Formation of point shocks for 3D compressible Euler

 

 

Formation of point shocks for 3D compressible Euler

Tristan Buckmaster*

Steve Shkoller†

Vlad Vicol‡

Abstract

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide
a constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum
regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove
that for an open set of Sobolev-class initial data which are a small L8 perturbation of a constant state,
there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The
blow up time and location can be explicitly computed, and solutions at the blow up time are smooth
except for a single point, where they are of cusp-type with H¨older C 1{3 regularity. Our proof is based on
the use of modulated self-similar variables that are used to enforce a number of constraints on the blow
up profile, necessary to establish global existence and asymptotic stability in self-similar variables.

Contents

1

Introduction

2 Self-similar shock formation

3 Main results

4 Bootstrap assumptions

5 Constraints and evolution of modulation variables

6 Closure of bootstrap estimates for the dynamic variables

7 Preliminary lemmas

8 Bounds on Lagrangian trajectories

9 L8 bounds for ˚ζ and S

10 Closure of L8 based bootstrap for Z and A

11 Closure of L8 based bootstrap for W

12 9H k bounds

A Appendices

13 Conclusion of the proof of the main theorems

*Department of Mathematics, Princeton University, Princeton, NJ 08544, buckmaster@math.princeton.edu
†Department of Mathematics, UC Davis, Davis, CA 95616, shkoller@math.ucdavis.edu.
‡Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, vicol@cims.nyu.edu.

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Formation of points shocks for 3D Euler

1 Introduction

A fundamental problem in the analysis of nonlinear partial differential equations concerns the finite-time
breakdown of smooth solutions and the nature of the singularity that creates this breakdown. In the context
of gas dynamics and the compressible Euler equations which model those dynamics, the classical singularity
is a shock. When the initial disturbance to a constant state is sufficiently strong, created for example by
explosions, supersonic projectiles, or a kingfisher shot out of a cannon, violent pressure changes lead to a
progressive self-steepening of the wave, which ends in a shock.

Our main goal is to give a detailed characterization of this shock formation process leading to the first
singularity, for the isentropic compressible Euler equations in three space dimensions. Specifically, we shall
give a precise description of the initial data from which smooth solutions to the Euler equations evolve,
steepen, and form a stable generic shock in finite time, in which the gradient of velocity and gradient of
density become infinite at a single point, while the velocity, density, and vorticity remain bounded. In the
process, we shall provide the exact blow up time, blow up location, and regularity of the three-dimensional
generic blow up profile. Away from this single blow up point, the solution remains smooth.

Let us now introduce the mathematical description. The three-dimensional isentropic compressible

Euler equations are written as

Btpρuq ` divxpρu b uq ` ∇xppρq “ 0 ,
Btρ ` divxpρuq “ 0 ,

(1.1a)

(1.1b)

where x “ px1, x2, x3q P R3 and t P R are the space and time coordinates, respectively. The unknowns are
the velocity vector field u : R3 ˆ R Ñ R3, the strictly positive density scalar field ρ : R3 ˆ R Ñ R`, and
the pressure p : R3 ˆ R Ñ R`, which is defined by the ideal gas law

ppρq “ 1

γ ργ ,

γ ą 1 .

a

The sound speed cpρq “
2 . The Euler equations (1.1) are
a system of conservation laws: (1.1a) is the conservation of momentum and (1.1b) is conservation of mass.
Defining the scaled sound speed by σ “ 1

α ρα, (1.1) can be equivalently written as the system

Bp{Bρ is then given by c “ ρα where α “ γ´1

We let ω “ curlx u denote the vorticity vector and we shall refer to the vector ζ “ ω
which satisfies the vector transport equation

ρ as the specific vorticity,

Btu ` pu ¨ ∇xqu ` ασ∇xσ “ 0 ,
Btσ ` pu ¨ ∇xqσ ` ασ divx u “ 0 .

Btζ ` pu ¨ ∇xqζ ´ pζ ¨ ∇xq u “ 0 .

(1.2a)

(1.2b)

(1.3)

Our proof of shock formation relies upon a transformation of the problem from the original space-time
variables px, tq to modulated self-similar space-time coordinates py, sq, and on a change of unknowns from
pu, σq to a set of geometric Riemann-like variables pW, Z, Aq in the self-similar coordinates. The singularity
model is characterized by the behavior near y “ 0 of the stable, stationary solution W “ W py1, y2, y3q
(described in Section 2.7 and shown in Figure 1) of the 3D self-similar Burgers equation
˘

`

´ 1

2 W `

3
2 y1 ` W

By1W ` 1

2 y2By2W ` 1

2 y3By3W “ 0 .

(1.4)

For a fixed T , the vector v “ pv1, v2, v3q given by

v1px1, x2, x3, tq “ pT ´ tq

1
2 W

˜

¸

x1
pT ´ tq

3
2

,

x2
pT ´ tq

1
2

,

x3
pT ´ tq

1
2

, v2 ” 0 , v3 ” 0 ,

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Formation of points shocks for 3D Euler

is the solution of the 3D Burgers equation in original variables, Btv ` pv ¨ ∇xqv “ 0, forming a shock at a
single point at time t “ T . An explicit computation shows that the Hessian matrix By1∇2
yW |y“0 is strictly
positive definite. This property ensures that the blow up profile W is generic in the sense described by
Christodoulou in equation (15.2) of [6]. This genericity condition, in turn, provides stability of the shock
below.
profile

as we will

the Euler

equations

solutions

explain

detail

for

to

in

A precise description of shock formation necessitates explicitly defin-
ing the set of initial data which lead to a finite-time singularity, or shock.
Additionally, from the initial datum alone, one has to be able to infer
the following properties of the solution at the first shock: (a) the geom-
etry of the shock set, i.e., to classify whether the first singularity occurs
along either a point, multiple points, a line, or along a surface; (b) the
precise regularity of the solution at the blow up time; (c) the explicitly
computable space-time location of the first singularity; (d) the stability
of the shock. For the last condition (d), by stability, we mean that for
any small, smooth, and generic (meaning outside of any symmetry class)
perturbation of the given initial data, the Euler dynamics yields a smooth
solution which self-steepens and shocks in finite time with the same shock
set geometry, with a shock location that is a small perturbation, and with the same shock regularity; that is,
properties (a)–(c) are stable.

Figure 1: The stable generic shock
profile (shown in 2D).

As an example, the solution W shown in Figure 1 is stable: the shock occurs at a single point, and any
small generic perturbation of W (as we will prove) also develops a shock at only a single point, and with
the same properties as those satisfied by W . On the other hand, a simple plane wave solution of the Euler
equations that travels along the x1 axis and is constant in px2, x3q produces a finite-time shock along an
entire plane, but a small perturbation of this simple plane wave solution can produce a very different shock
geometry (any of the sets from condition (a) are possible). Our main result can be roughly stated as follows:

Theorem 1.1 (Rough statement of the main theorem). For an open set of smooth initial data without vac-
uum, with nontrivial vorticity, and with a maximally negative gradient of size Op1{εq, for ε ą 0 sufficiently
small, there exist smooth solutions of the 3D Euler equations (1.1) which form a shock singularity within
time Opεq. The first singularity occurs at a single point in space, whose location can be explicitly computed,
along with the precise time at which it occurs. The blow up profile is shown to be a cusp with C 1{3 regularity,
and the singularity is given by an asymptotically self-similar shock profile which is stable with respect to the
H kpR3q topology for k ě 18.

A precise statement of the main result will be given below as Theorem 3.1

1.1 Prior results on shock formation for the Euler equations

In one space dimension, the isentropic Euler equations are an example of a 2 ˆ 2 system of conservation
laws, which can be written in terms of the Riemann invariants z “ u ´ c{α and w “ u ` c{α introduced in
[28]; the functions z and w are constant along the characteristics of the two wave speeds λ1 “ u ´ c and
λ2 “ u ` c. Using Riemann invariants, Lax [20] proved that finite-time shocks can form from smooth data
for general 2 ˆ 2 genuinely nonlinear hyperbolic systems. The proof showed that the derivative of w must
become infinite in finite time, but the nature of the proof did not permit for any classification of the type of
shock that forms. Generalizations and improvements of Lax’s result were obtained by John [17], Liu [21],
and Majda [23], for the 1D Euler equations. Again, these proofs showed that either a slope becomes infinite
in finite time or that (equivalently) the distance between nearby characteristics approaches zero, but we note

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Formation of points shocks for 3D Euler

that a precise description of the shock was not given. See the book of Dafermos [12] for a more extensive
bibliography of 1D results.

For the 3D Euler equations, Sideris [29] formulated a proof by contradiction (based on virial identities)
that C1 regular solutions to (1.1) have a finite lifespan; in particular, he showed that Opexpp1{εqq is an
upper bound for the lifespan (of 3D flows) for data of size ε. The proof, however, did not reveal the type of
singularity that develops, but rather, that some finite-time breakdown of smooth solutions must occur.

The first proof of shock formation for the compressible Euler equations in the multi-dimensional setting
was given by Christodoulou [6] for relativistic fluids and with the restriction of irrotational flow, and later
by Christodoulou-Miao [9] for non-relativistic, irrotational flow.1 This geometric method uses an eikonal
function (see also [8], [18]), whose level sets correspond to characteristic surfaces; it is shown that in finite
time, the distance between nearby characteristics tends to zero. For irrotational flows, the isentropic Euler
equations can be written as a scalar second-order quasilinear wave equation. The first results on shock
formation for 2D quasilinear wave equations which do not satisfy Klainerman’s null condition [19] were
established by Alinhac [1, 2], wherein a detailed description of the blow up was provided. The first proof
of shock formation for fluid flows with vorticity was given by Luk-Speck [22], for the 2D isentropic Euler
equations. Their proof uses Christodoulou’s geometric framework and develops new methods to study the
vorticity transport. In [6, 9, 22], solutions are constructed which are small perturbations of simple plane
waves. It is shown that there exists at least one point in spacetime where a shock must form, and a bound
is given for this blow up time; however, since the construction of the shock solution is a perturbation of
a simple plane wave, there are numerous possibilities for the type of singularity that actually forms. In
particular, their method of proof does not distinguish between these different scenarios. To be precise, a
simple plane wave solution of the 2D isentropic Euler equations that travels along the x1 axis and is constant
in x2 produces a finite-time shock along a line, but a small perturbation of this simple plane wave solution
can produce a very different singular set, with blow up occurring on different spatial sets such as one point,
multiple points, or a line.

In our earlier work [3], we considered solutions to the 2D isentropic Euler equations with Op1q vorticity
and with azimuthal symmetry. Using modulated self-similar variables, we provided the first construction of
shock solutions that completely classify the shock profile: the shock is an asymptotically self-similar, stable,
a generic 1D blow up profile, with explicitly computable blow up time and location, and with a precise
description of the C 1{3 H¨older regularity of the shock. Azimuthal symmetry allowed us to use transport-type
L8 bounds which simplified the technical nature of the estimates, but the proof already contained some of
the fundamental ideas required to study the full 3D Euler equations with no symmetry assumptions.

1.2 The variables used in the analysis and strategy of the proof

We now introduce the variables used in the analysis of shock formation. For convenience we first rescale
time t ÞÑ t, as described in (2.1). Associated to certain modulation functions (described in Section 1.3 be-
low), are a succession of transformations for both the independent variables and the dependent variables. In
order to dynamically align the blow up direction with the e1 direction, a time-dependent rotation and trans-
lation are made in (2.5) which maps x to rx, with u, σ, and ζ transformed to ru, rσ, and rζ via (2.6) and (2.8).
Fundamental to the analysis of stable shock formation, we make a further coordinate transformation rx ÞÑ x
given by (2.15); this mapping modifies the rx1 variable by a function f prx2, rx3, tq “ 1
2 φνγptqrxν rxγ which is
quadratic in space and dynamically modulated by φνγptq. The parameterized surface pf prx2, rx3, tq, rx2, rx3q
can be viewed as describing the steepening shock front near x “ 0, and provides a time-dependent or-
thonormal basis along the surface, given by the vectors the unit normal vector Npˇx, tq and the two unit

1For the restricted shock development problem, in which the Euler solution is continued past the time of first singularity but

vorticity production is neglected, see the discussion in Section 1.6 of [7].

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tangent vectors T2pˇx, tq, and T3pˇx, tq defined in (2.14) and (2.13). Together with the coordinate transfor-
mation rx ÞÑ x, the functions ru, rσ, and rζ are transformed to ˚u, ˚σ, and ˚ζ using (2.16) and (2.20). Moreover,
the Riemann variables w “ ˚u ¨ N ` ˚σ and z “ ˚u ¨ N ´ ˚σ, as well as the tangential components of velocity
aν “ ˚u ¨ Tν are introduced in (2.22).

Finally, we map px, tq to the modulated self-similar coordinates py, sq using the transformation (2.25).
The variables ˚u, ˚σ, and ˚ζ are mapped to their self-similar counterparts U , S, and Ω via (2.32a), (2.32b), and
(2.35), while w, z, and aν are mapped to the self-similar variables e´ s

2 W ` κ, Z, and Aν in (2.26).

As a consequence of this sequence of coordinate and variable changes, the Euler equations in the original
variables (1.2) for the unknowns pupx, tq, σpx, tqq become the self-similar evolution (2.34) for the unknowns
pU py, sq, Spy, sqq. Of crucial importance for our analysis is the evolution of the self-similar Riemann type
variables pW py, sq, Zpy, sq, Apy, sqq in (2.28), which encode the full Euler dynamics in view of (2.33).
The key insight to our analysis is that the self-similar Lagrangian trajectories associated to the W equation
escape exponentially fast towards spatial infinity if their starting label is at a fixed (small) distance away
from the blowup location y “ 0, whereas the Lagrangian trajectories for Z and A escape towards infinity
independently of their starting label, spending at most an Op1q time near y “ 0. This exponential escape
towards infinity is what allows us to transfer information about spatial decay of various derivatives of W into
integrable temporal decay for several damping and forcing terms, when viewed in Lagrangian coordinates.
As opposed to our earlier work [3], these pointwise estimates for pW, Z, Aq do not close by themselves,
as there is a loss of a ˇ∇ derivatives when the equations are analyzed in L8. This difficulty is overcome
by using the energy structure of the 3D compressible Euler system, which translates into a favorable 9H k
estimate for the self-similar variables pU, Sq, for k sufficiently large (e.g. k ě 18 is sufficient).

Coupled to the pW, Z, Aq evolution we have a nonlinear system of 10 ODEs which describe the evolution
of our 10 dynamic modulation variables κ, τ, n2, n3, ξ1, ξ2, ξ3, φ22, φ23, φ33, whose role is to dynamically
enforce constraints for W, ∇W and ∇2W at y “ 0, cf. (5.1).

For all s ă 8, or equivalently, t ă T˚, the above described transformations are explicitly invertible.
Therefore, our main result, Theorem 3.1, is a direct consequence of Theorem 3.4, which establishes the
global-in-self-similar-time stability of the solution pW, Z, Aq, in a suitable topology near the blowup profile
pW , 0, 0q, along with the stability of the 10 ODEs for the modulation parameters. In turn, this is achieved
by a standard bootstrap argument: fix an initial datum with certain quantitative properties; then postulate
that these properties worsen by a factor of at most K, for some sufficiently large constant K; to conclude
the proof, we a-posteriori show that in fact the solutions’ quantitative properties worsen by a factor of at
most K{2. Invoking local well-posedness of smooth solutions [23] and continuity-in-time, we then close
the bootstrap argument, yielding global-in-time solutions bounded by K{2.

The global existence of solutions pW, Z, Aq in self-similar variables, together with the stability of the
W , leads to a precise description of the blow up of a certain directional derivative of w. For the dynamic
modulations functions mentioned above, the function τ ptq converges to the blow up time T˚, the vector ξptq
converges to the blow up location ξ˚, and the normal vector Npt, ¨q converges to N˚ as t Ñ T˚. Moreover,
we will show that

pNpt, ξ2ptq, ξ3ptqq ¨ ∇xqwpξptq, tq “ esBy1W p0, sq “ ´ 1

τ ptq´t Ñ ´8

as

t Ñ T˚ .

(1.5)

Thus, it is only the directional derivative of w in the N direction that blows up as t Ñ T˚, while the tangen-
tial directional derivatives pT2pt, ξ2ptq, ξ3ptqq ¨ ∇xqwpξptq, tq and pT3pt, ξ2ptq, ξ3ptqq ¨ ∇xqwpξptq, tq remain
uniformly bounded as t Ñ T˚. Additionally, we prove that the directional derivative Npt, ξ2ptq, ξ3ptqq ¨ ∇x
of z and a remain uniformly bounded as t Ñ T˚. Thus, (1.5) shows that the wave profile steepens along the
N direction, leading to a single point shock at the space time location pξ˚, T˚q.

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1.3 Modulation variables and the geometry of shock formation

The symmetries of the 3D Euler equations lead to dynamical instabilities in the space-time vicinity of the
shock, which are amplified when considering self-similar variables [14]. Our analysis relies crucially on the
size of this invariance group. We recall that the 3D Euler equations are invariant under the 10 dimensional
Lie group of Galilean transformations consisting of rotations, translations, and rigid motions of spacetime,
as well as the 2 dimensional group of rescaling symmetries. Explicitly, given a time shift t0 P R, a space
shift x0 P R3, a velocity shift (Galilean boost) v0 P R3, a rotation matrix R P SOp3q, a hyperbolic scaling
parameter λ P R`, a temporal scaling parameter µ P R`, and a solution pu, σq of the 3D compressible Euler
system (1.2), where as before σ “ p1{αqρα, the pair of functions

unewpx, tq “

σnewpx, tq “

σ

ˆ

1
µ
1
µ

RT u
ˆ

Rpx ´ x0 ´ tv0q
λ

,

t ´ t0
λµ
˙

Rpx ´ x0 ´ tv0q
λ

,

t ´ t0
λµ

˙

` v0

also solve the 3D Euler system (1.2), and hence, these transformations define the 12 dimensional group of
symmetries of the 3D Euler equations. For simplicity we sacrifice 5 of these 12 of these degrees of freedom:
we fix a temporal rescaling since we choose to prove that an initial slope of size (negative) 1{ε causes a
blowup in time ε ` Opε2q (just as for the 1D Burgers equation); we discard the degree of freedom provided
by hyperbolic scaling since it is not necessary for our analysis to fix the determinant of By1∇2
y W to be
constant in time; we also only utilize two of the three degrees of freedom in the rotation matrix R P SOp3q
since we choose a particular basis for the plane orthogonal to the shock direction; lastly, we discard two
Galilean boosts as we do not need to modulate Aνp0, sq to be constant in time. This leaves us with a 7
dimensional group of symmetries which we use at the precise shock location. Additionally, since in self-
similar coordinates our blow up is modeled by the shear flow in the x1 direction, using a quadratic-in-ˇx
shift function, we are also able to modulate translational instabilities away from the shock in the directions
orthogonal to the shock.

A fundamental aspect of our analysis is to show that there is a correspondence between the instabilities
of the Euler solution and the symmetries discussed above. Thus, in order to develop a theory of stable
shock formation, it is of paramount importance to be able to modulate away these instabilities. This idea
was successfully used in [24–26] in the context of the Schr¨odinger equation, and in [27] for the nonlinear
heat equation. We also note here recent applications of modulated self-similar blowup techniques in fluid
dynamics: [10, 11, 13] for the Prandtl equations and [5, 15, 16] for the incompressible 3D Euler equation
with axisymmetry.

In the aforementioned works, the role of the modulation variables is to enforce certain orthogonality con-
ditions which prohibit the self-similar dynamics from evolving toward the unstable directions of a suitably
defined weighted energy space. Rather than enforcing orthogonality conditions, we shall instead employ
a generalization of the idea that we previously introduced in [3] in the setting of the 2D Euler equations
with azimuthal symmetry, in which the modulation functions are used to dynamically enforce pointwise
constraints at precisely the blow up location for a Riemann-type function W . For the 2D Euler equations
with azimuthal symmetry, we required only three modulation functions to enforce constraints on W and its
first two derivatives. In the 3D case considered herein, for which no symmetry assumptions are imposed, the
7 remaining invariances of 3D Euler correspond to 7 modulation functions κ, τ P R, ξ P R3, ˇn P R2, whose
role is to enforce 7 pointwise constraints for a 3D Riemann-type function W py, sq and its first-order and
second-order partial derivatives at y “ 0. We describe the one-to-one correspondence between symmetries
and pointwise constraints at y “ 0 as follows:

• The amplitude of the Riemann variable W is modulated via the unknown κptq by a Galilean boost of the

type pκptq, 0, 0q, whose role is to enforce the constraint W p0, sq “ 0.

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• The time-shift invariance of the equations is modulated via the unknown τ ptq, which allows us to pre-
cisely compute the time at which the shock occurs. This modulation function enforces the constraint
B1W p0, sq “ ´1.

• The invariance of the equations under the remaining two dimensional orthogonal rotation symmetry group
is modulated via the modulation vector ˇnptq “ pn2ptq, n3ptqq, allowing us to precisely compute the direc-
tion of the shock and its orthogonal plane. This modulation vector enforces the constraint ˇ∇yW p0, sq “ 0.
• The space-shift invariance of the equations is modulated via the vector ξptq, thereby allowing us to pre-
cisely compute the location of the shock. Dynamically, the modulation vector ξ enforces the constraint
B1∇Wyp0, sq “ 0.

The remaining 3 modulation functions φ22ptq, φ23ptq, φ33ptq P R which correspond to px2, x3q-dependent
spatial shifts, are used to enforce the constraint ˇ∇2
y W p0, sq “ 0. Geometrically, these 3 functions modulate
the second fundamental form of the shock profile in the directions orthogonal to the shock direction.

1.4 Outline

The remainder of the paper is structured as follows:

• In Section 2, we describe the changes of variables which transform the Euler system from its original
form (1.1) to its modulated self-similar version in Riemann-type variables (2.28). Certain tedious aspects
of this derivation are postponed to Appendix A.2. Herein, we also introduce the self-similar Lagrangian
flows used for the remainder of the paper, we define the self-similar blow up profile W and collect its
principal properties, and we record the evolution equations for higher-order derivatives of the pW, Z, Aq
variables.

• In Section 3, we state the assumptions on the initial datum in the original space-time variables and then
state (in full detail) the main result of our paper, Theorem 3.1. We emphasize that the set of assumptions
Instead, in Theorem 3.2, we show that the
on the initial datum stated here is not the most general.
set of allowable initial data can be taken from an open neighborhood in the H 18 topology near that
datum described in Theorem 3.1. In this section, we also state the self-similar version of our main result,
Theorem 3.4.

• In Section 4, we state the pointwise self-similar bootstrap assumptions which imply Theorem 3.4, as
discussed above. Note that these bootstraps are strictly worse than the initial datum assumptions discussed
in Section 3. We also state a few consequences of our bootstrap assumptions, chief among which is the
global in time 9H k energy estimate of Proposition 4.3, whose proof is postponed to Section 12.

• In Section 5, we show how the dynamic constraints of W, ∇W and ∇2W at p0, sq translate precisely into
a system of 10 coupled nonlinear ODEs for the time-dependent modulation parameters κ, τ, nν, ξi, φνµ,
given by polynomials and rational functions with coefficients obtained from the derivatives of the func-
tions pW, Z, Aq evaluated at y “ 0, cf. (5.30) and (5.31).

• In Section 6, we improve the bootstrap assumptions (4.1a) and (4.1b) for our dynamic modulation vari-

ables. The analysis in this section crucially uses the explicit formulas derived earlier in Section 5.

• In Section 7, we collect a number of technical estimates to be used later in the proof. These include bounds
for the y1 velocity components pgW , gZ, gU q defined in (2.29), the yν velocity components phW , hZ, hU q
given by (2.30), the pW, Z, Aq forcing terms from (2.31), and also the forcing terms arising in the evolu-
tion of ĂW “ W ´ W .

• In Section 8, we close the bootstrap on the spatial support of our solutions, cf. (4.4). Additionally, prove
a number of Lagrangian estimates which are fundamental to our analysis in L8 or weighted L8 spaces

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for the pW, Z, Aq system. We single out Lemma 8.2 which proves that trajectories of the (transport
velocity of the) W evolution, which start a small distance away from the origin, escape exponentially fast
towards infinity. Additionally, Lemma 8.3 proves that the flows of the transport velocities in the Z and U
equations, are swept towards infinity independently of their starting point, and spend very little time near
y “ 0.

• In Section 9, we establish pointwise estimates on the self-similar specific vorticity ˚ζ and the scaled sound
speed S. The bounds on ˚ζ rely on the structure of the equations satisfied by the geometric components
˚ζ ¨ N, ˚ζ ¨ T2, and ˚ζ ¨ T3.

• In Section 10, we improve the bootstrap assumptions for Z and A stated in (4.11) and (4.12). The most
delicate argument required is for the bound of B1A; we note in Lemma 10.1 that this vector may be
computed from the specific vorticity vector, the sound speed, and quantities which were already bounded
in view of our bootstrap assumptions.

• In Section 11, we improve on the bootstrap assumptions for W and ĂW , cf. (4.6) and (4.7a)–(4.9). This
analysis takes advantage of the forcing estimates established in Section 7 and the Lagrangian trajectory
estimates of Section 8.

• In Section 12, we give the proof of the 9H k energy estimate stated earlier in Proposition 4.3. As opposed to
the analysis which precedes this section and which relied on pointwise estimates for the pW, Z, Aq system,
for the energetic arguments presented here, it is convenient to work directly with the self-similar velocity
variable U and the scaled sound speed S, whose evolution is given by (2.38) and whose derivatives
satisfy (12.3). It is here that the good energy structure of the Euler system is fundamental. In our proof,
we use a weighted Sobolev norm to account for binomial coefficients, and appeal to some interpolation
inequalities collected in Appendix A.3.

• In Section 13, we use the above established bootstrap estimates to conclude the proofs of Theorem 3.4,
and as a consequence of Theorem 3.1. Herein, we provide the definition of the blow up time and location,
establish the H¨older 1{3 regularity of the solution at the first singular time, and show that the vorticity is
nontrivial at the shock. Additionally, we give the detailed proof of the statement that the set of initial
conditions for which Theorem 3.1 holds contains an open neighborhood in the H 18 topology, as claimed
in Theorem 3.2.

2 Self-similar shock formation

Prior to stating the main theorem (cf. Theorem 3.1 below), we describe how starting from the 3D Euler
equations (1.1) for the unknowns pu, ρq, which are functions of the spatial variable x P R3 and of the time
variable t P I Ă R, we arrive at the equations for the modulated self-similar Riemann variables pW, Z, Aνq,
which are functions of y P R3 and s P r´ log ε, 8q. This change of variables is performed in the following
three subsections, with some of the computational details provided in Appendix A.2.

2.1 A time-dependent coordinate system

In this section we switch coordinates, from the original space variable x to a new space variable rx, which is
obtained from a rigid body rotation and a translation. It is convenient for our subsequent analysis to perform
and α-dependent rescaling of time, by letting

Throughout the rest of the paper we abuse notation and denote the time variable defined in (2.1) still by t.

t ÞÑ 1`α

2 t “ t .

(2.1)

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