Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 10, pp. 1–60.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.10

A PRIORI ESTIMATES FOR THE LINEARIZED RELATIVISTIC

EULER EQUATIONS WITH A PHYSICAL VACUUM

BOUNDARY AND AN IDEAL GAS EQUATION OF STATE

BRIAN B. LUCZAK

Abstract. In this article, we will provide a result on the relativistic Euler
equations for an ideal gas equation of state and a physical vacuum boundary.

More specifically, we will prove a priori estimates for the linearized system

in weighted Sobolev spaces. Our focus will be on choosing the correct ther-
modynamic variables, developing a weighted book-keeping scheme, and then

proving energy estimates for the linearized system.

Contents

1. Introduction 1
1.1. Physical vacuum boundary with a barotropic equation of state 2
1.2. The case of an ideal gas 3
1.3. Main Result 5
1.4. Linearized system in terms of entropy and sound speed (squared) 6
1.5. Function spaces and Energies 9
1.6. Basic energy estimate 11
2. Creating a book-keeping scheme 16
2.1. Defining order for free-boundary terms 17
2.2. Solving for time derivatives 21
3. Estimates for the transport energy 25
3.1. Entropy estimates 25
3.2. Vorticity estimates 27
4. Elliptic and div-curl Estimates 32
4.1. Elliptic estimates for r̃ 32
4.2. Elliptic estimates for ũ 35
4.3. Higher order commutators with the convective derivative 39
4.4. Commutators with weighted spatial derivatives 41
5. Energy equivalence 48
6. Higher order wave energy estimates 50
7. Main theorem 54
8. Appendix 55

2020 Mathematics Subject Classification. 35Q75, 35Q35, 35Q31, 35R37.
Key words and phrases. Relativistic Euler equations; physical vacuum boundary;

ideal gas equation of state.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted November 27, 2024. Published January 27, 2025.
1



2 B. B. LUCZAK EJDE-2025/10

Acknowledgments 58
References 59

1. Introduction

The starting point for our work is the relativistic Euler equations which describe
the motion of a relativistic fluid in a Minkowski background (see [3] for historical
context). The equations of motion are given by

uµ∂µϱ+ (p+ ϱ)∂µu
µ = 0, (1.1a)

(p+ ϱ)uµ∂µu
α + παµ∂µp = 0, (1.1b)

gµνu
µuν = −1, (1.1c)

where ϱ is the fluid’s (energy) density, u is the fluid’s (four-)velocity, p is the
fluid’s pressure given by an equation of state (whose choice depends on the nature
of the fluid, see below), g is the Minkowski metric, and παβ := gαβ + uαuβ is
the projection onto the space orthogonal to u. This problem can be studied for
a general background metric, but the Minkowski metric already contains all the
important mathematical features. Coupling to Einstein’s equations, on the other
hand, is an entirely different (and much harder) problem.

Above and throughout, we employ standard Cartesian coordinates {xα}3α=0 with
t := x0 denoting a time coordinate and x := (x1, x2, x3) denoting spatial coordi-
nates, the sum convention is adopted, Greek indices vary from 0 to 3 and Latin
indices from 1 to 3, and indices are raised and lowered with g.

Remark 1.1. We note that (1.1a) is the conservation of energy for the fluid, (1.1b)
is the conservation of momentum, and (1.1c) is a normalization condition where
the velocity is assumed to be a forward time-like vector field (and this constraint
is propagated by the flow).

We are interested in the case where the fluid is confined within a domain that
is not fixed but is allowed to move with the fluid motion, such as, e.g., the motion
of a star. Fluids of this type are called free-boundary fluids. We further consider
the situation where the pressure and density vanish on the boundary which is often
referred to as the gas case. The liquid case where the density does not vanish on
the boundary is a different problem.

Denote the region containing the fluid at time t by Ωt. Then, Ωt can be described
as

Ωt = {(τ, x) ∈ R1+3 : τ = t, ϱ(t, x) > 0}.
Equations (1.1) hold in the spacetime region

D := ∪0≤t<T {t} × Ωt, (1.2)

for some T > 0, known as the moving domain. The fluid’s free-boundary at time
t is defined as Γt := ∂Ωt, and the fluid’s free boundary, also called the moving
boundary, free interface, or vacuum boundary, is defined as

Γ := ∪0≤t<T {t} × Γt = ∪0≤t<T {t} × ∂Ωt.

Sometimes we also call Ωt and Γt the moving domain and the free boundary, re-
spectively. In free-boundary problems, understanding the dynamics of Γ is crucial,



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 3

as it is the fact that Γt moves with time that distinguishes such problems from
a standard initial-boundary value problem where the boundary of the domain is
fixed. Note that, according to the foregoing, we have

p = ϱ = 0 on Γ. (1.3)

1.1. Physical vacuum boundary with a barotropic equation of state. Be-
fore discussing our work for a non-barotropic equation of state, let us consider the
barotropic setting which will motivate our discussion. Consider the equation of
state:

p = p(ϱ) = ϱκ+1, κ > 0, (1.4)

where κ is constant. Equation (1.1b) then becomes

(ϱκ+1 + ϱ)uµ∂µu
α + c2sπ

αµ∂µϱ = 0, (1.5)

where

c2s := p′(ϱ) = (κ+ 1)ϱκ (1.6)

is the fluid’s sound speed [3] and the second equality in (1.6) is valid for the equation
of state (1.4). Since ϱ vanishes on the free-boundary, we also have

c2s
∣∣
Γ
= 0. (1.7)

Remark 1.2. Observe that (1.7) follows immediately from (1.3) and (1.6). But
typically one would still impose (1.7) for other equations of state than (1.4) when
(1.3) holds, since sound waves should not be allowed to propagate into vacuum
across a region where the medium (the density) vanishes.

The decay rate of c2s near the free-boundary plays a key role in this problem. One
needs to consider decay rates that allow for Γt to move with a bounded non-zero
acceleration. We have from (1.5) and (1.6) that the fluid’s (four-)acceleration is

aα := uµ∂µu
α = − (κ+ 1)ϱκ−1

1 + ϱκ
παµ∂µϱ.

Since ϱ ∼ 0 near the free-boundary, we have that 1+ϱ = O(1) so, near the boundary,

aα ∼ ϱκ−1παµ∂µϱ ∼ ∂ϱκ ∼ ∂c2s,

using ∂ to denote a generic spacetime derivative. At this point, we need to make
some assumption on the decay rate of c2s. In view of the finite speed propagation
property, away from the boundary the motion of the fluid is essentially the same
as in the case without a free-boundary. Thus, a natural scale in this problem
which allows us to separate the bulk and boundary behaviors is the distance to the
boundary:

d ≡ d(t, x) = distance from x to Γt.

Therefore, a natural assumption to make on c2s is that it decays like a power of d,
i.e., c2s ∼ dβ for some β > 0. Under this assumption, the acceleration becomes

a ∼ ∂dβ = βdβ−1∂d ∼ dβ−1,

where we used that ∂d = O(1). From this, we see that

a
∣∣
Γt

=


0, β > 1,

∞, 0 < β < 1,

finite ̸= 0, β = 1.



4 B. B. LUCZAK EJDE-2025/10

Hence, we expect a realistic dynamics only in the case β = 1. Therefore, we assume
that

c2s(t, x) ≈ d(t, x) for x near Γt, (1.8)

i.e., c2s is comparable to the distance to the boundary. Condition (1.8) is known
as the physical vacuum boundary condition. One should understand (1.8) as a
constraint, i.e., something that is imposed on the initial data and then propagated
by the flow. The local well-posedness of the relativistic Euler equations with (1.4) as
equation of state and initial data satisfying the physical vacuum boundary condition
was obtained in [4] using Eulerian coordinates. Additionally, a-priori estimates for
the same problem with equation of state (1.4) were obtained in [6] using Lagrangian
coordinates. In [8], the authors proved a priori estimates using an equation of state
where the pressure is assumed to be a power law of the fluid’s baryon density n
(see (1.10) below).

1.2. The case of an ideal gas. We would like to extend the results of [4] to the
case of a non-barotropic equation of state. In addition, we would like, if possible,
to treat situations that are of direct relevance to physicists working in numerical
simulations of star evolution. A typical equation of state used by physicists in their
simulations of stars is that of an ideal gas (see [15] Section 2.4),

p = p(n, ε) = nε(γ − 1), (1.9)

where n is the fluid’s baryon density, ε is the fluid’s internal energy, and γ > 1 is a
constant. We recall that n satisfies the equation

uµ∂µn+ n∂µu
µ = 0, (1.10)

and ε is defined through the relation

ϱ = n(1 + ε). (1.11)

Equation (1.10) is known as the conservation of baryon density [3]. This provides
a closed system in terms of the unknowns (ε, n, uµ).

Since ε ≥ 0, equation (1.11) implies that n also vanishes on the free boundary. In
addition, since we must have ε = 0 in a vacuum, we also need to impose that ε = 0
on Γt. We need to find conditions that play a similar role to the physical vacuum
boundary condition (1.8) employed for barotropic fluids. As in Section 1.1, we try
to determine such conditions by ensuring a finite non-zero boundary acceleration.

We now adopt (ε, n, u) as primitive variables. In this situation, using (1.9),
equation (1.1b) becomes

1 + εγ

γ − 1
nuµ∂µu

α + επαµ∂µn+ nπαµ∂µε = 0. (1.12)

Inspired by the discussion of Section 1.1, we suppose that

n ∼ dβ , ε ∼ dσ, β > 0, σ > 0.

Using this Ansatz into equation (1.12) and proceeding as in Section 1.1, we find

a ∼ dσ−1.

An interesting observation is that dβ cancels out and we are left with

a
∣∣
Γt

=


0, σ > 1,

∞, 0 < σ < 1,

finite ̸= 0, σ = 1.



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 5

Therefore, we conclude that in the case of an equation of state of an ideal gas given
by (1.9), the physical vacuum boundary conditions should be

ε(t, x) ≈ d(t, x) and n(t, x) ≈ (d(t, x))β , β > 0, for x near Γt. (1.13)

Using (1.9) and (1.11), in terms of (ε, n, u), equations (1.1) and (1.10) read

uµ∂µε+ (γ − 1)ε∂µu
µ = 0, (1.14a)

1 + εγ

γ − 1
nuµ∂µu

α + επαµ∂µn+ nπαµ∂µε = 0, (1.14b)

uµ∂µn+ n∂µu
µ = 0, (1.14c)

gµνu
µuν = −1. (1.14d)

Remark 1.3. We observe that (1.13) is what we obtained solely from an analysis
of the boundary acceleration. However, from the thermodynamic relations and the
equation of state (see (1.22) below), we have that the entropy s is given by

s =
1

γ − 1
log

( ε

nγ−1

)
+ s0.

where s0 is a constant. From the physical boundary conditions, near the free
boundary we would have that

s ∼ 1

γ − 1
log

( d

d(γ−1)β

)
+ s0 ∼ log(d1−β(γ−1)) + s0.

Thus, as we approach the free boundary, d → 0 and s would behave like

s →


−∞ , for β < 1

γ−1

finite value, for β = 1
γ−1

∞, for β > 1
γ−1 .

(1.15)

Thus, β = 1
γ−1 is a natural condition for the decay rate of n in our problem.

However, in what follows, our analysis will be solely with respect to the variables
s, u, and r, where r is a multiple of c2s that we introduce in Section 1.4. We make this
remark to indicate that there is a natural choice for the physical vacuum boundary
condition in the case of an ideal gas.

1.3. Main Result. Here, we summarize our main result which is based on the
linearized version of system (1.14) with variables s, u and r (where r is defined in
(1.27) below). See (1.35) below for a rewriting of (1.14) with our new variables,
and (1.36) for the full linearized system. We remark that Theorem 1.4 below is a
statement amount the linearized problem, where the non-linear variables (s, r, u)
serve as background data.

Theorem 1.4 (Sobolev estimates for the linearized system). Let (s, r, u) be a
smooth solution to (1.35) that exists on some time interval [0, T ], and for which
the physical vacuum boundary condition (1.13) holds. Let (s̃0, r̃0, ũ0) be initial data
to system (1.36). Then, there exists a constant C depending only on s, r, u, and T

such that, if (s̃, r̃, ũ) ∈ C∞(D) is a solution to (1.36) on [0, T ], then

∥(s̃, r̃, ũ)∥H2k(Ωt) ≲ C∥(s̃0, r̃0, ũ0)∥H2k(Ω0) (1.16)

where H2k(Ωt) is defined in (1.46).

See Theorem 7.1 in Section 7 for a proof of our main result.



6 B. B. LUCZAK EJDE-2025/10

Remark 1.5. In Theorem 1.4, we assume smooth solutions for simplicity. Our
quantitative bounds depend only on finite regularity norms as indicated in Theorem
1.4, and in similar theorems.

Free boundary fluids have been studied in a variety of contexts since the 1930s,
see [14, 17, 18]. For more recent papers, we refer the reader to [1, 5, 8, 9, 10, 11,
16, 19, 20]. For recent work on free-boundary problems from relativity, we refer the
reader to [12, 13]. For a review of recent developments in mathematical aspects of
relativistic fluids, see [3, 4]

Remark 1.6. We will use the following schematic notation when discussing deriva-
tives in the arguments that follow. We will use the symbol

• ∂ to denote a generic spatial derivative, i.e. ∂1, ∂2, or ∂3, and
• ∂ to denote a generic spacetime derivative, i.e. ∂t,∂1,∂2, or ∂3.

In view of subsection 2.2, it is important to note that there is significant overlap
between these two symbols. Namely, one can use our primary system (1.35) to solve
for the time derivative of a given quantity in terms of the spatial derivatives in a
way that, as it turns out, does not introduce problematic terms into our estimates.

1.4. Linearized system in terms of entropy and sound speed (squared).
Our goal is to rewrite (1.14) so that we can apply energy estimate techniques. For
our purposes, it is more effective to consider the unknowns (s, r, uµ) where s denotes
the entropy per particle, r is a multiple of the sound speed (squared), and u is the
usual (four)-velocity. We will often abuse language and simply refer to r as the
sound speed (squared) or just the sound speed.

We begin by rewriting (1.14) in terms of p, and then introducing our new vari-
ables which will be used in the remainder of the work.

First, using (1.9) and (1.14a), we observe that

uµ∂µp = (γ − 1)nuµ∂µε+ (γ − 1)εuµ∂µn

= −(γ − 1)2nε∂µu
µ − (γ − 1)nε∂µu

µ

= −γp∂µu
µ.

(1.17)

Then, rewriting (1.14b) in terms of ε and p, our system (1.14) takes the form

uµ∂µε+ (γ − 1)ε∂µu
µ = 0 (1.18a)

1 + εγ

γ − 1
puµ∂µu

α + επαµ∂µp = 0 (1.18b)

uµ∂µp+ γp∂µu
µ = 0. (1.18c)

In the following lines, we will use several identities (1.19) (1.20), and (1.21),
which can be found in [15, Section 2].

Recall that the enthalpy per particle h is defined by

h =
p+ ϱ

n
=

nε(γ − 1) + n(1 + ε))

n
= εγ + 1 (1.19)

which has been simplified using (1.9) and (1.11). Using the well known thermody-
namic relation

ndh− dp = nθ ds, (1.20)

where θ is the temperature for the fluid, and the fact that

p = nθ (1.21)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 7

for an ideal fluid, we can solve for s by

nγ dε− (γ − 1)(ndε+ ε dn) = nε(γ − 1) ds

=⇒ s =
1

γ − 1
log

( ε

nγ−1

)
+ s0 ,

(1.22)

where s0 is a constant (and we will often set s0 = 0 for convenience). Additionally, a
quick computation (such as in [3]) shows that (1.14c) along with the thermodynamic
relations implies

uµ∂µs = 0. (1.23)

which is the conservation of entropy along flow lines. Moreover, (1.22) allows us to
solve for ε in terms of s and p. We see that e(γ−1)s = ε

nγ−1 = 1
γ−1 ·

p
nγ , which leads

to

ε =
e

γ−1
γ s

(γ − 1)
γ−1
γ

p
γ−1
γ (1.24)

Now, substituting for ε in (1.18b) and multiplying by (γ − 1)
γ−1
γ , we obtain

C1pu
µ∂µu

α + e
γ−1
γ sp

γ−1
γ παµ∂µp = 0 , (1.25)

where

C1 =
1

(γ − 1)1/γ
+ e

γ−1
γ sp

γ−1
γ

γ

γ − 1
.

Note that C1 is O(1) as we approach the free boundary since p → 0.
Before proceeding, we can quickly verify that the acceleration of the free bound-

ary is O(1) using (1.13). Near the free boundary, ε ∼ d where d is the distance to

the boundary. Using (1.24), we obtain p ∼ d
γ

γ−1 . In this case,

aα = uµ∂µu
α ∼ p−1/γ∂p ∼

(
d

γ
γ−1

)−1/γ
d

1
γ−1 = O(1). (1.26)

Because, d ∼ p
γ−1
γ , this motivates the definition of our new variable

r := p
γ−1
γ (1.27)

with the idea that r ∼ d and r will serve as our weight in the energy for the free
boundary problem. After computing the sound speed squared given by c2s = dp

dϱ |s,
we further note that r is comparable to the sound speed (squared) for the fluid
and it will play a similar role as in the work from [4]. Using (1.18c) above, we can
quickly verify that

uµ∂µr =
γ − 1

γ
p−1/γuµ∂µp = −(γ − 1)r∂µu

µ (1.28)

and thus (1.18c) becomes

uµ∂µr + (γ − 1)r∂µu
µ = 0 . (1.29)

To complete the system in terms of (s, r, uµ) we must further rewrite the second
equation (1.25). Observing that ∂µp = γ

γ−1p
1/γ∂µr, we obtain

C1u
µ∂µu

α + e
γ−1
γ sp−1/γπαµ∂µp = C1u

µ∂µu
α + e

γ−1
γ s γ

γ − 1
παµ∂µr . = 0 (1.30)

Then, calling

C2 := C1 ·
γ − 1

γ
e−

γ−1
γ s, (1.31)



8 B. B. LUCZAK EJDE-2025/10

we obtain the form of the system in terms of the unknowns (s, r, uµ):

uµ∂µs = 0 (1.32a)

uµ∂µu
α +

1

C2
παµ∂µr = 0 (1.32b)

uµ∂µr + (γ − 1)r∂µu
µ = 0 , (1.32c)

where C2 = (γ−1)
γ−1
γ

γe
γ−1
γ

s
+ r. For simplicity, let us define

Γ := Γ(s) =
(γ − 1)

γ−1
γ

γe
γ−1
γ s

(1.33)

so that C2 = Γ(s) + r.
Finally, using the convective derivative which we denote as

Dt = uµ∂µ, (1.34)

our new system in terms of the unknowns (s, r, uα) takes the form

Dts = 0, (1.35a)

Dtr + (γ − 1)r∂µu
µ = 0, (1.35b)

Dtu
α +

1

Γ + r
παµ∂µr = 0, (1.35c)

gαβu
αuβ = −1 . (1.35d)

where Γ = Γ(s) is defined according to (1.33).
Now that the form of our system is complete, we will compute the full linearized

equations using the standard procedure with linearized variables. Consider a one-
parameter family of solutions {sτ , rτ , uτ}τ for the main system (1.35) such that
(sτ , rτ , uτ )|τ=0 = (s, r, u). We can now formally define the function δ = d

dτ |τ=0

and linearized variables (s̃, r̃, ũ) := (δs, δr, δu). These variables will solve the lin-
earized system that we now compute by taking δ of the equations in (1.35). After
a computation, we can write the linearized equations in the form:

Dts̃ = f, (1.36a)

Dtr̃ + ũµ∂µr + (γ − 1)r∂µũ
µ = g, (1.36b)

Dtũ
α +

1

Γ + r
παµ∂µr̃ = hα, (1.36c)

ũαu
α = 0 , (1.36d)

where Γ is defined according to (1.33), and

f = −ũµ∂µs, g = −(γ − 1)r̃∂µu
µ

hα = −ũµ∂µu
α − 1

Γ + r
(ũαuµ + uαũµ)∂µr +

Γ′

(Γ + r)2
s̃παµ∂µr

+
1

(Γ + r)2
r̃παµ∂µr .

(1.37)

Remark 1.7. We note that the terms in blue will later combine to form a perfect
derivative that will assist in our energy estimates. Additionally, each of the terms on
the RHS contains undifferentiated (s̃, r̃, ũ) with coefficients of the form (∂s,∂r,∂u).



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 9

To clarify the exposition, we will make the following assumption which can be
used when handling the relativistic Euler equations with a physical vacuum bound-
ary (see [4] and [6] for similar examples).

Assumption 1.8 (Uniform smallness of r). Without loss of generality, we assume
that for each t ∈ [0, T ], r is uniformly small in Ωt:

∥r∥L∞(Ωt) ≤ ε̂ ≪ 1

2
,

where ε̂ is a small positive constant that we fix for the remainder of the paper. By
‘≪ 1

2 ’, we mean, in particular, that there exists a large positive constant C such

that Cε̂ < 1
2 . We will fix this constant C for the remainder of the paper.

Recall from the physical vacuum boundary conditions and (1.27) that this as-
sumption will be verified in a small neighborhood of the free boundary Γ. Because
of the finite speed of propagation, the behavior of the fluid away from the boundary
is non-degenerate and a priori estimates in Sobolev spaces can be handled using
methods from the standard relativistic Euler equations. The removal of Assump-
tion 1.8 requires a partition of unity argument in which one must separate the fluid
into its bulk behavior away from the free boundary and its behavior near the free
boundary. By using Assumption 1.8, we are able to focus on the essential difficulties
posed by the degenerate nature of the free boundary where ∥r∥L∞(Ωt) ≤ ε̂ ≪ 1

2 .

Remark 1.9. In what follows, we will often refer to the “smallness” of r near the
free boundary which will allow certain terms to make much smaller contributions.
We will use ε̂ when appropriate to reference Assumption 1.8.

Remark 1.10. Additionally, when handling these “small” terms (see Remark 2.5
in Section 6), we will often use the Cauchy-Schwarz inequality with ε. Thus, it is
useful to fix the interplay between ε̂ and ε so as to not cause any confusion. Given
that ε̂ ≪ 1

2 is fixed by Assumption 1.8, and we also have 1
C−1 < 1

2 where C is the
fixed positive constant from Assumption 1.8. We will use ε in applications of the
Cauchy-inequality-with- ε to be a small positive constant such that

0 <
1

C − 1
< ε <

1

2
(1.38)

Note that the previous definitions make the following inequalities true

0 < ε̂ <
1

2C
<

1

C − 1
< ε <

1

2
,(

1 +
1

ε

)
ε̂ < Cε̂ .

(1.39)

Remark 1.11 (The symbol ≲). Throughout our work, the symbol ≲ will be used
in the usual fashion, i.e.

A ≲ B ⇐⇒ A ≤ DB ,

where D is a constant depending on the the fixed data of the problem. In our case,
D will depend on the background solution (s, r, u) along with γ and T . By slight
abuse of notation, we will sometimes write A ≲ DB or A ≲ D(∂l(s, r, u))B if we
want to highlight, for example, that D depends on l derivatives of our fixed data
(s, r, u). This is also done to assist the reader in following our estimates where these
quantities are often removed from an integral with the L∞ norm.



10 B. B. LUCZAK EJDE-2025/10

1.5. Function spaces and Energies. Here, we define the function spaces and
energies that will play a key role in our work. We denote the weighted L2 spaces
with weight h as L2(h) and equip them with the norm

∥f∥2L2(h) =

∫
Ωt

h|f |2 dx. (1.40)

We will assume throughout that r is a positive function on Ωt which vanishes
simply on the free boundary Γt. Thus, r will be comparable to the distance to Γt.
For pairs of functions in our system defined on Ωt, we will use the base Hilbert
space

H = L2(r
2−γ
γ−1 )× L2(r

1
γ−1 ) (1.41)

with the usual norm depending only on the weight r. However, we will often use
an equivalent norm which is compatible with our energies in the problem. Let Gαβ

be the following two form
Gαβ := gαβ + 2uαuβ (1.42)

which, by [6], has been shown to be a Riemannian metric in spacetime.
Then, we define the norm

∥(r̃, ũ)∥2H̃ =

∫
Ωt

r
2−γ
γ−1

( 1

γ − 1
r̃2 + (Γ + r)r|ũ|2

)
dx , (1.43)

where |ũ|2 = |ũ|2G = Gαβ ũ
αũβ ≥ 0.

Remark 1.12. We observe that the two norms ∥ · ∥H and ∥ · ∥H̃ are equivalent
on the space of pairs (r̃, ũ) due to a key fact about the weights appearing in the
energy. For x sufficiently close to ∂Ωt, we have that r → 0, s approaches a finite
value Cs by (1.15), and the weight

(Γ + r) → (γ − 1)
γ−1
γ

γe
γ−1
γ Cs

> 0 (1.44)

which is a finite positive constant bounded away from 0 (as γ > 1 is fixed, and Cs

fixed). Hence, ∥ · ∥H and ∥ · ∥H̃ are equivalent near the free boundary.

We also define the higher order weighted Sobolev spaces Hj,σ with integer j ≥ 0
and σ > − 1

2 to be the space of all distributions in Ωt whose norm

∥f∥2Hj,σ =
∑
|α|≤j

∥rσ∂αf∥2L2 (1.45)

is finite. For higher regularity, we will use the following higher order Sobolev spaces
where the powers of r are linked to the number of derivatives. Similar to [4], we
will define higher order function spaces H2k on triplets (s̃, r̃, ũ) in Ωt with the norm

∥(s̃, r̃, ũ)∥2H2k =

2k∑
|α|=0

k∑
a=0

|α|−a≤k

(
∥r

2−γ
2(γ−1)

+ 1
2+a∂αs̃∥2L2

+ ∥r
2−γ

2(γ−1)
+a∂αr̃∥2L2 + ∥r

2−γ
2(γ−1)

+ 1
2+a∂αũ∥2L2

)
,

(1.46)

where we note that the r weight in the r̃ norm is 1
2 less than the weights on s̃

and ũ (which is a direct consequence of the powers of r in (1.36)). Additionally,
these higher order function spaces are based around an even number of derivatives
due to the underlying wave-like operator that governs the second order evolution of



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 11

(1.36). Taking Dt of (1.36) illustrates this underlying operator which is a variable
coefficient version of D2

t − r∆. We can show that the H2k norm is equivalent to

the H2k, 1
2(γ−1)

+k ×H2k, 2−γ
2(γ−1)

+k ×H2k, 1
2(γ−1)

+k norm.

Remark 1.13. Similar to [4] and [7], we can use embedding theorems to show that

the H2k norm is equivalent to the H2k, 1
2(γ−1)

+k×H2k, 2−γ
2(γ−1)

+k×H2k, 1
2(γ−1)

+k norm.

Remark 1.14. We remark that the H̃ norm will be used to control convective
derivatives of r̃ and ũ which satisfy a system of wave equations (see (1.47) for the
definition of E2k

wave). Meanwhile, ũ will be decomposed into its vorticity part which,
alongside s̃, satisfies a transport equation that we can estimate directly (see (1.48)
for the definition of E2k

transport).

Here, we introduce energies that will be used in the higher order analysis. First,
we have the wave energy

E2k
wave(s̃, r̃, ũ) =

k∑
j=0

∥(D2j
t r̃, D2j

t ũ)∥2H̃ (1.47)

recalling (1.43). Note that the definition of H̃ guarantees that |D2j
t ũ|2 = |D2j

t ũ|2G ≥
0.

Additionally, we will use the transport energy

E2k
transport(s̃, r̃, ũ) = ∥ω̂∥2

H
2k−1,k+ 1

2(γ−1)
+ ∥s̃∥2

H
2k,k+ 1

2(γ−1)
, (1.48)

where ω̂ is the reduced linearized vorticity defined in (3.14), and this will be ad-
dressed in Section 3.2. Then, the final linearized energy is given by

E2k(s̃, r̃, ũ) = E2k
wave(s̃, r̃, ũ) + E2k

transport(s̃, r̃, ũ). (1.49)

Remark 1.15. Although both components of E2k
transport satisfy transport equa-

tions, we remark that they will be used slightly differently in the arguments that
follow. In particular, the estimate for ω̂ will be used to complete div-curl estimates
for ũ, whereas s̃ can be estimated directly with the H2k norm from above and below
as in Section 3.1.

1.6. Basic energy estimate. In this section, we will prove a basic energy estimate
for the quantities s̃, r̃, and ũ in weighted L2 spaces. Although the higher order
estimates in Sobolev spaces require additional techniques, this section serves as a
starting point in our analysis. As discussed in Section 1.5, the s̃ equation will later
be treated separately (see section 3.1) as it satisfies a transport equation as opposed
to the wave equations satisfied by r̃ and ũ (see Section 6).

After computing the linearized equations (1.36), multiply (1.36a) by r
1

γ−1 s̃, the

second equation (1.36b) by 1
γ−1r

2−γ
γ−1 r̃ and contract the third equation (1.36c) with

(Γ + r)r
1

γ−1Gαβ ũ
β . Using (1.35d), (1.36d), and identities like ũαπ

αµ = ũµ =
Gαβ ũ

βπαµ, we obtain

1

2
r

1
γ−1Dts̃

2 = −r
1

γ−1 s̃ũµ∂µs (1.50a)

1

2(γ − 1)
r

2−γ
γ−1Dtr̃

2 +
1

γ − 1
r

2−γ
γ−1 r̃ũµ∂µr + r

1
γ−1 r̃∂µũ

µ = −r̃2r
2−γ
γ−1 ∂µu

µ (1.50b)



12 B. B. LUCZAK EJDE-2025/10

Γ + r

2
r

1
γ−1Dt|ũ|2 + r

1
γ−1 ũµ∂µr̃ = −(Γ + r)r

1
γ−1 ũαũ

µ∂µu
α − r

1
γ−1 |ũ|2uµ∂µr

+
Γ′

Γ + r
r

1
γ−1 s̃ũµ∂µr +

1

Γ + r
r

1
γ−1 r̃ũµ∂µr.

(1.50c)

Thus, after adding the equations together, the blue terms combine to form a perfect
derivative:

1

γ − 1
r

2−γ
γ−1 r̃ũµ∂µr + r

1
γ−1 r̃∂µũ

µ + r
1

γ−1 ũµ∂µr̃ = ∂µ(r
1

γ−1 r̃ũµ) (1.51)

which will be handled using integration be parts. Combining the equations together,
we obtain

1

2
r

1
γ−1Dts̃

2 +
1

2(γ − 1)
r

2−γ
γ−1Dtr̃

2 +
Γ + r

2
r

1
γ−1Dt|ũ|2 + ∂µ(r

1
γ−1 r̃ũµ)

= −r
1

γ−1 s̃ũµ∂µs− r̃2r
2−γ
γ−1 ∂µu

µ − (Γ + r)r
1

γ−1 ũαũ
µ∂µu

α − r
1

γ−1 |ũ|2uµ∂µr

+
Γ′

Γ + r
r

1
γ−1 s̃ũµ∂µr +

1

Γ + r
r

1
γ−1 r̃ũµ∂µr.

(1.52)

Next, we integrate with respect to x over Ωt and use the moving domain formula

(noting that the fluid particles on the boundary move with velocity ui

u0 ):

d

dt

∫
Ωt

f dx =

∫
Ωt

∂tf dx+

∫
Ωt

∂i
(
f
ui

u0

)
dx

=

∫
Ωt

1

u0
Dtf dx+

∫
Ωt

f∂i
( ui

u0

)
dx.

(1.53)

For the blue terms, we have r = 0 on ∂Ωt, so integrating by parts yields

∫
Ωt

∂µ(r
1

γ−1 r̃ũµ) dx =

∫
Ωt

∂i(r
1

γ−1 r̃ũi) dx+

∫
Ωt

∂t(r
1

γ−1 r̃ũ0) dx

= −
∫
∂Ωt

r
1

γ−1 r̃ũiνi dS +

∫
Ωt

∂t(r
1

γ−1 r̃ũ0) dx

=

∫
Ωt

∂t(r
1

γ−1 r̃ũ0) dx

(1.54)

For the energy terms, we plan to apply (1.53) to the function

f(t, x) =
1

2
r

1
γ−1 s̃2 +

1

2(γ − 1)
r

2−γ
γ−1 r̃2 +

Γ + r

2
r

1
γ−1 |ũ|2 (1.55)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 13

First, recall using the r equation that Dtr = −(γ − 1)r∂µu
µ. We then obtain

Dtf(t, x)

=
1

2
r

1
γ−1Dts̃

2 +
1

2(γ − 1)
r

2−γ
γ−1Dtr̃

2 +
Γ + r

2
r

1
γ−1Dt|ũ|2

− 1

2
r

1
γ−1 ∂µu

µs̃2 − 2− γ

2(γ − 1)
r

2−γ
γ−1 ∂µu

µr̃2 − 1

2
r

1
γ−1 (Γ + γr)∂µu

µ|ũ|2

= −∂µ(r
1

γ−1 r̃ũµ)− r
1

γ−1 s̃ũµ∂µs− r̃2r
2−γ
γ−1 ∂µu

µ

− (Γ + r)r
1

γ−1 ũαũ
µ∂µu

α − r
1

γ−1 |ũ|2uµ∂µr

+
Γ′

Γ + r
r

1
γ−1 s̃ũµ∂µr +

1

Γ + r
r

1
γ−1 r̃ũµ∂µr −

1

2
r

1
γ−1 ∂µu

µs̃2

− 2− γ

2(γ − 1)
r

2−γ
γ−1 ∂µu

µr̃2 − 1

2
r

1
γ−1 (Γ + γr)∂µu

µ|ũ|2

(1.56)

using that DtΓ = 0 and computing several more expressions.
After gathering terms, we apply (1.53) to f(t, x) and integrate over Ωt. This

yields

d

dt

∫
Ωt

f(t, x) dx

=

∫
Ωt

f(t, x)∂i
( ui

u0

)
dx

−
∫
Ωt

1

u0

(1
2
r

1
γ−1 s̃2 +

γ

2(γ − 1)
r

2−γ
γ−1 r̃2 +

1

2
r

1
γ−1 (Γ + γr)|ũ|2

)
∂µu

µ dx

−
∫
Ωt

1

u0
∂t(r

1
γ−1 r̃ũ0) dx−

∫
Ωt

1

u0
r

1
γ−1 s̃ũµ∂µs dx

−
∫
Ωt

1

u0
(Γ + r)r

1
γ−1 ũαũ

µ∂µu
α +

1

u0
r

1
γ−1 |ũ|2uµ∂µr dx

+

∫
Ωt

1

u0

Γ′

Γ + r
r

1
γ−1 s̃ũµ∂µr +

1

u0

1

Γ + r
r

1
γ−1 r̃ũµ∂µrdx

(1.57)

Thus, we define the squared base energy (k = 0) to be

E0(s̃, r̃, ũ)[t] := ∥(r̃, ũ)∥2H̃ +
1

2
∥s̃∥2

H
0, 1

2(γ−1)

=
1

2

∫
Ωt

r
2−γ
γ−1

( 1

γ − 1
r̃2 + (Γ + r)r|ũ|2 + rs̃2

)
dx.

(1.58)

Remark 1.16. Observe the correspondence between (1.58) and the higher order
energy (1.49) where we have a wave part and a transport part. We only make the
distinction here between s̃ and (r̃, ũ) to highlight the differences in the higher order
estimates. In Section 3.1, we can simply differentiate the s̃ equation with spatial
derivatives ∂2k. However, in Section 6, we must differentiate the r̃ and ũ equations
with the convective derivative Dt.

This will allow us to prove the following basic energy estimate.

Proposition 1.17 (Basic Energy Inequality). Let (s, r, u) be a solution to (1.35)
that exists on some time interval [0, T ]. Assume that s, r, u and their first order



14 B. B. LUCZAK EJDE-2025/10

derivatives are bounded in the L∞(Ωt) norm for each t ∈ [0, T ] and r vanishes sim-
ply on the free boundary. Then, the following estimate holds for solutions (s̃, r̃, ũ)
to (1.36):

E0[t] ≲ E0[0] exp
(∫ t

0

C(∥∂s, ∂r, ∂u, s, r, u∥L∞(Ωτ )) dτ
)
, (1.59)

where E0[t] is given by (1.58) and C is a function depending on the L∞(Ωτ ) norms
of s, r, u and their first order derivatives.

Before giving the proof, we provide the following remarks which are critical in
understanding the methods used throughout this paper.

Remark 1.18 (The symbol ≲). Throughout our work, the symbol ≲ will be used
in the usual fashion, i.e.

A ≲ B ⇐⇒ A ≤ DB,

where D is a constant depending on the the fixed data of the problem. In our case,
D will depend on the background solution (s, r, u) along with γ and T .

Remark 1.19. In view of (1.44), we will often use inequalities of the following
form when handling weights that appear in our integrals:∫

Ωt

r
1

γ−1 |ũ|2 dx ≤ ∥ 1

Γ + r
∥L∞(Ωt)

∫
Ωt

(Γ + r)r
1

γ−1 |ũ|2 dx ≲ E0 , (1.60)

where we note that 1
Γ+r tends to a finite positive constant depending on γ when

we are sufficiently close to the boundary (see Remark 1.12). When we refer to an
expression consisting of s, r, u and γ as O(1) near the free boundary, we simply are
referring to the fact that when r is sufficiently small, this weight tends to a finite
constant bounded away from zero. Thus, it can be absorbed into the ≲ symbol
without having much effect on the primary estimate.

Remark 1.20. Additionally, there are many situations in which we would like to
estimate expressions involving spacetime derivatives ∂µ over the spatial region Ωt.
However, this can be done with the help of section 2.2. The primary idea is that
we can use system (1.35) directly to solve for ∂t(s, r, u) and ∂t(s̃, r̃, ũ) in terms of
the spatial part ∂(s, r, u) and ∂(s̃, r̃, ũ) respectively, plus additional terms that can
be estimated. A key identity is

ũ0 =
uj

u0
ũj (1.61)

which follows from the orthogonality of u and ũ in (1.36d). With section 2.2 taken
for granted, one can treat ∂µ as a generic first order spatial derivative ∂ in the
following proof with the understanding that all time derivatives will be eventually
written in terms of purely spatial derivatives.

Proof of Proposition 1.17. Using Remarks 1.19 and 1.20, the first two terms in

(1.57) on the RHS containing ∂µu
µ and ∂i(

ui

u0 ) can be quite easily estimated using

E0 and an expression depending on the L∞(Ωt) norms of u and ∂u. We arrive at



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 15

the inequality

d

dt
E0 ≲ C0(∥∂u∥L∞(Ωt), ∥u∥L∞(Ωt))E

0

−
∫
Ωt

1

u0
∂t(r

1
γ−1 r̃ũ0) dx−

∫
Ωt

1

u0
r

1
γ−1 s̃ũµ∂µs dx

−
∫
Ωt

1

u0
(Γ + r)r

1
γ−1 ũαũ

µ∂µu
α +

1

u0
r

1
γ−1 |ũ|2uµ∂µr dx

+

∫
Ωt

1

u0

Γ′

Γ + r
r

1
γ−1 s̃ũµ∂µr +

1

u0

1

Γ + r
r

1
γ−1 r̃ũµ∂µrdx

=: ∥(∂u, u)∥L∞(Ωt)E
0 + I1 + I2 + I3 + I4 ,

(1.62)

where each lower order term I1, . . . , I4 will be estimated in the following way.
The first term I1 is unique in that we plan to combine it with the LHS of our

inequality after integrating in time. We leave this term for now and will return to
it later.

For I2, we estimate it using the Cauchy Schwartz inequality. We have

|I2| ≤
∫
Ωt

r
1

γ−1

∣∣(u0)−1
∣∣ |s̃ũµ∂µs|

≤ ∥∂s∥L∞(Ωt)∥(u
0)−1∥L∞(Ωτ )

(∫
Ωt

r
1

γ−1 |s̃|2
)1/2(∫

Ωt

r
1

γ−1 |ũ|2
)1/2

≲ C2(∥∂s∥L∞(Ωt), ∥u∥L∞(Ωτ ))E
0 ,

(1.63)

where we used remarks 1.19-1.20, and that G is a Riemannian metric, hence |ũ|2δ ≲
|ũ|2G where δ is the Euclidean metric in spacetime.

For I3, we simplify Dtr = −(γ − 1)r∂µu
µ and obtain a similar inequality

|I3| ≤
∫
Ωt

(Γ + r)r
1

γ−1

∣∣(u0)−1
∣∣ |ũ|2|∂µuµ| dx

+

∫
Ωt

r
1

γ−1

∣∣(u0)−1
∣∣ |ũ|2(γ − 1)r|∂µuµ| dx

≲ C3(∥∂u∥L∞(Ωt), ∥u∥L∞(Ωτ ))E
0 ,

(1.64)

where we used Remark 1.19.
For I4, we observe that Γ

′(s) = −γ−1
γ Γ(s) using (1.33), and we can apply similar

arguments along with the Cauchy-Schwarz inequality to obtain

|I4| ≤
∫
Ωt

(γ − 1

γ

) ∣∣(u0)−1
∣∣ 1

Γ + r
r

1
γ−1 |s̃ũµ∂µr| dx

+

∫
Ωt

1

Γ + r
r

1
γ−1

∣∣(u0)−1
∣∣ |r̃ũµ∂µr| dx

≲ C4(∥∂r∥L∞(Ωt), ∥u∥L∞(Ωτ ))E
0.

(1.65)

Thus, for the energy defined in (1.58) with t replaced by τ , we combine (1.63)-
(1.65) and I1 to obtain the inequality

d

dτ
E0 ≲ C(∥∂s∥, ∥∂r∥, ∥∂u∥, ∥u∥)E + ∥u∥

∫
Ωτ

∂τ (r
1

γ−1 r̃ũ0) dx , (1.66)



16 B. B. LUCZAK EJDE-2025/10

where the constant depends only on γ and the norms are on L∞(Ωτ ). Integrating
in time from 0 to t ∈ [0, T ], this implies

E0[t] ≲ E0[0] +

∫ t

0

C(∥∂s∥, ∥∂r∥, ∥∂u∥, ∥u∥)E0[τ ] dτ

+

∫ t

0

∫
Ωτ

∂τ (r
1

γ−1 r̃ũ0) dxdτ ,

(1.67)

where the constant depends on γ and the L∞ norm of u which is bounded on [0, T ].
Then, from a quick computation using the moving domain formula, we obtain∫ t

0

∫
Ωτ

∂τ (r
1

γ−1 r̃ũ0) dxdτ

=

∫ t

0

( d

dτ

∫
Ωτ

r
1

γ−1 r̃ũ0 dxdτ −
∫
Ωτ

∂i(r
1

γ−1 r̃ũ0 u
i

u0
) dxdτ

)
=

∫ t

0

d

dτ

∫
Ωτ

r
1

γ−1 r̃ũ0 dxdτ

=

∫
Ωt

(r(t, x))
1

γ−1 r̃(t, x)ũ0(t, x) dx−
∫
Ω0

(r(0, x))
1

γ−1 r̃(0, x)ũ0(0, x) dx ,

(1.68)

where we integrate by parts for one of the terms and used the fact that r vanishes
on the free boundary. Thus, we can substitute this into our inequality to obtain

E0[t] ≲ E0[0] +

∫ t

0

C(∥∂s∥, ∥∂r∥, ∥∂u∥, ∥u∥)E0[τ ] dτ

+

∫
Ωt

(r(t, x))
1

γ−1 r̃(t, x)ũ0(t, x) dx−
∫
Ω0

(r(0, x))
1

γ−1 r̃(0, x)ũ0(0, x) dx,

where the constant depends on γ and ∥u∥L∞(Ωτ ) which is bounded.
Then, for x sufficiently close to the boundary, we have estimates of the form∣∣∫

Ωt

r
1

γ−1 r̃ũ0 dx
∣∣ ≤ (∫

Ωt

r
1

γ−1 |r̃|2
)1/2(∫

Ωt

r
1

γ−1 |ũ|2
)1/2

≲ ∥r∥1/2L∞(Ωτ )

(∫
Ωt

r
2−γ
γ−1 |r̃|2

)1/2(∫
Ωt

r
1

γ−1 |ũ|2
)1/2

≲ ε̂1/2E0[t],

(1.69)

where we applied Assumption 1.8 and used the smallness of r so that ε̂1/2 ≪ 1√
2
< 1.

A similar estimate also holds for the Ω0 term. Thus, we can move the ε̂1/2E0[t]
term to the LHS which will yield

E0[t] ≲
1

1− ε̂1/2

(
E0[0](1 + ε̂1/2)

+

∫ t

0

(∥∂s∥L∞(Ωτ ) + ∥∂r∥L∞(Ωτ ) + ∥∂u∥L∞(Ωτ ))E
0[τ ] dτ

)
.

(1.70)

Thus, by Gronwall’s inequality, we have

E0[t] ≲ E0[0] exp
(∫ t

0

C(∥∂s∥, ∥∂r∥, ∥∂u∥, ∥s∥, ∥r∥, ∥u∥) dτ
)

(1.71)

with a constant depending only on γ and our distance to the free boundary. This
completes our proof of the basic energy estimate. □



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 17

2. Creating a book-keeping scheme

In works such as [4], a book-keeping scheme was created using the equations’
scaling law for the leading order dynamics near the free boundary. However, no
such scaling seems available for (1.36); thus, we will need to develop several key
ideas around the notion of order which is first defined in Definition 2.4. After
differentiating the equations, we plan to create a book-keeping scheme which ac-
counts for the number of derivatives and the powers of our free boundary weight
r. Along the way, any derivatives of the background quantities s, r, u will serve as
L∞ coefficients for each of the terms that we will encounter.

Using system (1.36), we record the computations when taking D2k
t of the equa-

tions (where D0
t is the identity). Since the s̃ equation will be treated separately in

Section 3.1, we put only the r̃ and ũ equations here for convenience. Writing the
equations in commutator notation and keeping only the higher order terms on the
LHS, we obtain the system

Dt

(
D2k

t r̃
)
+
(
D2k

t ũµ
)
∂µr + (γ − 1)r∂µ

(
D2k

t ũµ
)
= B2k (2.1a)

Dt

(
D2k

t ũα
)
+

1

Γ + r
παµ∂µ

(
D2k

t r̃
)
= Cα

2k, (2.1b)

where

B2k = D2k
t g −

2k−1∑
i=0

(
Di

tũ
µ∂µ

(
D2k−i

t r
)
−D2k−i

t r∂µ
(
Di

tũ
µ
))

−
2k−1∑
i=0

Di
tũ

µ[D2k−i
t , ∂µ]r − (γ − 1)

2k∑
i=1

D2k−i
t r[Di

t, ∂µ]ũ
µ

Cα
2k = D2k

t hα −
2k−1∑
i=0

D2k−i
t

( 1

Γ + r
παµ

)
∂µ

(
Di

tr̃
)

−
2k∑
i=1

D2k−i
t

( 1

Γ + r
παµ

)
[Di

t, ∂µ]r̃

(2.2)

and g and hα are defined in (1.37). Additionally, we removed terms from the
summation where the commuator was zero. Each of the terms in (2.2) will be
included in the higher order wave equation estimates in Section 6.

Using several lemmas from [4] on embedding theorems combined with Assump-
tion 1.8, we are able to prove the following lemmas which motivate our definition
of order defined in Remark 2.5.

Lemma 2.1. Suppose that σ ≥ 0, r vanishes simply on the free boundary, and
f ∈ Hk,σ. Then f ∈ Hk,σ+ 1

2 , and more specifically

∥f∥2
Hk,σ+1

2 (Ωt)
≤ ε̂∥f∥2Hk,σ(Ωt)

,

where the integrals are interpreted with respect to Remark 1.9.

Lemma 2.2 (Free boundary embedding lemma). Let H l,m be the corresponding
Sobolev space for the free-boundary problem with weight r that vanishes simply near
the free boundary, and l,m ≥ 0. If k ≥ 0 is provided, then

H2k,k+ 1
2 ⊂ H l,m

provided l ≤ 2k and m− l + k − 1
2 ≥ 0.



18 B. B. LUCZAK EJDE-2025/10

By following the proof, we also have that natural corollary that H2k,k ⊂ H l,m

provided m − l + k ≥ 0. By combining the above lemmas we have the following
useful corollary.

Corollary 2.3 (Free-boundary trading derivatives for weight). Suppose that f ∈
Hj+1,σ+1 and j, σ ≥ 0. Then, we have

∥f∥Hj,σ ≲ ε̂∥f∥Hj+1,σ , (2.3)

where ε̂ is the small positive constant defined in Assumption 1.8.

Proof. From [4], we see that ∥f∥Hj,σ ≲ ∥f∥Hj+1,σ+1 with a constant depending
on σ. Then, we can simply pull out a power of r in the L∞(Ωt) norm and use
Assumption 1.8. □

2.1. Defining order for free-boundary terms.

Definition 2.4 (H2k-critical, subcritical, and supercritical terms). Let m, l ∈ N.
In view of Lemma 2.2, terms of the form

rm∂lũ, rm∂ls̃ or rm∂lr̃

with 0 ≤ l ≤ 2k andm ≥ 0 will be calledH2k-critical provided thatm−l+k− 1
2 = 0,

or respectively m − l + k = 0. Additionally, terms such that m − l + k − 1
2 > 0

or respectively m− l+ k > 0 will be called H2k-subcritical. Similarly, terms where
m − l + k − 1

2 < 0, m − l + k < 0 will be denoted H2k-supercritical. In view of

subsection 2.2, a similar classification can also be used for the terms rm∂lũ, rm∂ls̃,
and rm∂lr̃.

Remark 2.5. We can also refer to the H2k-order of a given term by computing
the value O := m − l + k − 1

2 (or O := m − l + k for terms involving r̃) and the
sign of O will determine if the term is subcritical or supercritical. As an example,
a term of the form rk∂2kũ will be called H2k-supercritical of order − 1

2 . The value
O can be interpreted as an indicator for how well that term can be estimated using
the H2k norm. Critical terms have the exact number of derivatives and powers of
r to be estimated, whereas supercrtical terms are lacking key additional powers of
r. Also, note that the order O depends on a particular derivative level, i.e. what
value of k we are using in order to estimate terms using the H2k norm. Also, note
that that the weights in the H2k norm depend not only on k, but also on γ > 1
which is a fixed constant. For clarity, one can interpret this notion of order as in
the simplified setting where γ = 2 since our starting Lemma 2.2 is built around the
function spaces H2k,k+ 1

2 and H2k,k. In that case, the H2k-order of a term T can
be interpreted as a statement about the powers of r present/needed in the term T
in order for the inequality

∥T∥2L2 ≲ ∥(s̃, r̃, ũ)∥2H2k ≈ ∥s̃∥2
H2k,k+1

2
+ ∥r̃∥2H2k,k + ∥ũ∥2

H2k,k+1
2

to hold, and the symbol ≈ is used to represent the norm equivalence from Remark

1.13. For arbitrary γ > 1, we will instead use the function spaces H2k,k+ 1
2+

2−γ
2(γ−1)

and H2k,k+ 2−γ
2(γ−1) . In this setting, the H2k-order for a term T is a statement about

the inequality

∥T∥2
L2(r

2−γ
γ−1 )

≲ ∥(s̃, r̃, ũ)∥2H2k

≈ ∥s̃∥2
H

2k,
2−γ

2(γ−1)
+k+1

2
+ ∥r̃∥2

H
2k,

2−γ
2(γ−1)

+k
+ ∥ũ∥2

H
2k,

2−γ
2(γ−1)

+k+1
2

(2.4)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 19

which we summarize as follows:

• Suppose that T is H2k subcritical. We claim that inequality (2.4) holds as
well as

∥T∥2
L2(r

2−γ
γ−1 )

≲ ε̂∥(s̃, r̃, ũ)∥2H2k (2.5)

using the previous Lemmas (also, note that (2.5) implies (2.4)). To see this
fact, let us assume that T consists of r̃ terms so that T = rm∂lr̃ (although
ũ and s̃ terms are similar). By Definition 2.4, we have that m− l + k > 0.
If l = 2k, then we must have m > k (extra power of r) and we can apply
Lemma 2.1 to obtain the desired inequality (2.5) with ε̂. If l < 2k, i.e.
l = 2k − b with 0 < b ≤ 2k, then we must have m > k − b (and m = 0 for
b > k). We would obtain

∥rm∂2k−br̃∥2
H

0,
2−γ

2(γ−1)

≲ ∥r̃∥
H

2k−b,m+
2−γ

2(γ−1)
≤ ∥r̃∥

H
2k,m+b+

2−γ
2(γ−1)

. (2.6)

Then, since m+ b > k we can apply Lemma 2.1 to finish the proof of (2.5).
• If T is H2k critical, then (2.4) holds by Lemma 2.2, but (2.5) in general
does not since we do not have any additional powers of r to spare.

• If T is H2k supercritical with order O < 0, then (2.4) does not hold (and
subsequently (2.5) also does not), but (2.4) is true with T replaced by

T̂ = r−OT as T is lacking key powers of r. To see this fact, we simply
observe that the order of the term T̂ will be (m−O) + l − k = 0 which is
critical.

Lemma 2.6 (Sum of H2k orders). Suppose that T1 and T2 are free boundary terms
(i.e. T1 and T2 are of the form rm∂l(ũ, r̃, s̃) for some m ≥ 0 and 0 ≤ l ≤ 2k)
that have H2k orders O1 and O2 respectively. If O1 + O2 ≥ 0, then ∥T1T2∥L1

≲
∥(s̃, r̃, ũ)∥2H2k , i.e the H2k order of a product is the sum of its orders.

Proof. Without loss of generality, the proof can be obtained by splitting into cases:

(1) T1 = rm1∂l1 ũ and T2 = rm2∂l2 ũ, m1,m2 ≥ 0, 0 ≤ l1, l2 ≤ 2k;
(2) T1 = rm1∂l1 r̃ and T2 = rm2∂l2 r̃, m1,m2 ≥ 0, 0 ≤ l1, l2 ≤ 2k;
(3) T1 = rm1∂l1 r̃ and T2 = rm2∂l2 ũ, m1,m2 ≥ 0, 0 ≤ l1, l2 ≤ 2k.

Then correctly distributing the powers of r as needed. □

Remark 2.7. We note that Lemma 2.6 is especially useful in energy estimates
where we would like to compute the order of a product of two terms, but we are
estimating in L1 instead of L2 and utilizing the Cauchy-Schwarz inequality. Using
Lemma 2.6, we can extend the notion of order to a product of terms T1T2 by
replacing inequalities (2.4) and (2.5) with

∥T1T2∥
L1(r

2−γ
γ−1 )

≲ ∥(s̃, r̃, ũ)∥2H2k , (2.7)

∥T1T2∥
L1(r

2−γ
γ−1 )

≲ ε̂∥(s̃, r̃, ũ)∥2H2k (2.8)

With this new definition for products, Lemma 2.6 states that if the T1 has H2k-
order O1 and T2 has H2k-order O2, then T1T2 has H2k-order O1 + O2 provided
that O1 + O2 ≥ 0. When referring to the order of products T1T2, we will always
compute the order of T1 and T2 separately before using Lemma 2.6 to make a
statement about the order of T1T2.



20 B. B. LUCZAK EJDE-2025/10

Remark 2.8 (Definition of the ≃ symbol). Now that we have described the no-
tion of order, we will often use the symbol ≃ to mean the following. Suppose
that Ã(ũ, r̃, s̃) and B̃(ũ, r̃, s̃) are expressions involving powers of r and derivatives

(∂,∂, Dt) of our linearized variables. Using Remark 2.5, suppose that à and B̃
both contain terms of order O at worst (i.e. there are no terms with order strictly
more negative than O). Then, we will write

à ≃ B̃ ⇐⇒ C1(∂
2k+1(u, r, s), u, s)Ã = C2(∂

2k+1(u, r, s), u, s)B̃ + P , (2.9)

where C1 and C2 are expressions containing (u, s) and up to 2k + 1 derivatives
of (u, r, s), and P contains terms with order J > O (i.e. strictly better terms).
Also, note that C1 and C2 must not contain undifferentiated r, since powers of r
would contribute to the calculated order for à and B̃. For illustrative purposes
(and by slight abuse of notation), we will sometimes write A ≃ B + P instead of

just à ≃ B̃ if we want to highlight the terms in P and calculate their order which
will be strictly greater that B̃.

As a final remark, we stated that ∂(r, s, u) or ∂(r, s, u) does not, in general,
contribute to the order of given term. However, there is one exception which can
be seen by analyzing (1.35b) above. For situations where we have Dtr, we see that
a full power of r is obtained since Dtr = −(γ−1)r∂µu

µ and powers of r are needed
for calculating order. Thus, it can be useful to add the fact that

Dtr ≃ r (2.10)

even though in (2.9) we used the symbol ≃ when referring to terms involving
(ũ, r̃, s̃).

It remains to show several lemmas and commutator identities. We will ultimately
use our scheme to simplify the B2k and Cα

2k terms that appear in (2.2). Then, we
can estimate these terms using the H2k norm in Section 6. The proof of these
lemmas is a routine application of the commutator identities:

[∂,DN
t ]ϕ = [∂,Dt](D

N−1
t ϕ) +Dt([∂,D

N−1
t ]ϕ)

[Dt, ∂
N ]ϕ = [Dt, ∂]∂

N−1ϕ+ ∂
(
[Dt, ∂

N−1]ϕ
) (2.11)

Lemma 2.9 (First commutator lemma). Let N ∈ N. For the convective derivative
Dt = uµ∂µ, generic spatial derivative ∂, and ϕ sufficiently smooth, the following
identities hold:

[∂,Dt]ϕ = (∂uν)∂νϕ,

[∂,D2
t ]ϕ = 2(∂uν)∂ν(Dtϕ) + [Dt(∂u

ν)− (∂uµ)(∂µu
ν)]∂νϕ,

. . . ,

[∂,DN
t ]ϕ ≃

N−1∑
i=0

Cν
i ∂ν(D

i
tϕ),

(2.12)

where Cν
i := Ci(∂u,Dtu) for i ∈ {0, . . . , N − 1} are coefficients depending on the

spatial and convective derivatives of u.

We note that Lemma 2.9 is used extensively in Theorem 6.2 in order to simplify
the commutator terms from (2.2). For a discussion of the commutators used in
elliptic estimates, see Section 4. We also have an important commutator lemma
which will be used in our higher order estimates.



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 21

Lemma 2.10 (Second commutator lemma). Let N ∈ N. For the convective de-
rivative Dt = uµ∂µ, generic N -th order spatial derivative ∂N , and ϕ sufficiently
smooth, the following identities hold:

[Dt, ∂]ϕ = −(∂uν)∂νϕ,

[Dt, ∂
2]ϕ = −2(∂uν)∂ν(∂ϕ)− (∂2uν)∂νϕ,

. . . ,

[Dt, ∂
N ]ϕ ≃

N−1∑
i=0

Bν
i ∂ν(∂

iϕ),

(2.13)

where Bν
i := Bi(∂u) for i ∈ {0, . . . , N − 1} are coefficients depending on the spatial

derivatives of u.

Let us continue the book-keeping process for higher order convective derivatives
Di

ts̃, D
i
tr̃, and Di

tũ. The following Lemma can be taken with Section 2.2 and remark
2.13 in mind that we preview here. Namely, we use a generic spacetime derivative
with the understanding that time derivatives will eventually be solved for in terms
of spatial derivatives plus additional terms that are more sub-critical (see Lemma
2.12 and Remark 2.13 below). For now, we will write Lemma 2.11 using the generic
spacetime derivative symbol ∂ from Remark 1.6.

Lemma 2.11 (Book-keeping Di
t derivatives). The following simplifications hold

when counting derivatives and powers of r. For i = 1, we are left with the terms

Dts̃ ≃ ũ,

Dtr̃ ≃ r∂ũ+ ũ,

Dtũ ≃ ∂r̃ + ũ+ s̃,

(2.14)

where we only ignored ∂s,∂r,∂u, s, u, and π, and we listed only Hi critical or
supercritical terms. Moreover, for i ≥ 2, we obtain the following

Di
ts̃ ≃ Di−1

t ũ,

Di
tr̃ ≃

{∑i/2
l=0 r

l∂l+i/2r̃, if i is even,∑(i+1)/2
l=0 rl∂l+(i−1)/2ũ, if i is odd,

Di
tũ ≃

{∑i/2
l=0 r

l∂l+i/2ũ+
∑i/2−1

j=0 rj∂j+i/2r̃, if i is even,∑(i−1)/2
l=0 rl∂l+(i+1)/2r̃ + rl∂l+(i−1)/2(ũ+ s̃), if i is odd,

(2.15)

Proof. This lemma is proved inductively using that Dtr ≃ r, Lemma 2.9, and our
assumptions on key terms that can be ignored at each step. □

2.2. Solving for time derivatives. There are many instances in which we would
like to estimate the spacetime derivatives ∂µ of a particular quantity using only the
spatial derivatives ∂. To do so, we can can solve for the time derivatives in terms
of spatial derivatives by using system (1.35) directly and the orthogonality identity
(1.61).

Lemma 2.12 (Solving for time derivatives). The following computations and sim-
plifications hold under the book-keeping scheme from Section 2:

(u0)2 = 1 + ujuj , ũ0 =
uj

u0
ũj ≃ ũ,



22 B. B. LUCZAK EJDE-2025/10

∂iũ
0 =

uj

u0
∂iũ

j + ũj∂i
uj

u0
≃ ∂ũ+ ũ, ∂ts = − ui

u0
∂is,

∂ts̃ = − ui

u0
∂is̃+

uiũ0 − u0ũi

(u0)2
∂is ≃ ∂s̃+ ũ,

∂tu
0 = Ci

1∂iu
0 + Ci

2∂ir + C3r∂iu
i,

∂tũ
0 = Ci

1∂iũ
0 + Ci

2∂ir̃ + C3r∂iũ
i + C3r̃∂iu

i + C̃i
1∂iu

0 + C̃i
2∂ir + C̃3r∂iu

i

≃ ∂ũ+ ∂r̃ + r∂ũ+ (ũ+ s̃+ r̃) + r(ũ+ s̃+ r̃) + r2(ũ+ s̃+ r̃),

∂tr = − ui

u0
∂ir −

γ − 1

u0
r∂iu

i − γ − 1

u0
r[∂tu

0],

∂tr̃ = − ui

u0
∂ir̃ −

γ − 1

u0
r∂iũ

i − γ − 1

u0
r[∂tũ

0] +
uiũ0 − u0ũi

(u0)2
∂ir

+
( (γ − 1)

(u0)2
ũ0r − γ − 1

u0
r̃
)
∂iu

i +
( (γ − 1)

(u0)2
ũ0r − γ − 1

u0
r̃
)
[∂tu

0]

≃ ∂r̃ + r(∂ũ+ ∂r̃) + (ũ+ r̃) + r(ũ+ s̃+ r̃) + r2(ũ+ s̃+ r̃) + r3(ũ+ s̃+ r̃),

∂tu
j = − ui

u0
∂iu

j − 1

(Γ + r)u0
πj0 [∂tr]−

1

(Γ + r)u0
πji∂ir,

∂tũ
j = − ui

u0
∂iũ

j − 1

(Γ + r)u0
πj0 [∂tr̃]−

1

(Γ + r)u0
πji∂ir̃

+
uiũ0 − u0ũi

(u0)2
∂iu

j +
(uj(Γ′s̃+ r̃) + (Γ + r)ũj

(Γ + r)2

)
[∂tr]

+
( (δji + ujui)[(Γ + r)ũ0 + u0(Γ′s̃+ r̃)]− (Γ + r)u0(ũjui + uj ũi)

(Γ + r)2(u0)2

)
[∂ir]

≃ ∂ũ+ ∂r̃ + r(∂ũ+ ∂r̃) + (ũ+ s̃+ r̃) + r(ũ+ s̃+ r̃)

+ r2(ũ+ s̃+ r̃) + r3(ũ+ s̃+ r̃)

with

a1 = u0 − (γ − 1)((u0)2 − 1)

(Γ + r)u0
r,

ã1 = ũ0 − (γ − 1)((u0)2 − 1)

(Γ + r)u0
r̃

− r
(2(Γ + r)(γ − 1)u0ũ0 − (γ − 1)((u0)2 − 1)[(Γ′s̃+ r̃)u0 + (Γ + r)ũ0]

(Γ + r)2(u0)2

)
≃ ũ+ r̃ + r(ũ+ s̃+ r̃)

Ci
1 := −ui

a1
, C̃i

1 =
uiã1 − a1ũ

i

(a1)2
≃ ũ+ r̃ + r(ũ+ s̃+ r̃),

Ci
2 :=

1

(Γ + r)u0
Ci

1,

C̃i
2 =

1

(Γ + r)u0
C̃i

1 −
[(Γ′s̃+ r̃)u0 + (Γ + r)ũ0]

(Γ + r)2(u0)2
Ci

1 ≃ ũ+ s̃+ r̃ + r(ũ+ s̃+ r̃),

C3 :=
(γ − 1)((u0)2 − 1)

ui
Ci

2,



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 23

C̃3 =
(γ − 1)((u0)2 − 1)

ui
C̃i

2 +
ui2(γ − 1)ũ0 − (γ − 1)((u0)2 − 1)ũi

(ui)2
Ci

2

≃ ũ+ s̃+ r̃ + r(ũ+ s̃+ r̃) ,

where, namely, we only ignored ∂s, ∂r, ∂u, s, u and O(1) coefficients in the simplfi-
cation ≃.

Remark 2.13. In general, we have that

∂ts̃ ≃ ∂s̃+ (terms which are more subcritical than ∂s̃),

∂tũ ≃ ∂ũ+ (terms which are more subcritical than ∂ũ),

∂tr̃ ≃ ∂r̃ + (terms which are more subcritical than ∂r̃)

(2.16)

with coefficient expressions that have at most one spatial derivative of s, r, or u. The
point is that any time derivative of our linearized variables (s̃, r̃, ũ) can be written in
terms of spatial derivatives (with good coefficients) plus additional subcritical terms
that can be easily estimated. When estimating terms such as in Sections 3.2 and
6, we will often apply Lemma 2.12 proactively so as to not introduce unnecessary
subcritical terms that obscure the argument. Thus, going forward, one can treat
the spacetime expressions below with the generic spatial derivative symbol ∂,

∂µs̃ ≃ ∂s̃, ∂µr̃ ≃ ∂r̃, ∂µũ
ν ≃ ∂ũ

from the point of view of their H2k order, as opposed to Remark 1.6 where ∂ would
be used.

Recall that when referring to the order of a free boundary term, we are doing
so with respect to Definition 2.4 and Remark 2.5 where terms with order O = 0
are denoted as critical, O < 0 are supercritical, and O > 0 are subcritical. For
example, it is crucial in our expression (2.16) for ∂tr̃ that the parentheses do not
contain terms of the form ∂ũ. This is because, at a given level say k = 1, the H2

order of a term ∂r̃ is 0 (i.e. critical), but the order of a term like ∂ũ is − 1
2 (i.e.

supercritical) with order strictly worse. If this were the case, we could not reliably
argue that we could replace ∂tr̃ with ∂r̃ since we would be introducing terms that
require additional powers of r in order to be estimated with the H2k norm.

For Lemma 2.12, we have computation details in the Appendix. To finish this
section, we prove a useful Lemma which expands on the notion of order O to each
of our key operations on free boundary terms. This Lemma is especially helpful in
proving higher order analogs of our elliptic and div-curl estimates using induction.
The last proposition in Lemma 2.14 is helpful in jumping from the base case H2 to
the general case H2k.

Lemma 2.14 (Orders of each operation). Let k ∈ N be fixed. Suppose that T is a
free boundary term with H2k order O (i.e. T is of the form rm∂l(s̃, r̃, ũ) for some
m, l ∈ N, m ≥ 0, 0 ≤ l ≤ 2k, ). Then the following holds:

(1) the term rT has H2k order O + 1,
(2) the term ∂T has H2k order O − 1,
(3) the term DtT has a minimum H2k order O − 1

2 ,

(4) If K ∈ N with K ≥ k, then T has H2K order O +K − k.

Proof. Since the case with s̃ is identical to ũ, it suffices to consider the following
cases for our free boundary term T :

(a) T1 = rm1∂l1 r̃ which has H2k order m1 − l1 + k = O



24 B. B. LUCZAK EJDE-2025/10

(b) T2 = rm2∂l2 ũ which has H2k order m2 − l2 + k − 1
2 = O

For Part 1, observe that for cases (a) and (b), we simply have an extra power of
r,

rT1 = r
(
rm1∂l1 r̃

)
= rm1+1∂l1 r̃

=⇒ rT1 has H2k order m1 + 1− l1 + k = O + 1,

rT2 = r
(
rm2∂l2 ũ

)
= rm2+1∂l2 ũ

=⇒ rT2 has H2k order m+ 1− l + k − 1

2
= O + 1.

(2.17)

For Part 2, we see that there are two sub-cases when m1,m2 ≥ 1 and m1 = 0 =
m2. If m1 = 0 = m2, then

∂T1 = ∂(∂l1 r̃) = ∂l1+1r̃

=⇒ ∂T1 has H2k order m1 − (l1 + 1) + k = O − 1,

∂T2 = ∂(∂l2 ũ) = ∂l2+1ũ

=⇒ ∂T2 has H2k order m2 − (l2 + 1) + k − 1

2
= O − 1.

(2.18)

If m1,m2 ≥ 1, then we obtain

∂T1 = ∂(rm1∂l1 r̃) = m1r
m1−1∂l1 r̃ + rm1∂l1+1r̃

=⇒ ∂T1 has H2k order O − 1,

∂T2 = ∂(rm2∂l2 ũ) = m2r
m2−1∂l2 ũ+ rm2∂l2+1ũ

=⇒ ∂T2 has H2k order O − 1

(2.19)

since both terms have H2k order O − 1, noting that we either lose a power of r or
gain a derivative (and both operations decrease the order from O to O − 1).

For Part 3, we will first consider when l1 = 0 = l2. For case with T1, we have

DtT1 = Dt(r
m1 r̃)

= m1r
m1−1(Dtr)r̃ + rm1Dtr̃

≃ m1r
m1 r̃ + rm1(r∂ũ+ ũ)

= rm1+1∂ũ+ rm1 ũ+m1r
m1 r̃,

(2.20)

where we used (2.10) and Lemma 2.11. Then, computing the H2k order (and noting
that we now have some terms with ũ instead of r̃), we see that the first two terms
have order O − 1

2 , while the last one has order O. Thus, DtT1 has a minimum

H2k order O − 1
2 (i.e. it has order O − 1

2 at worst). For T2, we obtain a similar
computation using (2.10) and Lemma 2.11

DtT2 = Dt(r
m2 ũ)

= m2r
m2−1(Dtr)ũ+ rm2Dtũ

≃ m2r
m2 ũ+ rm2(∂r̃ + ũ+ s̃)

= rm2∂r̃ + rm2 ũ+m2r
m2 ũ+ rm2 s̃ ,

(2.21)

where the first term has order m2 − 1+ k = (m2 − 0+ k− 1
2 )−

1
2 = O− 1

2 and the
remaining terms have order O.



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 25

For Part 3, it remains to consider when l1, l2 ≥ 1. We will additionally make use
of Lemma 2.10. For the T1 case, we have

DtT1 = Dt(r
m1∂l1 r̃)

≃ rm1Dt(∂
l1 r̃) +m1r

m1∂l1 r̃

≃ rm1∂l1(Dtr̃) + rm1 [Dt, ∂
l]r̃ +m1r

m1∂l1 r̃

≃ rm1∂l1(r∂ũ+ ũ) + rm1

( l1−1∑
i=0

Bν
i ∂ν(∂

ir̃)
)
+m1r

m1∂l1 r̃

≃ rm1∂l1 ũ+ rm1∂r∂l1 ũ+ rm1+1∂l1+1ũ+
( l1−1∑

i=0

rm1∂i+1r̃
)
+m1r

m1∂l1 r̃,

where we note that ∂l1(r∂ũ) only produces critical terms when all ∂l1 derivatives
hit ∂ũ, or one derivative hits r and ∂l1−1 derivatives hit ∂ũ. For the summation
terms, we make use of Remak 2.13 to simplify ∂ν∂

ir̃ as ∂i+1r̃. Then, counting
the H2k order for each of the terms, we see that the first three have order O − 1

2 ,
whereas the remaining terms have order O at worst (with additional terms in the
summation having order O + l1 − i with 0 ≤ i < l1).

For the T2 case with l1, l2 ≥ 1, we perform a similar computation and keep track
of whether the terms have r̃ or ũ when computing order,

DtT2 = Dt(r
m2∂l2 ũ)

≃ rm2Dt(∂
l2 ũ) +m2r

m2∂l2 ũ

≃ rm2∂l2(Dtũ) + rm2 [Dt, ∂
l]ũ+m2r

m2∂l2 ũ

≃ rm2∂l2(∂r̃ + ũ+ s̃) + rm2

( l2−1∑
i=0

Bν
i ∂ν(∂

iũ)
)
+m2r

m2∂l2 ũ

≃ rm2∂l2+1r̃ + rm2∂l2 ũ+ rm2∂l2 s̃+
( l1−1∑

i=0

rm2∂i+1ũ
)
+m2r

m2∂l2 ũ,

(2.22)

where the first term has H2k order m2−(l2+1)+k = (m2− l2+k− 1
2 )−

1
2 = O− 1

2 ,
and the remaining terms have order O at worst. Thus DtT only produces terms
that have a minimum order of O − 1

2 , and all other terms have strictly O order or

better at the H2k level. This completes the proof of Part 3.
Lastly, we provide the proof of Part 4. In the previous parts, each operation

changed the number of powers of r, the number of derivatives, or whether we are
using r̃ or ũ and each of these has an effect on the computed order. In Part 4, we
are only seeing what happens if we raise the level from k to some K ≥ k, and the
proof can be seen by noticing that for T1,

m1 − l1 +K = (m1 − l1 + k) +K − k = O +K − k

and for T2,

m2 − l2 +K − 1

2
= (m2 − l2 + k − 1

2
) +K − k = O +K − k.

This completes the proof of Part 4. The proof is complete. □

Remark 2.15. A natural corollary of this Lemma is the following. Suppose that
T has H2k-order O. Then, D2

tT has H2k-order O − 1, but this implies that D2
tT



26 B. B. LUCZAK EJDE-2025/10

has H2(k+1)-order O (if we increase the level k by 1). Hence, if a given term is

H2 subcritical (with k = 1), applying D
2(k−1)
t (for k ≥ 2) will keep the term

H2k-subcritical at the corresponding k level.

3. Estimates for the transport energy

3.1. Entropy estimates. Recall the definition of E2k
transport in (1.48) which consists

of both the entropy and the vorticity. Our goal in this section is to prove estimates
for the entropy, whereas the voriticity will be handled separately in Section 3.2.
Instead of applying D2k

t to the s̃ equation, we can simply take ∂2k and estimate
s̃ using the H2k norm directly. This can be done because s̃ satisfies a transport
equation instead of a wave equation that would require the intermediate E2k

wave

energy. We have the following proposition which is routine by envoking out previous
Lemmas on the order of terms.

Proposition 3.1 (Entropy estimates).

d

dt
∥s̃∥2

H
2k,k+ 1

2(γ−1)
≲ B2∥(s̃, r̃, ũ)∥2H2k , (3.1)

where B2 depends on L∞ norms of up to 2k + 1 derivatives of (s, r, u).

Proof. Using equation (1.36a) Lemma 2.10 for the commutator [Dt, ∂
2k], we obtain

Dt(∂
2ks̃) = [Dt, ∂

2k]s̃+ ∂2kf

=

2k−1∑
i=0

Bν
i ∂ν(∂

is̃) + ∂2k(ũµ∂µs)

≃
2k∑
i=0

Bi∂
is̃+ ∂2k(ũµ∂µs) ,

(3.2)

where we recall the definition of f from (1.37), and we used Remark 2.13 to combine
∂ν∂

is̃ and write everything in terms of spatial derivatives. Then, multiplying the

equation by r2k+
1

γ−1 ∂2ks̃, we obtain

1

2
r2k+

1
γ−1Dt(|∂2ks̃|2) ≃ r2k+

1
γ−1 ∂2ks̃

( 2k∑
i=0

Bi∂
is̃
)
+ r2k+

1
γ−1 ∂2ks̃∂2k(ũµ∂µs),

1

2
Dt(|rk+

1
2(γ−1) ∂2ks̃|2)

≃ −1

2
Dt(r

2k+ 1
γ−1 )|∂2ks̃|2 + r2k+

1
γ−1 ∂2ks̃

( 2k∑
i=0

Bi∂
is̃
)
+ r2k+

1
γ−1 ∂2ks̃∂2k(ũµ∂µs)

≃
(
k(γ − 1) +

1

2

)
r2k+

1
γ−1 |∂2ks̃|2 + r2k+

1
γ−1 ∂2ks̃

( 2k∑
i=0

Bi∂
is̃
)

+ r2k+
1

γ−1 ∂2ks̃∂2k(ũµ∂µs).



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 27

Integrating, this leads to

d

dt

1

2

∫
Ωt

|rk+
1

2(γ−1) ∂2ks̃|2 dx ≃
∫
Ωt

1

u0

(
k(γ − 1) +

1

2

)
r2k+

1
γ−1 |∂2ks̃|2 dx

+

∫
Ωt

r2k+
1

γ−1 ∂2ks̃
( 2k∑

i=0

Bi∂
is̃
)
dx

+

∫
Ωt

1

u0
r2k+

1
γ−1 ∂2ks̃∂2k(ũµ∂µs) dx

+
1

2

∫
Ωt

|rk+
1

2(γ−1) ∂2ks̃|2∂i
( ui

u0

)
dx

=: J1 + J2 + J3 + J4.

(3.3)

For the J1 term, we note that u and its derivatives have been absorbed in ≃
symbol (for example when we simplifying Dt(r

2k+ 1
γ−1 )). Thus, we can remove any

derivatives of u and quickly estimate

|J1| ≲ ∥∂u∥L∞(Ωt)

(∫
Ωt

|rk+
1

2(γ−1) ∂2ks̃|2dx
)
≲ ∥∂u∥L∞(Ωt)∥(s̃, r̃, ũ)∥

2
H2k (3.4)

recalling the definition of H2k from (1.46), and Remark 1.13. The estimate for

J4 follows similarly since ∂i

(
ui

u0

)
is an expression depending on up to one spatial

derivative of u. For the J2 and J3 terms, we simply use Lemmas 2.2 and 2.12 as
needed while keeping track of subcritical terms. □

3.2. Vorticity estimates. We also plan to derive estimates for the vorticity which
will be used in connection with the elliptic estimates. We recall from [2] that the
relativistic vorticity is given by

ωαβ = ∂αvβ − ∂βvα. (3.5)

where v is the enthalpy current defined by

vα = huα, h =
Γ + r

Γ
, Γ = Γ(s) =

(γ − 1)
γ−1
γ

γ
e−

γ−1
γ s (3.6)

Similar to [15], we obtain an expression for uαωαβ using the following steps. From
the fact that vαvα = −h2, we obtain

−vα∂βvα = h∂βh. (3.7)

From (1.35c) rewritten in terms of v, we obtain

1

h2
vα∂αvβ = − 1

Γ + r
∂βr (3.8)

Then, we can compute the expression

1

h
(uα∂αvβ − uα∂βvα) =

r

(Γ + r)

γ − 1

γ
∂βs. (3.9)

Multiplying by h, we obtain

uαωαβ =
r

γΓ
(γ − 1)∂βs (3.10)

which agrees with [15] as

ε =
r

γΓ
(3.11)



28 B. B. LUCZAK EJDE-2025/10

after checking (1.24). To find the evolution equation satisfied by ωαβ , we can first
compute that

∂αε =
1

γΓ

(
∂αr +

γ − 1

γ
r∂αs

)
.

Using similar computations as [15] and applying (3.11), we have

uµ∂µωαβ + ∂αu
µωµβ + ∂βu

µωαµ =
γ − 1

γΓ
(∂αr∂βs− ∂βr∂αs) (3.12)

which is the evolution equation for the vorticity ωαβ in terms of the unknowns
(s, r, u).

Next, we introduce the full linearized voricity ω̃αβ by

ω̃αβ = ∂α(h̃uβ)− ∂β(h̃uα)

= ∂α(hũβ)− ∂β(hũα) + ∂α(h̃uβ)− ∂β(h̃uα)

:= ω̂αβ + ωαβ ,

(3.13)

where

ω̂αβ := ∂α(hũβ)− ∂β(hũα) (3.14)

will be called the reduced linearized vorticity and ωαβ is the h̃-scaled vorticity.
The plan is to derive estimates for the full linearized vorticity ω̃, and then relate

this back to the reduced linearized vorticity ω̂ by ∥ω̂∥ ≲ ∥ω̃∥+ ∥ω∥. Our reasoning
is a matter of convenience, since estimates for ω̃ are more readily available in view
of (3.16). Additionally, ω̂ better resembles curl ũ in connection with our elliptic
estimates (see Section 4 and Lemma 4.4).

Throughout this section, we will use the following facts similar to the bookkeep-
ing system from Lemma 2.11. We put the order simplifications here for reference

ωαβ ≃ ∂h̃+ h̃

h̃ ≃ r̃ + rs̃

∂αh̃ ≃ ∂r̃ + r∂s̃+ r̃ + s̃+ rs̃ ,

(3.15)

where we also make use of Lemma 2.12 in the expression for ∂αh̃, i.e. ∂th̃ will have a
slightly different expression, just with more subcritical terms coming from ∂ts̃ ∼ ∂s̃
and ∂tr̃ ∼ ∂r̃.

From linearizing (3.12), we have the evolution equation for ω̃ given by

uµ∂µω̃αβ + ∂αu
µω̃µβ + ∂βu

µω̃αµ

= −ũµ∂µωαβ − ∂αũ
µωµβ − ∂β ũ

µωαµ +
(γ − 1)2

γ2Γ
s̃(∂αr∂βs− ∂βr∂αs)

+
γ − 1

γΓ
(∂αr̃∂βs+ ∂αr∂β s̃− ∂β r̃∂αs− ∂βr∂αs̃)

(3.16)

For the following lemma, recall Remark 1.13 where we have by embedding the-

orems that H2k ∼ H2k,k+ 1
2(γ−1) ×H2k,k+ 1

2(γ−1)
− 1

2 ×H2k,k+ 1
2(γ−1) where the weight

on r̃ is less by 1
2 .

Lemma 3.2 (Estimates for ω̃). For a solution ω̃ to (3.16) on an interval [0, T ]
defined in this section, we have the estimate

d

dt
∥ω̃∥2

H
2k−1,k+ 1

2(γ−1)
≲ C∥(s̃, r̃, ũ)∥2H2k , (3.17)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 29

where C depends on the L∞(Ωt) norms of up to 2k + 1 derivatives of s, r, and u.

Proof. We begin by fixing a spatial multiindex l such that |l| ≤ 2k − 1. We recall

the notation |∂lω̃|2G = Gα
γG

β
δ ∂

lω̃γδ∂lω̃αβ (where l is not summed over) and G is

the Riemannian metric defined in (1.42). Then, we apply ∂l to (3.16) and contract

against the expression r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδ .
When contracting, we will use the quick observation that

r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδuµ∂µ∂
lω̃αβ

=
1

2
r2k+

1
(γ−1)Dt(|∂lω̃|2G)− r2k+

1
(γ−1)

1

2
Dt(G

α
γG

β
δ )∂

lω̃γδ∂lω̃αβ

=
1

2
Dt(r

2k+ 1
(γ−1) |∂lω̃|2G)−

1

2
(2k +

1

γ − 1
)r2k+

1
(γ−1) ∂µu

µ|∂lω̃|2G

− r2k+
1

(γ−1)
1

2
Dt(G

α
γG

β
δ )∂

lω̃γδ∂lω̃αβ .

(3.18)

and only the first term in (3.18) will be kept on the LHS. This leads to the long
expression

1

2
Dt(r

2k+ 1
(γ−1) |∂lω̃|2G)

=
1

2
(2k +

1

γ − 1
)r2k+

1
(γ−1) ∂µu

µ|∂lω̃|2G

+ r2k+
1

(γ−1)
1

2
Dt(G

α
γG

β
δ )∂

lω̃γδ∂lω̃αβ

− r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδ∂l (∂αu
µω̃µβ + ∂βu

µω̃αµ)

− r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδ∂l (ũµ∂µωαβ)

− r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδ∂l (∂αũ
µωµβ)

− r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδ∂l (∂β ũ
µωαµ)

+ r2k+
1

(γ−1)Gα
γG

β
δ ∂

lω̃γδ∂l
( (γ − 1)2

γ2Γ
s̃(∂αr∂βs− ∂βr∂αs)

)
+ r2k+

1
(γ−1)Gα

γG
β
δ ∂

lω̃γδ∂l
(γ − 1

γΓ
(∂αr̃∂βs+ ∂αr∂β s̃− ∂β r̃∂αs− ∂βr∂αs̃)

)
:=

8∑
i=1

Ri,

(3.19)

where each term on the RHS of (3.19) will be handled separately. Then, using the

function f(x, t) = r2k+
1

(γ−1) |∂lω̃|2G, we have

d

dt

∫
Ωt

f dx =

∫
Ωt

f∂i

(
ui

u0

)
dx+

∫
Ωt

1

u0

8∑
i=1

Ri dx. (3.20)

We plan to estimate each of the terms on the RHS of (3.20) by the H2k norm which
is also used in Theorem 6.2. This is done by analyzing the H2k order (see Remark
2.5) and making use of Lemma 2.12 as needed for any time derivatives that appear.

We start by observing that the first term on the RHS of (3.20) is handled in an
identical way as the term J4 from (3.3) in the entropy estimates. We can use (3.13)



30 B. B. LUCZAK EJDE-2025/10

and (3.15) to obtain the additional comment that

ω̃ := ω̂ + ω ≃ ∂ũ+ ∂r̃ + r∂s̃+ s̃+ r̃ + rs̃ (3.21)

noting that, from an H2k order perspective, ω̃ contains terms with the same order
as ∂ũ or better. These terms primarily come from the reduced linearized vorticity
ω̂, whereas ω ∼ ∂h̃+ h̃ and h̃ contains terms that are like r̃ at worst.

To finish estimating the first term on the RHS of (3.20), we have∣∣ ∫
Ωt

r2k+
1

(γ−1) |∂lω̃|2G∂i
( ui

u0

)
dx

∣∣
≲ ∥∂u∥L∞(Ωt)

∫
Ωt

r2k+
1

(γ−1)
∣∣Gα

γG
β
δ ∂

lω̃γδ∂lω̃αβ

∣∣ dx
≲ ∥∂u∥L∞(Ωt)

∫
Ωt

r
2−γ
γ−1

∣∣rk+ 1
2 ∂l(∂ũ+ ∂r̃ + r∂s̃+ s̃+ r̃ + rs̃)

∣∣2 dx,
(3.22)

where we applied (3.21). The point is that if we greatly expand the expression

Gα
γG

β
δ ∂

lω̃γδ∂lω̃αβ , only the terms with the worst H2k order need to be checked,
since terms that are more subcritical have plenty of powers of r to be estimated
using the H2k norm. In this integral, the terms with the worst order are rk+

1
2 ∂l+1ũ.

Since |l| ≤ 2k − 1, we see that the order of this term is exactly 0 (critical) when
|l| = 2k − 1 and subcritical when |l| < 2k − 1. Thus, it can be estimated using the
H2k norm. For any cross terms in the expansion of |∂l(∂ũ+∂r̃+r∂s̃+ s̃+ r̃+rs̃)|2,
we simply apply Lemma 2.6 where the order of a product of is the sum of the orders

of each term. For example, in the product |(rk+
1

2(γ−1) ∂l+1ũ)(rk+
1

2(γ−1) ∂l+1r̃)|, we
see that the order (i.e. when γ = 2) is 0+1/2 = 1/2 ≥ 0 at worst when |l| = 2k−1,
so Lemma 2.6 implies that∫

Ωt

|(rk+
1

2(γ−1) ∂l+1ũ)(rk+
1

2(γ−1) ∂l+1r̃)| dx ≲ ∥(s̃, r̃, ũ)∥2H2k . (3.23)

All other cross terms in |∂l(∂ũ+ ∂r̃+ r∂s̃+ s̃+ r̃+ rs̃)|2 are handled using Lemma
2.6. Thus, we obtain∣∣ ∫

Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂lω̃|2G∂i
( ui

u0

)
dx

∣∣ ≲ C0∥(s̃, r̃, ũ)∥2H2k , (3.24)

where C0 depends on the L∞(Ωt) norms of up to 2k derivatives of s, r, and u.
We will proceed through each remaining term on the RHS of (3.20). For the

integrals containing R1 and R2, the estimate is quite similar to the first term since

derivatives of u will be removed, and Dt(G
α
γG

β
δ ) = Dt((g

α
γ + 2uαuγ)(g

β
δ + 2uβuδ))

is another expression containing derivatives of u. We have

|R1| ≲ ∥∂u∥L∞(Ωt)

∫
Ωt

r
2−γ
γ−1

(
k +

1

2(γ − 1)

)
|rk+ 1

2 ∂lω̃|2G dx ≲ C1∥(s̃, r̃, ũ)∥2H2k

and

|R2| ≲
∫
Ωt

r2k+
1

(γ−1)
1

2
|Dt(G

α
γG

β
δ )∂

lω̃γδ∂lω̃αβ | dx

≲ ∥∂u∥L∞(Ωt)

∫
Ωt

r
2−γ
γ−1

∣∣rk+ 1
2 ∂l(∂ũ+ ∂r̃ + r∂s̃+ s̃+ r̃ + rs̃)

∣∣2 dx
≲ C2∥(s̃, r̃, ũ)∥2H2k ,

(3.25)

where we handled it in the same way as for (3.22).



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 31

For the R3 integral, distributing ∂l (∂αu
µω̃µβ + ∂βu

µω̃αµ) leads to terms that
contain ∂lω̃ at worst along with up to 2k derivatives of u. We can write this as

∂l (∂αu
µω̃µβ + ∂βu

µω̃αµ) ≃
|l|∑

n=0

∂nω̃∂|l|−n+1u (3.26)

so that, upon estimating, we will have

|R3| ≲ ∥∂2ku∥L∞(Ωt)

∫
Ωt

r
2−γ
γ−1

|l|∑
n=0

|rk+ 1
2 ∂lω̃||rk+ 1

2 ∂nω̃| dx ≲ C3∥(s̃, r̃, ũ)∥2H2k

since the worst term in the summation only contains 2k − 1 derivatives of ω̃.
For the R4 integral, we have ∂l(ũµ∂µωαβ) which will lead to terms with ∂2k−1ũ

at worst and coefficients containing up to 2k + 1 derivatives of u, r, and s coming
from ∂l(∂µωαβ). Since we have previously estimated up to ∂2kũ, the terms in R4

will be more subcritical (and thus easier to estimate) than R1 through R3 which
all contain 2k − 1 derivatives of ω̃. We can write it as

|R4| ≲ ∥∂2k+1(u, r, s)∥L∞(Ωt)

∫
Ωt

r
2−γ
γ−1 |(rk+ 1

2 ∂lω̃)(rk+
1
2 ∂lũ)| dx ≲ C4∥(s̃, r̃, ũ)∥2H2k

For the R5 and R6 integrals, the argument is quite similar since we obtain up to
2k derivatives of ũ and up to 2k derivatives of ω.

For the R7, integral, we will have terms containing up to 2k − 1 derivatives of s̃
and 2k derivatives of r and s. However, we know from computing H2k orders that
∂2k−1s̃ has the same order as ∂2k−1ũ, so it can be estimated in the same way as
R4.

Finally, for the R8 integral, we will have terms containing up to 2k derivatives
of s̃ and r̃, and up to 2k derivatives of s and r. However, we know that the order
of these terms will be comparable to ∂2kũ at worst (and in fact, terms with ∂2kr̃
are more subcritical by an order value of 1/2).

Thus, by estimating each of the terms in (3.20) separately and removing expres-
sions involving L∞ norms of up to 2k+1 derivatives of s, r, and u, we combine the
above estimates with (3.20) to obtain

d

dt

∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂lω̃|2G dx ≲ C∥(s̃, r̃, ũ)∥2H2k , (3.27)

where C depends on up to 2k + 1 derivatives of s, r, and u. Summing over the
multiindex l achieves the desired vorticity estimate using the H2k norm. □

Lemma 3.2 leads nicely into the next theorem, and we note how this theorem
compares with the transport energy defined in (1.48).

Theorem 3.3 (Linearized vorticity, Transport energy estimates). The following
estimate holds for the linearized vorticity defined in (3.13).

∥ω̂∥2
H

2k−1,k+ 1
2(γ−1) (Ωt)

≲ ∥ω̃0∥2
H

2k−1,k+ 1
2(γ−1) (Ω0)

+ ε∥(s̃, r̃, ũ)∥2H2k(Ωt)
+

∫ t

0

C∥(s̃, r̃, ũ)∥2H2k(Ωτ )
dτ ,

(3.28)

where ω̃0 is the initial linearized vorticity and C is from Lemma 3.2.



32 B. B. LUCZAK EJDE-2025/10

Proof. We plan to use Lemma 3.2 and the identity ω̂ = ω̃−ω to split the estimate:

∥ω̂∥2
H

2k−1,k+ 1
2(γ−1) (Ωt)

≲ ∥ω̃∥2
H

2k−1,k+ 1
2(γ−1) (Ωt)

+ ∥ω∥2
H

2k−1,k+ 1
2(γ−1) (Ωt)

(3.29)

Now, from the definition of ω and the simplifications outlined in (3.15), we obtain
the following when applying a multiindex l with |l| ≤ 2k − 1:

∂lω ≃ ∂l(∂(r̃ + rs̃) + r̃ + rs̃)

≃ ∂l+1r̃ + r∂l+1s̃+ ∂ls̃,
(3.30)

where the additional terms are even more subcritical than the ones listed. Then,

using the L2(r2k+
1

(γ−1) ) norm, we have∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂lω|2G dx

≲
∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂l+1r̃|2 dx+

∫
Ωt

r
2−γ
γ−1 |rk+ 1

2+1∂l+1s̃|2 dx+

∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂ls̃|2 dx

≲ ε̂

∫
Ωt

r
2−γ
γ−1 |rk∂l+1r̃|2 dx+ ε̂

∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂l+1s̃|2 dx

+ ε̂

∫
Ωt

r
2−γ
γ−1 |rk−1+ 1

2 ∂ls̃|2 dx,

where we used the smallness of r from Remark 1.9. Summing over |l| ≤ 2k −
1 and counting the order for each term (where the worst terms take the form

rk∂2kr̃, rk+
1
2 ∂2kũ, and rk+

1
2−1∂2k−1s̃), we obtain the estimate

∥ω∥2
H

2k−1,k+ 1
2(γ−1)

≲ ε̂∥(s̃, r̃, ũ)∥2H2k (3.31)

By integrating Lemma 3.2, we have

∥ω̃∥2
H

2k−1,k+ 1
2(γ−1) (Ωt)

≲ ∥ω̃0∥2
H

2k−1,k+ 1
2(γ−1) (Ω0)

+

∫ t

0

C∥(s̃, r̃, ũ)∥2H2k(Ωτ )
dτ , (3.32)

where ω̃0 can be solved for using the initial data for the linearized equation (s̃0, r̃0, ũ0).
Combining the above equations together yields the desired result. □

Theorem 3.3 will be used in our main result, Theorem 7.1.

4. Elliptic and div-curl Estimates

Our goal is to bound the total energy (1.49) from above and below by the H2k

norm (1.46). In this section, we focus on bounding (1.49) from below, and one of
the key ingredients for the wave part (1.47) is elliptic estimates involving r̃ and the
spatial divergence of ũ. The proofs in this section involve isolating a “good” spatial
elliptic operator similar to [4], with the added difficulty in that we must explicitly
split off time derivatives while isolating the critical terms and using Section 2.2.
Additionally, we will often make use of Lemma 2.11 where identities with Dt serve
as another method for counting order.

Our analysis will focus on isolating the “good” spatial elliptic operators, as one
can readily compare that the structure of these operators closely resembles the
elliptic operators already treated in [4].



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 33

4.1. Elliptic estimates for r̃. Upon taking Dt of equation (1.36b) and applying
Theorem 2.11 as needed, we can rewrite D2

t r̃ in the following way:

D2
t r̃ ≃ L1r̃ + additional terms (4.1)

with

L1r̃ :=
γ − 1

Γ + r
πµν(r∂µ∂ν r̃ +

1

γ − 1
∂µr∂ν r̃)

=
γ − 1

Γ + r
πij(r∂i∂j r̃ +

1

γ − 1
∂ir∂j r̃) +

γ − 1

Γ + r
π00(r∂2

t r̃ +
1

γ − 1
∂tr∂tr̃)

+
2(γ − 1)

Γ + r
πi0(r∂i∂tr̃) +

1

Γ + r
π0j(∂tr∂j r̃) +

1

Γ + r
πi0(∂ir∂tr̃)

(4.2)

serving as the “spacetime” elliptic operator for r̃, and we have split the time deriva-
tives from the spatial derivatives. To see how the additional terms in (4.1) will be
treated, as well as the higher order elliptic estimates, we refer the reader to section
4.3 where these additional terms are shown to be subcritical using our book-keeping
scheme from Section 2.

It is important to note that since we are estimating the wave energy from below
by the H2k norm, we cannot simply cite section 2.2 for the time derivatives since
the structure of the terms is quite important. Instead, we will make use of the
following additional facts from the definition of Dt:

∂tr̃ =
1

u0
Dtr̃ −

ui

u0
∂ir̃

r∂2
t r̃ =

1

(u0)2
rD2

t r̃ −
uiuj

(u0)2
r∂i∂j r̃ − 2

ui

u0
r∂i∂tr̃ −

1

(u0)2
Dtu

0r∂tr̃ −
1

(u0)2
Dtu

ir∂ir̃

=
1

(u0)2
rD2

t r̃ +
uiuj

(u0)2
r∂i∂j r̃ − 2

ui

(u0)2
r∂i(Dtr̃) + 2

ui

u0
∂i
uj

u0
r∂j r̃

− 1

(u0)2
Dtu

0r∂tr̃ −
1

(u0)2
Dtu

ir∂ir̃,

(4.3)
where, by Remark 2.5, the only critical terms (for k = 1) are in red. For the
remaining terms, we can write

1

Γ + r
π00∂tr∂tr̃ =

1

Γ + r
π00∂tr

( 1

u0
Dtr̃ −

ui

u0
∂ir̃

)
=

1

(Γ + r)u0
π00∂trDtr̃−

1

(Γ + r)u0
π00∂tru

i∂ir̃

(4.4)

and

2(γ − 1)

Γ + r
πi0(r∂i∂tr̃) =

2(γ − 1)

Γ + r
uiu0r∂i

( 1

u0
Dtr̃ −

uj

u0
∂j r̃

)
=

2(γ − 1)

Γ + r
rui∂i(Dtr̃)−

2(γ − 1)

Γ + r
ruiuj∂i∂j r̃

+
2(γ − 1)

Γ + r
uiu0r

[
Dtr̃∂i

1

u0
− ∂i

uj

u0
∂j r̃

]
(4.5)



34 B. B. LUCZAK EJDE-2025/10

and

1

Γ + r
πi0(∂ir∂tr̃) =

1

Γ + r
πi0∂ir

( 1

u0
Dtr̃ −

uj

u0
∂j r̃

)
=

1

(Γ + r)u0
πi0∂irDtr̃−

1

(Γ + r)u0
πi0∂iru

j∂j r̃.

(4.6)

So, if we gather all the red critical terms (which look like r∂2r̃ or ∂r̃), we will get
precisely,

L1r̃ =
γ − 1

Γ + r

(
πij +

((u0)2 − 1)uiuj

(u0)2
− 2uiuj

)
r∂i∂j r̃

+
1

Γ + r
gij∂ir∂j r̃ +

1

Γ + r

(
u0uj − ((u0)2 − 1)

u0
uj
)
∂tr∂j r̃

+
γ − 1

Γ + r

( 1

(u0)2
rD2

t r̃ − 2
ui

(u0)2
r∂i(Dtr̃) + 2

ui

u0
∂i
uj

u0
r∂j r̃

− 1

(u0)2
Dtu

0r∂tr̃ −
1

(u0)2
Dtu

ir∂ir̃
)
+

1

(Γ + r)u0
π00∂trDtr̃

+
2(γ − 1)

Γ + r
rui∂i(Dtr̃) +

2(γ − 1)

Γ + r
uiu0r

[
Dtr̃∂i

1

u0
− ∂i

uj

u0
∂j r̃

]
+

1

(Γ + r)u0
πi0∂irDtr̃

(4.7)

and we can check that each of the black terms is subcritical with the help of Lemma
2.11 and Remark 2.5. For a discussion of how these subcritical terms are handled,
see section 4.3 below. Finally, we use (4.3) to simplify ∂tr in the red critical terms
(which splits into a subcritical and a critical term). Combining, we end up with

L1r̃ =
γ − 1

Γ + r
Hij(r∂i∂j r̃ +

1

γ − 1
∂ir∂j r̃) + (black subcritical terms) , (4.8)

where

Hij := δij − uiuj

(u0)2
. (4.9)

If we set the “good” spatial elliptic operator to be

L̃1r̃ :=
γ − 1

Γ + r
Hij(r∂i∂j r̃ +

1

γ − 1
∂ir∂j r̃), (4.10)

then we obtain the expression:

L1r̃ = L̃1r̃ + (black subcritical terms) , (4.11)

where the term in red is critical, and all the black sub-critical terms are equal to

1

Γ + r

1

(u0)2
Dtru

j∂j r̃ +
γ − 1

Γ + r

( 1

(u0)2
rD2

t r̃ − 2
ui

(u0)2
r∂i(Dtr̃)

+ 2
ui

u0
∂i
uj

u0
r∂j r̃ −

1

(u0)2
Dtu

0r∂tr̃ −
1

(u0)2
Dtu

ir∂ir̃
)

+
1

(Γ + r)u0
π00∂trDtr̃ +

2(γ − 1)

Γ + r
rui∂i(Dtr̃)

+
2(γ − 1)

Γ + r
uiu0r

[
Dtr̃∂i

1

u0
− ∂i

uj

u0
∂j r̃

]
+

1

(Γ + r)u0
πi0∂irDtr̃



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 35

under the book-keeping scheme of Section 2 (where we can use Remark 2.5 and
Lemma 2.11 as needed to see that these terms have order O = 1

2 at worst). See
section 4.3 below for how these terms are handled.

Lemma 4.1 (Ellptic estimates for r̃). The following estimates hold for r̃ and the

“good” spatial elliptic part L̃1r̃:

∥r̃∥
H

2, 1
2(γ−1)

+ 1
2
≲ ∥L̃1r̃∥

H
0, 1

2(γ−1)
− 1

2
+ ∥r̃∥

L2(r
2−γ
γ−1 )

(4.12)

Proof. After examining the structure of L̃1r̃, we see that the proof will closely follow
[4] by proving two related inequalities:

∥r̃∥
H

2, 1
2(γ−1)

+ 1
2
≲ ∥L̃1r̃∥

H
0, 1

2(γ−1)
− 1

2
+ ∥r̃∥

H
1, 1

2(γ−1)
− 1

2

∥r̃∥
H

1, 1
2(γ−1)

− 1
2
≲ ∥L̃1r̃∥

H
0, 1

2(γ−1)
− 1

2
+ ∥r̃∥

L2(r
2−γ
γ−1 )

(4.13)

Instead of repeating the proof of [4, Lemma 5.3], we illustrate a few of the
primary differences. When derivatives hit the weight 1

Γ+r , we see by (1.33) that

∂k

( 1

Γ + r

)
=

1

(Γ + r)2

(γ − 1

γ
Γ∂ks− ∂kr

)
(4.14)

which only contains derivatives of r and s. Similarly, when derivatives hit the
metric Hij , we obtain an expression that only contains derivatives of u (and this
metric also appears in [4]).

To show the next inequality in (4.13), we use a similar method as the proof of
[4, Corollary 5.5]. □

4.2. Elliptic estimates for ũ. Next, we handle the ũ equation. From (2.1b), we
note that D2

t ũ
α will satisfy an equation involving one derivative of the spacetime

divergence ∂ν ũ
ν (see (4.18) for the definition of operator L2). In view of Section

2.2, we would like to solve for time derivatives in terms of spatial derivatives and
relate our analysis back to the spatial divergence of ũ which we denote by

−→
divũ := ∂j ũ

j (4.15)

To estimate our energies from below with the H2k norm, we will need to do elliptic

estimates involving
−→
divũ. After extracting the spatial part, we will arrive at the

“good” operator L̃2 (see (4.26)) which is properly suited for div-curl estimates. To
complete the estimates, we will need to examine the spatial curl whose components
are given by:

(
−−→
curlũ)ij := ∂iũj − ∂j ũi. (4.16)

We plan to pair L̃2 with the corresponding operator L̃3 (see (4.28) below), and we

note that L̃3 is roughly one spatial derivative of
−−→
curlũ with a weight r.

Our goal is Lemma 4.3, but we will begin by highlighting key steps in the follow-
ing computation to show how we correctly isolate the spatial components. Similar
to the r̃ equation, we start by isolating the spacetime elliptic piece of D2

t ũ
α which

will produce the following upon taking Dt of (2.1b):

D2
t ũ

α ≃ (L2ũ)
α
+ additional terms, (4.17)



36 B. B. LUCZAK EJDE-2025/10

where

(L2ũ)
α
:=

γ − 1

Γ + r
παµ

(
∂µ(r∂ν ũ

ν) +
1

γ − 1
∂µũ

ν∂νr
)

=
γ − 1

Γ + r
παi

(
∂i(r∂ν ũ

ν) +
1

γ − 1
∂iũ

ν∂νr
)

+
γ − 1

Γ + r
πα0

(
∂t(r∂ν ũ

ν) +
1

γ − 1
∂tũ

ν∂νr
) (4.18)

and the additional terms will be shown to be subcritical (see section 4.3 for a
discussion of these terms). Then, solving for

∂t =
1

u0
Dt −

ui

u0
∂i (4.19)

and gathering up similar terms, we have the useful identity

παµ∂µ = παi∂i + πα0∂t := Bαi∂i + πα0 1

u0
Dt , (4.20)

where we defined:

Bαi := gαi − gα0
ui

u0
. (4.21)

Further, note the explicit identity

GαβB
αiBβj = Hij , (4.22)

where Gαβ is defined in (1.42), and Hij is defined in (4.9).
Using these identities, we arrive at the equivalent expression

(L2ũ)
α
=

γ − 1

Γ + r
Bαi

(
∂i(r∂ν ũ

ν) +
1

γ − 1
∂iũ

ν∂νr
)

+
γ − 1

Γ + r
πα0 1

u0

(
Dt(r∂ν ũ

ν) +
1

γ − 1
Dtũ

ν∂νr
)
.

(4.23)

Remark 4.2. If we suppose that k = 1 for computing order (see Remark 2.5), note
that the terms in red have order − 1

2 which are supercritical, but the terms in black
have order 0 which is a strictly better order. In the context of estimating L2ũ

α, we
will use the norm H0, 12 in which case we will have an extra 1/2 power of r in order
to safely move these black terms to the other side. Thus, the terms in black can be
treated in a very similar way as (4.11) in the r̃ elliptic estimates.

At this stage, we still have an expression for L2ũ
α involving the spacetime diver-

gence ∂ν ũ
ν . To further isolate the spatial divergence, we make use of the following

identities by repeatedly splitting and removing time derivatives as needed. We
apply (1.61) and (4.19) to obtain

∂ν ũ
ν = Hjk∂j ũk +

uj

(u0)2
Dtũj + ũj∂t

uj

u0
. (4.24)

Similarly, we can apply the same identities such as (1.61) to replace any instances
of ũ0 with a spatial component in the expression ∂iũ

ν∂νr. Splitting the sum over
ν and solving for ∂tr using (4.19), we arrive at

∂iũ
ν∂νr

= ∂iũ
k∂kr +

(
−ulu

m

(u0)2
∂iũ

l∂mr +
ul

(u0)2
∂iũ

lDtr −
ũlum

u0
∂i

ul

u0
∂mr +

ũl

u0
∂i

ul

u0
Dtr

)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 37

= Hkm∂iũk∂mr +
( ul

(u0)2
∂iũ

lDtr −
ũlum

u0
∂i

ul

u0
∂mr +

ũl

u0
∂i

ul

u0
Dtr

)
Plugging these identities into (4.23), we have

(L2ũ)
α
=

γ − 1

Γ + r
BαiHjk

(
∂i(r∂j ũk) +

1

γ − 1
∂jr∂iũk

)
+

γ − 1

Γ + r
Bαi∂i

(
uj

(u0)2
rDtũj + rũj∂t

uj

u0

)
+

1

Γ + r
Bαi

(
ul

(u0)2
∂iũ

lDtr −
ũlum

u0
∂i

ul

u0
∂mr +

ũl

u0
∂i

ul

u0
Dtr

)
.

(4.25)

If we define the “good” spatial elliptic part for the divergence of ũ to be(
L̃2ũ

)α

:=
γ − 1

Γ + r
BαiHjk

(
∂i(r∂j ũk) +

1

γ − 1
∂jr∂iũk

)
, (4.26)

and observe Remark 4.2, we obtain the nice expression

(L2ũ)
α
=

(
L̃2ũ

)α

+ (black critical and subcritical terms) (4.27)

Next, we will combine the
(
L̃2ũ

)α

with a corresponding curl term in order to

prove the desired div-curl estimates. Consider the term(
L̃3ũ

)α

:=
γ − 1

Γ + r
Bαir−

1
γ−1Hml∂l

(
r1+

1
γ−1 (∂mũi − ∂iũm)

)
(4.28)

Lemma 4.3 (div-curl estimates for ũ). The following estimates hold for ũ and the

“good” spatial elliptic parts
(
L̃2ũ

)α
and

(
L̃3ũ

)α
:

∥ũ∥
H

2, 1
2(γ−1)

+1 ≲ ∥(L̃2 + L̃3)ũ∥
H

0, 1
2(γ−1)

+ ∥ũ∥
L2(r

1
γ−1 )

(4.29)

Proof. This proof is done in two steps similar to those in Lemma 4.1, and it closely
follows [4, Lemma 5.3] where one should compare with our definitions of L̃2 and

L̃3. Additionally, we make explicit use of identity (1.61) since ũαuα = 0, and we
refer the reader to Remark 1.20.

We start by writing

∥(L̃2 + L̃3)ũ∥2
H

0, 1
2(γ−1)

=

∫
Ωt

r
1

γ−1Gαβ [(L̃2 + L̃3)ũ]
α[(L̃2 + L̃3)ũ]

β dx (4.30)

Then, using identity (4.22), we obtain the following collection of terms in the inte-
gral: (γ − 1

Γ + r

)2

r
1

γ−1HiaHjkH ln
[
r∂i∂j ũk + ∂ir∂j ũk

+
1

γ − 1
∂jr∂iũk + r−

1
γ−1 ∂k

(
r1+

1
γ−1 (∂j ũi − ∂iũj)

) ][
r∂a∂lũn

+ ∂ar∂lũn +
1

γ − 1
∂lr∂aũn + r−

1
γ−1 ∂n

(
r1+

1
γ−1 (∂lũa − ∂aũl)

) ] (4.31)



38 B. B. LUCZAK EJDE-2025/10

which can be expanded to obtain(γ − 1

Γ + r

)2

r
1

γ−1HiaHjkH ln
[
r∂i∂j ũk + ∂ir∂j ũk +

1

γ − 1
∂jr∂iũk

+ r∂k∂j ũi − r∂k∂iũj +
γ

γ − 1
(∂kr∂j ũi − ∂kr∂iũj)

][
r∂a∂lũn + ∂ar∂lũn

+
1

γ − 1
∂lr∂aũn + r∂n∂lũa − r∂n∂aũl +

γ

γ − 1
(∂nr∂lũa − ∂nr∂aũl)

]
,

(4.32)

where each of the terms in red take the form r∂2ũ. Then, we group the red terms
and integrate by parts while using the symmetry present in the expression

Dia,jk,ln :=
(γ − 1

Γ + r

)2

HiaHjkH ln. (4.33)

In this form, we can apply the same ideas as [4] where we first prove

∥(L̃2 + L̃3)ũ∥2
H

0, 1
2(γ−1)

≳ ∥ũ∥2
H

2, 1
2(γ−1)

+1
− ∥ũ∥2

H
1, 1

2(γ−1)
, (4.34)

and then integrate by parts with ∂3ũa to prove the second inequality

∥(L̃2 + L̃3)ũ∥2
H

0, 1
2(γ−1)

≳ ∥ũ∥2
H

1, 1
2(γ−1)

− ∥ũ∥2
L2(r

1
γ−1 )

(4.35)

in which case (4.34) and (4.35) will combine to prove the desired result. □

For the last lemma in this section, we ultimately need to connect the spacetime

two-form ω̂ back to
−−→
curlũ which is a key quantity in our div-curl estimates for ũ.

Lemma 4.4 (Relating vorticity to spatial curl). The following estimate holds for
the reduced linearized vorticity:

∥ω̂∥2
H

2k−1,k+ 1
2(γ−1)

≳ ∥
−−→
curlũ∥2

H
2k−1,k+ 1

2(γ−1)
− ε̂2∥ũ∥2

H
2k,k+ 1

2(γ−1)
, (4.36)

where ε̂ is small positive constant defined in Assumption 1.8, and
−−→
curlũ is defined

in 4.16.

Proof. Recalling the definition of ω̂ in (3.14) which is a spacetime two-form, we

observe the following. First, we define the spatial reduced linearized vorticity
−→
ω̂ to

be a spatial two-form with components

ω̂ij = ∂i(hũj)− ∂j(hũi) = h(∂iũj − ∂j ũi) + ũj∂ih− ũi∂jh,

Then, if we denote by | · |2
δ(3)

the spatial Euclidean norm (squared), we have

|
−→
ω̂ |2δ(3) := δijδkmω̂ikω̂jm. (4.37)

For a generic multiindex l with |l| ≤ 2k − 1, we claim that the following inequality
holds:

|∂l−→ω̂ |2δ(3) ≲ |∂lω̂|2G (4.38)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 39

Indeed, since G is a Riemannian metric on R4, it equivalent to δ on R4 with
constants depending on u. This gives

|∂lω̂|2G
≳

∣∣∂lω̂
∣∣2
δ(4)

= δαγδβδ∂lω̂αβ∂
lω̂γδ

= δiγδβδ∂lω̂iβ∂
lω̂γδ + δ0γδβδ∂lω̂0β∂

lω̂γδ

= δijδβδ∂lω̂iβ∂
lω̂jδ + δi0δβδ∂lω̂iβ∂

lω̂0δ + δβδ∂lω̂0β∂
lω̂0δ

= δijδkδ∂lω̂ik∂
lω̂jδ + δijδ0δ∂lω̂i0∂

lω̂jδ + δβδ∂lω̂0β∂
lω̂0δ

= δijδkm∂lω̂ik∂
lω̂jm + δijδk0∂lω̂ik∂

lω̂j0 + δij∂lω̂i0∂
lω̂j0 + δβδ∂lω̂0β∂

lω̂0δ

=
∣∣∣∂l−→ω̂

∣∣∣2
δ(3)

+ δij∂lω̂i0∂
lω̂j0 + δij∂lω̂0i∂

lω̂0j ,

(4.39)

where we split indices, the red terms are zero, and the blue term is zero when β = 0
or δ = 0 using the definition of ω̂. Then, using ω̂ij = −ω̂ji, we have

|∂l−→ω̂ |2δ(3) + δij∂lω̂i0∂
lω̂j0 + δij∂lω̂0i∂

lω̂0j = |∂l−→ω̂ |2δ(3) +
3∑

i=1

(∂lω̂i0)
2

≥ |∂l−→ω̂ |2δ(3)

(4.40)

which completes the proof of (4.38). Multiplying (4.38) by the weight r2k+
1

γ−1 ,
integrating, and summing over the multiindex l with |l| ≤ 2k − 1 produces the
inequality

∥ω̂∥2
H

2k−1,k+ 1
2(γ−1)

≳ ∥
−→
ω̂ ∥2

H
2k−1,k+ 1

2(γ−1)
. (4.41)

To complete the proof of Lemma 4.4, we simply observe from (4.16) that

δijδkm∂lω̂ik∂
lω̂jm

= δijδkm∂l[h(
−−→
curlũ)ik + ũk∂ih− ũi∂kh]∂

l[h(
−−→
curlũ)jm + ũm∂jh− ũj∂mh]

≃ δijδkm(h∂l(
−−→
curlũ)ik + additional terms)

(
h∂l(

−−→
curlũ)jm + additional terms

)
(4.42)

We note that any terms where less than |l| derivatives hit
−−→
curlũ will be less critical.

For each additional term, we will always have 2k − 2 or less derivatives of ũ, and
thus we can can apply Corollary 2.3 to gain an extra weight and apply the standard
smallness arguments with Cauchy-Schwarz as needed for cross terms (see Lemma
4.3 where similar cross terms are handled). Moreover, since h ∼ O(1) near the free
boundary, and ∂ih = 1

Γ∂ir+
γ−1
γΓ r∂is does not contribute to the order, we can pull

out derivatives or r and s in the L∞(Ωt) norm. Thus, we have

∥ω̂∥2
H

2k−1,k+ 1
2(γ−1)

≳ ∥
−−→
curlũ+ (smallness terms)∥2

H
2k−1,k+ 1

2(γ−1)
. (4.43)

which produces the desired inequality (4.36) after removing the terms with ε̂. □

4.3. Higher order commutators with the convective derivative. In this
subsection, we adapt the elliptic estimates to higher values of k and show that
all lower order and commutator terms can be treated using the correct weighted
Sobolev norm. First, we will prove Lemma 4.5 which handles the additional terms



40 B. B. LUCZAK EJDE-2025/10

that appear when taking multiple convective derivatives of our system (1.36) for r̃
and ũ. In Section 4.4, we will focus on commuting with weighted spatial derivatives
where the r̃ equation will be discussed in detail. For ũ, the method is quite similar,
and we state the important result in Lemma 4.14.

Earlier in section 4, recall that there were many “additional terms” or black
subcritical terms that appeared when defining our “good” operators L̃1, L̃2, and
L̃3. In the following summarizing lemma, we will indicate how each of these terms
is subcritical, as well as the interaction between Dt and these operators so that we
can eventually prove higher order elliptic estimates.

Lemma 4.5 (Commutators between Dt and L̃1, L̃2, L̃3). Using the Book-keeping
scheme of Lemma 2.11, the following identities hold

D2k
t r̃ ≃ L̃1(D

2k−2
t r̃) + (terms with H2k-order

1

2
at worst), (4.44)

D2k
t ũα ≃ L̃2(D

2k−2
t ũα) + (terms with H2k-order 0 at worst), (4.45)

D2k−2
t (L̃3ũ)

α ≃ L̃3(D
2k−2
t ũα) + (terms with H2k-order 0 at worst), (4.46)

where the order of the terms has been computed with Remark 2.5. Moreover,
[L̃3, D

2k−2
t ]ũ contains terms which are critical/subcritical at worst.

Proof. First, we collect the “additional terms” that appear in (4.1). We begin by
writing the structure of (4.1) in a manner similar to Lemma 2.11 where derivatives
and powers of r are examined. We obtain

D2
t r̃ ≃ L1r̃ + r∂(s̃+ r̃ + ũ) + r[Dt, ∂]ũ+ ũ

≃ L1r̃ + r∂ũ+ r∂s̃+ ũ+ r∂r̃,
(4.47)

where L1 is the operator defined in (4.2). If we account for the black subcritical
terms that appear in (4.11), we will have

D2
t r̃ ≃ L̃1r̃ + (H2-order

1

2
terms at worst) + r∂ũ+ r∂s̃+ ũ+ r∂r̃. (4.48)

If we compute the order of each of these terms at the j = 1 level, we see that the
blue terms have order O = 1

2 > 0 and the blue terms have order O = 1 > 0. By

Lemma 2.14 and Remark 2.15, applying D2k−2
t to the blue and blue terms will

produce corresponding subcritical terms with H2k-orders O = 1
2 at worst for level

k. Thus, we will take D2k−2
t of (4.48) to obtain

D2k
t r̃ ≃ D2k−2

t (L̃1r̃) + (O =
1

2
terms at worst)

= L̃1(D
2k−2
t r̃) + [D2k−2

t , L̃1]r̃ + (O =
1

2
terms at worst).

(4.49)

To complete the proof of (4.44), it remains to check the commutator [D2k−2
t , L̃1]r̃

which can be done with the help of Lemma 2.10 and an induction argument. We
start by computing the first commutator with the help of Lemma 2.10.

[Dt, L̃1]ϕ = [Dt,
γ − 1

Γ + r
Hijr∂i∂j ]ϕ+ [Dt,

1

Γ + r
Hij∂ir∂j ]ϕ

≃ [Dt, r∂
2]ϕ+ [Dt, ∂]ϕ

≃ r∂2ϕ+ ∂ϕ+ r∂ϕ,

(4.50)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 41

recalling that ∂ indicates a generic spatial derivative, and we are ordering terms
from the worst to the best order. We have

[D2
t , L̃1]ϕ = Dt[Dt, L̃1]ϕ+ [Dt, L̃1]Dtϕ

≃ r∂2(Dtϕ) + ∂(Dtϕ) + r∂(Dtϕ) + r∂2ϕ+ ∂ϕ+ r∂ϕ.
(4.51)

We also note by Remark 2.15 and when ϕ = r̃, we will obtain

[D2
t , L̃1]r̃ ≃ r∂2(r∂ũ+ ũ) + ∂(r∂ũ+ ũ) + r∂(r∂ũ+ ũ) + r∂2r̃ + ∂r̃ + r∂r̃

≃ (r2∂3ũ+ r∂2ũ+ ∂ũ) + r∂2r̃ + ∂r̃ + r∂r̃.
(4.52)

Then since, the “k” value when computing order at this level is 2, (i.e. from Remark
2.5) we see that each one of these terms has H4-order 1

2 , 1, and 2 respectively (and

they are all subcritical with O = 1
2 at worst). To complete the induction, suppose

that for some j ≥ 2, we have [D2j−2
t , L̃1] is subcritical (with order O = 1

2 ) at level
j. Then, we compute using the commutator identities that

[D
2(j+1)−2
t , L̃1]r̃ = [D2j

t , L̃1]r̃

= [D2
tD

2j−2
t , L̃1]r̃ = D2

t [D
2j−2
t , L̃1]r + [D2

t , L̃1]D
2j−2
t r̃.

(4.53)

Since [D2j−2
t , L̃1] is subcritical at level j, we can conclude using Remark 2.15 that

the first term in (4.53) must have order O = 1/2 at the j + 1 level. For the second
term in (4.53), it is easiest to compute the order at the j + 1 level by counting
derivatives and powers of r. We will have using Lemma 2.11 that

[D2
t , L̃1]D

2j−2
t r̃ ≃ r∂2(DtD

2j−2
t r̃) + ∂(DtD

2j−2
t r̃)

+ r∂(DtD
2j−2
t r̃) + r∂2D2j−2

t r̃ + ∂D2j−2
t r̃ + r∂D2j−2

t r̃

≃ r∂2(rj∂2j−1ũ) + ∂(rj∂2j−1ũ) + r∂(rj∂2j−1ũ)

+ r∂2(rj−1∂2j−2r̃) + ∂
(
rj−1∂2j−2r̃

)
+ r∂

(
rj−1∂2j−2r̃

) (4.54)

noting that there is a collection of terms in the simplification for D2j−1
t r̃ which all

have the same order. Thus, computing the total order at the j +1 level, we obtain
that each of the terms has order O = 1

2 at worst. We have completed the induction

and shown that [D2k−2
t , L̃1] has order O = 1

2 at worst for level k, and this completes
the proof of (4.44). The proofs of (4.45) and (4.46) are similar. □

Remark 4.6. When we go to estimate E2k
wave from below, we will observe that

these subcritical/ critical terms will not cause any issue. First, consider estimating
terms in the r̃ equation which are subcritical. In Lemma 4.12 below, there terms
are ultimately handled with smallness arguments combined with Remark 2.5. A
similar situation also happens for the ũ equation. Although the expression for D2k

t ũ
contains terms which are critical as well (and not just subcritical), we recall the
useful fact that the ũ equation gets a multiplier of order 1/2 when doing energy
estimates. Subsequently, the E2k

wave energy has an extra r1/2 weight and this will be
perfect for applying the same smallness arguments for these extra terms.

4.4. Commutators with weighted spatial derivatives. Our goal is to extend
Lemmas 4.1 and 4.3 to higher order Sobolev spaces. To do so, we will need to
compute commutators between our good elliptic operators L̃1, L̃2, L̃3, and the
weighted spatial derivatives rk−j∂2(k−j).



42 B. B. LUCZAK EJDE-2025/10

Let k ∈ N be fixed, and j ∈ N with 1 ≤ j ≤ k. To begin, we will use the following
notation

m := k − j, La,b := ra∂b, r̃2j := D2j
t r̃ (4.55)

and we recall from (4.10) that

L̃1 =
γ − 1

Γ + r
Hij(r∂i∂j +

1

γ − 1
∂ir∂j).

We plan to handle L̃1 first, before moving on to L̃2 and L̃3. By Lemma 4.5, we
have that

r̃2j ≃ L̃1r̃2j−2 + P , (4.56)

where P are terms that have an order of 1/2 or better at level j (i.e. at the H2j

level). We plan to ultimately handle these terms using smallness arguments after

we apply Lm,2m to (4.56) and analyze with the H0, 2−γ
2(γ−1) norm. Applying Lm,2m

to both sides (where Lm,2m is defined in (4.55)), we have

Lm,2mr̃2j ≃ L̃1L
m,2mr̃2j−2 + [Lm,2m, L̃1]r̃2j−2 + Lm,2mP (4.57)

Remark 4.7. Observe what will happen to the order of the terms in P when we
apply Lm,2m. At the start, P has terms with H2j order 1

2 (at worst). However, by

repeatedly applying Lemma 2.14, we see that ∂2mP has H2j-order 1
2 − 2m. Then,

rm∂2mP has H2j-order 1
2 − 2m +m = 1

2 −m = 1
2 − (k − j). Then, if we want to

calculate the order at level k instead of level j (and note that k ≥ j), we see by
Part 4 in Lemma 2.14 that rm∂2mP must have H2k-order 1

2 − (k− j)+(k− j) = 1
2 .

By Remark 4.7, this means that at the H2k level, Lm,2mP will still be subcritical
and we plan to handle these terms with smallness arguments. Thus, let us use the
notation

PO≥1/2 := (terms with H2k order O ≥ 1

2
) (4.58)

which we will use to collect any perturbative/smallness terms along the way that
can handled easily with respect to the H2k norm. Using this notation, we have

Lm,2mr̃2j ≃ L̃1L
m,2mr̃2j−2 + [Lm,2m, L̃1]r̃2j−2 + PO≥1/2 (4.59)

and the bulk of our analysis concerns the proper handling of the commutator
[Lm,2m, L̃1]r̃2j−2.

Since the commutator is trivial for m = 0, let us assume that m ≥ 1, and we
plan to apply an induction argument on m. As we will see in the coming pages, the
critical terms in the commutator need to be handled in a particular way where we
absorb a critical term into the elliptic operator L̃1, and then the rest of the terms
are subcritical. Note that if we had arrived at (4.59) with Lm−1,2(m−2) instead of
Lm,2m, this would produce

Lm−1,2(m−1)r̃2j ≃ L̃1L
m−1,2(m−1)r̃2j−2+[Lm−1,2(m−1), L̃1]r̃2j−2+PO≥3/2 , (4.60)

where the other terms, at worst, have H2k-order 1 (instead of critical in the m
case). For the inductive hypothesis, we need to assume that Lm−1,2(m−1) “behaves

well” when we commute it with L̃1 in that we have already absorbed the order 1
terms. We will make the following inductive assumption (recalling from (4.55) that
k = m+ j):



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 43

Assumption 4.8 (Inductive hypothesis on m− 1). (1) All the terms in

[Lm−1,2(m−1), L̃1]r̃2j−2 have H2(m−1+j) order 1
2 , except for key order 0 terms which

have already been handled by absorbing into L̃1L
m−1,2(m−1)r̃2j−2 with the analog

of Proposition 4.10 below.
(2) The above assumption also holds for L̃1 replaced by L̃1 + b γ−1

Γ+rH
ij∂ir∂j with

b ≥ 0.

Remark 4.9. There are several consequences of Assumption 4.8 that we summa-
rize:

(1) The terms in [Lm−1,2(m−1), L̃1]r̃2j−2 have H2(m+j) = H2k order 3
2 , i.e. they

belong to PO≥3/2,

(2) The terms in [Lm−1,2(m−1), L̃1]∂r̃2j−2 belong to PO≥1/2,
(3) Suppose that m ≥ 2. We claim that Assumption 4.8 on the m − 1 case

also provides information on lower order commutators. Indeed, we compute the
following using two different ways:

[r∂
(
Lm−2,2m−3

)
, L̃1]φ ≃ [Lm−1,2m−2 + Lm−2,2m−3,

L̃1]φ = [Lm−1,2m−2, L̃1]φ+ [Lm−2,2m−3, L̃1]φ

[r∂
(
Lm−2,2m−3

)
, L̃1]φ = r∂[Lm−2,2m−3, L̃1]φ+ [r∂, L̃1]L

m−2,2m−3φ

Combining terms, we have

[Lm−1,2m−2, L̃1]φ ≃ r∂[Lm−2,2m−3, L̃1]φ+[r∂, L̃1]L
m−2,2m−3φ+[Lm−2,2m−3, L̃1]φ.

Thus, since [Lm−1,2(m−1), L̃1]r̃2j−2 belongs to PO≥3/2 and [Lm−1,2(m−1), L̃1]∂r̃2j−2

belongs to PO≥1/2, we can conclude that [Lm−2,2m−3, L̃1]r̃2j−2 belongs to PO≥3/2

and [Lm−2,2m−3, L̃1]∂r̃2j−2 belongs to PO≥1/2. By repeated use of this argument,

a similar statement also holds for commutators involving Ll,m+l−1 where 1 ≤ l ≤
m− 1.

Now that we have finished discussing the consequences of our inductive hypoth-
esis, we proceed with the commutator by computing the following

[Lm,2m, L̃1]r̃2j−2 = Lm,2mL̃1r̃2j−2 − L̃1L
m,2mr̃2j−2 (4.61)

and

Lm,2mL̃1r̃2j−2 = rm∂2m
(γ − 1

Γ + r
Hij(r∂i∂j r̃2j−2 +

1

γ − 1
∂ir∂j r̃2j−2)

)
≃ γ − 1

Γ + r
Hijrm∂2m

(
r∂i∂j r̃2j−2 +

1

γ − 1
∂ir∂j r̃2j−2

)
+ PO≥1/2 ,

where we note that any terms where derivatives hit γ−1
Γ+rH

ij get absorbed into

PO≥1/2 (since these derivatives are not hitting either r or r̃2j−2 which are critical
terms at worst). For analyzing the remaining terms, we continue

rm∂2m
(
r∂i∂j r̃2j−2 +

1

γ − 1
∂ir∂j r̃2j−2

)
≃ rm

2m∑
l=0

∂lr∂2m−l∂i∂j r̃2j−2 +
1

γ − 1
∂l∂ir∂

2m−l∂j r̃2j−2,

(4.62)

where we absorbed combinatorial constants with ≃. Now, we plan to separate out
any of the terms that belong to PO≥1/2 from the critical terms which must be



44 B. B. LUCZAK EJDE-2025/10

handled. When ∂2m hits L̃1, we observe that the worst terms only occur when 2m
derivatives hit r̃2j−2, or 1 derivative hits r (in the first term) and 2m−1 derivatives
hit r̃2j−2. The point is that after the first power of r is removed, any subsequent
derivatives that hit ∂r do not actually contribute to the order. In fact, they are
only taking away derivatives that could have hit r̃2j−2. Indeed, we can check by
computing the order. When l = 0, we have that

rmr∂2m∂i∂j r̃2j−2 ≃ rm+1∂2m+2(D2j−2
t r̃) is H2k-critical (O = 0),

1

γ − 1
rm∂ir∂

2m∂j r̃2j−2 ≃ rm∂2m+1(D2j−2
t r̃) is H2k-critical (O = 0)

(4.63)

using Lemma 2.14. When l = 1, we see that

rm∂r∂2m−1∂i∂j r̃2j−2 ≃ rm∂2m+1(D2j−2
t r̃) is H2k-critical (O = 0),

1

γ − 1
rm∂∂ir∂

2m−1∂j r̃2j−2 ≃ rm∂2m(D2j−2
t r̃) is H2k-subcritical (O = 1).

(4.64)
Thus, in the summation when l ≥ 2, we will obtain

rm∂lr∂2m−l∂i∂j r̃2j−2 ≃ rm∂≤2m(D2j−2
t r̃) is H2k-subcritical (O ≥ 1),

1

γ − 1
rm∂l∂ir∂

≤2m−l∂j r̃2j−2 ≃ rm∂≤2m−1(D2j−2
t r̃) is H2k-subcritical (O ≥ 2).

Summarizing our order analysis, we see that many of the terms in the summation
will be absorbed into PO≥1/2, and we have

Lm,2mL̃1r̃2j−2 ≃ γ − 1

Γ + r
Hij

(
rm+1∂2m∂i∂j r̃2j−2 +

1

γ − 1
rm∂ir∂

2m∂j r̃2j−2

+ rm∂r∂2m−1∂i∂j r̃2j−2

)
+ PO≥1/2 .

(4.65)

For the other side of the commutator, we have

L̃1

(
Lm,2mr̃2j−2

)
≃ γ − 1

Γ + r
Hij

(
r∂i∂j(r

m∂2mr̃2j−2) +
1

γ − 1
∂ir∂j(r

m∂2mr̃2j−2)
)

≃ γ − 1

Γ + r

(
Hij

(
rm+1∂i∂j∂

2mr̃2j−2 + 2r∂i(r
m)∂j∂

2mr̃2j−2 + r∂i∂j(r
m)∂2mr̃2j−2

)
+

1

Γ + r
Hij∂ir

(
rm∂j∂

2mr̃2j−2 + ∂j(r
m)∂2mr̃2j−2

))
.

(4.66)

Now, if we compute the order of the blue term, we see that

γ − 1

Γ + r
Hijr∂i∂j(r

m)∂2mr̃2j−2 ≃ rm−1∂2mr̃2j−2 is H2k-subcritical (O ≥ 2)

and thus, we can absorb it into PO≥1/2. Observing that the blue terms cancel in
(4.65) and (4.66), we have

[Lm,2m, L̃1]r̃2j−2 =
γ − 1

Γ + r
Hij

(
∂rLm,2m−1∂i∂j r̃2j−2 − 2m∂irL

m,2m∂j r̃2j−2

− 1

γ − 1
∂ir∂jrL

m−1,2mr̃2j−2

)
+ PO≥1/2 ,

(4.67)

where the first two terms take the form Lm,2m+1r̃2j−2.



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 45

Then, combining with (4.59), and applying the H0, 2−γ
2(γ−1) norm, we have

∥L̃1L
m,2mr̃2j−2∥

H
0,

2−γ
2(γ−1)

≲ ∥Lm,2mr̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm,2m+1r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥Lm−1,2mr̃2j−2∥
H

0,
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

(4.68)

For the LHS, we can apply Lemma 4.1 to obtain

∥L̃1L
m,2mr̃2j−2∥

H
0,

2−γ
2(γ−1)

≳ ∥Lm,2mr̃2j−2∥
H

2,1+
2−γ

2(γ−1)
− ∥Lm,2mr̃2j−2∥

H
0,

2−γ
2(γ−1)

Then, adding the blue term to the other side of (4.68), we have

∥Lm,2mr̃2j−2∥
H

2,1+
2−γ

2(γ−1)

≲ ∥Lm,2mr̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm,2m+1r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥Lm−1,2mr̃2j−2∥
H

0,
2−γ

2(γ−1)

+ ∥Lm,2mr̃2j−2∥
H

0,
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

.

(4.69)

Now, by adding the following quantity on both sides of our inequality:

∥r̃2j−2∥
H

0,
2−γ

2(γ−1)
+

m−1∑
l=0

∥Ll,m+lr̃2j−2∥
H

2,1+
2−γ

2(γ−1)
(4.70)

we see using the help or Remark 1.13 that the LHS will be equivalent to

∥r̃2j−2∥
H

2m+2,m+1+
2−γ

2(γ−1)
. (4.71)

For the RHS, we will get the following collection of terms which we number

∥Lm,2mr̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm,2m+1r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥Lm−1,2mr̃2j−2∥
H

0,
2−γ

2(γ−1)

+ ∥Lm,2mr̃2j−2∥
H

0,
2−γ

2(γ−1)

m−1∑
l=0

∥Ll,m+lr̃2j−2∥
H

2,1+
2−γ

2(γ−1)
+ ∥r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)

=: ∥Lm,2mr̃2j∥
H

0,
2−γ

2(γ−1)
+

6∑
a=0

R6

(4.72)

using R1 through R6.
For the terms R1, R2, and R3, we first observe that

∥Lm,2m+1r̃2j−2∥
H

0,
2−γ

2(γ−1)
+ ∥Lm−1,2mr̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥Lm,2mr̃2j−2∥
H

0,
2−γ

2(γ−1)

≲ ∥Lm−1,2m−1r̃2j−2∥
H

2,1+
2−γ

2(γ−1)
= ∥Lm−1,2m−2∂r̃2j−2∥

H
2,1+

2−γ
2(γ−1)

Then, we claim that the following proposition.

Proposition 4.10.

∥Lm−1,2m−2∂r̃2j−2∥
H

2,1+
2−γ

2(γ−1)



46 B. B. LUCZAK EJDE-2025/10

≲ ∥Lm−1,2m−2∂r̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm−1,2m−2∂r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)

Proof. To prove this proposition, we rely on the smallness of ∂3r (similar to the

work in [4]) as well as a key step in which we we must absorb a term into L̃1,

creating the updated elliptic operator ̂̃L1. Similar to [4], we rely on localizing in
the neighborhood of a boundary point such that

|∂′r| ≲ A, |∂3r − 1| ≲ A, |H3,i′ | ≲ A , (4.73)

where A ≪ 1 is small constant, and primed indices will range over x1 and x2

(i.e. ∂′r ≃ ∂1r, ∂2r). Note that a key assumption is present on the off-diagonal
components of H (and a similar simplification can also be found in [4]). The proof
is quite similar, and we only need to show the related inequalities

∥Lm−1,2m−2∂3r̃2j−2∥
H

2,1+
2−γ

2(γ−1)

≲ ∥Lm−1,2m−2∂3r̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm−1,2m−2∂′r̃2j−2∥

H
2,1+

2−γ
2(γ−1)

+ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)

(4.74)

and

∥Lm−1,2m−2∂′r̃2j−2∥
H

2,1+
2−γ

2(γ−1)

≲ ∥Lm−1,2m−2∂′r̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm−1,2m−2∂′r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)
.

(4.75)

□

From Proposition 4.10 we have a corollary.

Corollary 4.11. Let l ∈ N with 1 ≤ l ≤ m− 1. Then

∥Ll,m+l−1∂r̃2j−2∥
H

2,1+
2−γ

2(γ−1)

≲ ∥Ll,m+l−1∂r̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Ll,m+l−1∂r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)

Proof. To prove this corollary, we simply need to follow the proof of Proposition
4.10 combined with the last consequence in Remark 4.9. Anywhere the inductive
hypothesis is used on the commutator Lm−1,2m−2, we know that a similar fact holds
for Ll,m+l−1 by Remark 4.9. □

Now, we can return to (4.68)-(4.72) and summarize our results for what we have
on the LHS and RHS now that we have estimated the terms R1, R2, and R3 using
Proposition 4.10. Simplifying Lm−1,2m−2∂ = Lm−1,2m−1, this yields

∥r̃2j−2∥
H

2m+2,m+1+
2−γ

2(γ−1)

≲ ∥Lm,2mr̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm−1,2m−1r̃2j∥

H
0,

2−γ
2(γ−1)

+ ∥Lm−1,2m−1r̃2j−2∥
H

0,
2−γ

2(γ−1)
+

m−1∑
l=0

∥Ll,m+lr̃2j−2∥
H

2,1+
2−γ

2(γ−1)

+ ∥r̃2j−2∥
H

0,
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

,

(4.76)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 47

where only the red terms need to be handled, and we will be keeping R5 and R6

from before on the RHS. For the first red term, we have

∥Lm−1,2m−1r̃2j−2∥
H

0,
2−γ

2(γ−1)
≲ ∥Lm−1,2m−2r̃2j−2∥

H
1,

2−γ
2(γ−1)

≲ ∥L̃1(L
m−1,2m−2r̃2j−2)∥

H
0,

2−γ
2(γ−1)

,
(4.77)

where we used the elliptic estimate for L̃1 in the last line. Then we can use (4.8)
to commute with Lm−1,2(m−1) which will only produce additional terms in PO≥1/2.

Finally, we use (4.56) to express L̃1r̃2j−2 in terms of r̃2j plus additional P terms.
We have

∥Lm−1,2m−1r̃2j−2∥
H

0,
2−γ

2(γ−1)

≲ ∥Lm−1,2m−2(L̃1r̃2j−2) + PO≥1/2∥
H

0,
2−γ

2(γ−1)

≲ ∥Lm−1,2m−2(r̃2j + P )∥
H

0,
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

≲ ∥Lm−1,2m−2r̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

≲ ∥r̃2j∥
H

2m,m+
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

,

(4.78)

which is an important estimate for the first red term.
For the final red term, we start by rewriting and then applying Corollary 4.11.

m−1∑
l=0

∥Ll,m+l−1∂r̃2j−2∥
H

2,1+
2−γ

2(γ−1)

≲
m−1∑
l=0

(
∥Ll,m+l−1∂r̃2j∥

H
0,

2−γ
2(γ−1)

+ ∥Ll,m+l−1∂r̃2j−2∥
H

0,
2−γ

2(γ−1)

)
≲

m−1∑
l=0

(
∥Ll,m+lr̃2j∥

H
0,

2−γ
2(γ−1)

+ ∥Ll,m+lr̃2j−2∥
H

0,
2−γ

2(γ−1)

)
≲ ∥r̃2j∥

H
2m,m+

2−γ
2(γ−1)

+

m−1∑
l=0

∥Ll,m+l−1r̃2j−2∥
H

1,
2−γ

2(γ−1)
,

(4.79)

where, in the last line, we used an inequality similar to 4.77 for the r̃2j−2 terms.

For the remaining sum, it remains to apply the elliptic estimate for L̃1 and then
commute L̃1 with Ll,m+l−1 using Remark 4.9. We have

m−1∑
l=0

(
∥Ll,m+l−1r̃2j−2∥

H
1,

2−γ
2(γ−1)

)
≲

m−1∑
l=0

∥L̃1(L
l,m+l−1r̃2j−2)∥

H
0,

2−γ
2(γ−1)

≲ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)
+

m−1∑
l=0

∥Ll,m+l−1(r̃2j + P )∥
H

0,
2−γ

2(γ−1)

≲ ∥r̃2j∥
H

2m,m+
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

,

(4.80)

where we followed the argument in (4.78) with Lm−1,2m−1 replaced by Ll,m+l−1.



48 B. B. LUCZAK EJDE-2025/10

Finally combining (4.76)- (4.80), we are now ready to prove our higher order
elliptic estimates for the r̃ equation.

Lemma 4.12 (Higher order elliptic estimates for r̃). Under the conditions for
Lemma 4.1, we have

∥r̃∥
H

2k,k+
2−γ

2(γ−1)
≲

k∑
j=0

∥D2j
t r̃∥

H
0,

2−γ
2(γ−1)

+ Cε̂∥(s̃, r̃, ũ)∥H2k , (4.81)

where Cε̂ ≪ 1 depends on ε̂.

Proof. Combining (4.76)- (4.80) and summarizing our work in this section, we ob-
tain the inequality

∥r̃2j−2∥
H

2m+2,m+1+
2−γ

2(γ−1)

≲ ∥Lm,2mr̃2j∥
H

0,
2−γ

2(γ−1)
+ ∥Lm−1,2m−1r̃2j∥

H
0,

2−γ
2(γ−1)

+ ∥r̃2j∥
H

2m,m+
2−γ

2(γ−1)

+ ∥r̃2j−2∥
H

0,
2−γ

2(γ−1)
+ ∥PO≥1/2∥

H
0,

2−γ
2(γ−1)

≲ ∥r̃2j∥
H

2m,m+
2−γ

2(γ−1)
+ ∥r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ∥PO≥1/2∥
H

0,
2−γ

2(γ−1)

≲ ∥r̃2j∥
H

2m,m+
2−γ

2(γ−1)
+ ∥r̃2j−2∥

H
0,

2−γ
2(γ−1)

+ ε̂∥(s̃, r̃, ũ)∥H2k

(4.82)

which we showed using an induction argument on m ≥ 1, and the PO≥1/2 terms are
treated using (4.58) and Remark 2.5. Then, we recall from (4.55) that m = k − j,
so when m = k − 1, we have j = 1 and

∥r̃∥
H

2k,k+
2−γ

2(γ−1)
≲ ∥D2

t r̃∥
H

2k−2,k−1+
2−γ

2(γ−1)
+ ∥r̃∥

H
0,

2−γ
2(γ−1)

+ ε̂∥(s̃, r̃, ũ)∥H2k (4.83)

Iterating, we obtain successive inequalities all the way up to ∥D2k
t r̃∥

H
0,

2−γ
2(γ−1)

. Com-

bining together, we have

∥r̃∥
H

2k,k+
2−γ

2(γ−1)
≲

k∑
j=0

∥D2j
t r̃∥

H
0,

2−γ
2(γ−1)

+ Cε̂∥(s̃, r̃, ũ)∥H2k , (4.84)

where Cε̂ ≪ 1 depends on ε̂. This completes the proof. □

Remark 4.13. We observe that the RHS of the inequality in Lemma 4.12 closely
resembles the higher order wave energy E2k

wave found in (1.47). After obtaining the

matching estimates for D2j
t ũ, we will have the full E2k

wave energy on the RHS and
we will almost have the proof of Theorem 5.1 on the equivalence between our full
energy (1.49) and the H2k norm (1.46).

Next, we apply a similar argument for the ũ equation and the operators L2 with
“good” spatial parts L̃2 and L̃3. We observe that much of the analysis will be quite

similar with L̃1 replaced by L̃2, and H0, 2−γ
2(γ−1) replaced by H0, 12+

2−γ
2(γ−1) = H0, 1

2(γ−1) .
Following the arguments of Section 4.4, we will need to also incorporate the L̃3

operator at each step in order to apply Lemma 4.3. We will state the desired
Lemma here for convenience.

Lemma 4.14 (Higher order div-curl estimates for ũ). Under the conditions for
Lemma 4.3, the following inequality holds

∥ũ∥
H

2k,k+ 1
2(γ−1)

≲
( k∑

j=0

∥D2j
t ũ∥

H
0, 1

2(γ−1)

)
+∥

−−→
curlũ∥

H
2k−1,k+ 1

2(γ−1)
+Dε̂∥(s̃, r̃, ũ)∥H2k ,



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 49

where Dε̂ ≪ 1 depends on ε̂.

Remark 4.15. We observe that the RHS of the inequality in Lemma 4.14 closely
resembles the higher order wave energy E2k

wave found in (1.47) plus an additional

term that depends on
−−→
curl ũ. However, in view of Lemma 4.4, this term is precisely

bounded by ∥ω̂∥
H

2k−1,k+ 1
2(γ−1)

which is the key vorticity piece of E2k
transport. We note

that Lemma 4.14 is used in conjuction with Lemma 4.12 to prove Theorem 5.1 on
the equivalence between our full energy (1.49) and the H2k norm (1.46).

5. Energy equivalence

Theorem 5.1 (Equivalence between E2k and H2k norms). Let (s̃, r̃, ṽ) be smooth
functions in Ω. If r is positive and uniformly non-degenerate on Γ, then

E2k(s̃, r̃, ũ) ≈ ∥(s̃, r̃, ũ)∥2H2k ,

where the equivalence depends on γ and up to 2k derivatives of s, r, and u.

Proof. Let us begin with the ≲ direction. Using our book-keeping scheme from
Section 2, it will be quite straightforward to track the order of each of the terms
as they appear. We plan to use Remark 1.13 in which the H2k norm for (s̃, r̃, ũ)

is shown to be equivalent to the H2k, 2−γ
2(γ−1)

+ 1
2+k ×H2k, 2−γ

2(γ−1)
+k ×H2k, 2−γ

2(γ−1)
+ 1

2+k

norm.
We start with E2k

transport, which we recall as

E2k
transport(s̃, r̃, ũ) = ∥ω̂∥2

H
2k−1,

2−γ
2(γ−1)

+ 1
2
+k

+ ∥s̃∥2
H

2k,
2−γ

2(γ−1)
+ 1

2
+k

(5.1)

First, it is clear from Remark 1.13 that

∥s̃∥2
H

2k,
2−γ

2(γ−1)
+ 1

2
+k

≲ ∥(s̃, r̃, ũ)∥2H2k (5.2)

For the vorticity part of E2k
transport, we look at the order of each of the terms and

observe that

ω̂αβ = ∂α(hũβ)− ∂β(hũα)

= h(∂αũβ − ∂β ũα) + ũβ∂αh− ũα∂βh

≃ ∂ũ+ ũ

(5.3)

using our book-keeping scheme and recalling that h = Γ+r
Γ which is O(1) near the

free boundary, as well as ∂h. We plan to estimate ω̂ using a similar argument as
the estimate for ω in (3.31). Applying a multiindex l with |l| ≤ 2k−1 and counting
the number of derivatives and powers of r, we have

∂lω̂ ≃ ∂l+1ũ+ ∂lũ (5.4)

Applying the L2(r
2−γ
γ−1+2k+1) = H0, 2−γ

2(γ−1)
+ 1

2+k norm, we have∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂lω̂|2G dx ≲
∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂l+1ũ|2 dx+

∫
Ωt

r
2−γ
γ−1 |rk+ 1

2 ∂lũ|2 dx (5.5)

Summing over l, we obtain the H2k−1, 2−γ
2(γ−1)

+ 1
2+k norm, and we will get terms that

are only critical at worst, i.e. they will have the form rk+
1
2 ∂2kũ modulo coefficients

depending on derivatives of s and r, and the second intergral produces subcritical



50 B. B. LUCZAK EJDE-2025/10

terms of the form rk+
1
2 ∂2k−1ũ at worst. Each of these can easily be estimated via

the H2k norm, so we have

∥ω̂∥2
H

2k−1,
2−γ

2(γ−1)
+ 1

2
+k

≲ ∥(s̃, r̃, ũ)∥2H2k . (5.6)

and combining with (5.2) yields

E2k
transport(s̃, r̃, ũ) ≲ ∥(s̃, r̃, ũ)∥2H2k . (5.7)

For E2k
wave, we first use Remark 1.12 to simplify

E2k
wave(s̃, r̃, ũ) =

k∑
j=0

∥(D2j
t r̃, D2j

t ũ)∥2H̃

≃
k∑

j=0

(
∥D2j

t r̃∥2
H

0,
2−γ

2(γ−1)

+ ∥D2j
t ũ∥2

H
0,

2−γ
2(γ−1)

+ 1
2

) (5.8)

using that norms for H̃, (L2(r
2−γ
γ−1 )×L2(r

2−γ
γ−1+1)) and (H0, 2−γ

2(γ−1) ×H0, 2−γ
2(γ−1)

+ 1
2 ) are

equivalent. Then, we make frequent use of Lemma 2.11 which allows us to greatly
simplify convective derivatives. Taking 0 ≤ j ≤ k, we have

∥D2j
t r̃∥2

H
0,

2−γ
2(γ−1)

≃
∥∥ j∑

l=0

rl∂l+j r̃
∥∥2
H

0,
2−γ

2(γ−1)
≲ ∥r̃∥2

H
2j,

2−γ
2(γ−1)

+j
(5.9)

noting that each term appearing in the summation will be critical at the j level.
Then, further applying our embedding lemmas, we obtain

∥r̃∥2
H

2j,
2−γ

2(γ−1)
+j

≲ ∥r̃∥2
H

2j+k−j,
2−γ

2(γ−1)
+j+k−j

= ∥r̃∥2
H

k+j,
2−γ

2(γ−1)
+k

≲ ∥r̃∥2
H

2k,
2−γ

2(γ−1)
+k

≲ ∥(s̃, r̃, ũ)∥2H2k

(5.10)

noting that 0 ≤ j ≤ k and we can apply Corollary 2.3 as needed when j < k. Thus,
the r̃ part of E2k

wave is bounded by the desired H2k norm. For the ũ part, we can
similarly invoke Lemma 2.11 again, where the only difference is that we now have
terms involving both ũ and r̃. However, the extra r1/2 weight is exactly what we
need to handle any of the terms that appear:

∥D2j
t ũ∥2

H
0,

2−γ
2(γ−1)

+ 1
2
≃

∥∥ j∑
l=0

rl∂l+j ũ+

j−1∑
i=0

ri∂i+j r̃
∥∥2
H

0,
2−γ

2(γ−1)
+ 1

2

≲ ∥(s̃, r̃, ũ)∥2H2k

(5.11)

and we observe that the r̃ terms actually have an extra r1/2 in terms of bounding
by the H2k norm. Combining (5.8)- (5.11) produces

E2k
wave(s̃, r̃, ũ) ≲ ∥(s̃, r̃, ũ)∥2H2k . (5.12)

Thus, we combine with (5.7) and the ≲ direction for Theorem 5.1 is complete.



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 51

For the ≳ direction, we make use of many previously established lemmas. First,
by the norm equivalence from Remark 1.13, and Lemmas 4.12, and 4.14, we have

∥(s̃, r̃, ũ)∥2H2k ≈ ∥r̃∥2
H

2k,
2−γ

2(γ−1)
+k

+ ∥ũ∥2
H

2k,
2−γ

2(γ−1)
+ 1

2
+k

+ ∥s̃∥2
H

2k,
2−γ

2(γ−1)
+ 1

2
+k

≲
k∑

j=0

(
∥D2j

t r̃∥2
H

0,
2−γ

2(γ−1)

+ ∥D2j
t ũ∥2

H
0,

2−γ
2(γ−1)

+ 1
2

)
+ ∥

−−→
curlũ∥2

H
2k−1,k+ 1

2(γ−1)
+ ∥s̃∥2

H
2k,

2−γ
2(γ−1)

+ 1
2
+k

+ (C2
ε̂ +D2

ε̂)∥(s̃, r̃, ũ)∥2H2k ,

(5.13)

where Cε̂, Dε̂ ≪ 1. Then, we observe by examining E2k
wave in (1.47) that

k∑
j=0

(
∥D2j

t r̃∥2
H

0,
2−γ

2(γ−1)

+ ∥D2j
t ũ∥2

H
0,

2−γ
2(γ−1)

+ 1
2

)

≲
k∑

j=0

∥(D2j
t r̃, D2j

t ũ)∥2H̃ = E2k
wave(s̃, r̃, ũ)

(5.14)

as any weights involving Γ can be removed in view of Remark 1.12. Then, combining

(5.13), (5.14), and applying Lemma 4.4 for the term with
−−→
curl ũ, we have

∥(s̃, r̃, ũ)∥2H2k ≲ E2k
wave(s̃, r̃, ũ) + ∥ω̂∥2

H
2k−1,

2−γ
2(γ−1)

+ 1
2
+k

+ ∥s̃∥2
H

2k,
2−γ

2(γ−1)
+ 1

2
+k

+ ε̂2∥ũ∥2
H

2k,
2−γ

2(γ−1)
+ 1

2
+k

+ (C2
ε̂ +D2

ε̂)∥(s̃, r̃, ũ)∥2H2k

≲ E2k
wave(s̃, r̃, ũ) + E2k

transport(s̃, r̃, ũ) + (C2
ε̂ +D2

ε̂ + ε̂2)∥(s̃, r̃, ũ)∥2H2k

Absorbing the constants (C2
ε̂ +D2

ε̂ + ε̂2) ≪ 1 into the LHS produces

E2k(s̃, r̃, ũ) = E2k
wave(s̃, r̃, ũ) + E2k

transport(s̃, r̃, ũ) ≳ ∥(s̃, r̃, ũ)∥2H2k (5.15)

which completes the proof of the ≳ direction. Thus, the proof of Theorem 5.1 is
complete. □

6. Higher order wave energy estimates

Using the book-keeping scheme from Section 2, let us estimate each of the terms
on the RHS of (2.1) with the following Theorem. Note that the B2k equation will

get multiplied by 1
γ−1r

2−γ
γ−1D2k

t r̃, and the Cα
2k equation will get contracted with

(Γ + r)r
1

γ−1GαβD
2k
t ũβ .

We plan to integrate and estimate using the H2k norm which we recall:

∥(s̃, r̃, ũ)∥2H2k

=

2k∑
|α|=0

k∑
a=0

|α|−a≤k

∥r
2−γ

2(γ−1)
+ 1

2+a∂αs̃∥2L2 + ∥r
2−γ

2(γ−1)
+a∂αr̃∥2L2 + ∥r

2−γ
2(γ−1)

+ 1
2+a∂αũ∥2L2 .

The following book-keeping remark is useful when taking convective derivatives
of the many expressions that appear.



52 B. B. LUCZAK EJDE-2025/10

Remark 6.1. Using that Dt(Γ(s)) = 0 from (1.35a), we have

Dt

( 1

Γ + r

)
= − 1

(Γ + r)2
Dtr ≃

( 1

Γ + r

)2

r,

where we applied (2.10). Thus, taking multiple convective derivatives of our weight
1

Γ+r leads to gaining additional powers of r, and thus makes the terms more subcrit-
ical. In view of Remarks 2.4 and 2.8, we will often ignore these terms and use the
≃ symbol so that we can highlight the key critical/supercritical terms that cause
difficulty when estimating.

Theorem 6.2 (Higher order energy estimates). Let (s, r, u) be a solution to (1.35)
that exists on some time interval [0, T ]. Assume that s, r, u and their 2k + 1 order
derivatives are bounded in the L∞(Ωt) norm for each t ∈ [0, T ] and r vanishes sim-
ply on the free boundary. Then, the following estimate holds for solutions (s̃, r̃, ũ)
to the higher order linearized equations (2.1):∣∣ d

dt
E2k

wave(s̃, r̃, ũ)
∣∣ ≲ B∥(s̃, r̃, ũ)∥2H2k , (6.1)

where B is a function that depends on up to 2k + 1 derivatives of s, r, and u.

Proof. The proof relies on Section 1.6 and the basic energy estimate in Proposition
1.17. Along the way, we will make use of Lemmas 2.6, 2.9, 2.11, and 2.14.

We multiply (2.1a) and (2.1b) by their respective multipliers 1
γ−1r

2−γ
γ−1D2k

t r̃ and

(Γ + r)r
1

γ−1GαβD
2k
t ũβ . Then, we compare with (1.50b) and (1.50c) and observe

that we will be following the Proof of Proposition 1.17 with r̃ and ũ replaced by
D2k

t r̃ and D2k
t ũ. Integrating over Ωt and applying the moving domain formula

(1.53), it remains to show that all of the terms on the RHS can be estimated using
the H2k norm. More specifically, we plan to show that∫

Ωt

r
2−γ
γ−1 |B2k

1

γ − 1
D2k

t r̃| dx ≲ C1∥(s̃, r̃, ũ)∥2H2k , (6.2a)∫
Ωt

r
2−γ
γ−1 |Cα

2k(Γ + r)rGαβD
2k
t ũβ | dx ≲ C2∥(s̃, r̃, ũ)∥2H2k , (6.2b)

where B2k and Cα
2k are long expressions defined in (2.2). Recall from Remarks 2.5

and 2.7 that when calculating the order of a given term where γ > 1 is arbitrary,

we simply ignore powers of r coming from the weight r
2−γ
γ−1 , since our estimates are

built around the weighted space L2(r
2−γ
γ−1 ) = H0, 2−γ

2(γ−1) . Also, recall Lemma 2.6 and

Remark 2.7 where estimates involving L1(r
2−γ
γ−1 ) are used extensively.

We will begin by showing (6.2a). First, observe that the multiplier D2k
t r̃ is H2k

critical since by Lemma 2.11, we have

D2k
t r̃ ≃ (rk∂2kr̃ + rk−1∂2k−1r̃ + · · ·+ ∂kr̃) + . . . ,

where one can quickly check the order of the terms listed as being H2k critical, and
all remaining terms are subcritical.

For the first term in (6.2a) after expanding B2k using (2.2), we obtain

D2k
t r̃D2k

t g = D2k
t r̃D2k

t (r̃∂µu
µ) = D2k

t r̃

2k∑
i=0

Di
tr̃D

2k−i
t (∂µu

µ) (6.3)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 53

Then, by computing the order of terms in Lemma 2.11, we see that Di
tr̃ only

containsH2k-critical terms when i = 2k. After integrating in L1(r
2−γ
γ−1 ), the product

can estimated by making use of Lemma 2.6 since we have the product of two terms
which are both critical:∫

Ωt

r
2−γ
γ−1

∣∣∣D2k
t r̃

2k∑
i=0

Di
tr̃D

2k−i
t (∂µu

µ)
∣∣∣ dx ≲ C1∥(s̃, r̃, ũ)∥2H2k , (6.4)

where C1 depends on the L∞ norms of up to 2k + 1 derivatives of (s, r, u). Notice

that in the coefficient expression D2k−i
t (∂µu

µ), we can use Lemma 2.12 to solve for
the time derivative of u in terms of spatial derivatives. Additionally, we can solve
for D2k−i

t (∂u) in terms of spatial derivatives of u with additional powers of r in a
similar way as Lemma 2.11 be returning to (1.35c).

For the second term in (6.2a), we obtain

D2k
t r̃

2k−1∑
i=0

(
Di

tũ
µ∂µ

(
D2k−i

t r
)
−D2k−i

t r∂µ
(
Di

tũ
µ
))

, (6.5)

where each piece will be handled separately. For the first piece with Di
tũ, we see

by Lemma 2.9 that Di
tũ only contains critical terms at worst when i = 2k− 1, and

all other terms are subcritical. Thus, the product can be estimated using Lemma
2.6 since D2k

t r̃ is also critical:∫
Ωt

r
2−γ
γ−1

∣∣∣D2k
t r̃

2k−1∑
i=0

Di
tũ

µ∂µ
(
D2k−i

t r
) ∣∣∣ dx ≲ C2,1∥(s̃, r̃, ũ)∥2H2k . (6.6)

For the second piece, with ∂µ(D
i
tũ

µ) we see that D2k−i
t r contains one power of r

by (2.10), and up to 2k − i derivatives of r and u. Similar to the first piece, it

suffices to check when i = 2k − 1 in which case D2k−1
t ũ contains terms that are

H2k critical, and terms that are subcritical with order α = 1/2. Simplifying the
expression when i = 2k − 1 and solving for time derivatives in terms of spatial
derivatives with Lemma 2.12, we will obtain

Dtr∂µ(D
2k−1
t ũµ)

≃ r∂
(
(terms with order α = 0) + (terms with order α =

1

2
)
)
.

(6.7)

Then, by applying Lemma 2.14, we see that applying r∂ to any free boundary term
will produce a new term with the same order, i.e. here we have α = 0 and α = 1

2
respectively. Thus, the second piece can be estimated using Lemma 2.6∫

Ωt

r
2−γ
γ−1

∣∣∣D2k
t r̃

2k−1∑
i=0

D2k−i
t r∂µ

(
Di

tũ
) ∣∣∣ dx ≲ C2,2∥(s̃, r̃, ũ)∥2H2k . (6.8)

For the third term in (6.2a), the estimate is quite similar to before as Di
tũ only

contains critical terms when i = 2k − 1. By Lemma 2.9, we obtain

[∂µ, D
2k−i
t ]r = Cν

2k−i−1∂ν(D
2k−i−1
t r) + Cν

2k−i−2∂ν(D
2k−i−2
t r)

+ · · ·+ Cν
1 ∂ν(Dtr) + Cν

0 ∂νr
(6.9)



54 B. B. LUCZAK EJDE-2025/10

which consists of derivatives of u and r. Then, by Lemma 2.6, the estimate will
look like∫

Ωt

r
2−γ
γ−1

∣∣∣D2k
t r̃

2k−1∑
i=0

Di
tũ

µ[D2k−i
t , ∂µ]r

∣∣∣ dx ≲ C3∥(s̃, r̃, ũ)∥2H2k , (6.10)

since we have the product of terms that are both H2k-critical at worst.
To estimate the G2k fourth term, we will use the commutator identities from

Lemma 2.9:

D2k
t r̃

2k∑
i=1

D2k−i
t r[Di

t, ∂µ]ũ
µ ≃ D2k

t r̃

2k∑
i=1

D2k−i
t r

( i−1∑
j=0

Cν
j ∂ν(D

j
t ũ

µ)
)
. (6.11)

Now, the worst possible terms only occur when i = 2k, in which case we will
have D0

t r = r and D2k
t r̃

(
r∂(D2k−1

t ũ)
)
. This term can be handled in a similar way

as (6.7) since we obtain the product of terms which are both critical. In fact, when
i ≤ 2k − 1, we obtain that all terms are subcritical and easily estimated. Thus, we
have ∫

Ωt

r
2−γ
γ−1

∣∣∣D2k
t r̃

2k∑
i=1

D2k−i
t r[Di

t, ∂µ]ũ
µ
∣∣∣ dx ≲ C4∥(s̃, r̃, ũ)∥2H2k . (6.12)

Combining the previous inequalities proves (6.2a).
Let us turn now to the Cα

2k terms occurring in (6.2b). Recall from (6.2b) that
the Cα

2k has the multiplier (Γ+r)rGαβD
2k
t ũβ . In general, we observe using Lemma

2.11 that the multiplier rD2k
t ũβ has H2k subcritical terms with order 1/2 at worst

since

D2k
t ũ ≃

k∑
l=0

rl∂l+kũ+

k−1∑
j=0

rj∂j+kr̃

≃ (α = −1/2 supercritical terms) + (α = 0 critical terms)

=⇒ rD2k
t ũ ≃

k∑
l=0

rl+1∂l+kũ+

k−1∑
j=0

rj+1∂j+kr̃

≃ (α = 1/2 subcritical terms) + (α = 1 subcritical terms) .

For the first term in (6.2b), we obtain

(Γ + r)rGαβD
2k
t ũβD2k

t hα (6.13)

and we recall the from (1.37) that

D2k
t hα

= D2k
t

(
− ũµ∂µu

α − 1

Γ + r
(ũαuµ + uαũµ)∂µr

+
Γ′

(Γ + r)2
s̃παµ∂µr +

1

(Γ + r)2
r̃παµ∂µr

)
≃ −

2k∑
i=0

Di
tũ

µD2k−i
t (∂µu

α)− 1

Γ + r

2k∑
i=0

Di
t(ũ

αuµ + uαũµ)D2k−i
t (∂µr)

+
Γ′

(Γ + r)2

2k∑
i=0

Di
ts̃D

2k−i
t (παµ∂µr) +

1

(Γ + r)2

2k∑
i=0

Di
tr̃D

2k−i
t (παµ∂µr) ,

(6.14)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 55

where we recall that D2k
t

(
1

Γ+r

)
≃ − 1

(Γ+r)2D
2k
t r using Remark 6.1, and this only

contributes additional powers of r. After observing D2k
t hα, we see that it contains

terms of the form D2k
t (ũ, s̃, r̃) which are order − 1

2 supercritical at worst. However,

those terms all get multiplied by rD2k
t ũ which has order 1

2 . Thus, by Lemma

2.6, these terms are also estimated using the H2k norm and we can solve for any
expressions like D2k−i

t (παµ∂µr) in terms of spatial derivatives of u, r, and s.
For the second term in (6.2b), we obtain

(Γ + r)rGαβD
2k
t ũβ

2k−1∑
i=0

D2k−i
t

( 1

Γ + r
παµ

)
∂µ

(
Di

tr̃
)
. (6.15)

Now, when i = 2k − 1, D2k−1
t r̃ contains terms that have order 1

2 at worst by

Remarks 2.4 and 2.15. This implies that ∂(D2k−1
t r̃) will contain terms that are

order − 1
2 at worst by Lemma 2.14. However, the multiplier rD2k

t ũhas order 1
2 ,

which is just enough to allow us to apply Lemma 2.6. Thus, we can estimate this
term using the H2k norm.

For the third and final term in (6.2b), we can follow the analysis of the second
term and use Lemma 2.9 to obtain

(Γ + r)rGαβD
2k
t ũβ

2k∑
i=1

D2k−i
t

( 1

Γ + r
παµ

)
[∂µ, D

i
t]r̃

≃ (Γ + r)GαβrD
2k
t ũβ

2k∑
i=1

D2k−i
t

( 1

Γ + r
παµ

)( i−1∑
j=0

Cν
j ∂ν(D

j
t r̃)

)
.

(6.16)

Since the worst terms appear when i = 2k, we see that once again D2k−1
t r̃ has

terms with order 1
2 at worst, thus, ∂(D2k−1

t r̃) has terms with − 1
2 order. However,

the multiplier has order 1
2 , and this allows us to apply Lemma 2.6 as usual. This

completes estimate (6.2b), and all higher order terms on the RHS in (2.1) have
been handled. □

7. Main theorem

Combining the previous sections, we have reached our final result.

Theorem 7.1 (Estimates in H2k). Let (s, r, u) be a smooth solution to (1.35) that
exists on some time interval [0, T ], and for which the physical vacuum boundary
condition (1.13) holds. Let (s̃0, r̃0, ũ0) be initial data to system (1.36). Then, there

exists a constant C depending only on s, r, u, and T such that, if (s̃, r̃, ũ) ∈ C∞(D)
is a solution to (1.36) on [0, T ], then

∥(s̃, r̃, ũ)∥H2k(Ωt) ≲ C∥(s̃0, r̃0, ũ0)∥H2k(Ω0) , (7.1)

where H2k(Ωt) is defined in (1.46).

Proof. Combining Proposition 3.1 and Theorem 6.2, we obtain

d

dt

(
∥s̃∥2

H
2k,k+

2−γ
2(γ−1)

+ 1
2
+ E2k

wave(s̃, r̃, ũ)
)
≲ C1∥(s̃, r̃, ũ)∥2H2k , (7.2)

where C1 is a constant depending on L∞(Ωt) norm of up to 2k + 1 derivatives of
s, r, and u. Then, after adding d

dt∥ω̂∥
2

H
2k−1,k+

2−γ
2(γ−1)

+ 1
2

to both sides and recalling



56 B. B. LUCZAK EJDE-2025/10

(1.48) and (1.49), we have

d

dt
E2k

total(s̃, r̃, ũ) ≲ C1∥(s̃, r̃, ũ)∥2H2k +
d

dt
∥ω̂∥2

H
2k−1,k+

2−γ
2(γ−1)

+ 1
2
. (7.3)

Integrating both sides and using Theorem 5.1 on the energy equivalence, we have

∥(s̃, r̃, ũ)∥2H2k(Ωt)
− ∥(s̃0, r̃0, ũ0)∥2H2k(Ω0)

≲
∫ t

0

C1∥(s̃, r̃, ũ)∥2H2k(Ωτ )
dτ + ∥ω̂∥2

H
2k−1,k+

2−γ
2(γ−1)

+ 1
2 (Ωt)

.
(7.4)

Applying Theorem 3.3 for the ω̂ term and rearranging, we obtain

∥(s̃, r̃, ũ)∥2H2k(Ωt)

≲ ∥(s̃0, r̃0, ũ0)∥2H2k(Ω0)
+

∫ t

0

C1∥(s̃, r̃, ũ)∥2H2k(Ωτ )
dτ

+ ∥ω̃0∥2
H

2k−1,k+ 1
2(γ−1) (Ω0)

+ ε̂∥(s̃, r̃, ũ)∥2H2k(Ωt)
+

∫ t

0

C2∥(s̃, r̃, ũ)∥2H2k(Ωτ )
dτ.

Then, combining the initial data ω̃0 with the initial data term ∥(s̃0, r̃0, ũ0)∥2H2k(Ω0)
,

and soaking the term with ε̂ to the LHS, we arrive at

∥(s̃, r̃, ũ)∥2H2k(Ωt)
≲

1

1− ε̂

(
∥(s̃0, r̃0, ũ0)∥2H2k(Ω0)

+

∫ t

0

C3∥(s̃, r̃, ũ)∥2H2k(Ωτ )
dτ

)
.

Finally, the desired result is a straightforward application of Gronwall’s inequality.
□

8. Appendix

Here, we record some of the computations for proving Lemma 2.12. Using (1.35)
directly, we have

∂ts = − ui

u0
∂is,

∂tr = − ui

u0
∂ir −

γ − 1

u0
r∂tu

0 − γ − 1

u0
r∂iu

i,

∂tu
α = − ui

u0
∂iu

α − 1

(Γ + r)u0
πα0∂tr −

1

(Γ + r)u0
παi∂ir .

(8.1)

Starting from (8.1), we can further solve for ∂tu
0 by plugging in α = 0 into the last

equation. If we set a1 = u0 − γ−1
(Γ+r)u0 rπ

00, then

∂tu
0 =

1

a1

(
− ui∂iu

0 +
γ − 1

(Γ + r)u0
rπ00∂iu

i +
1

(Γ + r)u0
π00ui∂ir −

1

Γ + r
π0i∂ir

)
= − 1

a1
ui∂iu

0 − 1

a1(Γ + r)u0
ui∂ir +

(γ − 1)((u0)2 − 1)

a1(Γ + r)u0
r∂iu

i

=: Ci
1∂iu

0 + Ci
2∂ir + C3r∂iu

i

≃ ∂u0 + ∂r + r∂u

implies

∂tũ
0 = Ci

1∂iũ
0 + Ci

2∂ir̃ + C3r∂iũ
i + C3r̃∂iu

i + C̃i
1∂iu

0 + C̃i
2∂ir + C̃3r∂iu

i (8.2)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 57

We have the computations:

a1 = u0 − (γ − 1)((u0)2 − 1)

(Γ + r)u0
r

implies

ã1 = ũ0 − (γ − 1)((u0)2 − 1)

(Γ + r)u0
r̃

− r
(2(Γ + r)(γ − 1)u0ũ0 − (γ − 1)((u0)2 − 1)[(Γ′s̃+ r̃)u0 + (Γ + r)ũ0]

(Γ + r)2(u0)2

)
,

and

Ci
1 := −ui

a1
=⇒ C̃i

1 =
uiã1 − a1ũ

i

(a1)2
,

Ci
2 :=

1

(Γ + r)u0
Ci

1 =⇒ C̃i
2 =

1

(Γ + r)u0
C̃i

1 −
[(Γ′s̃+ r̃)u0 + (Γ + r)ũ0]

(Γ + r)2(u0)2
Ci

1,

C3 :=
(γ − 1)((u0)2 − 1)

ui
Ci

2

=⇒ C̃3 =
(γ − 1)((u0)2 − 1)

ui
C̃i

2 +
ui2(γ − 1)ũ0 − (γ − 1)((u0)2 − 1)ũi

(ui)2
Ci

2.

Note that after analyzing all of the terms (with k = 1 here), we see that a large
number are subcritical, and we can simplify:

∂tũ
0 ≃ ∂ũ0 + ∂r̃ + r∂ũ+ r̃∂u

≃ (H2 super-critical of order −1

2
)

+ (H2 critical) + (H2 sub-critical of order
1

2
)

Similarly, we can simplify ∂tr and ∂tr̃ as follows:

∂tr = − ui

u0
∂ir −

γ − 1

u0
r∂iu

i − γ − 1

u0
r
(
Ci

1∂iu
0 + Ci

2∂ir + C3r∂iu
i
)

implies

∂tr̃ = − ui

u0
∂ir̃ −

γ − 1

u0
r∂iũ

i − γ − 1

u0
r
(
∂tũ

0
)

+
uiũ0 − u0ũi

(u0)2
∂ir +

( (γ − 1)

(u0)2
ũ0r − γ − 1

u0
r̃
)
∂iu

i +
( (γ − 1)

(u0)2
ũ0r − γ − 1

u0
r̃
)
∂tu

0

≃ ∂r̃ + r∂ũ+ r∂ũ0 + r∂r̃ + r2∂ũ+ (additional subcritical terms)

and

∂tu
j = − 1

(Γ + r)u0
πj0 [∂tr]−

ui

u0
∂iu

j − 1

(Γ + r)u0
πji∂ir ≃ −[∂tr]− ∂u− ∂r .

The expressions for ∂ts and ∂ts̃ take the easiest form

∂ts = − ui

u0
∂is,

∂ts̃ = − ui

u0
∂is̃+

uiũ0 − u0ũi

(u0)2
∂is.

(8.3)



58 B. B. LUCZAK EJDE-2025/10

We also see that (u0)2 = 1 + ujuj , so

ũ0 =
uj

u0
ũj ,

∂iũ
0 =

uj

u0
∂iũ

j + ũj∂i
uj

u0
.

(8.4)

Thus, linearizing our expression for ∂tu
j , we obtain

∂tũ
j ≃ [∂tr̃] + ∂ũ+ ∂r̃

≃ ∂ũ+ ∂r̃ + (r∂ũ+ r∂r̃ + r2∂ũ+ (additional subcritical terms))

Starting from (8.1), we can further solve for ∂tu
0 by plugging in α = 0 into the

last equation. If we set a1 = u0 − γ−1
(Γ+r)u0 rπ

00, then

∂tu
0 =

1

a1

(
− ui∂iu

0 +
γ − 1

(Γ + r)u0
rπ00∂iu

i +
1

(Γ + r)u0
π00ui∂ir −

1

Γ + r
π0i∂ir

)
= − 1

a1
ui∂iu

0 − 1

a1(Γ + r)u0
ui∂ir +

(γ − 1)((u0)2 − 1)

a1(Γ + r)u0
r∂iu

i

=: Ci
1∂iu

0 + Ci
2∂ir + C3r∂iu

i

≃ ∂u0 + ∂r + r∂u

implies

∂tũ
0 = Ci

1∂iũ
0 + Ci

2∂ir̃ + C3r∂iũ
i + C3r̃∂iu

i + C̃i
1∂iu

0 + C̃i
2∂ir + C̃3r∂iu

i

We have the computations

a1 = u0 − (γ − 1)((u0)2 − 1)

(Γ + r)u0
r

implies

ã1 = ũ0 − (γ − 1)((u0)2 − 1)

(Γ + r)u0
r̃

− r

(
2(Γ + r)(γ − 1)u0ũ0 − (γ − 1)((u0)2 − 1)[(Γ′s̃+ r̃)u0 + (Γ + r)ũ0]

(Γ + r)2(u0)2

)
and

Ci
1 := −ui

a1
=⇒ C̃i

1 =
uiã1 − a1ũ

i

(a1)2
,

Ci
2 :=

1

(Γ + r)u0
Ci

1 =⇒ C̃i
2 =

1

(Γ + r)u0
C̃i

1 −
[(Γ′s̃+ r̃)u0 + (Γ + r)ũ0]

(Γ + r)2(u0)2
Ci

1,

C3 :=
(γ − 1)((u0)2 − 1)

ui
Ci

2

=⇒ C̃3 =
(γ − 1)((u0)2 − 1)

ui
C̃i

2 +
ui2(γ − 1)ũ0 − (γ − 1)((u0)2 − 1)ũi

(ui)2
Ci

2.

Note that after analyzing all of the terms (with k = 1 here), we see that a large
number are subcritical, and we can simplify:

∂tũ
0 ≃ ∂ũ0 + ∂r̃ + r∂ũ+ r̃∂u

≃ (H2 super-critical of order −1

2
)



EJDE-2025/10 RELATIVISTIC EULER EQUATIONS 59

+ (H2 critical) + (H2 sub-critical of order
1

2
)

Similarly, we can simplify ∂tr and ∂tr̃ in the following way:

∂tr = − ui

u0
∂ir −

γ − 1

u0
r∂iu

i − γ − 1

u0
r
(
Ci

1∂iu
0 + Ci

2∂ir + C3r∂iu
i
)

implies

∂tr̃ = − ui

u0
∂ir̃ −

γ − 1

u0
r∂iũ

i − γ − 1

u0
r
(
∂tũ

0
)
+

uiũ0 − u0ũi

(u0)2
∂ir

+
( (γ − 1)

(u0)2
ũ0r − γ − 1

u0
r̃
)
∂iu

i +
( (γ − 1)

(u0)2
ũ0r − γ − 1

u0
r̃
)
∂tu

0

≃ ∂r̃ + r∂ũ+ r∂ũ0 + r∂r̃ + r2∂ũ+ (additional subcritical terms)

and

∂tu
j = − 1

(Γ + r)u0
πj0 [∂tr]−

ui

u0
∂iu

j − 1

(Γ + r)u0
πji∂ir ≃ −[∂tr]− ∂u− ∂r

The expressions for ∂ts and ∂ts̃ take the easiest form

∂ts = − ui

u0
∂is,

∂ts̃ = − ui

u0
∂is̃+

uiũ0 − u0ũi

(u0)2
∂is .

(8.5)

We also see that (u0)2 = 1 + ujuj , so

ũ0 =
uj

u0
ũj ,

∂iũ
0 =

uj

u0
∂iũ

j + ũj∂i
uj

u0
.

(8.6)

Thus, linearizing our expression for ∂tu
j , we obtain

∂tũ
j ≃ [∂tr̃] + ∂ũ+ ∂r̃

≃ ∂ũ+ ∂r̃ + (r∂ũ+ r∂r̃ + r2∂ũ+ (additional subcritical terms))

Acknowledgments. B. B. Luczak gratefully acknowledges support from NSF
grant DMS-2107701.

References

[1] Luis A. Caffarelli, Avner Friedman; Regularity of the free boundary for the one-dimensional
flow of gas in a porous medium, Amer. J. Math. 101 (1979), no. 6, 1193–1218. MR 548877

[2] Yvonne Choquet-Bruhat; General relativity and the Einstein equations, Oxford Mathematical
Monographs, Oxford University Press, Oxford, 2009. MR 2473363

[3] Marcelo M. Disconzi; Recent developments in mathematical aspects of relativistic fluids, 2023.

[4] Marcelo M. Disconzi, Mihaela Ifrim, Daniel Tataru; The relativistic Euler equations with a

physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continua-
tion criterion, Archive for Rational Mechanics and Analysis (2022), 55 Pages.

[5] Marcelo M. Disconzi, Igor Kukavica, Amjad Tuffaha; A Lagrangian interior regularity result
for the incompressible free boundary Euler equation with surface tension, SIAM J. Math.

Anal. 51 (2019), no. 5, 3982–4022. MR 4019189

[6] Mahir Hadžić, Steve Shkoller, Jared Speck; A priori estimates for solutions to the relativistic
Euler equations with a moving vacuum boundary, Comm. Partial Differential Equations 44

(2019), no. 10, 859–906. MR 3974126



60 B. B. LUCZAK EJDE-2025/10

[7] Mihaela Ifrim, Daniel Tataru; The compressible euler equations in a physical vacuum: A com-

prehensive eulerian approach, Annales de l’Institut Henri Poincaré C, Analyse non linéaire

(2020).
[8] Juhi Jang, Philippe G. LeFloch, Nader Masmoudi; Lagrangian formulation and a priori

estimates for relativistic fluid flows with vacuum, J. Differential Equations 260 (2016), no. 6,

5481–5509. MR 3448785
[9] Philippe G. LeFloch and Seiji Ukai, A symmetrization of the relativistic Euler equations with

several spatial variables, Kinet. Relat. Models 2 (2009), no. 2, 275–292. MR 2507449

[10] Tetu Makino, Seiji Ukai; Local smooth solutions of the relativistic Euler equation. II, Kodai
Math. J. 18 (1995), no. 2, 365–375. MR 1346915

[11] Šárka Matušocircu-Nečasová, Mari Okada, Tetu Makino; Free boundary problem for the equa-
tion of spherically symmetric motion of viscous gas. III, Japan J. Indust. Appl. Math. 14

(1997), no. 2, 199–213. MR 1454522

[12] Shuang Miao, Sohrab Shahshahani; Well-posedness for the free boundary hard phase model
in general relativity, Advances in Mathematics 443 (2024), 109614.

[13] Shuang Miao, Sohrab Shahshahani, Sijue Wu; Well-posedness of the free boundary hard phase

fluids in minkowski background and its newtonian limit, arXiv:2003.02987 [math.AP] (2020).
[14] J. R. Oppenheimer, G. M. Volkoff; On Massive neutron cores, Phys. Rev. 55 (1939), 374–381.

[15] Luciano Rezzolla, Olindo Zanotti; Relativistic hydrodynamics, Oxford University Press, New

York, 2013.
[16] Jalal Shatah, Chongchun Zeng; Geometry and a priori estimates for free boundary problems

of the Euler equation, Comm. Pure Appl. Math. 61 (2008), no. 5, 698–744. MR 2388661
[17] Richard C. Tolman; Effect of imhomogeneity on cosmological models, Proc. Nat. Acad. Sci.

20 (1934), 169–176.

[18] Richard C. Tolman; Static solutions of Einstein’s field equations for spheres of fluid, Phys.
Rev. 55 (1939), 364–373.

[19] Yuri Trakhinin; Local existence for the free boundary problem for nonrelativistic and rela-

tivistic compressible Euler equations with a vacuum boundary condition, Comm. Pure Appl.
Math. 62 (2009), no. 11, 1551–1594. MR 2560044

[20] Ping Zhang, Zhifei Zhang; On the free boundary problem of three-dimensional incompressible

Euler equations, Comm. Pure Appl. Math. 61 (2008), no. 7, 877–940. MR 2410409

Brian B. Luczak

University of Chicago Laboratory Schools, 1362 E. 59th St., Chicago, IL 60637, USA

Email address: bluczak@uchicago.edu


	1. Introduction
	1.1. Physical vacuum boundary with a barotropic equation of state
	1.2. The case of an ideal gas
	1.3. Main Result
	1.4. Linearized system in terms of entropy and sound speed (squared)
	1.5. Function spaces and Energies
	1.6. Basic energy estimate

	2. Creating a book-keeping scheme
	2.1. Defining order for free-boundary terms
	2.2. Solving for time derivatives

	3. Estimates for the transport energy
	3.1. Entropy estimates
	3.2. Vorticity estimates

	4. Elliptic and div-curl Estimates
	4.1. Elliptic estimates for 
	4.2. Elliptic estimates for 
	4.3. Higher order commutators with the convective derivative
	4.4. Commutators with weighted spatial derivatives

	5. Energy equivalence
	6. Higher order wave energy estimates
	7. Main theorem
	8. Appendix
	Acknowledgments

	References