices i, j, etc.
correspond to components with respect to the rectangular spatial coordinates x1 , x2 .
That is, lowercase Greek indices vary over 0, 1, 2 and lowercase Latin indices vary over
1, 2. All lowercase Greek indices are lowered and raised with the spacetime metric g
and its inverse g −1 , and not with the Minkowski metric.
• We use Einstein’s summation convention in that repeated indices are summed over
their respective ranges.
Stable Shock Formation
28
• We sometimes use · to denote the natural contraction between two tensors (and
thus raising or lowering indices with a metric is not needed). For example, if ξ is a
spacetime one-form and V is a spacetime vectorfield, then ξ · V := ξα V α .
• If ξ is a one-form and V is a vectorfield, then ξV := ξα V α . Similarly, if W is a
vectorfield, then WV := Wα V α = g(W, V ). We use similar notation when contracting
higher-order tensorfields against vectorfields. Similarly, if Γακβ are the rectangular
Christoffel symbols (2.6.6), then ΓU V W := U α V κ W β Γακβ .
• If ξ is an `t,u −tangent one-form (as defined in Subsect. 2.8), then ξ # denotes its
g/−dual vectorfield, where g/ is the
on `t,u by g. Simi-
Riemannian metric induced
0
#
larly, if ξ is a symmetric type 2 `t,u −tangent tensor, then ξ denotes the type 11
`t,u −tangent tensor formed by raising one index with g/−1 and ξ ## denotes the type
2
`t,u −tangent tensor formed by raising both indices with g/−1 .
0
• Unless otherwise indicated, all quantities in our estimates that are not explicitly
under an integral are viewed as functions of the geometric coordinates (t, u, ϑ) of
Def. 2.8. Unless otherwise indicated, quantities under integrals have the functional
dependence established below in Def. 2.24.
• If Q1 and Q2 are two operators, then [Q1 , Q2 ] = Q1 Q2 − Q2 Q1 denotes their commutator.
• A . B means that there exists C > 0 such that A ≤ CB.
• A = O(B) means that |A| . |B|.
• Constants such as C and c are free to vary from line to line. Explicit and implicit
constants are allowed to depend in an increasing, continuous fashion on
the data-size parameters δ̊ and δ̊−1
∗ from Sects. 1.8 (in the plane symmetry)
and 2.23 (in the nearly plane symmetric case) . However, the constants
can be chosen to be independent of the parameters ˚
and ε whenever ˚
and ε are sufficiently small relative to δ̊−1 and δ̊∗ .
2.6. Assumptions on the nonlinearities. We recall that we are studying Cauchy problems of the form
g(Ψ) Ψ = 0,
(2.6.1a)
(Ψ|Σ0 , ∂t Ψ|Σ0 ) = (Ψ̊, Ψ̊0 ),
(2.6.1b)
where (Ψ|Σ0 , ∂t Ψ|Σ0 ) = (Ψ̊, Ψ̊0 ) are compactly supported in Σ10 , g(Ψ) := (g −1 )αβ Dα Dβ denotes the covariant wave operator of the spacetime metric g, and D denotes the Levi-Civita
connection of g. To obtain a shock-formation result along the lines of Theorem 1.1, we must
make assumptions on the nonlinearities.
We assume that relative to the rectangular coordinates {xα } on R × Σ (with Σ = R × T)
introduced in Subsect. 1.2, we have
(Small)
gµν = gµν (Ψ) := mµν + gµν
(Ψ),
(µ, ν = 0, 1, 2),
(2.6.2)
where
mµν = diag(−1, 1, 1)
(2.6.3)
J. Speck
29
(Small)
is the standard Minkowski metric on R × Σ and gµν
with
(Ψ) is a given smooth function of Ψ
(Small)
(0) = 0.
gµν
(2.6.4)
Relative to the rectangular coordinates, (2.6.1a) takes the form35
(g −1 )αβ ∂α ∂β Ψ − (g −1 )αβ (g −1 )κλ Γακβ ∂λ Ψ = 0.
(2.6.5)
The Γακβ are the fully lowered Christoffel symbols36 of g relative to rectangular coordinates
and can be expressed as
1
Γακβ = Γακβ (Ψ, ∂Ψ) := {∂α gκβ + ∂β gακ − ∂κ gαβ }
(2.6.6)
2
1
= {Gκβ ∂α Ψ + Gακ ∂β Ψ − Gαβ ∂κ Ψ} ,
2
where
d
Gαβ = Gαβ (Ψ) :=
gαβ (Ψ).
(2.6.7)
dΨ
Exercise 2.1. Derive equation (2.6.5) and the last “=” in (2.6.6).
We now describe our assumptions on the tensorfield Gαβ (Ψ = 0), which can be viewed
as a 3 × 3 matrix with constant entries relative to rectangular coordinates. We could prove
the existence37 of stable shock-forming solutions whenever there exists a Minkowski-null
vectorfield L(F lat) (that is, mαβ Lα(F lat) Lβ(F lat) = 0) such that
Gαβ (Ψ = 0)Lα(F lat) Lβ(F lat) 6= 0.
(2.6.8)
The assumption (2.6.8) holds for most nonlinearities and is equivalent to the failure of
Klainerman’s classic null condition [16]. We will explain this fact in more detail at the
end of this section.
We recall that the main results that we present in these notes rely on the existence of a
family of plane symmetric shock-forming solutions. The existence of the family is based on
the following assumption: there exists a vectorfield L(F lat) ∈ span{∂t , ∂1 } such that (2.6.8)
holds. We may then perform a Lorentz transformation on the t, x1 coordinates if necessary
in order put L(F lat) into the following form, which we assume throughout the remainder of
these notes:
L(F lat) = ∂t + ∂1 .
(2.6.9)
Note that under the above assumptions, LHS (2.6.8) is equal to the non-zero constant
G00 (Ψ = 0) + 2G01 (Ψ = 0) + G11 (Ψ = 0).
35Alternatively,
one can express LHS (2.6.1a)
following well-known formula: relative to an
p using the
1
−1 αβ
√
arbitrary coordinate system, g Ψ =
∂α ( |detg|(g ) ∂β Ψ).
|detg |
36Our Christoffel symbol index conventions are such that for vectorfields V , we have D V β = ∂ V β +
α
α
Γαβ λ V λ , where Γαβ λ := (g −1 )βκ Γακλ .
37The condition (2.6.8) would be sufficient for allowing us to prove the existence of stable large-data
shock-forming solutions. However, in order to handle the set of data corresponding to the data treated in
Theorem 1.1 (which includes some small data), we need the additional assumption (2.6.9).
Stable Shock Formation
30
Remark 2.1 (Genuinely nonlinear systems). Our assumption that the vectorfield (2.6.9)
verifies (2.6.8) is reminiscent of the well-known genuine nonlinearity condition for first-order
strictly hyperbolic systems. In particular, for plane symmetric solutions with Ψ sufficiently
small, the assumption ensures that there are quadratic Riccati-type terms38 in the wave
equation, which is the main mechanism driving the singularity formation in the 2 × 2 strictly
hyperbolic genuinely nonlinear systems studied by Lax [20].
By rescaling the metric by the scalar function 1/(g −1 )00 (Ψ), we may assume without loss
of generality39 that
(g −1 )00 (Ψ) ≡ −1.
(2.6.10)
The assumption (2.6.10) simplifies many of our formulas. Note that (2.6.10) is satisfied by
the model metric (1.4.1) that we studied in the previous section.
Remark 2.2. In total, our assumptions on the nonlinearities imply the term 21 GLL X̆Ψ on
RHS (2.17.1), which lies at the heart of our analysis, is sufficiently strong to drive µ to 0 in
the regime under study.
We now return to the connection between the condition (2.6.8) and Klainerman’s classic
null condition. To flesh out the connection, we first Taylor expand the nonlinear terms in
equation (2.6.5) around (Ψ, ∂Ψ) = (0, 0) and denote the quadratic parts of the principal
and semilinear terms respectively by ΨAαβ ∂α ∂β Ψ and N αβ ∂α Ψ∂β Ψ, where the rectangular
components Aαβ and N αβ are constants.
Definition 2.1 (Klainerman’s classic null condition). We say that the nonlinearities in
equation (2.6.5) (which is written relative to the rectangular coordinates) verify the classic
null condition if for every covector ` = (`0 , `1 , `2 ) satisfying (m−1 )αβ `α `β := −`20 + `21 + `22 = 0
(with (m−1 )αβ = diag(−1, 1, 1) the standard inverse Minkowski metric), we have the following
identities involving the tensorfields Aαβ and N αβ from the previous paragraph:
Aαβ `α `β = N αβ `α `β = 0.
Remark 2.3. Def. 2.1 can be extended to equations of type (1.3.3a) and also to systems of
wave equations. The definition is slightly more cumbersome to state in these cases, so we do
not present it here; see [16, 18].
Exercise 2.2. Show that if the classic null condition does not hold for equation (2.6.5),
then there exists a Minkowski-null vector40 ` such that
0 6= Gαβ (Ψ = 0)`α `β ,
38The
(2.6.11)
vectorfield frame that we construct in fact leads to the cancellation of the Riccati-type term. We
already saw this in plane symmetry; see Exercise 1.13.
39Technically, rescaling the metric introduces a semilinear term proportional to (g −1 )αβ (Ψ)∂ Ψ∂ Ψ in
α
β
the covariant wave equation corresponding to the rescaled metric. However, this term makes only negligible
contribution to the dynamics due to its special null structure, and we therefore ignore it for simplicity.
Specifically, if one decomposes µ times this term (see Remark 2.14) relative to the rescaled frame (2.7.14a)
introduced below, then one finds that in each product, the transversal derivative X̆Ψ appears at most linearly.
That is, each product contains a Pu −tangent derivative of Ψ, which remains small all the way up to the
shock (under the smallness assumptions that we introduce in Sects. 2.21-2.23).
40That is, m `α `β = 0.
αβ
J. Speck
31
with Gαβ as in (2.6.7).
Next, under the assumption that the null condition fails, do the following. Extend the
vector ` from above to a vectorfield L(F lat) on R × Σ by setting, relative to the rectangular
coordinates, Lα(F lat) := `α (at all spacetime points). Assume that L(F lat) can and has been
brought into the form (2.6.9) via Lorentz transformations. Show that if one Taylor expands
the nonlinear terms in equation (2.6.5) around (Ψ, ∂Ψ) = (0, 0) to obtain their quadratic part
and then decomposes the quadratic part relative to the flat frame {L(F lat) := ∂t +∂1 , L(F lat) :=
∂t − ∂1 , ∂2 }, then there are non-zero terms in the decomposition that are proportional to
ΨL(F lat) L(F lat) Ψ and (L(F lat) Ψ)2 . Moreover, show that the coefficients of ΨL(F lat) L(F lat) Ψ
and (L(F lat) Ψ)2 are in fact proportional to Gαβ (Ψ = 0)Lα(F lat) Lβ(F lat) 6= 0.
Hence, in the context of shock formation, failure of the null condition is relevant because
it means that there is a term in the evolution equation for µ that is strong enough41 to drive
it to 0 in finite time; see42 Remark 2.2.
Exercise 2.3 (The null condition is exceptional). Show that for the equations g(Ψ) Ψ =
0, a necessary and sufficient condition for the nonlinearities to verify the classic null condition
of Def. 2.1 is that up to cubic terms, g(Ψ) = (1+f (Ψ))m, where m is the standard Minkowski
metric and f (0) = 0. Consequently, for these equations, the classic null condition is very
restrictive and is satisfied only in special cases.
2.7. The eikonal function and related constructions. We start by recalling the definition of the eikonal function in the present context.
Definition 2.2 (The eikonal function). The eikonal function u is the solution to the
following hyperbolic initial value problem:
(g −1 )αβ (Ψ)∂α u∂β u = 0,
∂t u > 0,
u|Σ0 = 1 − x1 .
(2.7.1a)
(2.7.1b)
In the following definition, we provide analogs of the spacetime subsets that we referred
to in Sect. 1 in our study of plane symmetric solutions.
Definition 2.3 (Subsets of spacetime). We define the following spacetime subsets, which
are depicted below in Figure 2:
Σt0 := {(t, x1 , x2 ) ∈ R × R × T | t = t0 },
u0
t0
Σ := {(t, x1 , x2 ) ∈ R × R × T | t = t0 , 0 ≤ u(t, x1 , x2 ) ≤ u0 },
t0
u0
P := {(t, x1 , x2 ) ∈ R × R × T | 0 ≤ t ≤ t0 , u(t, x1 , x2 ) = u0 },
t0
u0
u0
t0
`t0 ,u0 := P ∩ Σ = {(t, x1 , x2 ) ∈ R × R × T | t = t0 , u(t, x1 , x2 ) = u0 },
t0
Mt0 ,u0 := ∪u∈[0,u0 ] Pu ∩ {(t, x1 , x2 ) ∈ R × R × T | 0 ≤ t < t0 }.
41At
(2.7.2a)
(2.7.2b)
(2.7.2c)
(2.7.2d)
(2.7.2e)
least in the solution regime under consideration in these notes.
note here that the difference between the term GLL from Remark 2.2 and the term Gαβ (Ψ =
α
0)L(F lat) Lβ(F lat) tied to the failure of the null condition is a small error term in the solution regime under
study.
42We
Stable Shock Formation
32
We refer to the Σt and Σut as “constant time slices,” the Put as “null hyperplanes,” and the
`t,u as “curves.” Note that the condition (2.7.1b) implies that the `0,u are straight lines. For
t > 0, the `t,u are typically curves, except when u = 0. We sometimes use the notation Pu in
place of Put when we are not concerned with the truncation time t. We restrict our attention
to spacetime regions with 0 ≤ u ≤ U0 , where we recall (see (1.3.2)) that 0 < U0 ≤ 1 is a
parameter.
Σut
`t,u `t,0
Put
x2 ∈ T
`0,u
Σu0
P0t
Mt,u
`0,0
Ψ≡0
x1 ∈ R
Figure 2. The spacetime region and various subsets.
The following definition for the inverse foliation density is an analog of equation (1.6.10)
for the model wave equation (1.4.7).
Definition 2.4 (Inverse foliation density).
µ :=
−1
(g −1 )αβ (Ψ)∂α t∂β u
> 0.
(2.7.3)
As in plane symmetry, shock formation is exactly characterized by µ → 0.
Exercise 2.4. Show that in plane symmetry under the model metric (1.4.1), the quantity
µ from definition (1.6.10) coincides with the one from definition (2.7.3).
Many of our constructions are based on the following geodesic gradient vectorfield L(Geo) .
Definition 2.5 (Null geodesic vectorfield). We define
Lα(Geo) := −(g −1 )αβ ∂β u.
(2.7.4)
Exercise 2.5. Show that L(Geo) is g−null, that is, that
g(L(Geo) , L(Geo) ) = 0.
(2.7.5)
J. Speck
33
Also, using the basic property Dg = Dg −1 = 0, show that L(Geo) verifies the geodesic
equation:
DL(Geo) Lα(Geo) = 0.
(2.7.6)
Definition 2.6 (Rescaled null vectorfield). Relative to the rectangular coordinates, we
define the rescaled null (see (2.7.5)) vectorfield
Lα := µLα(Geo) = −µ(g −1 )αβ ∂β u.
(2.7.7)
g(L, L) = 0,
(2.7.8a)
Lu = 0,
(2.7.8b)
Lt = 1.
(2.7.8c)
Exercise 2.6. Show that
Remark 2.4 (We are studying a coupled system). Having introduced the eikonal
function, we now stress the following important point: the basic equations that we need to
study are the wave equation (2.6.1a) coupled to the nonlinear transport equations (2.7.6)
for the three rectangular components Lν(Geo) , which are consequences of the eikonal equation
(2.7.1a). In practice, it is convenient to replace the transport equations for Lν(Geo) with
the equivalent transport equations for µ and Li , (i = 1, 2) provided in Lemma (2.10) the
advantage is that unlike Lν(Geo) , µ and Li remain bounded in L∞ all the way up to the
shock (in the solution regime under study). We may therefore think of Ψ, µ, L1 , L2 as the
fundamental unknowns to be solved for.43 We remark that although the form µg(Ψ) Ψ = 0
(see Remark 2.14) of the wave equation is useful for deriving energy estimates, the equivalent
form (2.19.3) is fundamental for understanding the behavior of the rescaled quantities.
Definition 2.7 (Geometric torus coordinate ϑ and the corresponding vectorfield
Θ). Along Σ10 , we define ϑ(t = 0, x1 , x2 ) = x2 . We extend ϑ to regions of the form Mt,u by
solving the transport equation Lϑ = 0 with ϑ subject to the above initial conditions along
Σ10 .
∂
∂
We define Θ = ∂ϑ
:= ∂ϑ
|t,u to be the vectorfield corresponding to partial differentiation
with respect to ϑ at fixed t and u.
Definition 2.8 (Geometric coordinates). We refer to (t, u, ϑ) as the geometric coordinates.
Remark 2.5 (µ is connected to the Jacobian determinant of the change of variables
map). We note here another important role played by µ: it is not too difficult to show that
Υ
the Jacobian determinant of Υ, the change of variables map (t, u, ϑ) → (t, x1 , x2 ) from
geometric to rectangular coordinates, is proportional to µ; see Exercise 2.9.
In particular, in the solution regime under consideration one can show that the Jacobian
determinant det dΥ vanishes precisely at the points where µ vanishes. Hence, solutions that
are regular relative to the geometric coordinates can have rectangular derivatives that blow
up at the locations where µ vanishes because of the degeneracy of Υ.
We now define some additional vectorfields that we will use in our analysis of solutions.
43Recall
that L0 ≡ 1 by construction.
Stable Shock Formation
34
Definition 2.9 (X, X̆, and N ). We define X to be the unique vectorfield that is Σt −tangent,
g−orthogonal to the `t,u , and normalized by
g(L, X) = −1.
(2.7.9)
X̆ := µX.
(2.7.10)
N := L + X.
(2.7.11)
L := L + 2X.
(2.7.12)
L̆ := µL + 2X̆.
(2.7.13)
We define
We define
We define
We define
Remark 2.6. In our analysis away from plane symmetry, we will not often use the vectorfields L and L̆ defined in (2.7.12)-(2.7.13), which are analogs of the null vectorfields L and
L̆ from plane symmetry (see (1.6.1) and (1.6.13)). Instead, we will typically use X and X̆.
Even though X and X̆ are not null, they are transversal to the Pu , which is the property of
greatest relevance. The advantage of X and X̆ is that they are Σt −tangent, which is useful
in various parts of the analysis.
In addition to the geometric coordinates, we also use two vectorfield frames adapted to
the characteristics.
Definition 2.10 (Two frames). We define, respectively, the rescaled frame and the nonrescaled frame as follows:
{L, X̆, Θ},
Rescaled frame,
(2.7.14a)
{L, X, Θ},
Non-rescaled frame.
(2.7.14b)
J. Speck
35
µ small
L
Θ
X̆
L
Θ
X̆
P1t
µ≈1
Put
P0t
Ψ≡0
Figure 3. The rescaled vectorfield frame at two distinct points in Put , where
0 < u < 1.
Lemma 2.1 (Basic properties of X, X̆, L, N , L, and L̆). The following identities hold:
Lu = 0,
Lt = L0 = 1,
(2.7.15a)
X̆u = 1,
X̆t = X̆ 0 = 0,
(2.7.15b)
g(X, X) = 1,
g(L, X) = −1,
g(X̆, X̆) = µ2 ,
g(L, X̆) = −µ,
(2.7.16a)
(2.7.16b)
g(L, L) = g(L̆, L̆) = 0,
(2.7.16c)
g(L, L) = −2,
(2.7.16d)
g(L, L̆) = −2µ.
(2.7.16e)
Moreover, relative to the geometric coordinates, we have
∂
L= .
(2.7.17)
∂t
In addition, there exists an `t,u −tangent vectorfield Ξ = ξΘ (where ξ is a scalar function)
such that
∂
∂
X̆ =
−Ξ=
− ξΘ.
(2.7.18)
∂u
∂u
The vectorfield N defined in (2.7.11) is future-directed, g−orthogonal to Σt and is normalized by
g(N, N ) = −1.
(2.7.19)
Stable Shock Formation
36
Moreover, relative to rectangular coordinates, we have (for ν = 0, 1, 2):
N ν = −(g −1 )0ν .
(2.7.20)
Finally, the following identities hold relative to the rectangular coordinates (for ν = 0, 1, 2):
Xν = −Lν − δν0 ,
X ν = −Lν − (g −1 )0ν ,
(2.7.21)
where δν0 is the standard Kronecker delta.
Proof. We first prove (2.7.15a). We begin by using (2.7.1a), (2.7.4), and (2.7.7) to deduce
that Lu = Lα ∂α u = −µ(g −1 )αβ ∂α u∂β u = 0 as desired. The fact that Lt = 1 is a simple
consequence of (2.7.4), (2.7.3), and (2.7.7).
We now prove (2.7.15b). We begin by using (2.7.4), (2.7.3), (2.7.9), and (2.7.10) to deduce
that X̆u = µX α ∂α u = −X α Lα = −g(X, L) = 1. The fact that X̆t = 0 is an immediate
consequence of the fact that by construction, X̆ is Σt −tangent.
∂
and Θ span the tangent
(2.7.18) then follows easily from (2.7.15b) and the fact that ∂u
space of Σt at each point.
(2.7.16b) is an easy consequence of (2.7.9) and (2.7.10).
(2.7.16c)-(2.7.16e) are easy consequences of definitions (2.7.12)-(2.7.13), (2.7.16a)-(2.7.16b),
and the fact that L is g−null.
To derive the properties of N , we consider the vectorfield V ν := −(g −1 )0ν , which is g−dual
to the one-form with rectangular components −δν0 and therefore g−orthogonal to Σt . By
(2.6.10), g(V, V ) = (g −1 )αβ δα0 δβ0 = −1, so V is future-directed, timelike, and unit-length.
In particular, V belongs to the g−orthogonal of `t,u , a space spanned by {L, X}. Thus,
there exist scalars a, b such that V = aL + bX. Since V t = V 0 = 1 = Lt = L0 and since
Xt = X 0 = 0, we find that a = 1, that is, that V = L + bX. Taking the inner product of
this expression with X and using (2.7.16b) together with the fact that X is Σt −tangent (and
hence g−orthogonal to V ), we find that 0 = −1+bg(X, X). Similarly, using (2.7.16b) the fact
that L is null, and the previous identity, we compute that −1 = g(V, V ) = −2b+b2 g(X, X) =
−2b + b = −b. It follows that V = L + X := N and g(X, X) = 1. We have thus obtained
the properties of N and obtained (2.7.19), (2.7.20), and the first identity in (2.7.16a). The
second identity in (2.7.16a) follows easily from the first one and definition (2.7.10). (2.7.21)
follows from the definition (2.7.11) of N and from lowering the indices in (2.7.20) with g.
To obtain (2.7.17), we simply use (2.7.15a) and the fact that by construction, we have
Lϑ = 0 (see Def. 2.7).
2.8. Projections. A great deal of our analysis is based on projecting various tensorfields
onto the Σt and `t,u .
Definition 2.11 (Projection tensorfields). We define the Σt projection tensorfield Π and
the `t,u projection tensorfield Π
/ relative to rectangular coordinates as follows:
Πνµ := δνµ − Nν N µ = δνµ + δν0 Lµ + δν0 X µ ,
(2.8.1a)
Π
/ νµ := δνµ + Xν Lµ + Lν (Lµ + X µ ) = δνµ − δν0 Lµ + Lν X µ .
(2.8.1b)
J. Speck
37
Definition 2.12 (Projections of tensorfields). Given any spacetime tensorfield ξ, we
define its Σt projection Πξ and its `t,u projection Π
/ ξ as follows:
···e
µm
···µm
:= Πννe11 · · · Πννenn Πµeµ11 · · · Πµeµmm ξνeµe11···e
(Πξ)µν11···ν
νn ,
n
(2.8.2a)
···e
µm
···µm
/ µeµmm ξνeµe11···e
/ µeµ11 · · · Π
/ ννenn Π
:= Π
/ ννe11 · · · Π
(Π
/ ξ)µν11···ν
νn .
n
(2.8.2b)
We say that a spacetime tensorfield ξ is Σt −tangent (respectively `t,u −tangent) if Πξ = ξ
(respectively if Π
/ ξ = ξ).
Definition 2.13 (`t,u projection notation). If ξ is a spacetime tensor, then we define
/ξ := Π
/ ξ.
If ξ is a symmetric type
define
0
2
(2.8.3)
spacetime tensor and V is a spacetime vectorfield, then we
/ξ V := Π
/ (ξV ),
(2.8.4)
where ξV is the spacetime one-form with rectangular components ξαν V α , (ν = 0, 1, 2).
To obtain estimates for the higher derivatives of the solution, we work with projected Lie
derivatives. We now recall the standard definition of Lie derivatives.
···µm
is a type
Definition
2.14 (Lie derivatives). If V µ is a spacetime vectorfield and ξνµ11···ν
n
m
44
spacetime
tensorfield,
then
relative
to
the
arbitrary
coordinates,
the
Lie
derivative
of ξ
n
m
with respect to V is the type n spacetime tensorfield LV ξ with the following components:
···µm
···µm
LV ξνµ11···ν
:= V α ∂α ξνµ11···ν
−
n
n
m
X
a=1
···µa−1 αµa+1 ···µm
ξνµ11···ν
∂α V µa +
n
n
X
···µm
ξνµ11···ν
∂ V α.
b−1 ανb+1 ···νn νb
b=1
(2.8.5)
In addition, when V and W are both vectorfields, we often use the standard Lie bracket
notation [V, W ] := LV W .
It is a standard fact that Lie differentiation obeys the Leibniz rule as well as the Jacobitype identity
LV LW ξ − LW LV ξ = L[V,W ] ξ = LLV W ξ.
(2.8.6)
Moreover, it is a standard fact based on the torsion-free property45 of D that RHS (2.8.5) is
invariant upon replacing all coordinate partial derivatives ∂ with covariant derivatives D.
In our analysis, we will apply the Leibniz rule for Lie derivatives to contractions of tensor
products of `t,u −tensorfields. Due in part to the special properties (see, for example, Remark 2.15) of the vectorfields that we use to differentiate, the non-`t,u components of the
differentiated factor in the products typically cancel. This motivates the following definition
of projected Lie derivatives.
44It
is well-known that RHS (2.8.5) is coordinate invariant.
property is equivalent to the invariance of the Christoffel symbols (2.6.6) under interchanges of α
45This
and β.
Stable Shock Formation
38
Definition 2.15 (`t,u and Σt −projected Lie derivatives). Given a tensorfield ξ and a
vectorfield V , we define the Σt −projected Lie derivative LV ξ of ξ and the `t,u −projected Lie
derivative L
/V ξ of ξ as follows:
LV ξ := ΠLV ξ,
L
/V ξ := Π
/ LV ξ.
(2.8.7)
Many of our geometric formulas involve the geometric torus differential.
Definition 2.16 (Geometric torus differential). If f is a scalar function on `t,u , then
d/f := ∇
/f = Π
/ Df , where Df is the gradient one-form associated to f .
The above definition avoids potentially confusing notation such as ∇
/ Li by replacing it with
i
i
d/L ; the latter notation clarifies that L is to be viewed as a scalar rectangular component
function.
2.9. First and second fundamental forms and covariant differential operators. In
this section, we define the first and second fundamental forms and introduce various covariant
differential operators.
Definition 2.17 (First fundamental forms). We define the first fundamental form g of
Σt and the first fundamental form g/ of `t,u as follows:
g := Πg,
g/ := Π
/ g.
(2.9.1)
We define the corresponding inverse first fundamental forms by raising the indices with
g :
−1
(g −1 )µν := (g −1 )µα (g −1 )νβ g αβ ,
(g/−1 )µν := (g −1 )µα (g −1 )νβ g/αβ .
(2.9.2)
Note that g is the Riemannian metric on Σt induced by g and that g/ is the Riemannian
metric on `t,u induced by g. Moreover, a straightforward calculation shows that (g −1 )µα g αν =
Πνµ and (g/−1 )µα g/αν = Π
/ νµ .
Definition 2.18 (Differential operators associated to the metrics). We use the following notation for various differential operators associated to the spacetime metric g, the
Minkowski metric m, and the Riemannian metric g/ induced on the `t,u .
• D denotes the Levi-Civita connection of the spacetime metric g.
• ∇
/ denotes the Levi-Civita connection of g/.
• If ξ is an `t,u −tangent one-form, then div
/ ξ is the scalar-valued function div
/ ξ :=
−1
g/ · ∇
/ ξ.
• Similarly, if V is an `t,u −tangent vectorfield, then div
/ V := g/−1 · ∇
/ V[ , where V[ is the
one-form g/−dual to V .
• If ξ is a symmetric type 02 `t,u −tangent tensorfield, then div
/ ξ is the `t,u −tangent
−1
one-form div
/ ξ := g/ · ∇
/ ξ, where the two contraction indices in ∇
/ ξ correspond to the
operator ∇
/ and the first index of ξ.
Definition 2.19 (Covariant wave operators and Laplacians). We use the following
standard notation.
2
• g := (g −1 )αβ Dαβ
denotes the covariant wave operator corresponding to the spacetime metric g.
• ∆
/ := g/−1 · ∇
/ 2 denotes the covariant Laplacian corresponding to g/.
J. Speck
39
Definition 2.20 (Second fundamental
forms). We define the second fundamental form
0
k of Σt , which is a symmetric type 2 Σt −tangent tensorfield, by
1
k := LN g.
(2.9.3)
2
We define the null second fundamental form χ of `t,u , which is a symmetric type 02
`t,u −tangent tensorfield, by
1
/ g/.
(2.9.4)
χ := L
2 L
Exercise 2.7. Show that the following alternate expressions hold for the tensorfields from
Def. 2.20:
1
1
k = LN g,
χ= L
/ g.
(2.9.5)
2
2 L
We now provide some identities that we use later.
Lemma 2.2 (Alternate expressions for the second fundamental forms). We have
the following identities:
χΘΘ = g(DΘ L, Θ),
(2.9.6a)
/kXΘ = g(DΘ L, X).
(2.9.6b)
Proof. We prove only (2.9.6b) since the proof of (2.9.6a) is similar. Using (2.9.5), we compute
that 2k
/XΘ = (LN g)XΘ = (LN g)XΘ = g(DX N, Θ) + g(DΘ N, X). Since g(X, X) = 1 and
N = L + X, we see that g(DΘ N, X) = g(DΘ L, X). Thus, to complete the proof, we need
only to show that g(DX N, Θ) = g(DΘ L, X). To proceed, we note that since g(N, X) = 0 and
g(X, X) = 1, we have g(DΘ N, X) = −g(DΘ X, N ) = −g(DΘ X, L). Then since g(X, L) = −1,
we conclude that −g(DΘ X, L) = g(DΘ L, X) as desired.
2.10. Useful expressions for the null second fundamental form. We now provide
some identities for χ. The identities are useful for estimating its below-top-order derivatives.
Lemma 2.3 (Expressions for χ). We have the following identities:
1
χ = gab (d/La ) ⊗ d/xb + G
/ LΨ,
(2.10.1a)
2
1
trg/ χ = gab g/−1 · (d/La ) ⊗ d/xb + g/−1 · G
/ LΨ.
(2.10.1b)
2
where χ is the `t,u −tangent tensorfield defined by (2.9.4) υ is the metric component from
Def. 2.21.
Remark 2.7 (It is convenient to think of χ as an independent quantity). Note that
(2.10.1a) shows that χ is an auxiliary quantity expressible in terms of the frame derivatives
of Ψ and the frame derivatives of the rectangular components Li . However, because of its
importance, to be clarified below, it is convenient to think of χ as an independent quantity.
Proof. To prove (2.10.1a), we use (2.9.6a) and (2.6.6) to compute, relative to rectangular
coordinates, that χΘΘ = gab (ΘLa )Θb + ΓΘΘL = gab (ΘLa(Small) )Θb + 21 GΘΘ LΨ. Noting that
Θxb = Θ · d/xb , we easily conclude (2.10.1a). To deduce (2.10.1b), we simply take the g/−trace
of (2.10.1a).
Stable Shock Formation
40
2.11. Expressions for the metrics. In this section, we decompose g relative to the nonrescaled frame and relative to the geometric coordinates. We then provide expressions for
various induced volume, area, and length forms relative to the geometric coordinates and for
the change of variables map from geometric to rectangular coordinates.
Lemma 2.4 (Expressions for g and g −1 in terms of the non-rescaled frame). We
have the following identities:
gµν = −Lµ Lν − (Lµ Xν + Xµ Lν ) + g/µν ,
−1 µν
(g )
µ
ν
µ
ν
µ
ν
−1 µν
= −L L − (L X + X L ) + (g/ ) .
(2.11.1a)
(2.11.1b)
Proof. It suffices to prove (2.11.1a) since (2.11.1b) then follows from raising the indices of
(2.11.1b) with g −1 .
To verify the formula (2.11.1a), we contract each side against the rectangular coordinates
of pairs of elements of the frame {L, X, Θ} and check that both sides agree. This of course
requires that we know the inner products of all pairs of elements of the frame, some of
which follow from the basic properties of the frame vectorfields, and some of which were
established in Lemma 2.1. As an example, we note that contracting the LHS against Lµ Θν
yields g(L, Θ) = 0, while contracting the RHS yields −g(L, L)g(L, Θ) − g(L, L)g(X, Θ) −
g(X, L)g(L, Θ) + g/(L, Θ) = 0 + 0 + 0 + 0 = 0 as desired. As a second example, we note
that contracting the LHS against Lµ X ν yields g(L, X) = −1, while contracting the RHS
yields −g(L, L)g(L, X) − g(L, L)g(X, X) − g(X, L)g(L, X) + g/(L, X) = 0 + 0 − 1 + 0 = −1
as desired.
The following scalar function captures the `t,u part of g.
Definition 2.21 (The metric component υ). We define the scalar function υ > 0 by
υ 2 := g(Θ, Θ) = g/(Θ, Θ).
(2.11.2)
It follows that relative to the geometric coordinates, we have
g/−1 = υ −2 Θ ⊗ Θ.
(2.11.3)
Exercise 2.8 (Expressions for g and g −1 in terms of the geometric coordinate
frame). Show that relative to the geometric coordinate (t, u, ϑ), we have
g = −2µdt ⊗ du + µ2 du ⊗ du + υ 2 (dϑ + ξdu) ⊗ (dϑ + ξdu),
(2.11.4)
∂
∂
∂
∂
∂
∂
∂
∂
g −1 = − ⊗
− µ−1 ⊗
− µ−1
⊗
− µ−1 ξ ⊗ Θ − µ−1 ξΘ ⊗
+ υ −2 Θ ⊗ Θ,
∂t ∂t
∂t ∂u
∂u ∂t
∂t
∂t
(2.11.5)
where the scalar functions ξ and υ from above are defined respectively in (2.7.18) and
(2.11.2).
Exercise 2.9 (Basic properties of the change of variables map). Let Υ : [0, T ) ×
[0, U0 ] × T → Mt,u , Υ(t, u, ϑ) = (t, x1 , x2 ) be the change of variables map from geometric to
rectangular coordinates. Show that


1
0
0
0
1
2
∂Υ
∂(x , x , x )  1
L X̆ 1 + Ξ1 Θ1  .
:=
=
(2.11.6)
∂(t, u, ϑ)
∂(t, u, ϑ)
2
2
2
2
L X̆ + Ξ Θ
J. Speck
Then show that the Jacobian determinant of Υ can be expressed as
∂(x0 , x1 , x2 )
= µ(detg ij )−1/2 υ,
det
∂(t, u, ϑ)
41
(2.11.7)
where υ is the metric component from Def. 2.21 and (detg ij )−1/2 is a smooth function of Ψ in
a neighborhood of 0 with (detg ij )−1/2 (Ψ = 0) = 1. In (2.11.7), g is viewed as the Riemannian
metric on ΣUt 0 defined by (2.9.1) and detg ij is the determinant of the corresponding 2 × 2
matrix of components of g relative to the rectangular spatial coordinates.
Remark 2.8. Note the factor of µ on RHS (2.11.7), which is familiar from the earlier
Exercise 1.9 in plane symmetry.
2.12. Energy-momentum tensor. To derive energy
estimates, we rely on the energy0
momentum tensor Q, which is the symmetric type 2 tensor
1
Qµν = Qµν [Ψ] := Dµ ΨDν Ψ − gµν (g −1 )αβ Dα ΨDβ Ψ.
(2.12.1)
2
In the next lemma, we exhibit the basic divergence property of Q; we omit the proof,
which is a simple calculation.
Lemma 2.5 (Basic divergence property of Q). For solutions to µg Ψ = F, we have
µDα Qαν = FD ν Ψ.
(2.12.2)
Exercise 2.10. Establish Lemma 2.5
In the next lemma, we provide the components of Q relative to the rescaled frame.
Lemma 2.6 (The frame components of Q). The components of the energy-momentum
tensor Q relative to the rescaled frame {L, X̆, Θ} can be expressed as follows:
1
1
QLL [Ψ] = (LΨ)2 ,
QLX̆ [Ψ] = − µ(LΨ)2 + µ|∇
/ Ψ|2 ,
(2.12.3a)
2
2
1
1
QX̆ X̆ [Ψ] = µ2 (LΨ)2 + (X̆Ψ)2 + µ(LΨ)X̆Ψ − µ2 |∇
/ Ψ|2 ,
(2.12.3b)
2
2
Q
/ L [Ψ] = (LΨ)∇
/ Ψ,
Q
/ X̆ [Ψ] = (X̆Ψ)∇
/ Ψ,
(2.12.3c)
1
1
Q
/ [Ψ] = (LΨ)2 g/ + µ−1 (LΨ)(X̆Ψ)g/ + |∇
/ Ψ|2 g/.
(2.12.3d)
2
2
Proof. The lemma is a simple consequence of the formula (2.12.1) and the frame decompositions of g and g −1 provided by (2.11.1a) and (2.11.1b).
We derive our energy estimates with the help of the vectorfield T defined just below. We
present the details on how to use T to derive energy estimates in Sect. 3.2.1.
Definition 2.22 (The timelike multiplier vectorfield T ). We define the multiplier vectorfield T by
T := (1 + 2µ)L + 2X̆.
(2.12.4)
A simple calculation yields that g(T, T ) = −4µ(1 + µ). Thus, T is g−timelike whenever
µ > 0. This property is important because it leads to coercive energy identities.
Stable Shock Formation
42
2.13. The definition of a deformation tensor. The error terms in our energy estimates
involve the deformation tensors of various vectorfields.
Definition 2.23 (Deformation tensor of a vectorfield V ). If V is a spacetime vectorfield,
(V )
then
its deformation tensor π (relative to the spacetime metric g) is the symmetric type
0
tensor
2
παβ := LV gαβ = Dα Vβ + Dβ Vα ,
(V )
(2.13.1)
where the last equality in (2.13.1) is a standard consequence of (2.8.5) and the torsion-free
property of the connection D (see Footnote 45).
We define our geometric integrals in terms of length, area, and volume forms that remain
non-degenerate throughout the evolution, all the way up to the shock.
Definition 2.24 (Non-degenerate forms and related integrals). We define the length
form dλg/ on `t,u , the area form d$ on Σut , the area form d$ on Put , and the volume form d$
on Mt,u as follows (relative to the geometric coordinates):
dλg/ := υ(t, u, ϑ) dϑ,
d$ := dλg/ (t, u0 , ϑ)du0 ,
d$ := dλg/ (t0 , u0 , ϑ0 )dt0 ,
d$ := dλg/ (t0 , u0 , ϑ0 )du0 dt0 ,
where υ is the scalar function from Def. 2.21.
If f is a scalar function, then we define
Z
Z
f (t, u, ϑ) υ(t, u, ϑ)dϑ,
f dλg/ :=
ϑ∈T
`t,u
Z
Z u Z
f d$ :=
f (t, u0 , ϑ) υ(t, u0 , ϑ)dϑdu0 ,
Σu
t
u0 =0 ϑ∈T
Z t Z
Z
f d$ :=
t0 =0
Put
Z
Z
t
ϑ∈T
Z u
f (t0 , u, ϑ) υ(t0 , u, ϑ)dϑdt0 ,
Z
f d$ :=
Mt,u
t0 =0
u0 =0
f (t0 , u0 , ϑ) υ(t0 , u0 , ϑ)dϑdu0 dt0 .
(2.13.2)
(2.13.3a)
(2.13.3b)
(2.13.3c)
(2.13.3d)
ϑ∈T
Remark 2.9 (Canonical forms). The canonical forms from differential geometry that are
typically associated to g and g are respectively µd$ and µd$.
2.14. Norms. In our analysis, we primarily estimate scalar functions and `t,u −tangent tensorfields. We always use the metric g/ when taking the pointwise norm of `t,u −tangent
tensorfields, a concept which we make precise in the next definition.
···µm
Definition 2.25 (Pointwise norms). If ξνµ11···ν
`t,u −tangent tensor, then we
is a type m
n
n
define the norm |ξ| ≥ 0 by
···e
µm
···µm µ
ξνee11···e
|ξ|2 := g/µ1 µe1 · · · g/µm µem (g/−1 )ν1 νe1 · · · (g/−1 )νn νen ξνµ11···ν
νn .
n
Our analysis relies on the following L2 and L∞ norms.
(2.14.1)
J. Speck
43
Definition 2.26 (L2 and L∞ norms). In terms of the non-degenerate forms of Def. 2.24,
we define the following norms for `t,u −tangent tensorfields:
Z
Z
2
2
2
|ξ| dλg/ ,
kξkL2 (Σut ) :=
|ξ|2 d$,
kξkL2 (`t,u ) :=
(2.14.2a)
u
`t,u
Σt
Z
2
|ξ|2 d$,
kξkL2 (Put ) :=
Put
kξkL∞ (`t,u ) := ess supϑ∈T |ξ|(t, u, ϑ),
kξkL∞ (Σut ) := ess sup(u0 ,ϑ)∈[0,u]×T |ξ|(t, u0 , ϑ),
(2.14.2b)
kξkL∞ (Put ) := ess sup(t0 ,ϑ)∈[0,t]×T |ξ|(t0 , u, ϑ).
2.15. Commutation vectorfields. To obtain higher-order estimates for Ψ and the eikonal
function quantities along `t,u , we commute various evolution equations with an `t,u −tangent
vectorfield. A natural candidate commutator is the geometric coordinate partial derivative
vectorfield Θ, which solves the transport equation L
/L Θ = 0. In rectangular components, the
i
i
transport equation reads LΘ = Θ · d/L and thus Θi is one degree less differentiable than Li
in directions transversal to L. It turns out that this loss introduces technical complications
into the analysis. It is an open question whether or not it is possible to overcome those
complications. Hence, we choose to instead commute with a more convenient `t,u −tangent
vectorfield Y , which we obtain by projecting a rectangular coordinate vectorfield Y(F lat)
onto the `t,u . It turns out that Y has just enough regularity to allow us to close our energy
estimates at the top-order and the delicate argument needed to close leads to µ−1 −degenerate
top-order estimates; see Sects. 2.3 and 3.1.
Definition 2.27 (The vectorfields Y(F lat) and Y ). We define the rectangular components
of the Σt −tangent vectorfields Y(F lat) and Y as follows (i = 1, 2):
i
i
Y(F
lat) := δ2 ,
i
Y :=
a
Π
/ ai Y(F
lat)
(2.15.1)
=
Π
/ 2i ,
(2.15.2)
where Π
/ is the `t,u projection tensorfield defined in (2.8.1b).
To prove our main theorem, we commute the equations with the elements of the following
set of vectorfields.
Definition 2.28 (Commutation vectorfields). We define the commutation set Z as
follows:
Z := {L, X̆, Y },
(2.15.3)
where L, X̆, and Y are respectively defined by (2.7.7), (2.7.10), and (2.15.2).
We define the Pu −tangent commutation set P as follows:
P := {L, Y }.
(2.15.4)
The rectangular spatial components of L, X, and Y deviate from their flat values by a
small amount captured in the following definition.
Stable Shock Formation
44
Definition 2.29 (Perturbed part of the various vectorfields). For i = 1, 2, we define
the following scalar functions:
i
Y(Small)
:= Y i − δ2i .
i
:= X i + δ1i ,
X(Small)
Li(Small) := Li − δ1i ,
(2.15.5)
The vectorfields L, X, and Y in (2.15.5) are defined in (2.7.7), 2.9, and 2.27.
i
Remark 2.10. From (2.6.2), (2.6.4), (2.7.21), and (2.15.5), we have that X(Small)
= −Li(Small) −
(g −1 )0i , where (g −1 )0i (Ψ = 0) = 0. We will use this simple fact later on.
In the next lemma, we characterize the discrepancy between Y(F lat) and Y .
Lemma 2.7 (Decomposition of Y(F lat) ). We can decompose Y(F lat) into an `t,u −tangent
vectorfield and a vectorfield parallel to X as follows: since Y is `t,u −tangent, there exists a
scalar function ρ such that
i
i
i
Y(F
lat) = Y + ρX ,
i
Y(Small)
(2.15.6a)
i
= −ρX .
(2.15.6b)
Moreover, we have
(Small)
a
b
a
ρ = g(Y(F lat) , X) = gab Y(F
lat) X = g2a X = g21
2
X 1 − g22 X(Small)
.
(2.15.7)
Proof. The existence of the decomposition (2.15.6a) follows from the fact that by construction, Y(F lat) and Y differ only by a vectorfield that is parallel to X (because the `t,u projection
tensorfield Π
/ annihilates the X component of the Σt −tangent vectorfield Y(F lat) while preserving its `t,u −tangent component).
The expression (2.15.6b) then follows from definition (2.15.5) and (2.15.6a).
To obtain (2.15.7), we contract (2.15.6a) against Xi and use (2.6.2)-(2.6.4), the identities
a
Y Xa = 0 and X a Xa = 1, and definition (2.15.5)
2.16. Deformation tensor calculations. In this section, we provide the frame components
of the deformation tensors of the commutation vectorfields and the multiplier vectorfield.
Lemma 2.8 (The frame components of (Z)π). The following identities are verified by
the deformation tensors (see Def. 2.23) of the elements Z of the commutation set Z defined
in (2.15.3):
(X̆)
πLL = 0,
(X̆)
πX̆X = 2X̆µ,
(X̆)
πLX̆ = −X̆µ,
(X̆)
π
/L = −d/µ − 2ζ(T rans−Ψ) − 2µζ(T an−Ψ) ,
(X̆)
π
/X̆ = 0,
π
/ = −2µχ + 2k
/ (T rans−Ψ) + 2µk
/ (T an−Ψ) ,
(X̆)
(L)
πLL = 0,
(L)
π
/L = 0,
(L)
π
/ = 2χ,
(L)
(2.16.1a)
(2.16.1b)
(2.16.1c)
(L)
(2.16.2a)
π
/X̆ = d/µ + 2ζ(T rans−Ψ) + 2µζ(T an−Ψ) ,
(2.16.2b)
πX̆X = 2Lµ,
(L)
πLX̆ = −Lµ,
(2.16.2c)
J. Speck
(Y )
45
(Y )
πX̆X = 2Y µ,
πLX̆ = −Y µ,
(2.16.3a)
1
1
1
(Y )
/ · Y )LΨ + ρG
/ X LΨ + (G
/ L · Y )∇
/ Ψ − ρGLX ∇
/ Ψ − ρGXX ∇
/ Ψ,
π
/L = −χ · Y + (G
2
2
2
(2.16.3b)
1
(Y )
π
/X̆ = µχ · Y + ρd/µ + ρG
/ X X̆Ψ − µρGXX ∇
/Ψ
(2.16.3c)
2
1
− µ(G
/ · Y )LΨ + µ(G
/ L · Y )∇
/ Ψ + µ(G
/ X · Y )∇
/ Ψ,
2
1
1
(Y )
π
/ = 2ρχ + (G
/ ·Y)⊗∇
/Ψ + ∇
/ Ψ ⊗ (G
/ · Y ) − ρG
/ LΨ
(2.16.3d)
2
2
+ ρG
/L ⊗ ∇
/ Ψ + ρ∇
/Ψ ⊗ G
/ L + ρG
/X ⊗ ∇
/ Ψ + ρ∇
/Ψ ⊗ G
/X .
πLL = 0,
(Y )
The scalar function ρ from above is as in Lemma 2.7, while the `t,u −tangent tensorfields
χ, ζ(T rans−Ψ) , /k (T rans−Ψ) , ζ(T an−Ψ) , and /k (T an−Ψ) from above are as in (2.9.4), (2.18.4a),
(2.18.4b), (2.18.5a), and (2.18.5a).
Remark 2.11 (Some key properties of the commutation vectorfield deformation
tensor components). Note that (Z)πLL = 0 for all three vectorfields Z ∈ Z . This property
is possible only because we have used a true eikonal function to construct the elements of
Z . The property is among the most important of all properties of the Z, for it leads
to the exact cancellation of terms on RHS (2.20.1) of the commuted wave equation that
are proportional to µ−1(Z)πLL X̆ X̆Ψ; if (Z)πLL were non-zero, then the factor of µ−1 in the
previous expression would introduce singular terms into the commuted wave equation at low
derivative levels. In turn, this would prevent us from exhibiting the regular behavior of Ψ
at the low Z −derivative levels, which is crucial to the whole proof. For similar reasons, the
property (Z)πLX̆ + Zµ = 0 is also of crucial importance.
Discussion of the proof of Lemma 2.8. The full proof is based on a series of somewhat delicate computations that rely on Lemma 2.11, Lemma 2.4, the second formula in (2.13.1),
and the fact that Dg = 0. We refer the reader to [12, Lemma 2.18] for the detailed calculations. Here, as a warm-up, we show that (Y )πLL = 0. To this end, we use (2.13.1),
the fact that Dg = 0, to deduce that (Y )πLL = g(DL Y, L) = −gY, DL L). In the last step,
we used the relation g(Y, L) = 0, the fact that Dg = 0, and the Leibniz rule to obtain
g(DL Y, L)g(Y, DL L) = 0. Now from (2.18.2a), we obtain that DL L is proportional to L and
hence g−orthogonal to Y . The desired relation (Y )πLL = 0 thus follows.
In the next lemma, we provide the frame components of (T )π. These are important for our
energy estimates because (T )π appears in our fundamental energy-flux identity (see Prop. 3.4).
The proof is similar to that of Lemma 2.8 so we do not provide the details.
Lemma 2.9. [12, Lemma 3.3; The frame components of (T )π] The components of the
deformation tensor (T )π (see Def. 2.23) of the multiplier vectorfield (2.12.4) can be expressed
Stable Shock Formation
46
as follows relative to the rescaled frame:
(T )
πLL = 0,
(T )
πLX̆
(2.16.4a)
n
o
= − Lµ + 4µLµ + 2X̆µ ,
(T )
πX̆X = 2(1 + 2µ)Lµ,
(T )
π
/L = −2d/µ − 4 ζ(T rans−Ψ) + µζ(T an−Ψ) ,
(T )
π
/X̆ = d/µ + 2(1 + 2µ) ζ(T rans−Ψ) + µζ(T an−Ψ) ,
n
o
(T rans−Ψ)
(T an−Ψ)
(T )
π
/ = 2χ + 4 /k
+ µk
/
.
(2.16.4b)
(2.16.4c)
(2.16.4d)
(2.16.4e)
(2.16.4f)
The `t,u tensorfields χ, ζ(T rans−Ψ) , /k (T rans−Ψ) , ζ(T an−Ψ) , and /k (T an−Ψ) from above are as in
(2.9.4), (2.18.4a), (2.18.4b), (2.18.5a), and (2.18.5a).
2.17. Transport equations for µ and Li . We now derive a transport equation for µ,
analogous to the equation (1.6.17) in our analysis in plane symmetry. We also provide a
similar transport equation for the rectangular spatial46 components Li , where L is defined in
(2.7.7). These are the basic evolution equations that we use to estimate µ and Li below top
order. We note that in plane symmetry, it was not necessary to derive evolution equations
for the Li because of the availability of the explicit formula (1.6.1).
Lemma 2.10 (Transport equation for µ). The inverse foliation density µ (defined in
(2.7.3)) and the rectangular spatial components Li verify the following transport equations:
1
Lµ : ω = GLL X̆Ψ + µError,
2
LLi = Error,
GLL := Gαβ (Ψ)Lα Lβ ,
(2.17.1)
(2.17.2)
where the terms Error are small error terms involving products containing the factor LΨ or
∇
/ Ψ.
Proof. We prove only (2.17.1) since the proof of (2.17.2) is similar. We first observe that relative to the rectangular coordinates, we have µ = 1/L0(Geo) . We now consider the (rectangular)
0 component of the geodesic equation (2.7.6):
L(Geo) L0(Geo) = −(g −1 )0γ Γαγβ Lα(Geo) Lβ(Geo) .
(2.17.3)
Multiplying (2.17.3) by µ3 , using the definition (2.7.7) of L, the decomposition (2.11.1b) ,
the identities (g/−1 )0γ = 0, L0 = 1, X 0 = 0, X̆ = µX, and equation (2.6.6), we deduce that
o
1n γ
2
0
γ
µL + X̆ {Gγβ ∂α Ψ + Gαγ ∂β Ψ − Gαβ ∂γ Ψ} Lα Lβ
(2.17.4)
µ LL(Geo) =
2
1
1
= µGLL LΨ + µGLX LΨ − GLL X̆Ψ.
2
2
Equation (2.17.1) now follows from equation (2.17.4), the relation µ2 LL0(Geo) = µ2 L( µ1 ) =
−Lµ, and incorporating the LΨ−containing products into the term Error.
46Recall
that L0 ≡ 1.
J. Speck
47
2.18. Connection coefficients of the rescaled frame. We now provide expressions for
the connection coefficients of the frame {L, X̆, Θ} in terms of Ψ, µ, L1 , L2 . We also decompose
some of the connection coefficients into “regular” pieces and pieces that have a “singular”
µ−1 factor.
Lemma 2.11 (Connection coefficients of the rescaled frame {L, X̆, Θ} and their
decomposition into µ−1 −singular and µ−1 −regular pieces). Let ζ be the `t,u −tangent
one-form defined by (see the identity (2.9.6b))
ζΘ := /kXΘ = g(DΘ L, X) = µ−1 g(DΘ L, X̆).
(2.18.1)
Then the covariant derivatives of the rescaled frame vectorfields can be expressed as follows,
where the tensorfields χ, k, and ω are defined in (2.9.4), (2.9.3), and (2.17.1):
DL L = µ−1 ωL,
(2.18.2a)
DX̆ L = −ωL + µζ# + d/ µ,
(2.18.2b)
DΘ L = −ζΘ L + trg/ χΘ,
(2.18.2c)
#
DL X̆ = −ωL − µζ# ,
n
o
DX̆ X̆ = µωL + µ−1 X̆µ + ω X̆ − µd/# µ,
(2.18.2d)
DΘ X̆ = µζΘ L + ζΘ X̆ + µ−1 (d/Θ µ)X̆ + µtrg//k Θ − µtrg/ χΘ,
(2.18.2f)
DL Θ = DΘ L,
(2.18.2g)
DΘ Θ = ∇
/ Θ Θ + /kΘΘ L + µ−1 χΘΘ X̆.
(2.18.2h)
(2.18.2e)
Furthermore, we can decompose the frame components of the `t,u −tangent tensorfields /k
and ζ into µ−1 −singular and µ−1 −regular pieces as follows:
ζ = µ−1 ζ(T rans−Ψ) + ζ(T an−Ψ) ,
(2.18.3a)
/k = µ−1/k (T rans−Ψ) + /k (T an−Ψ) ,
(2.18.3b)
where
1
/ X̆Ψ,
ζ(T rans−Ψ) := − G
2 L
1
/k (T rans−Ψ) := G
/ X̆Ψ,
2
(2.18.4a)
(2.18.4b)
and
1
1
1
ζ(T an−Ψ) := G
/ X LΨ − GLX ∇
/ Ψ − GXX ∇
/ Ψ,
(2.18.5a)
2
2
2
1
1
1
1
1
/k (T an−Ψ) := G
/ LΨ − G
/L ⊗ ∇
/Ψ − ∇
/Ψ ⊗ G
/L − G
/X ⊗ ∇
/Ψ − ∇
/Ψ ⊗ G
/ X . (2.18.5b)
2
2
2
2
2
Proof sketch. The proof is based on a series of long computations and we refer the reader
to [12, Lemma 2.13] for complete details. Here, we prove only (2.18.2c). We first expand
DΘ L = aL L + aX̆ X̆ + aΘ Θ. Taking the inner product of each side with L and using the
identities g(L, X̆) = −µ, g(L, Θ) = 0, and g(L, L) = 0, we find that aX̆ = 0 as desired.
Similarly, taking the inner product of each side with X̆, we find that −µaL = g(DΘ L, X̆) =
Stable Shock Formation
48
µζΘ as desired. Similarly, taking the inner product of each side with Θ and using (2.9.6a),
we find that aΘ g(Θ, Θ) = g(DΘ L, Θ) = χΘΘ , from which we easily conclude that aΘ = trg/ χ
as desired.
2.19. Frame decomposition of the wave operator. In this section, we decompose µg(Ψ)
relative to the rescaled frame. The factor of µ is important for our decompositions.
Proposition 2.12 (Frame decomposition of µg(Ψ) f ). Let f be a scalar function. Then
relative to the rescaled frame {L, X̆, Θ}, µg(Ψ) f can be expressed in either of the following
two forms:
µg(Ψ) f = −L(µLf + 2X̆f ) + µ∆
/ f − trg/ χX̆f − µtrg//k Lf − 2µζ# · d/f,
#
(2.19.1a)
#
= −(µL + 2X̆)(Lf ) + µ∆
/ f − trg/ χX̆f − ωLf + 2µζ · d/f + 2(d/ µ) · d/f,
(2.19.1b)
where the `t,u −tangent tensorfields χ, ζ, and /k can be expressed via (2.10.1a), (2.18.3a), and
(2.18.3b).
Proof. To derive (2.19.1a), we first use (2.11.1b) to decompose
2
2
f − 2Lα X̆ β Dαβ
f + (g/−1 ) · D 2 f
µg(Ψ) f = −µLα Lβ Dαβ
(2.19.2)
= −L(µLf + 2X̆f ) + (g/−1 ) · D 2 f
+ µ(DL Lα )Dα f + 2(DL X̆ α )Dα f + (Lµ)Lf.
2
f + (DΘ Θ)α Dα f = ∇
/ 2ΘΘ f + (∇
/ Θ Θ) ·
Next, we note that47 Θ(Θf ) = Θα Dα (Θβ Dβ f ) = DΘΘ
2
−1
2
d/f . Hence, by (2.18.2h), wenhave DΘΘ f = ∇
/ ΘΘ f − /kΘΘ Lf
o − µ χΘΘ X̆f . Consequently,
µ(g/−1 ) · D 2 f = (1/g(Θ, Θ)) µ∇
/ 2ΘΘ f − µk
/ΘΘ Lf − χΘΘ X̆f = µ∆
/ f − µtrg//k Lf − trg/ χX̆f .
We now substitute this identity into RHS (2.19.2). We also use Lemma 2.11 to substitute
for the terms µ(DL Lα )Dα f and 2(DL X̆ α )Dα f . The identity (2.19.1a) then follows from
straightforward calculations.
The proof of (2.19.1b) is similar and we omit the details.
From (2.19.1a), we deduce that we can write the wave equation g(Ψ) Ψ = 0 in the form
L̆Ψ
z
}|
{
L (µLΨ + 2X̆Ψ) = µ∆
/ Ψ + Error,
(2.19.3)
where the terms Error are products involving at most one Pu −transversal differentiation and
at least one Pu −tangent differentiation and therefore can be viewed as small error terms,
under the bootstrap assumptions outlined in Subsect. 2.22. The form of equation (2.19.3)
is important for proving some key estimates that are fundamental to the proof of shock
formation, as we describe in the proof sketch of Lemma 3.9.
and throughout, DV2 W := V α W β Dα Dβ . That is, under this notation, contractions are taken after
covariant differentiations with no derivatives falling on W . We use the same convention for the connection
∇
/.
47Here
J. Speck
49
Remark 2.12 (Special null structure of equation (2.19.3)). The special null structure
of the terms on RHS (2.19.3) mentioned above is central to the whole proof. In addition
to affording us small factors in the nonlinear products that we use to propagate the ˚
− δ̊
hierarchy all the way up to the shock (as we outlined in Subsect. 2.4), it implies the total
absence of products containing quadratic or higher powers of pure transversal derivatives
(such as (X̆Ψ)2 , (X̆Ψ)3 , etc.). This good structure is related to Klainerman’s classic null
condition (see Def. 2.1), but unlike in his condition, the structure of the cubic and higherorder terms matters. As such, the structure is a sensitive nonlinear one tied to the dynamic
metric g(Ψ), and it does not generally survive48 under perturbations of the wave equation
(2.6.1a) by the addition of semilinear terms (even cubic or higher-order ones).
Remark 2.13 (Rescaling by µ “removes” the dangerous semilinear term). Note that
we have brought a factor of µ under the outer L differentiation in equations (2.19.1a) and
(2.19.3). We have already seen the importance of this “rescaling by µ” in plane symmetry:
as we outlined in Exercise 1.13, it leads to the cancellation of a dangerous Riccati term
from the wave equation. Although a similar remark applies away from plane symmetry, it
is more difficult to see that the remaining error terms Error are negligible error terms. The
reason is that they involve, in addition to the first derivatives of Ψ, the second derivatives
of the eikonal function. Hence, to show that these error terms remain small throughout the
evolution, one must control the asymptotic behavior of the eikonal function. For example,
one of the error terms in equation (2.19.3) is trg/ χX̆Ψ. Using the data-size and bootstrap
assumptions outlined in Sects. 2.21-2.23 one can show that the norm k·kL∞ (Σut ) of the product
trg/ χX̆Ψ is of small size O(ε), consistent with the small size of the error term on RHS (1.6.18a)
in plane symmetry.
Remark 2.14 (One key property of the µ−weighted wave operator). Here is yet
another important fact concerning the covariant wave operator. It turns out that µg(Ψ)
has better commutation properties with the vectorfields belonging to the set Z (see (3.1.1))
than the unweighted operator g(Ψ) . See Prop. 2.13 for a more detailed statement. See
also Remark 2.16. An important property of Z is that we do not introduce any factors of
µ−1 when we repeatedly commute µg(Ψ) with Z ∈ Z ; generally, we would not be able to
control µ−1 factors. The moral reason behind these good properties can be discerned from
the form (2.19.3) of the wave equation. To see this, we recall that, relative to the geometric
∂
∂
coordinates, we have L = ∂t
and X̆ = ∂u
+ Error, where Error is an `t,u −tangent error
vectorfield (see Lemma 2.1). We can therefore rewrite (2.19.3) as
∂
∂
∂
∂
µ Ψ + 2 Ψ + µ∆
/ Ψ − 2 (∇
/
rΨ) + l.o.t.,
(2.19.4)
µg(Ψ) Ψ = −
∂t
∂t
∂u
∂t Error
where the terms l.o.t. depend on ≤ 1 derivative of Ψ. The right-hand side of (2.19.4) now
suggests that, for example, the differential operator X̆ ∈ Z can be commuted through
µg(Ψ) without introducing any dangerous factors of µ−1 .
48
As a crude caricature of this sensitivity, one may consider the Riccati ODE ẏ = y 2 and the perturbed
equation ẏ = y 2 + Ay 3 , with A a constant. Note how the long-time behavior of solutions can drastically
depend on whether A > 0, A = 0, or A < 0.
Stable Shock Formation
50
Remark 2.15 (The special null structure survives under commutations). We now
further refine the structure identified in Remark 2.14. To propagate the ˚
− δ̊ hierarchy
(as we outlined in Subsect. 2.4), we must exploit the fact that the special null structure of
equation (2.19.3) (described in Remark 2.12) survives under commutations. More precisely,
we have the following:
Repeatedly commuting the wave equation µg(Ψ) Ψ = 0 up to top order with Pu −tangent
vectorfields P ∈ P = {L, Y } produces commutator error term products that are quadratic and higher order in the derivatives of Ψ, µ, and Li with each product involving
no more than one X̆ derivative.
The above structure suggests that we can close the energy estimates by commuting only
with Pu −tangent vectorfields, since higher transversal derivatives of the solution are never
created by Pu −tangent commutations. Indeed, this is the case. Hence, our main energy
estimates (see Prop. 3.6) involve only Pu −tangent commutations.49
Since the above structure is central to the whole proof, we now sketch why the structure
is available. It is a consequence of the schematic structures [P1 , P2 ] ∼ P3 and [X̆, P1 ] ∼ P2 ,
where P1 , P2 , and P3 are arbitrary Pu −tangent vectorfields. These schematic commutator
relations are easy to see relative to the geometric coordinates (t, u, ϑ). To further explain
∂
∂
∂
these issues, we first note that X̆ = ∂u
− ξ ∂ϑ
, where Θ = ∂ϑ
and ξ is a scalar function
(see (2.7.18)). From these expressions, it easily follows that for Z1 , Z2 ∈ {X̆, P1 , P2 }, the
∂
∂
commutator [Z1 , Z2 ] belongs to span{ ∂t
, ∂ϑ
} and is therefore Put −tangent (note that the
∂
coefficient of ∂u
in the above expression for X̆ is a constant!). That is, we have shown
that [P1 , P2 ] ∼ P3 and [X̆, P1 ] ∼ P2 . Recalling the wave equation decomposition (2.19.4), we
easily obtain the structure for the commutators [µg(Ψ) , P ] highlighted in the above indented
sentence in the special case P ∈ {L, Y } relevant for our energy estimates.
2.20. The wave equation after one commutation. In this equation, we exhibit the basic
structure of the commutator error terms that appear after we commute the wave equation
one time with a commutation vectorfield Z ∈ Z . To keep the discussion short, we focus on
two of the most important error terms.
Proposition 2.13 (Important error terms generated by one commutation). Let
Z ∈ Z (see definition (2.15.3)) be a commutation vectorfield. Then we have the following
commutation identity, valid for solutions Ψ to µg(Ψ) Ψ = 0:
(Z)
µg(Ψ) (ZΨ) = −(X̆Ψ)div
/ (Z)π
/#
πLX̆ ∆
/Ψ + ··· ,
L +
(2.20.1)
where · · · additional terms that have the schematic form µD (Z)π · DΨ or µ(Z)π · D 2 Ψ and
that have the special null structure mentioned in Remark 2.14.
Moreover, in the case Z = Y (see Def. 2.27), we have
µg(Ψ) (Y Ψ) = (X̆Ψ)Y trg/ χ − (Y µ)∆
/Ψ + ··· ,
49We
(2.20.2)
note, however, that as we outline in Sects. 2.21 and 2.22, we need to obtain control of up to three
transversal derivatives of Ψ in L∞ to close the energy estimates. To this end, we can commute the wave
equation in the form (2.19.1a) up to 2 times with X̆ and treat it like a transport equation that loses a
derivative in view of the presence of the ∆
/ Ψ term present in the decomposition (2.19.1a). Readers may
consult [12, Section 9] for the details.
J. Speck
51
where · · · denotes a long list of error term products, also of the schematic form µD (Z)π · DΨ
or µ(Z)π · D 2 Ψ, that turn out to be easier to control than the two ones listed on RHS (2.20.2)
due in part to the special null structure mentioned in Remark 2.15.
Remark 2.16 (Critically important cancellations). Due to the special properties of the
vectorfields Z ∈ Z , there are critically important cancellations that occur on RHS (2.20.1)
and (2.20.2); see, for example, Remark 2.11.
Proof sketch for Prop. 2.13. The proof of (2.20.1) amounts to a series of lengthy computations whose details may be found in [12, Proposition 4.4]; we now sketch the proof. The
first step is deriving (2.20.1) is to derive the following commutation formula (see [12, Lemma
4.2]):
1 (Z) α
1
(Z) αβ
µg(Ψ) (ZΨ) = µDα
π Dβ Ψ − trg πD Ψ + Z(µg(Ψ) Ψ) + trg/ (Z)π
/(µg(Ψ) Ψ),
2
2
(2.20.3)
where trg (Z)π := (g −1 )αβ (Z)παβ . Note that the first term on RHS (2.20.3) is the divergence
of the spacetime vectorfield J α := (Z)π αβ Dβ Ψ − 21 trg (Z)πD α Ψ. The next step is to derive
the following identity for the divergence of a vectorfield J (see [12, Lemma 4.3]):
/ (µJ
/ ) − µtrg//k JL − trg/ χJX̆ ,
µDα J α = −L(µJL ) − L(JX̆ ) − X̆(JL ) + div
(2.20.4)
/ is the g−orthogonal projection of J onto
where JL := g(J , L), JX̆ := g(J , X̆), and J
`t,u . The next step (see the proof of [12, Proposition 4.4]) is to compute JL , JX̆ , and J
/
for the specific vectorfield J given above. One then arrives at the formula (2.20.1).
To obtain (2.20.2) from (2.20.1), we consider the term −(X̆Ψ)div
/ (Y )π
/#
L generated by the
first term on RHS (2.20.1). We aim to identity products involving the top-order derivatives
of the eikonal function, which in the present context means two derivatives of χ or two
derivatives of µ. To proceed, we decompose div
/ (Y )π
/#
L , using the formula (2.16.3b). The
only top-order eikonal function term is generated by the first term −χ · Y = −trg/ χY[ on
RHS (2.16.3b), where the equals in the previous identity holds only because the `t,u are onedimensional. It follows that the top-order term of interest in the product −(X̆Ψ)div
/ (Y )π
/#
L is
precisely (X̆Ψ)Y trg/ χ, as is indicated on RHS (2.20.2).
2.21. Size assumptions on the data. Recall that (Ψ|Σ0 , ∂t Ψ|Σ0 ) := (Ψ̊, Ψ̊0 ) and that we
assume (Ψ̊, Ψ̊0 ) ∈ He19 (Σ10 ) × He18 (Σ10 ). We assume that the data verify the following size
estimates:
≤17;3 [1,3] Z∗
Ψ ∞ 1 , Z∗≤19;3 Ψ 2 1 ≤ ˚
,
:= δ̊ > 0.
(2.21.1)
X̆ Ψ
L (Σ0 )
L (Σ0 )
L∞ (Σ10 )
In (2.21.1) and throughout, kZ∗≤N ;M ΨΨkL∞ (Σut ) , denotes the max of the norms k · kL∞ (Σut ) of
the up-to-order N derivatives of Ψ with respect to vectorfields belonging to the commutation
set Z defined in (2.15.3), where at most M X̆ derivatives are taken, and the pure transversal
derivatives X̆Ψ, X̆ X̆Ψ, · · · , X̆ M Ψ are excluded.
As in plane symmetry, the time of first shock formation is approximately δ̊−1
∗ , where δ̊∗ is
now given by the following definition.
Stable Shock Formation
52
Definition 2.30 (The quantity that controls the blow-up time). We define
h
i
1
δ̊∗ := sup GLL X̆Ψ ,
(2.21.2)
2 Σ10
−
where Gαβ is defined in (2.6.7).
Exercise 2.11 (Equivalence of the two definitions up to small error terms). Show
that the general definition (2.21.2) agrees with the one (1.7.3) defined in the case of the
model metric (1.4.1), up to an error term that is proportional to LΨ (we will show that
LΨ remains small throughout the evolution and hence the error term makes a negligible
contribution to the blow-up time).
2.22. An overview of the bootstrap assumptions. Recall that in proving Theorem 1.1
in plane symmetry, it was convenient to make bootstrap assumptions. Away from plane symmetry, we also find it convenient to make bootstrap assumptions. The full set of bootstrap
assumptions involve statements about the derivatives of µ, the rectangular components Li ,
and the `t,u tensorfield χ. Since the full set is lengthy and cumbersome to state, here we
state only the ones involving Ψ. Specifically, we assume that for (t, u) ∈ [0, T(Boot) ) × [0, U0 ],
we have
kP ≤11 ΨkL∞ (Σut ) ≤ ε,
(2.22.1a)
kZ∗≤10;1 ΨkL∞ (Σut ) ≤ ε1/2 ,
(2.22.1b)
kX̆ M ΨkL∞ (Σut ) ≤ kX̆ M ΨkL∞ (Σu0 ) + ε1/2 ,
(M = 1, 2, 3).
(2.22.1c)
In (2.22.1a), kP ≤11 ΨkL∞ (Σut ) , denotes up to 11 derivatives of Ψ with respect to vectorfields
belonging to Pu −tangents subset P. In (2.22.1b), Z∗≤10;1 Ψ denotes the max of the norms
k · kL∞ (Σut ) of the up-to-order 10 derivatives of Ψ with respect to vectorfields belonging to Z ,
where at most 1 X̆ derivative is taken, and the pure transversal derivative X̆Ψ is excluded.
Remark 2.17. In a complete proof, one would improve the bootstrap assumptions by first
proving that (2.22.1b)-(2.22.1c) hold with ε1/2 replaced by Cε and then that all factors of
ε can be replaced by C˚
(with ˚
as in (2.21.1)); clearly for υ̊ sufficiently small, this would
imply a strict improvement of the bootstrap assumptions. This last step would take place
at the very end of the proof, after the energy estimates have been derived, and would rely
on Sobolev embedding.
Remark 2.18 (Why X̆ X̆ X̆Ψ?). There is precisely one reason why we need the estimate
X̆µ
(2.22.1c): to control the energy estimate error term generated by the factor µ|∇
/ Ψ|2
in
µ
X̆µ
equation (3.2.17c). This analysis of the ratio
very technical, and we refer the reader to
µ
[12, Sect. 10] for the full details. Here, we note that in order to derive suitable estimates
X̆µ
for
, we rely on the estimate kX̆ X̆µkL∞ (Σut ) . 1. We derive this bound by commuting
µ
the evolution equation (2.17.1) for µ with up to two factors of X̆. Since RHS (2.17.1)
depends on X̆Ψ, in order to derive the desired bound, we clearly must obtain control over
kX̆ X̆ X̆ΨkL∞ (Σut ) .
J. Speck
53
2.23. Smallness assumptions. To close our proof in the general case, we make the same
smallness assumptions that we made in Subsect. 1.8 in the case of plane symmetry. Specifically, we make the following assumptions.
• The initial data size parameter ˚
from (2.21.1) is small relative50 to δ̊−1 , where δ̊ is
the data-size parameter from (2.21.1).
•˚
is small relative to the data-size parameter δ̊∗ from (1.7.3).
• We also assume that ˚
≤ ε, where ε is the bootstrap parameter from Subsect. 2.22.
3. Generalized energy estimates away from plane symmetry
In this section, we discuss the most difficult aspect of proving small-data shock formation
away from plane symmetry: the derivation of generalized energy estimates that hold up to
top order. Our discussion in this section applies to the geometric wave equation
g(Ψ) Ψ = 0
(3.0.1)
under the assumptions on the nonlinearities stated in Subsect. 2.6.
3.1. The basic strategy for deriving generalized energy estimates. The discussion
from the beginning of Sect. 2 suggests the following strategy for proving shock formation in
solutions to equation (3.0.1).
(1) With the help of the eikonal function u, one should construct commutation vectorfields
Z that have good commuting properties with the covariant wave operator g(Ψ)
corresponding to the wave equation (3.0.1). As we first mentioned in Subsect. 2.15,
a suitable set of commutation vectorfields is
Z := {L, X̆, Y },
(3.1.1)
where we recall that geometric torus vectorfield Y was constructed in Def. 2.27. A
basic property of Z is that it has span equal to span{∂α }α=0,1,2 at each spacetime
point where µ > 0. As we first described in Subsect. 2.3, for Z ∈ Z , the rectangular
components Z α depend on the first rectangular derivatives of u. This basic fact has
important implications for the regularity theory of the vectorfields Z ∈ Z , which is
at the heart of the difficulties one faces.
As we described in Remark 2.15, due to the special structure of the equations, we
can close the energy estimates by commuting only with vectorfields belonging to the
Pu −tangent subset
P := {L, Y }.
(3.1.2)
(2) To close the energy estimates, we rely on a collection of L∞ estimates for the low-level
Z −derivatives of Ψ, µ, and Li , which are used to control error terms and to obtain
sharp information about the behavior of µ. To this end, we make “fundamental”
bootstrap assumptions about the L∞ norms of various low-level derivatives of Ψ with
respect to vectorfields in P. These assumptions are non-degenerate in the sense that
they do not lead to infinite expressions even when µ = 0. Using them, we derive the
desired non-degenerate L∞ estimates for the low-level Z derivatives of Ψ, µ, and Li .
To obtain the estimates for µ, and Li , we use the transport equations of Lemma 2.10.
50We
use the phrase “small relative to” in the same way that we used it in Subsect. 1.8.
Stable Shock Formation
54
We also derive sharp estimates that precisely characterize the behavior of µ. The
estimates are analogs of the ones (1.10.5c) and (1.10.6b) that we obtained in plane
symmetry. In addition, we derive related sharp estimates for certain time-integrals
involving degenerate factors of 1/µ. The degenerate time integrals appear in the
Gronwall estimates that we use to derive a-priori energy estimates. These Gronwall
estimates are very delicate and lead to degenerate top-order energy estimates, as we
describe in the next step.
(3) Derive generalized energy estimates for a sufficiently large number of P -derivatives
of Ψ by commuting the equations, where P ∈ P. Commuting the equations with
well-chosen vectorfield operators is known as the commutator method. As in our
study of the plane symmetric case, the blow-up time is approximately δ̊−1
∗ , where
δ̊∗ is defined in (2.21.2). Hence, it suffices to obtain the desired energy estimates
for times t ≤ 2δ̊−1
∗ . The work of the authors in [12] showed (see Prop. 3.6) that it
suffices to commute the wave equation (3.0.1) up to 18 times with the elements of
P. As we shall see starting in Subsubsect. 3.2.5, the proof relies on a large number
of commutations because the high-order energy estimates are allowed to blow-up like
powers of µ−1
? due to the degenerate Gronwall estimate mentioned in the previous
step. In addition, to recover the fundamental L∞ bootstrap assumptions for the lowlevel derivatives of the solution via Sobolev embedding, we need to obtain several
orders of non-degenerate low-level energy estimates. Obtaining the non-degenerate
estimates is based on gaining powers of µ? via time integration, and it costs many
derivatives to descend to a non-degenerate level.
In addition to the commutator method, the derivation of generalized energy estimates also relies on the multiplier method, which we explain in more detail in Subsubsect. 3.2.1. In the absolute simplest settings, the multiplier method is implemented
by contracting the energy-momentum tensorfield (see (3.2.2)) against the following
multiplier vectorfield :
1
∂t =
(3.1.3)
L(F lat) + L(F lat) ,
2
where L(F lat) = ∂t +∂x and L(F lat) = ∂t −∂x are the standard null pair. In this simplest
setting, using the multiplier method is equivalent to multiplying the wave equation by
∂t Ψ and then integrating by parts over Σt . We remark that ∂t is Minkowski Killing.51
The vectorfield that we use in the shock-formation problem in the region Mt,u
(see (2.7.2e)) is the following modified version52 of the vectorfield in (3.1.3), which is
adapted to the eikonal function u:
T := (1 + µ)L + L̆ = (1 + 2µ)L + 2X̆,
(3.1.4)
T is designed to yield generalized energy quantities that are useful both in regions
where µ is large and where it is small (see Lemma 3.2). A basic fact, which follows
from simple computations based on Lemma 2.1, is that g(T, T ) = −4µ(1 + µ) < 0.
51Recall
that call a vectorfield X is Minkowski Killing if LX m = 0, while it is conformally Minkowski
Killing if LX m is a scalar function multiple of m.
52In many other works, the symbol T denotes the future-directed unit normal to Σ . In contrast, the
t
vectorfield T defined in (3.1.4) is not the future-directed unit normal to Σt .
J. Speck
55
Hence, T is future-directed and g−timelike, which are properties connected to the
coerciveness of our energies (see Subsubsect. 3.2.1). T is generally neither Killing
nor conformal Killing53 (relative to the dynamic metric g). Nonetheless, the energy
estimate error terms corresponding to the deformation tensor (as defined in (2.3.2))
of T , which is located on the right-hand side of the energy identity (3.2.7) (with
V := T ), is controllable in the region Mt,u .
Remark 3.1 ((T )π has a component with a good sign). Actually, as we will see
in Lemma 3.5, one of the deformation tensor error terms generated by T has a good
sign in regions where µ is small and is essential for controlling other error terms.
(4) As long as we can suitably bound the generalized energy quantities, which are constructed with the help of the commutation vectorfields P defined in (3.1.1) and the
multiplier vectorfield T defined in (3.1.4), we can also derive, via Sobolev embedding,
L∞ estimates for the low-order L and Y derivatives of Ψ. In particular, under a small
data assumption, we can recover the L∞ bootstrap assumptions (2.22.1a) for Ψ.
(5) The deformation tensors (Z)π (defined in (2.3.2)) for Z ∈ Z can be expressed in
terms of the covariant Hessian H = D 2 u, which verifies a transport equation of the
schematic form
LH + H 2 = R,
(3.1.5)
where, as we mentioned in Subsect. 2.3, the curvature term R depends on the upto-second-order derivatives of Ψ and the up-to-second-order derivatives of u. As we
explained in Subsect. 2.3, when we commute54 µg(Ψ) with a vectorfield Z ∈ Z , we
generate terms of the form D (Z)π · DΨ, which can be traced back, via the transport
equation (3.1.5), to one more derivative of Ψ than we are able to control by an
energy estimate at the same level. However, even though we lose a derivative in
equation (3.1.5), the special properties of the vectorfields in Z lead to error terms
in the commuted versions of (3.1.5) that do not involve55 any factors of µ−1 . This is
especially important at low derivative levels because factors of µ−1 would blow up as
we approach the expected singularity, which would in turn prevent us from showing
that the L∞ bootstrap assumptions for the low-order L and Y derivatives of Ψ are
valid. In other words, we can derive estimates for56 the derivatives of H that are
regular with respect to µ, but only at the expense of losing a derivative relative to
Ψ. This is key to understanding the overall strategy: only at the top level do we
combat derivative loss through renormalization (see the next item), which has as a
trade-off the introduction of a difficult factor of µ−1 into the energy estimates. In
contrast, at the lower derivative levels, we can avoid the µ−1 factor because the loss
53A
vectorfield V is said to be Killing (for g) if Lg V = 0. It is said to be conformal Killing if LV g is a
scalar function multiple of g.
54See Remark 2.14 for an explanation of why we work with the µ−weighted operator µ
g(Ψ) .
55To see this in detail, one must decompose (3.1.5) relative to the rescaled frame {L, X̆, Θ}. At one
derivative level lower, the µ-regular behavior can be seen in the transport equation (2.17.1) for µ, where
there are no factors of µ−1 present.
56More precisely for the components of the derivatives of H relative to the rescaled frame {L, X̆, Θ}.
Stable Shock Formation
56
of one derivative is permissible. This trade-off is where understanding the dynamic
geometry is most important.
(6) To control the top derivatives of Ψ when the loss of a derivative, due to (3.1.5), can
no longer be ignored, we use a renormalization procedure,57 which recovers the loss
of derivatives mentioned above at the expense of introducing a singular factor of µ−1
into the energy estimates; see Subsubsect. 3.4.3. This new difficulty of having to
derive a priori energy estimates in the presence of the singular factor µ−1 can be
handled with the help of a subtle Gronwall-type inequality. We illustrate the main
ideas with a model Gronwall inequality in Lemma 3.7 below.
(7) Another important part of the proof is showing that the top-order energy degeneracy
with respect to µ−1 does not propagate down to far to the lower derivative levels.
For if it did, then we would not be able to recover the fundamental L∞ bootstrap
assumptions that are needed to close the whole proof. In Subsect. 3.5, we explain
why the energy estimates become successively less degenerate as one descends below
top order, until one reaches a level where they no longer degenerate as all, even as
the shock forms!
3.2. Energy estimates via the multiplier method. We now present the multiplier
method for deriving generalized energy estimates for solutions to the inhomogeneous covariant wave equation
µg Ψ = F.
(3.2.1)
In Remark 2.14, we explain why we choose to include the factor of µ on the left-hand side
of (3.2.1).
3.2.1. A geometric version of the divergence theorem via the multiplier method. The geometric energy method that we use is tied to the spacetime metric g and the true characteristics
Pu . A key ingredient is the energy-momentum tensorfield Qµν [Ψ]:
1
Qµν [Ψ] = Qµν := Dµ ΨDν Ψ − gµν D α ΨDα Ψ.
(3.2.2)
2
It is straightforward to check that for solutions to (3.2.1), we have
µ Dα Qαν = FD ν Ψ.
(3.2.3)
Furthermore, for any pair of future-directed vectorfields V and W verifying58 g(V, V ), g(W, W ) ≤
0, we have the following well-known inequality,59 which motivates our construction, given
below, of coercive L2 −type quantities:
Qαβ V α W β ≥ 0.
(3.2.4)
We will apply the divergence theorem with the help of the following compatible currents,
which are useful for bookkeeping during integration by parts.
57The
procedure involves combining (3.1.5) with the wave equation g(Ψ) Ψ = 0. In n ≥ 3 spatial
dimensions, one also needs to use elliptic estimates on the n − 1-dimensional surfaces that are the analogs
of `t,u . This is similar to the approach taken in [5] and [19], though the difficulty of obtaining estimates in
regions where µ is small is not present in these works.
58Such vectorfields are said to be causal.
59In general relativity, inequality (3.2.4) is often referred to as the dominant energy condition.
J. Speck
57
Definition 3.1 (Compatible current). To an auxiliary “multiplier” vectorfield V , we
associate the following compatible current vectorfield (V )J ν [Ψ]:
(V ) ν
J [Ψ] := Qνα [Ψ]V α .
(3.2.5)
By (3.2.3), for solutions Ψ to (3.2.1), we have
1
(3.2.6)
µDα (V )J α [Ψ] = µQαβ [Ψ](V )παβ + (V Ψ)F,
2
where (V )παβ = Dα Vβ + Dβ Vα is the deformation tensor of V , as in (2.3.2).
To derive generalized energy estimates, we apply the divergence theorem to a current
(V ) ν
J [Ψ] for a well-chosen vectorfield V on the region Mt,u (see Fig. 2). This procedure is
precisely the multiplier method mentioned in SubSubsect. 3.1. We provide the main energy
identity in the next lemma.
Lemma 3.1. [23, Adapted version of Lemma 10.2.1; Divergence theorem] Let V
be a multiplier vectorfield and let (V )J be the compatible current (3.2.5). For solutions Ψ to
µg Ψ = F that vanish along60 P0 , we have
Q[Ψ](V,N )
Z
Q[Ψ](V,L)
Z
z }| {
(V )
µ g( J, N ) d$ +
Σu
t
Q[Ψ](V,N )
Z
z }| {
(V )
g( J, L) d$ =
z }| {
µ g((V )J, N ) d$
(3.2.7)
Σu
0
Put
Z
−
(V Ψ)F d$
Mt,u
1
−
2
Z
µQ[Ψ] · (V )π d$,
Mt,u
where Q[Ψ] · (V )π := Qαβ (V )παβ .
In (3.2.7), N = L + X is the future-directed unit-normal to Σut (see Lemma 2.1) and the
integration forms are as in Def. 2.24.
Proof sketch. We use the formula (1.4.12) relative to the geometric coordinates to compute
the covariant divergence of the vectorfield (V )J[Ψ] defined by (3.2.5). To show that |detg| =
µ2 υ 2 , a formula that plays a role in (1.4.12), we use the identity (2.11.4). We then integrate
the resulting identity with respect to dt0 du0 dϑ over the domain [0, t] × [0, u] × T. In applying
the formula (1.4.12), we must compute the components of (V )J[Ψ] relative to the geometric
∂
∂
coordinates. Specifically, we decompose J = J t ∂t
+ J u ∂u
+ J Θ Θ, where J t , J u , J Θ are scalar
∂
functions and we recall that Θ = ∂ϑ
. One can then show that
J u = −µ−1 QLT [Ψ],
t
u
J = µJ − JX = −QLT [Ψ] − QLX [Ψ] = −QN T [Ψ],
(3.2.8)
(3.2.9)
where N is the vectorfield defined in (2.7.11). Moreover, we can express RHS (3.2.8)-(3.2.9)
using Lemma 2.6. The identity (3.2.7) then follows from all of the above considerations and
a series of computations that relies in part on Fubini’s theorem.
60Recall
the data.
that the vanishing of Ψ along P0 is an easy consequence of our assumptions on the support of
Stable Shock Formation
58
Note that by the property (3.2.4), the first two integrands on the left-hand side of (3.2.7)
are non-negative for the future-directed timelike multiplier V = T ; see Prop. 3.3 for a more
precise account of the coerciveness of these terms.
3.2.2. Energies and fluxes. With the help of the above currents and the corresponding divergence theorem identities, we can now define the energies and fluxes that will enable us to
control Ψ and its derivatives in L2 .
Definition 3.2 (Energies and fluxes). Let T be the timelike multiplier vectorfield defined
in (3.1.4), let (T )J be the corresponding compatible current (3.2.5), and let N := L+X denote
the future-directed unit normal to Σut (see Lemma 2.1) We define the energy E[Ψ](t, u) and
the null flux F[Ψ](t, u) corresponding to T (see (3.1.4)) in terms of the rescaled volume forms
of Def. 2.24 as follows:
Q[Ψ](T,N )
Z
E[Ψ](t, u) :=
z }| {
µ g((T )J, N ) d$,
(3.2.10a)
Σu
t
Q[Ψ](T,L)
Z
F[Ψ](t, u) :=
z }| {
g((T )J, L) d$.
(3.2.10b)
Put
In the next lemma, we provide explicit expressions for E[Ψ] and F[Ψ] in terms of the
rescaled frame derivatives of Ψ.
Lemma 3.2 (Explicit expressions for the energy and null flux). The energy and null
flux from Def. 3.2 can be expressed as follows:
Z
1
1
E[Ψ](t, u) =
(1 + 2µ)µ(LΨ)2 + 2µ(LΨ)X̆Ψ + 2(X̆Ψ)2 + (1 + 2µ)µ|∇
/ Ψ|2 d$,
u
2
2
Σt
(3.2.11a)
Z
F[Ψ](t, u) :=
(1 + µ)(LΨ)2 + µ|∇
/ Ψ|2 d$.
(3.2.11b)
Put
Proof. The lemma follows from the identities
1
1
1
µQN T = (1 + µ)µ(LΨ)2 + (µLΨ + 2X̆Ψ)2 + (1 + 2µ)µ|∇
/ Ψ|2 ,
2
2
2
QLT = (1 + µ)(LΨ)2 + µ|∇
/ Ψ|2 ,
(3.2.12)
(3.2.13)
which are a simple consequence of Lemma 2.6 and the identities (2.7.11) and (2.12.4).
Note that Def. 3.2 and the dominant energy condition (3.2.4) suggest that our energies and
fluxes should be coercive. The following proposition provides an explicit characterization of
their coercive nature.
Proposition 3.3. [12, Adapted version of Lemma 12.2; Coerciveness of the energies
and fluxes] Under suitable smallness bootstrap assumptions, the energies and fluxes from
J. Speck
59
Def. 3.2 have the following coerciveness properties:
2
n1 √
1 √
2
2
, k µ∇
E[Ψ](t, u) ≥ max
k µLΨkL2 (Σu ) , X̆Ψ
/ ΨkL2 (Σu ) ,
u
t
t
2
L2 (Σt ) 2
o
C −1 kΨk2L2 (Σut ) , C −1 kΨk2L2 (`t,u ) ,
n
o
√
2
2
F[Ψ](t, u) ≥ max kLΨkL2 (Put ) , k µ∇
/ ΨkL2 (Put ) .
(3.2.14a)
(3.2.14b)
The L2 norms above are as in Def. 2.26.
Remark 3.2 (On the importance of the sharp constants). The sharp constants 1 and 12
2
2
√
and 1 µ∇
/ Ψ 2 u in the estimate (3.2.14a) influence
in front the quantities X̆Ψ
L2 (Σu
t)
2
L (Σt )
the blow-up rate of our top-order energy estimates. In turn, this affects the number of
derivatives that we need to close our estimates; see, for example, the derivation of inequality
(3.4.16) from inequality (3.4.15).
In the next proposition, we specialize Lemma 3.1 to the case V = T , which yields integral
identities for the energy and null flux.
Proposition 3.4 (Fundamental energy-flux identity). For solutions Ψ to
µg(Ψ) Ψ = F
that vanish along the outer null hyperplane P0 , we have the following identity involving the
energy and flux from Def. 3.2:
E[Ψ](t, u) + F[Ψ](t, u)
(3.2.15)
Z
Z
n
o
1
(1 + 2µ)(LΨ) + 2X̆Ψ F d$ −
= E[Ψ](0, u) −
µQαβ [Ψ](T )παβ d$.
2
Mt,u
Mt,u
Furthermore, with f+ := max{f, 0} and f− := max{−f, 0}, we have
5
(T )
1
1
[Lµ]− X (T )
P(i) [Ψ],
P[Ψ] := − µQαβ [Ψ](T )παβ = − µ|∇
/ Ψ|2
+
2
2
µ
i=1
(3.2.16)
where
1
1
(T rans−Ψ)
(T an−Ψ)
P(1) [Ψ] := (LΨ) − Lµ + X̆µ − µtrg/ χ − trg//k
− µtrg//k
, (3.2.17a)
2
2
n
o
(T )
P(2) [Ψ] := −(LΨ)(X̆Ψ) trg/ χ + 2trg//k (T rans−Ψ) + 2µtrg//k (T an−Ψ) ,
(3.2.17b)
(
)
1 [Lµ]+ X̆µ
1
(T )
P(3) [Ψ] := µ|∇
/ Ψ|2
+
+ 2Lµ − trg/ χ − trg//k (T rans−Ψ) − µtrg//k (T an−Ψ) ,
2 µ
µ
2
(3.2.17c)
#
(T )
P(4) [Ψ] := (LΨ)(d/ Ψ) · (1 − 2µ)d/µ + 2ζ(T rans−Ψ) + 2µζ(T an−Ψ) ,
(3.2.17d)
(T )
P(5) [Ψ] := −2(X̆Ψ)(d/# Ψ) · d/µ + 2ζ(T rans−Ψ) + 2µζ(T an−Ψ) .
(3.2.17e)
(T )
2
The tensorfields χ, ζ(T rans−Ψ) , /k (T rans−Ψ) , ζ(T an−Ψ) , and /k (T an−Ψ) from above are as in
(2.9.4), (2.18.4a), (2.18.4b), (2.18.5a), and (2.18.5a).
Stable Shock Formation
60
Proof sketch. Readers may consult [12, Proposition 3.5] for full details. (3.2.15) is a direct
consequence of the formula (3.2.7) with V := T . To obtain the decomposition (3.2.16)(3.2.17e), we first write the last term on RHS (3.2.7) as follows:
µQ[Ψ] · (V )π = µ(g −1 )αβ (g −1 )κλ Qακ [Ψ](T )πβλ .
We then decompose the RHS of the previous identity using (2.11.1b) and Lemma 2.9, and
[Lµ]−
[Lµ]+
we furthermore use the decomposition (Lµ)|∇
/ Ψ|2 = −µ|∇
/ Ψ|2
+ µ|∇
/ Ψ|2
.
µ
µ
3.2.3. The role of µ weights in the energies and null fluxes. Observe that the energies E
from Prop. 3.3 control only µ−weighted versions of LΨ and ∇
/ Ψ. Hence, for µ near 0, they
provide only very weak control over LΨ and ∇
/ Ψ. However, when bounding various error
integrals on the RHS (3.2.15), we encounter non−µ−weighted factors of LΨ and ∇
/ Ψ, which
cannot be controlled directly by E. We give an example of such an error term and describe
how to handle it in Subsubsect. 3.4.2. To handle the non−µ−weighted factors of LΨ when
µ is small, we will need to rely on the null flux F from Prop. 3.3, which provide control over
LΨ without any µ weights.
3.2.4. The need for the coercive spacetime integral. Note that Prop. 3.3 does not provide
any quantity that yields control of the non µ−weighted factors of ∇
/ Ψ when µ is small. To
obtain such control, we use a much more interesting and subtle estimate. A related version
of the estimate was first exploited by Christodoulou in [3] in the context of small solutions
on R × R3 with compact spatial support.
The main idea is that there is a subtly coercive term hiding in the last integral on
−
RHS (3.2.15). The integral is precisely the one corresponding to the term − 21 µ|∇
/ Ψ|2 [Lµ]
µ
on RHS (3.2.16). Note that this term generates a negative spacetime integral, denoted by
−K[Ψ](t, u), on RHS (3.2.15). For convenience, we now explicitly identify the corresponding
positive spacetime integral.
Definition 3.3 (Coercive spacetime integral).
Z
1
K[Ψ](t, u) :=
[Lµ]− |∇
/ Ψ|2 d$.
2 Mt,u
(3.2.18)
In (3.2.18), [Lµ]− := |Lµ| when Lµ < 0, and [Lµ]− := 0 otherwise.
The coerciveness of the integral (3.2.18) is provided by the following lemma.
Lemma 3.5 (Strength of the coercive spacetime integral). Under the data-size and
bootstrap assumptions outlined in Sects. 2.21-2.23, the following lower bound holds for (t, u) ∈
[0, T(Boot) ) × [0, U0 ]:
Z
1
K[Ψ](t, u) ≥ δ̊∗
1{µ≤1/4} |∇
/ Ψ|2 d$,
(3.2.19)
8
Mt,u
where δ̊∗ > 0 is the data-dependent constant from Def. 2.30 and 1{µ≤1/4} is the characteristic
function of the set {µ ≤ 1/4}.
Proof. The lemma follows easily from definition (3.2.18) and the estimate (1.10.6a), which
also holds away from plane symmetry; see (3.3.7).
J. Speck
61
The most important points concerning K[Ψ](t, u) are the following:
• −K[Ψ](t, u) appears on the right-hand side of (3.2.15) and hence we can bring K[Ψ]
to the left and obtain additional spacetime control over |∇
/ Ψ|2 .
• The integrand in K[Ψ](t, u) contains no µ weights, so it is significantly coercive even
in regions where µ is near 0.
3.2.5. Overview of the L2 hierarchy and the µ−1
? degeneracy. We are almost ready to provide
an overview of the main a priori energy-flux estimates. The estimates involve the following
important quantity, which captures the “worst-case” scenario for µ being small along Σut and
which is familiar from our previous study of the plane symmetric case.
Definition 3.4 (A modified minimum value of µ). We define the function µ? (t, u) as
follows:
µ? (t, u) := min{1, min
µ}.
u
Σt
(3.2.20)
Now that we have defined all of the quantities of interest, we can state a proposition that
provides the a priori energy-flux estimates that hold on spacetime domains of the form Mt,u .
Clearly there is no analog of this proposition in plane symmetry because in that setting, we
did not need to derive L2 estimates.
Proposition 3.6. [12, Proposition 14.1; Rough statement of the hierarchy of a
priori energy-flux estimates] Assume that the data for the covariant wave equation
g(Ψ) Ψ = 0 are compactly supported in Σ10 . Assume that Ψ exists classically in the region Mt,u defined by (2.7.2e). There exist large constants C > 0 and A∗ > 0 such that under
the data-size and bootstrap assumptions outlined in Sects. 2.21-2.23, if ˚
is sufficiently small,
then the following energy-flux estimates hold for the quantities from Defs. 3.2 and 3.3:
p
p
p
E[P 13+M Ψ](t, u) + F[P 13+M Ψ](t, u) + K[P 13+M Ψ](t, u)
(3.2.21a)
≤ C˚
µ?−(M +.9) (t, u),
p
p
p
E[P 1+M Ψ](t, u) + F[P 1+M Ψ](t, u) + K[P 13+M Ψ](t, u)
(0 ≤ M ≤ 5),
≤ C˚
,
(0 ≤ M ≤ 11).
(3.2.21b)
In the above estimates, P ≤K denotes an arbitrary differential operator of order ≤ K corresponding to repeated differentiation with respect to the commutation vectorfields belonging
to the Pu −tangential set P defined in (3.1.2). Similarly, P K denotes an arbitrary such
vectorfield operator of order precisely K.
Remark 3.3 (The µ−1
hierarchy). An important feature of Prop. 3.6 to notice is that
?
the top-order quantities are allowed to blow up like µ?−5.9 as µ? tends to 0. The power
−5.9 is a consequence of some delicate structural features of the equations. We explain
this below (see in particular Remark 3.6). Another important feature is that as we descend
below the top order, we see improvements in the µ−1
? blow-up rate until we reach the levels
(3.2.21b), in which the quantities no longer blow up when µ? vanishes. The non-degenerate
estimates can be used to show that the lower-order derivatives of Ψ, relative to the geometric
coordinates (t, u, ϑ), extend to the constant-time hypersurface of first shock formation as
bounded functions. These non-degenerate estimates are in fact the main ingredients needed
Stable Shock Formation
62
to establish the L∞ bootstrap assumptions of Subsect. 2.22. The precise features of this
hierarchy play a fundamental role in guiding the analysis.
3.3. Details on the behavior of µ. In order to explain how to derive the energy estimate
hierarchy of Prop. 3.6, we first need to provide some sharp information on the behavior of
µ. In the next three lemmas, we state the most relevant properties of µ and sketch some
of their proofs. See [12, Section 10] for more details. We emphasize that one needs very
detailed control on the blow-up behavior of µ−1 to close the energy estimates and that this
is a major difference from the spherically symmetric case.
The first lemma provides the main Gronwall estimate that leads to the degeneracy of
the top-order energy estimates (3.2.21a). As we will explain in Subsubsect. 3.4.3, integral
inequalities in the spirit of (3.3.1) appear when we try to close the energy estimates at the
top order. The lemma is a drastically simplified version of estimates derived in the proof of
[12, Proposition 14.1].
Lemma 3.7 (A difficult Gronwall estimate used at the top order). Let B > 0 be
a constant. There exists a constant C > 0 such that under the data-size and bootstrap
assumptions outlined in Sects. 2.21-2.23, for u ∈ [0, U0 ], solutions f (t) to the inequality
!
Z t
Lµ f (t) ≤ C˚
2 + B
sup f (t0 ) dt0
(3.3.1)
µ
Σu
t0 =0
t0
verify the estimate
f (t) ≤ C˚
2 µ?−(B+C
√
ε)
(t, u).
(3.3.2)
The second lemma is used to show that the below-top-order energy estimates are less
degenerate than the top-order ones. The main idea is that we can gain powers of µ? by
integrating in time.
Lemma 3.8. [12, Proposition 10.3; Gaining powers of µ? by time integration] Let
B > 1 be a constant. Under the data-size and bootstrap assumptions outlined in Sects. 2.212.23, we have the following estimate for u ∈ [0, U0 ]:
Z t
1
(t, u).
(3.3.3)
dt0 ≤ Cµ1−B
?
B
0 , u)
µ
(t
0
t =0 ?
Furthermore,
Z
t
1
9/10 0
(t , u)
t0 =0 µ?
dt0 ≤ C.
(3.3.4)
The third lemma plays a supporting role in establishing the previous two lemmas. In
addition, the estimate (3.3.7) is the ingredient used to show that the spacetime integral
(3.2.18) integral is coercive in the regions where µ is small; see Lemma 3.5.
Lemma 3.9. [12, Sections 10.2 and 10.3; Some key properties of µ.] Consider a fixed
point (t, u, ϑ) and let κt,u,ϑ := Lµ(t, u, ϑ). Under the data-size and bootstrap assumptions
J. Speck
63
outlined in Sects. 2.21-2.23, for 0 ≤ s ≤ t, we have the following estimates61:
Lµ(s, u, ϑ) ∼ κt,u,ϑ ,
µ(s, u, ϑ) ∼ 1 + δt,u,ϑ s.
(3.3.5)
(3.3.6)
Moreover, at any point (t, u, ϑ) with µ(t, u, ϑ) ≤ 1/4, we have
1
Lµ(t, u, ϑ) ≤ − δ̊∗ .
(3.3.7)
4
Discussion of the proof of Lemma 3.9. Thanks to the bootstrap assumptions outlined in Subsect. 2.22, Lemma 3.9 can be proved by using essentially the same arguments that we used
in plane symmetry in the proof of Theorem 1.1; see, for example, the estimates (1.10.6a),
(1.10.15a), (1.10.16), and (1.10.18). The additional terms present away from plane symmetry
involve Pu −tangential derivatives of Ψ, and hence, according to the bootstrap assumptions,
are of a small size ε and make only a negligible contribution to the inequalities. We stress
that the estimate (1.10.15a) can be proved as stated away from plane symmetry and that it
is of fundamental importance for obtaining the estimates of Lemma 3.9, for showing that µ
can vanish in finite time, and for showing that the first rectangular derivatives of Ψ blow-up
when µ vanishes. To obtain (1.10.15a) away from plane symmetry, we rely on the wave
equation in the form (2.19.3) and argue as in the plane symmetric case. The main point is
that all terms on RHS (2.19.3) make only a negligible contribution to the dynamics.
Discussion of the proof of Lemma 3.7. By the standard Gronwall inequality, we deduce
!
Z t
Lµ sup ds .
f (t) ≤ C˚
2 exp B
(3.3.8)
µ
s=0 Σu
s
We now need to pass from (3.3.8) to (3.3.2). The detailed proof is somewhat difficult because
of the presence of the sup on the right.
To reveal the main ideas behind the proof, we first use (3.3.5) and (3.3.6) to deduce that
for 0 ≤ s ≤ t, we have
Lµ
δt,u,ϑ
(s, u, ϑ) ∼
.
µ
1 + δt,u,ϑ s
(3.3.9)
The important point in (3.3.9) is that the same constant δt,u,ϑ appears in the numerator
and denominator. For the sake of illustration, let us simplify the analysis by assuming that
δt,u0 ,ϑ ≤ 0 for u0 ∈ [0, u], ϑ ∈ T.62 Using the fact that for a fixed a > 0, the function
x
f (x) = 1+ax
is increasing on the domain x ∈ (−a−1 , 0], we deduce (recall δt,u0 ,ϑ is nonpositive)
minu0 ∈[0,u],ϑ∈T δt,u0 ,ϑ
Lµ
(s, u, ϑ) ≥
+ Error.
µ
1 + minu0 ∈[0,u],ϑ∈T δt,u0 ,ϑ s
61For
(3.3.10)
the remainder of these notes, the notation A ∼ B indicates, in an imprecise fashion, that A is
well-approximated by B.
62This is indeed what holds for some (u0 , ϑ) close to the formation of the shock. The fact that in reality
it does not hold for all (u0 , ϑ), even close to the time of shock formation, leads to additional technical
complications which we suppress here.
Stable Shock Formation
64
We finally set
δt := 0 min δt,u0 ,ϑ u ∈[0,u],ϑ∈T
(3.3.11)
Lµ δt
sup ≤
+ Error.
µ
1 − δt s
Σu
s
(3.3.12)
and conclude
Note also that, in view of Def. 3.4 and (3.3.6), we deduce that for 0 ≤ s ≤ t, we have
µ? (s, u) ∼ 1 − δt s.
We now integrate (3.3.12) ds from s = 0 to t and use (3.3.13) to deduce that
Z t
Z t
Lµ δt
sup ds ∼
ds
µ
s=0 1 − δt s
s=0 Σu
s
= ln |1 − δt s| ∼ ln µ−1
(t,
u)
?
(3.3.13)
(3.3.14)
(recall that µ? (t, u) ≤ 1 by definition). The desired estimate63 (3.3.2) now easily follows from
(3.3.8) and (3.3.14).
Discussion of the proof of Lemma 3.8. The main idea is that the time integrals on the lefthand sides of (3.3.3)-(3.3.4) are easy to estimate once we have obtained estimates in the
spirit of (3.3.6). To keep the discussion short, we sketch only the proof of (3.3.3) and
only in the case u = 1. A complete proof requires some rather detailed analysis in regions
where µ? is small, so this is really only a proof sketch. We first note that the estimate
(1.10.5c) from plane symmetry can also be obtained away from plane symmetry by using
essentially the same arguments; the additional terms present away from plane symmetry
involve Pu −tangential derivatives of Ψ, and hence, according to the bootstrap assumptions,
are of a small size ε and make only a negligible contribution to the inequalities. Ignoring the
last term on RHS (1.10.5c) in order to simplify the discussion, we obtain µ? (t, 1) ∼ 1 − δ̊∗ t.
With the help of this caricature estimate, we obtain
Z t
Z t
1
1
1
1−B
ds ∼
ds .
δ̊−1
(3.3.15)
∗ (1 − δ̊∗ t)
B
B
µ
(s,
1)
1
−
B
s=0 ?
s=0 (1 − δ̊∗ s)
. µ?1−B (t, 1),
where in the last step, we have used (1.10.5c) and our convention that we can soak factors
of δ̊−1
∗ into our implicit constants. We have thus sketched the main ideas behind proving
(3.3.3).
63Up
to the correction factor σ.
J. Speck
65
3.4. Details on the top-order energy estimates. We now explain some of the main
ideas behind the proof of Prop. 3.6. Throughout SubSubsect. 3.4, ˚
denotes the small datasize parameter from Subsect. 2.21 and ε denotes the small amplitude size corresponding to
the bootstrap assumptions outlined in Subsect. 2.22. For convenience, here we focus on only
the difficult top-order energy estimate (3.2.21a) (with M = 5). To illustrate the main ideas,
we might as well commute the wave equation with a single geometric torus vectorfield Y
(defined in Def. 2.27) pretend that we are at the highest derivative level, and show how to
avoid the derivative loss, as outlined in points (5) and (6) at the beginning of Subsect. 3.1.
We remark that we must also avoid, using similar arguments, the derivative loss when we
commute with L, though the difficulties are somewhat less severe in this case. To proceed,
we commute the equation µg(Ψ) Ψ = 0 with Y to arrive at the following wave equation for
Y Ψ, obtained in (2.20.2):
µg(Ψ) Y Ψ = (X̆Ψ)Y trg/ χ − (Y µ)∆
/Ψ + ··· ,
(3.4.1)
where · · · denotes other error terms that are similar or easier. We recall that the `t,u
tensorfield χ on the right-hand side of (3.4.1) is defined in (2.9.4) (see also (2.10.1a)).
Remark 3.4 (On the importance of terms that are not present). Another crucial
property of the commutation vectorfield set Z is that after commuting Z ∈ Z the through
the operator µg(Ψ) one time, we never produce terms of the form ∇
/ X̆µ or X̆ X̆µ. This
is important because we have no means to bound the top-order derivatives of these terms.
In contrast, as we will see in Subsubsect. 3.4.3, there is a procedure based on modified
quantities64 that allows us to bound the top-order derivatives of the term Y trg/ χ on the righthand side of (3.4.1). This discrepancy occurs even though ∇
/ X̆µ, Y trg/ χ, and X̆ X̆µ are all
third-order derivatives of the eikonal function u.
Our goal is to show how to derive energy estimates for solutions to (3.4.1) without losing
derivatives. In particular, we sketch a proof of how to derive a “top-order” estimate for Y Ψ
of the form
E1/2 [Y Ψ](t, u) + F1/2 [Y Ψ](t, u) ≤ C˚
µ−B
? (t, u),
in the spirit of the top-order estimate in (3.2.21a).. In our proof sketch, we highlight the role
played by Lemma 3.7. To begin, we use (3.4.1), (3.2.7), (3.2.10a), and (3.2.10b) to deduce
that65
Z
2
E[Y Ψ](t, u) + F[Y Ψ](t, u) ≤ C˚
−
(2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$
(3.4.2)
Mt,u
Z
+
(Y µ)(LY Ψ)∆
/ Ψ d$ + · · · .
Mt,u
We clarify that to obtain (3.4.2), we have used the divergence identity (3.2.7) with V :=
T = (1 + 2µ)L + 2X̆ (see (3.1.4)), Y Ψ in the role of Ψ, and F = (X̆Ψ)Y trg/ χ − (Y µ)∆
/Ψ+···
64The
procedure also relies on elliptic estimates in three or more space dimensions.
we mentioned above, the remaining error integrals · · · on the right-hand side of (3.4.2) are similar
or easier to estimate than the explicitly indicated ones, so we ignore them here.
65As
Stable Shock Formation
66
(that is, F is equal to the right-hand side of (3.4.1)). Furthermore, we have replaced the
integrand (T Y Ψ)F from (3.2.7) with the expression
2(X̆Ψ)(X̆Y Ψ)Y trg/ χ − (Y µ)(LY Ψ)∆
/Ψ + ...
with · · · denoting terms that are easier to treat, with the exception of one that we describe
below in Remark 3.10. We also remark that the C˚
2 term on the right-hand side of (3.4.2)
is a data term generated by the Σu0 integral on the right-hand side of (3.2.7), which has
quadratic dependence on the data.
Remark 3.5 (The necessity ofR relying on the coercive spacetime integral). We
have suppressed an error integral Mt,u (Y µ)(X̆Y Ψ)∆
/ Ψ d$ by relegating it to the . . . term
on the right-hand side of (3.4.2). This kind of error integral appears at all commuted orders.
Hence, in order to derive the non-degenerate low-level energy estimates (3.2.21b), we must
find a way to bound it without incurring any degeneracy relative to µ−1
? . One might expect
that this integral is more difficult to estimate than the second one written on the righthand side of (3.4.2) because it involves the transversal derivative factor X̆Y Ψ in place of
LY Ψ. However, the LY Ψ−involving error integral is actually more difficult to estimate
without incurring degeneracy relative to µ−1
? . To obtain a non-degenerate bound for the
LY Ψ−involving error integral, we have to use the null fluxes and the crucially important
coercive spacetime integral ; see the derivation of inequality (3.4.14) below. In contrast, the
arguments
given in Subsubsect. 3.4.2 can easily be modified to show that the error integral
R
(Y µ)(X̆Y Ψ)∆
/ Ψ d$ can be bounded in magnitude by
Mt,u
Z t Z
.ε
|X̆Y Ψ||∇
/ Y Ψ| d$ dt0 + · · ·
(3.4.3)
t0 =0
Z
.ε
Σu
t0
t
t0 =0
E[Y Ψ](t0 , u) dt0 ,
where ε is the small parameter appearing in the bootstrap assumptions of SubSubsect. 2.22,
and we have used Cauchy-Schwarz on Σut0 and Prop. 3.3 to pass to the final inequality in
−1/2
(3.4.3). We can handle the singular factor µ?
with the help of the estimate (3.3.4), and
it is therefore easy to show that the right-hand side of (3.4.3) is a harmless cubic error term
that does not create, at any derivative level, degeneracy of the energy estimates relative to
µ−1
? . In contrast, if we tried to handle the LY Ψ−involving error integral in the same way,
0
we would find the worse factor µ−1
? (t , u) in the integrand. That strategy would fail because
0
µ−1
? (t , u) is too singular to be handled by inequality (3.3.4); the strategy would lead to
an energy estimate that is allowed to blow up like ln µ−1
? . Though at the high derivative
levels this kind of blow-up is completely consistent with the estimates of Prop. 3.6, such
degeneracy would prevent us from deriving the non-degenerate low-level energy estimates
(3.2.21b). This would in turn completely invalidate our strategy since we have no means to
recover our non-degenerate L∞ bootstrap assumptions without the non-degenerate low-level
energy estimates.
3.4.1. Raychaudhuri-type identity. We now highlight the main technical hurdle in proving
Prop. 3.6, which we already mentioned in Subsect. 2.3: the only way that we know how
to estimate the factor Y trg/ χ on the right-hand side of (3.4.2) without losing a derivative is
J. Speck
67
by exploiting an important transport equation that is the exact analog of the well-known
Raychaudhuri equation [22] in General Relativity. The Raychaudhuri-type equation satisfied
by trg/ χ is (see, for example, the proof of [23, Corollary 11.2.1])
1
Lµ
Ltrg/ χ + (trg/ χ)2 + |χ̂|2 = −RicLL +
trg/ χ,
2
µ
(3.4.4)
where Ric is the Ricci curvature of g and χ̂ is the trace-free part of χ. Note that χ̂ ≡ 0 in
two space dimensions in view of the one-dimensional nature of the `t, u. Moreover, the Ricci
tensor (see [23, Corollary 11.1.3]) can be decomposed through a tedious but straightforward
calculation, which yields RicLL := Ricαβ Lα Lβ = − 12 GLL ∆
/ Ψ + Lµ
trg/ χ + · · · . Hence, we arrive
µ
at the transport equation
1
Ltrg/ χ = GLL ∆
/Ψ + ··· ,
2
(3.4.5)
where · · · denotes easier terms that we ignore here.66
As we first explained in Subsect. 2.3, the main difficulty is that after we commute (3.4.5)
with Y , we obtain the equation
1
LY trg/ χ = GLL ∆
/Y Ψ + ··· ,
2
which depends on three derivatives of Ψ, whereas the left-hand side of (3.4.2) only yields
control over two derivatives of Ψ (see Prop. 3.3). Hence, it seems that we are losing a
derivative in our estimates for Y trg/ χ. In Subsubsect. 3.4.3, we explain how to overcome this
difficulty.
3.4.2. The energy estimates ignoring derivative loss. Before we address how to circumvent
the loss in derivatives mentioned above, we first describe how the proof of Prop. 3.6 would
work if we did not have to worry about it. Our discussion will highlight the role of the
coercive spacetime integral inequality (3.2.19) in allowing us to bound certain error integrals
without incurring any energy estimate degeneracy relative to µ−1
? from them. In a complete
proof, this is important for deriving the non-degenerate energy estimates (3.2.21b). To begin,
we imagine that (3.4.1), that is, the equation
µg(Ψ) Y Ψ = (X̆Ψ)Y trg/ χ − (Y µ)∆
/Ψ + ··· ,
(3.4.6)
is the top-order equation and that we are trying to bound the right-hand side of (3.4.2) back
in terms of E, F, etc., so that we can apply a Gronwall-type argument. We will use Prop. 3.3
to connect various L2 norms back to E, F, etc. Here, we ignore the difficult error integral on
the right-hand side of (3.4.2) and instead focus on the second one
Z
(Y µ)(LY Ψ)∆
/ Ψ d$,
(3.4.7)
Mt,u
66In
three or more space dimensions, the analysis is more complicated. For example, to avoid derivative
loss, we need to use elliptic estimates to bound the top-order derivatives of |χ̂|2 , where χ̂ is the trace-free
part of χ; see Remark 3.8.
Stable Shock Formation
68
in which we do not have to worry about derivative loss. As we described in Remark 3.5,
to close the proof, we must derive a bound for (3.4.7) that does not involve any degeneracy
with respect to µ−1
? . To proceed, we will rely on the following pointwise estimate:
|Y µ| . ε.
(3.4.8)
The estimate (3.4.8) is easy to derive by commuting the evolution equation (2.17.1) for µ
with Y , using the bootstrap assumptions (see Subsect. 2.22) to bound the right-hand side,
∂
, and using the smallintegrating the resulting inequality along the integral curves of L = ∂t
data assumption; see [12, Proposition 8.10] for the details. We also use the following simple
comparison estimate (see [12, Lemma 8.2] for a proof):
|∆
/ Ψ| . |∇
/ Y Ψ| + |∇
/ Ψ|.
(3.4.9)
In view of (3.4.8) and (3.4.9), we see that the error integral (3.4.7) can be bounded as follows,
where we split it into the two regions {µ ≤ 1/4} and {µ > 1/4} and we denote the easier
integral generated the second lower-order term on RHS (3.4.9) by “· · · ”:
Z
.ε
1{µ≤1/4} |LY Ψ||∇
/ Y Ψ| d$ + · · ·
(3.4.10)
Mt,u
Z
+ε
1{µ>1/4} |LY Ψ||∇
/ Y Ψ| d$ + · · ·
Mt,u
The main difficulty is present in the first integral in (3.4.10). Indeed, it lacks a µ weight,
involves the geometric torus derivative term |∇
/ Y Ψ|, and a corresponds to a region where
µ can be very small. Hence, the geometric torus derivative coerciveness of the energy-flux
quantities, which is provided by Prop. 3.3, is not sufficient to control it. To overcome the
difficulty, we use the strength of the spacetime integral (3.2.18); see inequality (3.2.19). More
precisely, using Cauchy-Schwarz, we deduce that the first integral on the right-hand side of
(3.4.10) is
Z
Z
2
.ε
1{µ≤1/4} |LY Ψ| d$ + ε
1{µ≤1/4} |∇
/ Y Ψ|2 d$.
(3.4.11)
Mt,u
Mτ,u
Using (3.2.14b), we see that the first term on the right-hand side of (3.4.11) can be bounded
as follows:
Z
Z u
2
ε
1{µ≤1/4} |LY Ψ| d$ . ε
F[Y Ψ](t, u0 ) du0 .
(3.4.12)
u0 =0
Mt,u
In addition, the second term on the right-hand side of (3.4.11) can be bounded as follows:
Z
ε
1{µ≤1/4} |∇
/ Y Ψ|2 d$
(3.4.13)
Mτ,u
. εK[Y Ψ](τ, u),
where in passing to the last line of (3.4.13), we have used the key estimate (3.2.19) for the
coercive spacetime integral.
J. Speck
69
We then insert these estimates into the right-hand side of (3.4.2), ignore the (difficult)
first error integral, and find that
Z u
2
E[Y Ψ](t, u) + F[Y Ψ](t, u) ≤ C˚
+ε
F[Y Ψ](t, u0 ) du0
(3.4.14)
u0 =0
+ εK[Y Ψ](t, u) + · · · .
Clearly, the first integral on the right-hand side of (3.4.14) is treatable with Gronwall’s
inequality (recall that 0 < u < 1). Furthermore, the second term εK[Y Ψ](t, u) can be
treated as a harmless cubic term, consistent with the estimates of Prop. 3.6. Hence, assuming the data-size and bootstrap assumptions outlined in Sects. 2.21-2.23, we have provided
some indication of how to derive a non-degenerate a priori energy estimate of the form
E1/2 [Y Ψ](t,Ru) + F1/2 [Y Ψ](t, u) . ˚
if we did not have to worry about the dangerous error
integral − Mt,u (2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$. As we discuss in Subsubsect. 3.4.3, this dangerous
integral leads to a much worse top-order a priori energy estimate.
3.4.3. Avoiding top-order derivative loss via a Raychaudhuri-type identity. We now confront
the main difficulty in deriving the top-order energy estimate (3.2.21a) (with M = 5): the
potential derivative loss coming from the Y trg/ χ term in the error integral
Z
−
(2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$
Mt,u
on the right-hand side of (3.4.2). We are still imagining, for the sake of illustration, that
the second-order derivatives of Ψ represent the top order. The main point is that there is a
procedure, outlined below, that allows us to replace the error integral under consideration
with
Z
Lµ
−4
(X̆Y Ψ)2 d$ + · · · ,
(3.4.15)
Mt,u µ
where the · · · integrals are similar in nature
R or easier. We
R tcanRthen use the coerciveness property (3.2.14a) and the co-area formula Mt,u · · · d$ = t0 =0 Σu · · · d$dt0 to bound (3.4.15)
t0
in magnitude by
!
Z t
Lµ sup E[Y Ψ](t, u) dt0 .
(3.4.16)
≤4
µ
Σu
t0 =0
t0
Thus, recalling (3.4.2), we find that
E[Y Ψ](t, u) + F[Y Ψ](t, u) ≤ C˚
2 + 4
Z
t
t0 =0
!
Lµ sup E[Y Ψ](t0 , u) dt0 + · · · ,
µ
Σu
t0
(3.4.17)
where the constant “4” on the right-hand side of (3.4.17) is a “structural constant,” the · · ·
terms are similar in nature or easier, and ˚
is the size of the data.
We can now appeal to Lemma 3.7 to derive an priori estimate for E[Y Ψ](t, u)+F[Y Ψ](t, u).
That is, ignoring the · · · terms on the right-hand side of (3.4.17), and taking the square root
of the estimate provided by (3.3.2), we have
E1/2 [Y Ψ](t, u) + F1/2 [Y Ψ](t, u) ≤ C˚
µ?−(2+C
√
ε)
(t, u).
(3.4.18)
Stable Shock Formation
70
We have thus explained the main ideas behind the top-order estimate (3.2.21a) (with M = 5).
Remark 3.6 (The importance of the structural constants). Note that the structural
constant “4” that appears in (3.4.17) is independent of the number of times that we commute
the wave equation with vectorfield operators. This observation is important because the above
arguments, leading to (3.4.18), have shown that the structural constant affects the power
of µ−1
? appearing in the top-order energy estimates and hence the number of derivatives we
need to close the estimates.
Remark 3.7 (The rate of top-order energy blow-up in terms of powers of µ−1
? ).
In our argument above, we have ignored some other error integrals that can
be treated with
√
−(2+C ε)
similar ideas but that lead to a larger blow-up rate than the one µ?
(t, u) explicitly
indicated on the right-hand side of (3.4.18); see also Remark 3.10. For this reason, the
blow-up rate of the top-order energy on the right-hand side of (3.2.21a) is µ?−5.9 (t, u).
It remains for us to explain the procedure used above, which allowed us to replace the
integral
Z
−
(2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$
Mt,u
with (3.4.15). The procedure is based on a renormalized Raychaudhuri equation,67 which we
explain in Subsubsect. 3.4.4.
3.4.4. The renormalized Raychaudhuri equation. We begin by recalling equation (3.4.5):
1
Ltrg/ χ = GLL ∆
/Ψ + ··· .
(3.4.19)
2
To avoid the derivative loss, we need to take advantage of the wave equation in the form
(see (2.19.1a) and (2.19.3))
0 = µg(Ψ) Ψ = −L(µLΨ + 2X̆Ψ) + µ∆
/ Ψ + l.o.t.,
where l.o.t. denotes terms involving ≤ 1 derivative of Ψ. Hence, using the wave equation,
we can replace, up to a crucially important factor of µ−1 and l.o.t., the term 21 GLL ∆
/Ψ
1
in (3.4.19) with a perfect L derivative of 2 GLL (µLΨ + 2X̆Ψ) and then bring this perfect
L derivative over to the left-hand side of (3.4.19). Furthermore, one can show that the
remaining second derivatives of Ψ in the · · · terms on the right-hand side of (3.4.19) are
also perfect L derivatives, and thus we can bring those terms to the left as well. In total, at
the expense of the singular factor µ−1 , we can renormalize away all of the terms in equation
(3.4.19) that lose derivatives relative to Ψ, thereby obtaining an equation for a “modified”
version of trg/ χ of the form LModified) = l.o.t.
Hence, the important structure used by Christodoulou in [3] can be restated as follows: for
solutions to g(Ψ) Ψ = 0, the RicLL term in the Raychaudhuri equation (3.4.4) is, up to lowerorder terms, a perfect L derivative of the first derivatives of Ψ. To close our estimates, what
we really need are higher-order68 versions of this identity. In particular, we can commute the
Raychaudhuri-type identity with Y to obtain a transport equation equation for a “modified”
67The
68In
same idea was also used earlier, in a different context, in [19].
fact, we need only top-order versions of the identity.
J. Speck
71
version of Y trg/ χ that does not lose derivatives relative to Ψ. We make this precise in the
following definition, where (Y )X is the “modified” quantity.
Definition 3.5 (Modified version of Y trg/ χ). We define the modified quantity
follows:
X := µY trg/ χ + Y X,
1
1
X := −GLL X̆Ψ − µtrg/ G
/ LΨ − µGLL LΨ + µG
/L# · ∇
/ Ψ.
2
2
(Y )
X as
(Y )
(3.4.20a)
(3.4.20b)
The term G
/L# appearing in (3.4.20b) is the `t,u −tangent vectorfield formed by g−orthogonally
projecting the vectorfield with rectangular components Gαν Lα onto the `t,u .
In total, the strategy described above can be used to show that (Y )X verifies a transport
equation of the following delicate form.
Lemma 3.10. [12, 6.2; Transport equation for the modified quantity] The quantity
X defined in (3.4.20a) verifies the transport equation
Lµ
1
Lµ
(Y )
(Y )
L X − 2
X =
− trg/ χ
trg/ χ − 2
Y X + Error,
(3.4.21)
µ
2
µ
(Y )
where Error denotes error terms that involve ≤ 2 derivatives of Ψ and that do not involve
any factors of µ−1 .
We stress again that the advantage of (3.4.21) over the unmodified equation LY trg/ χ =
Ψ + · · · is that the right-hand side of equation (3.4.21) depends on ≤ 2 derivatives
of Ψ. Hence, equation (3.4.21) can be used to derive L2 estimates for (Y )X that do not lose
derivatives relative to Ψ.
1
G ∆
/Y
2 LL
Remark 3.8 (The need for elliptic estimates in three or more spatial dimensions).
In n ≥ 3 spatial dimensions, hiding in the terms Error in (3.4.21) lies another technical
headache that we will briefly mention but not dwell on. Specifically, there is a quadratically
small term, roughly of the form µχ̂ · L
/Y χ̂, that formally involves the same number of χ
(Y )
derivatives as the modified quantity X (that is, one) but that cannot be directly estimated
back in terms of (Y )X . Here, χ̂ is the trace-free part69 of the `t,u tensor (2.9.4) and L
/Y denotes
Lie differentiation with respect to Y followed by projection onto the `t,u . The term L
/Y χ̂
involves three derivatives of the eikonal function u. Thus, as we described in SubSubsect. 2.3,
it will lead to derivative loss if not properly handled. To derive suitable L2 estimates for
L
/Y χ̂, we derive a family of elliptic estimates on the n − 2 dimensional hypersurfaces that are
analogs of the `t,u . The main ideas behind this strategy can be traced back to ChristodoulouKlainerman’s proof of the stability of Minkowski spacetime [5]. Similar strategies were also
employed in [19] and [3]. The main point is that the elliptic estimates allow us to estimate
kµL
/Y χ̂kL2 (`t,u ) back in terms of kµY trg/ χkL2 (`t,u ) plus errors, and that µY trg/ χ can be controlled
in L2 by using the L2 estimates for (Y )X and the up-to-second-order L2 estimates for Ψ.
Remark 3.9. In a complete proof, one must invert the transport equation (3.4.21) and
obtain suitable L2 estimates for (Y )X . However, this is not an easy task; see the proof of
69Note
that in two space dimensions, χ̂ = 0 in view of the one-dimensional nature of the `t,u in this case.
Stable Shock Formation
72
n
o
[12, Lemma 14.8] for the details. The main difficulty is that the factors 2 Lµ
−
tr
χ
and
g
/
µ
o
n
1
tr χ − 2 Lµ
in (3.4.21) have a drastic effect on the behavior of (Y )X and require a careful
2 g/
µ
analysis.
We now return to our explanation of how we are able to replace the error integral
Z
−
(2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$
Mt,u
with (3.4.15). We first use the identity Y trg/ χ = µ−1(Y )X − µ−1 Y X, which follows easily
from Def. 3.5, to replace Y trg/ χ with two terms that do not lose derivatives. As we described
in Remark 3.9, the most difficult analysis corresponds to the error integral generated by the
term µ−1(Y )X . We do not want to burden the reader with the large number of technical
complications that arise in the analysis of this error
integral. Instead, we focus on the error
R
integral generated by the other term, namely Mt,u 2(X̆Ψ)µ−1 (Y X)X̆Y Ψ d$. The difficult
part of this error integral comes from the top-order part of Y applied to the first term
−GLL X̆Ψ on the right-hand side of (3.4.20b). That is, we focus on the following error
integral:
Z
1
−2
(X̆Ψ)GLL (X̆Y Ψ)2 d$.
(3.4.22)
µ
Mt,u
Though the integral (3.4.22) does not lose derivatives, it is nonetheless difficult to bound.
To handle it, we observe the following critically important structure, first observed by
Christodoulou in [3]: by using the transport equation Lµ = 12 GLL X̆Ψ+µError (see (2.17.1)),
we can rewrite (3.4.22) as
Z
Lµ
−4
(X̆Y Ψ)2 d$ + · · · ,
(3.4.23)
Mt,u µ
which is precisely the integral (3.4.15) that we successfully treated above. We have thus
sketched the main ideas behind the procedure that allows us to avoid losing derivatives.
Remark 3.10 (Other difficult top-order error integrals). The spacetime integral
Z
−
(X̆Ψ)(Y trg/ χ)LY Ψ d$
(3.4.24)
Mt,u
is also present on RHS (3.4.2). Like the dangerous error integral
Z
−
(2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$
Mt,u
that we discussed above, the error integral (3.4.24) also leads to µ−1
? degeneracy in the toporder energy estimates; it contributes to the large size of the exponent −5.9 corresponding to
the top-order estimate (3.2.21a) (with M = 5). To handle the integral (3.4.24), we rely on a
different argument than the one we described in Sect. 3.4.3. The main idea is to integrate by
parts in the integral (3.4.24) in order to trade the Y derivative on trg/ χ for the L derivative
on Y Ψ. The gain is that whenever the top-order derivative of an eikonal function quantity
such as trg/ χ involves an L derivative, we do not have to worry about losing derivatives
J. Speck
73
because we have a “direct expression” for these quantities based on the fact that they verify
a transport equation in the direction of L (see, for example, equation (3.4.5)). Hence, we
can avoid working with the fully modified quantities (3.4.20a)-(3.4.20b). However, moving
the L derivative generates some difficult Σut boundary error integrals that lead to top-order
degeneracy relative to µ−1
? , similar to the degeneracy we encountered in Lemma 3.7. In
deriving the desired estimates, although we do not need to use fully modified quantities of
the form (3.4.20a)-(3.4.20b), we do need to define and use a partially modified version both
trg/ χ in order to avoid the appearance of certain error integrals that have unfavorable sizes.
We do not want to further burden the reader with these technical details here, so we do not
pursue this issue further.
3.5. Descending below top order. If we applied the strategies of Subsubsects. 3.4.3 and
3.4.4 at all derivative levels, then all of the energy-flux estimates would degenerate in the same
way as the top-order estimate (3.2.21a) (with M = 5) with respect to µ−1
? . In particular, we
would not recover the non-degenerate estimates (3.2.21b). This would in turn prevent us
from recovering the L∞ bootstrap assumptions outlined in Subsect. 2.22, which are based
on (3.2.21b) and Sobolev embedding. Hence, we would not be able to show that the terms
we have deemed small errors are in fact small, and the entire proof would break down.
To overcome this difficulty, we note that below top order, we can allow the loss of one
derivative in the difficult error integral. In particular, there is no need to use the complicated
procedure that led to the difficult top-order integral (3.4.15). It turns out that in avoiding
this procedure, we can prove below-top-order energy estimates that enjoy less µ−1
? degeneracy
compared to the top-order estimate. As we shall see, the gain is a consequence of Lemma 3.8
and the availability of favorable powers of t.
As before, in the following discussion, ˚
denotes the small data-size parameter from Subsect. 2.21. To illustrate our strategy in some detail, let us imagine that three derivatives of
Ψ in L2 (which corresponds70 to E[P 2 Ψ], etc.) represents the top order. We also imagine,
consistent with (3.2.21a), that the top-order energy-flux quantities verify bounds of the form
E1/2 [P 2 Ψ](t, u) + F1/2 [P 2 Ψ](t, u) . ˚
µ−B
? (t, u),
(3.5.1)
for a positive constant B. We will show how the estimate (3.5.1) can be used to derive a
bound for the just-below-top-order quantities E1/2 [Y Ψ](t, u) + F1/2 [Y Ψ](t, u) with a smaller
(and therefore less degenerate) power of µ−1
? on the right-hand side.
To proceed, we commute equation (3.4.5) with Y to obtain
1
/Y Ψ + ··· .
LY trg/ χ = GLL ∆
2
From (3.5.2) and (3.4.9), we obtain
LY trg/ χ . |∆
/ Y Ψ| + · · · . |∇
/ Y Y Ψ| + · · · ,
(3.5.2)
(3.5.3)
where ∇
/ Y Y Ψ is a top-order term (that is, above the present order). Using the coerciveness
−1/2
property (3.2.14a) (which leads to the appearance of another factor of µ? (t, u)!) and the
and in (3.5.1), P 2 denotes an arbitrary second-order differential operator corresponding to differentiating with respect to two vectorfields belonging to the Pu −tangential set of commutators P.
70Here
Stable Shock Formation
74
top-order estimate (3.5.1), and ignoring the · · · terms, we deduce
LY trg/ χ 2 u . µ−1/2
(t, u)E1/2 [Y Y Ψ](t, u) . ˚
µ?−B−1/2 (t, u).
?
L (Σ )
(3.5.4)
t
∂
Recalling that L = ∂t
and taking into account the fact that the length form dλg/ on `t,u (see
(2.13.2)), which is inherent the norm k · kL2 (Σut ) , verifies dλg/ ≈ dϑ0 (with ϑ0 the geometric
torus coordinate induced on `t,u ), it is not too difficult (see [12, Lemma 14.3]) to integrate
(3.5.4) in time to deduce
Z t
Y trg/ χ 2 u . ˚
(3.5.5)
µ?−B−1/2 (t0 , u) dt0 + · · · .
L (Σ )
t
t0 =0
Using the estimate (3.3.3) of Lemma 3.8 to bound the time-integral on the right-hand side
−B+1/2
of (3.5.5) by . µ?
(t, u), we gain a power of µ? through the time integration:
Y trg/ χ 2 u . ˚
µ?−B+1/2 (t, u) + · · · .
(3.5.6)
L (Σ )
t
We now bound the first integral on the right-hand side of (3.4.2), that is, the integral
Z
(2X̆Ψ)(Y trg/ χ)X̆Y Ψ d$,
−
Mt,u
by using the estimate (3.5.6), the bootstrap assumption estimate |X̆Ψ| . 1 from SubSubsect. 2.22, the coerciveness property (3.2.14a), and Cauchy-Schwarz. It therefore follows
from (3.4.2) that the following bound holds:
sup {E[Y Ψ](s, u) + F[Y Ψ](s, u)}
(3.5.7)
s∈[0,t]
Z
t
µ−B+1/2
(t0 , u)E1/2 [Y Ψ](t0 , u) dt0 + · · ·
?
Z t
2
1/2
µ−B+1/2
≤ C˚
+ Cε˚
sup E [Y Ψ](s, u)
(t0 , u) dt0 .
?
2
≤ C˚
+ Cε˚
t0 =0
t0 =0
s∈[0,t]
Z
t
Using the estimate (3.3.3) to obtain
t0 =0
µ−B+1/2
(t0 , u) dt0 . µ−B+3/2
(t, u), we deduce from
?
?
(3.5.7) that for ε and ˚
sufficiently small, we have
E1/2 [Y Ψ](t, u) + F1/2 [Y Ψ](t, u) . ˚
µ?−B+3/2 (t0 , u) + · · · .
(3.5.8)
The inequality (3.5.8) has thus yielded the desired gain in µ? compared to the top-order
energy bound (3.5.1).
Remark 3.11. Inequality (3.5.8) is mildly misleading in the sense that in deriving it, we
have ignored some slightly worse error terms that only allow us to gain a single power of µ? ,
rather than the gain of a power of 3/2 suggested by (3.5.8).
We have thus explained the main ideas of how to descend one level below the top order in
the energy estimate hierarchy of Prop. 3.6. One can continue the descent, each time using
Lemma 3.8 to gain a power of µ? . Furthermore, the estimate (3.3.4) allows us to eventually
descend to the estimates (3.2.21b), which no longer degenerate at all, even as a shock forms!
J. Speck
75
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