POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER
EQUATIONS WITH
C
α
INITIAL VORTICITY FOR A LARGE
RANGE OF
α
. PART II: THE
N
-DIMENSIONAL CASE
THOMAS Y. HOU
∗
AND
SHUMAO ZHANG
†
Abstract.
In Part II of this sequence to our previous paper for the 3-dimensional Euler equations
[8], we investigate potential singularity of the
n
-diemnsional axisymmetric Euler equations with
C
α
initial vorticity for a large range of
α
. We use the adaptive mesh method to solve the
n
-dimensional
axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine
the potential blow-up and capture its self-similar profile. Our study shows that the
n
-dimensional
axisymmetric Euler equations with our initial data develop finite-time blow-up when the H ̈older
exponent
α < α
∗
, and this upper bound
α
∗
can asymptotically approach 1
−
2
n
. Moreover, we
introduce a stretching parameter
δ
along the
z
-direction. Based on a few assumptions inspired by
our numerical experiments, we obtain
α
∗
= 1
−
2
n
by studying the limiting case of
δ
→
0. For
the general case, we propose a relatively simple one-dimensional model and numerically verify its
approximation to the
n
-dimensional Euler equations. This one-dimensional model sheds useful light
to our understanding of the blowup mechanism for the
n
-dimensional Euler equations. As shown in
[8], the scaling behavior and regularity properties of our initial data are quite different from those of
the initial data considered by Elgindi in [6].
Key words.
n
-dimensional axisymmetric Euler equations, finite-time blow-up
AMS subject classifications.
35Q31, 76B03, 65M60, 65M06, 65M20
1. Introduction.
The question regarding the global regularity of the 3D incom-
pressible Euler equations with smooth initial data has been widely recognized as a
major open problem in partial differential equations (PDEs). Depsite a lot of previous
efforts, this question remains unresolved until very recently when Chen and Hou [2]
provided a rigorous justification of the Luo-Hou blow-up scenario with smooth initial
data and boundary [14, 15]. In 2021, Elgindi [6] showed that the 3D axisymmetric Eu-
ler equations with no swirl can develop finite-time singularity for a class of
C
α
initial
vorticity with very small
α >
0. Inspired by Elgindi’s work, we provided numerical ev-
idence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler
equations with no swirl and with
C
α
initial vorticity for a large range of
α
∈
(0
,
1
/
3)
in [8] using a different class of initial data.
In this paper, we extend the numerical study in [8] of the potential finite-time
singularity in axisymmetric Euler equations with no swirl from the 3-dimensional case
to the higher dimensional case. We will also provide more interesting observations
that shed useful light on the potential blowup mechanism. As in our previous paper
[8], we will use an adaptive mesh method and the dynamic rescaling formulation
[7, 1, 3] to study the self-similarity and scaling properties of the potential singularity.
By introducing a parameter
δ
to control the stretching of the computational domain
and the initial data in the
z
-axis, we find that the
n
-dimensional axisymmetric Euler
equations with
C
α
initial vorticity can develop potential finite-time blow-up when
α
is smaller than some critical value
α
∗
, and this critical value
α
∗
can asymptotically
approach 1
−
2
n
as
δ
→
0. This result supports Conjecture 8 of [5] and generalizes it
to the high-dimensional case.
More specifically, we denote by
ω
θ
as the angular vorticity and
ψ
θ
as the angular
∗
Department of Computing and Mathematical Sciences, California Institute of Technology,
Pasadena, CA (hou@cms.caltech.edu).
†
Department of Computing and Mathematical Sciences, California Institute of Technology,
Pasadena, CA (shumaoz@caltech.edu).
1
arXiv:2212.11924v1 [math.AP] 22 Dec 2022
2
THOMAS Y. HOU AND SHUMAO ZHANG
stream function. We will use the same change of variables as in [8]:
ω
θ
(
r,z
) =
r
α
ω
1
(
r,z
)
, ψ
θ
(
r,z
) =
rψ
1
(
r,z
)
.
We make the following self-similar blowup ansatz for
ω
1
and
ψ
1
:
ω
1
(
x,t
)
≈
1
(
T
−
t
)
c
ω
Ω
(
x
−
x
0
(
T
−
t
)
c
l
)
,
ψ
1
(
x,t
)
≈
1
(
T
−
t
)
c
ψ
Ψ
(
x
−
x
0
(
T
−
t
)
c
l
)
,
for some constants
c
ω
,
c
ψ
,
c
l
,
x
0
and
T
. Here
T
is the blow-up time, and
x
0
is
the location of the self-similar blow-up. The parameters
c
ω
,
c
ψ
,
c
l
are called scaling
factors. Based on our numerical observations, we make a few assumptions about the
potential blow-up, and study the limiting case of
δ
→
0. Using a formal asymptotic
analysis inspired by our numerical results, we derive the following scaling relationships
for
c
l
and
c
ω
as
δ
→
0:
c
l
=
n
−
1
n
−
2
−
nα
, c
ω
=
n
−
2
−
α
n
−
2
−
nα
.
In the limiting case of
δ
→
0, we obtain
α
∗
= 1
−
2
n
, which agrees with our numer-
ical results. Moreover, we propose a relatively simple one-dimensional model that
focuses on the behavior of the
n
-dimensional axisymmetric Euler equations along the
z
-axis. Our numerical experiments confirm that the one-dimensional model is a good
approximation of the original
n
-dimensional Euler equations, and can develop ap-
proximately the same potential finite-time blow-up as the original Euler equations.
This one-dimensional model can be used as a simplified model to study the finite-time
blow-up of the
n
-dimensional Euler equations rigorously.
There have been many previous studies of potential finite time singularity of
the 3D incompressible Euler equations with smooth initial data. These include both
analytic studies and numerical investigations. We refer to Part I of this sequence [8]
for more detailed discussions.
The rest of the paper is organized as follows. In Section 2, we describe the setup of
the problem and our numerical method. In Section 3, we present convincing numerical
evidences for a potential self-similar blowup. In Section 4, we study the relationship
between the H ̈older exponent
α
and the space dimension
n
in the potential self-similar
blowup. We investigate the possible mechanism leading to the potential finite time
blowup of the
n
-dimensional Euler equations in Section 5. Section 6 is devoted to a
one-dimensional model to study the potential self-similar blowup of the
n
-dimensional
axisymmetric Euler equations. Some concluding remarks are drawn in Section 7.
2. Problem set up and numerical method.
In this section, we first establish
the high-dimensional axisymmetric Euler equations with no swirl, and some basic
properties of the equations, then present the initial data, and the boundary conditions
that we use in our study.
2.1. High-dimensional axisymmetric Euler equations with no swirl.
To
start with, we introduce the
n
-dimensional axisymmetric Euler equations. Let
u
(
x,t
) :
R
n
×
[0
,T
)
→
R
n
,
and
p
(
x,t
) :
R
n
×
[0
,T
)
→
R
,
POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER EQUATIONS, PART II
3
be an
n
-D vector field of the velocity and an
n
-D scalar field of the pressure respec-
tively, where
x
= (
x
1
,x
2
,...,x
n
)
∈
R
n
. Then the
n
-dimensional Euler equations can
be written as
u
t
+
u
·∇
u
=
−∇
p,
(2.1a)
∇·
u
= 0
.
(2.1b)
Next, we consider a hyper-cylindrical coordinate system (
r,θ
1
,...,θ
n
−
2
,z
), which
is related to the Cartesian coordinate system (
x
1
,x
2
,...,x
n
) via the following relation
x
1
=
r
cos
θ
1
,
x
2
=
r
sin
θ
1
cos
θ
2
,
.
.
.
x
n
−
1
=
r
sin
θ
1
···
sin
θ
n
−
2
x
n
=
z.
The hyper-cylindrical coordinate system is simply the direct product of a (
n
−
1)-D
spherical coordinate system with a 1D Cartesian coordinate system. The frame of the
hyper-cylindrical coordinate system can be expressed in the Cartesian coordinate as
e
r
= (cos
θ
1
,
sin
θ
1
cos
θ
2
,...,
sin
θ
1
···
cos
θ
n
−
2
,
sin
θ
1
···
sin
θ
n
−
2
,
0)
,
e
θ
1
= (
−
sin
θ
1
,
cos
θ
1
cos
θ
2
,...,
cos
θ
1
···
cos
θ
n
−
2
,
cos
θ
1
···
sin
θ
n
−
2
,
0)
,
.
.
.
e
θ
n
−
2
= (0
,
0
,...,
−
sin
θ
n
−
2
,
cos
θ
n
−
2
,
0)
,
e
z
= (0
,
0
,...,
0
,
0
,
1)
.
Similar to the 3D case, we call an
n
-D vector field
v
to be axisymmetric if the
following ansatz applies
v
=
v
r
(
r,z
)
e
r
+
v
θ
1
(
r,z
)
e
θ
1
+
v
z
(
r,z
)
e
z
.
In other words,
v
r
,
v
θ
1
, and
v
z
are only functions of (
r,z
). For such vector field, the
calculus on the curvilinear coordinate [4] of (
r,θ
1
,...,θ
n
−
2
,z
) gives
∇·
v
=
v
r
r
+
n
−
2
r
v
r
+
(
n
−
3) cot
θ
1
r
v
θ
1
+
v
z
z
,
(
v
·∇
)
v
=
(
v
r
v
r
r
+
v
z
v
r
z
−
1
r
v
θ
1
v
θ
1
)
e
r
+
(
v
r
v
θ
1
r
+
v
z
v
θ
1
z
+
1
r
v
r
v
θ
1
)
e
θ
1
+ (
v
r
v
z
r
+
v
z
v
z
z
)
e
z
.
We can see that if there is non-zero “swirl”
u
θ
1
in the initial condition for dimen-
sion
n
6
= 3, then the incompressibility condition
∇·
u
= 0 will inevitably introduce
the dependence on
θ
1
for the equations, which implies that we cannot obtain a truly
axisymmetric Euler equations for dimension greater than 3. Note that when
n
= 3,
the incompressibility condition does not introduce any trouble since the third term in
∇·
v
vanishes exactly for
n
= 3 even if there is swirl.
Therefore, to derive the
n
-dimensional axisymmetric Euler equations with
n >
3, we need to impose the “no swirl” assumption
u
θ
1
= 0. Luckily, the “no swirl”
4
THOMAS Y. HOU AND SHUMAO ZHANG
assumption will be preserved dynamically by the
n
-D axisymmetric Euler equations.
We remark that the axisymmetric Euler equations offer tremendous computational
saving, which enables us to investigate potential finite time singularity for the general
n
-dimensional Euler equations using our current computational resources.
Thus, the axisymmetric
n
-D Euler equations with no swirl can be written in the
vorticity-stream function form as
ω
θ
t
+
u
r
ω
θ
r
+
u
z
ω
θ
z
=
n
−
2
r
u
r
ω
θ
,
(2.2a)
−
ψ
θ
rr
−
ψ
θ
zz
−
n
−
2
r
ψ
θ
r
+
n
−
2
r
2
ψ
θ
=
ω
θ
,
(2.2b)
u
r
=
−
ψ
θ
z
, u
z
=
n
−
2
r
ψ
θ
+
ψ
θ
r
,
(2.2c)
where we introduce the angular vorticity
ω
θ
and angular stream function
ψ
θ
as
ω
θ
=
u
r
z
−
u
z
r
and
−
∆
ψ
θ
=
ω
θ
, similar to the 3D axisymmetric Euler equations.
Since we plan to use
C
α
continuous initial data for the angular vorticity
ω
θ
, we
follow the same change-of-variables as in [8]:
ω
θ
(
r,z
) =
r
α
ω
1
(
r,z
)
, ψ
θ
(
r,z
) =
rψ
1
(
r,z
)
.
(2.3)
Using (
ω
1
,ψ
1
), an equivalent form of the
n
-D axisymmetric Euler equations with no
swirl is given below
ω
1
,t
+
u
r
ω
1
,r
+
u
z
ω
1
,z
=
−
(
n
−
2
−
α
)
ψ
1
,z
ω
1
,
(2.4a)
−
ψ
1
,rr
−
ψ
1
,zz
−
n
r
ψ
1
,r
=
ω
1
r
α
−
1
,
(2.4b)
u
r
=
−
rψ
1
,z
, u
z
= (
n
−
1)
ψ
1
+
rψ
1
,r
.
(2.4c)
Roughly speaking, the dimension
n
controls the strength of the vortex stretching term
−
(
n
−
2
−
α
)
ψ
1
,z
ω
and the
z
-advection speed
u
z
= (
n
−
1)
ψ
1
+
rψ
1
,r
. It also modifies
the Poisson equation for
ψ
1
. It seems natural to conjecture that the singularity
formation will be more likely in the high-dimensional case because of the stronger
vortex stretching term
−
(
n
−
2
−
α
)
ψ
1
,z
ω
.
Similar to the 3D case, the
n
-D axisymmetric Euler equations also admit the
conservation of the kinetic energy
E
, which is defined as
E
=
1
2
∫
1
0
∫
1
/
2
0
(
|
u
r
|
2
+
|
u
z
|
2
)
r
n
−
2
d
r
d
z.
2.2. Self-similar solution.
The self-similar blow-up solutions for nonlinear
PDEs are of particular interest in our study. A self-similar solution is when the
local profile of the solution remains nearly unchanged in time after rescaling the spa-
tial and the temporal variables of the physical solution. For example, for (2.4), the
self-similar profile is the ansatz
(2.5)
ω
1
(
x,t
)
≈
1
(
T
−
t
)
c
ω
Ω
(
x
−
x
0
(
T
−
t
)
c
l
)
,
ψ
1
(
x,t
)
≈
1
(
T
−
t
)
c
ψ
Ψ
(
x
−
x
0
(
T
−
t
)
c
l
)
,
for some constants
c
ω
,
c
ψ
,
c
l
,
x
0
and
T
. Here
T
is the blow-up time, and
x
0
is
the location of the self-similar blow-up. The parameters
c
ω
,
c
ψ
,
c
l
are called scaling
factors.
POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER EQUATIONS, PART II
5
Similar to the 3D case, the axisymmetric
n
-D Euler equations with no swirl (2.4)
admits the following scaling invariant property: if (
ω
1
,ψ
1
) is a solution of (2.4), then
{
1
λ
α
τ
ω
1
(
x
λ
,
t
τ
)
,
λ
τ
ψ
1
(
x
λ
,
t
τ
)}
(2.6)
is also a solution.
Thus if we assume the existence of the self-similar solution (2.5), then the new
solutions in (2.6) should also admit the same ansatz, regardless of the values of
λ
and
μ
. As a result, we must have
c
ω
= 1 +
αc
l
, c
ψ
= 1
−
c
l
.
(2.7)
Therefore, the self-similar profile (2.6) of (2.4) only has one degree of freedom, for
example
c
l
, in the scaling factors. And as a consequence, we have
‖
ω
θ
(
x,t
)
‖
L
∞
∼
1
T
−
t
,
‖
ψ
1
,z
(
x,t
)
‖
L
∞
∼
1
T
−
t
,
(2.8)
if the solution is self-similar.
2.3. Boundary conditions, symmetries and initial data.
For (2.4), we
impose a periodic boundary condition in
z
with period 1:
ω
1
(
r,z
) =
ω
1
(
r,z
+ 1)
, ψ
1
(
r,z
) =
ψ
1
(
r,z
+ 1)
.
(2.9)
and a pole condition for
ψ
1
at
r
= 0 and a no-flow boundary condition for
ψ
1
at
r
= 1:
ψ
1
,r
(0
,z
) = 0
, ψ
1
(1
,z
) = 0
.
(2.10)
This would allow us to only focus on the cylinder region
D
cyl
=
{
(
r,z
) : 0
≤
r
≤
1
}
.
Here at
r
= 0,
u
r
(0
,z
) =
−
rψ
0
,z
= 0, so there is no need for the boundary condition
for
ω
1
at
r
= 0. And according to [12, 13],
ψ
θ
=
rψ
1
is at least
C
2
-continuous, so
ψ
θ
must be an odd function of
r
, and thus we have the pole condition
ψ
1
,r
(0
,z
) = 0.
The condition
ψ
1
(1
,z
) = 0 implies that the no-flow boundary condition
u
r
(1
,z
) =
−
rψ
1
,z
= 0 is satisfied at
r
= 1. Therefore, there is no need to introduce a boundary
condition for
ω
1
at
r
= 1.
In addition to the boundary conditions, we enforce that (
ω
1
,ψ
1
) are odd functions
in
z
:
ω
1
(
r,z
) =
−
ω
1
(
r,
−
z
)
, ψ
1
(
r,z
) =
−
ψ
1
(
r,
−
z
)
.
(2.11)
And the
n
-D Euler equations (2.4) will preserve this symmetry dynamically.
Due to the periodicity and the odd symmetry along the
z
direction, the equations
(2.4) only need to be solved on the half-periodic cylinder
D
=
{
(
r,z
) : 0
≤
r
≤
1
,
0
≤
z
≤
1
/
2
}
because there is no transport of the flow across its boundaries. Indeed, we have
u
r
= 0
on
r
= 0 or 1
,
and
u
z
= 0
on
z
= 0 or 1
/
2
.
6
THOMAS Y. HOU AND SHUMAO ZHANG
Fig. 1
.
3D profiles of the initial value
−
ω
◦
1
and
−
ψ
◦
1
.
Fig. 2
.
The initial data for the angular vorticity
ω
θ
, when
α
= 0
.
5
.
Thus, the boundaries of
D
behave like “impermeable walls”.
Following to the choice in [8], we propose the initial data for
ω
1
in
D
as,
ω
◦
1
=
−
12000
(
1
−
r
2
)
18
sin(2
πz
)
1 + 12
.
5 sin
2
(
πz
)
.
(2.12)
Notice that (2.12) has a slightly different denominator compared to the initial data
used in [8]. In fact, both choices of the initial data will lead to the similar numerical
phenomenon in the 3D or
n
-D case, as observed in Section 6 of [8]. However, our
study shows that it takes less time for (2.12) to develop a potential blow-up, because
the initial vorticity will concentrate more near the origin, which helps reduce our
computational cost. Thus we use the above initial data in our study. We solve the
Poisson equation (2.4b) to get the initial value
ψ
◦
1
of
ψ
1
.
Fig. 3
.
Initial velocity fields
u
r
and
u
z
.
POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER EQUATIONS, PART II
7
Fig. 4
.
A heuristic diagram of the hyperbolic flow.
The 3D profiles of (
ω
◦
1
,ψ
◦
1
) can be found in Figure 1. Since most parts of
ω
◦
1
and
ψ
◦
1
are negative, we negate them for better visual effect when generating figures. In
Figure 2, we show the 3D profile and pseudocolor plot of the angular vorticity
ω
θ
at
t
= 0. We can see that there is a sharp drop to zero of
−
ω
θ
near
r
= 0, which is due
to the H ̈older continuous of
ω
θ
at
r
= 0.
We plot the initial velocity field
u
r
and
u
z
in Figure 3. We can see that
u
r
is essentially positive near
z
= 0 and negative near
z
= 1
/
2 when
r
is small, and
u
z
is mainly negative when
r
is small. Such a pattern suggests a hyperbolic flow
near (
r,z
) = (0
,
0) as depicted in the heuristic diagram Figure 4, which will extend
periodically in
z
.
Although the initial data are very smooth, the solutions of the
n
-dimensional
Euler equations quickly become very singular and concentrate in a rapidly shrinking
region. Therefore, we use the same adaptive mesh method to resolve the singular
profile of the solutions as we did in our previous paper [8]. A detailed description of
the adaptive mesh method can be found in [7, 16, 19].
In our algorithm, we adopt a second-order implementation for our adaptive mesh
method. In Section 3.3 of [8], we present resolution study to confirm the order of
accuracy of the adaptive mesh method.
3. Numerical evidence for a potential self-similar singularity.
We pro-
vide numerical evidence for the finite-time blow-up in the high-dimensional case. We
will use the setting
n
= 10,
α
= 0
.
5 in this section. More exploration of different
settings of
n
and
α
will be studied in the Section 5 and 6.
3.1. Evidence for a potential singularity.
On 1024
×
1024 spatial resolution,
we solve the equations (2.4) until
t
= 7
.
9582242
×
10
−
4
, where the solution becomes
too singular to be resolved by our numerical method. The zoomed-in profiles of
−
ω
1
,
−
ψ
1
, and the velocity fields
u
r
,
−
u
z
are shown in Figure 5. We can see that
−
ω
1
seems to primarily depend on
z
and is very flat as a function of
r
. In Figure 6, we show
the curves of important quantities of the solution. At the end of the computation,
‖
ω
1
‖
L
∞
has increased by a factor of around 6
.
5
×
10
6
, and
‖
ω
‖
L
∞
has increased by
a factor of around 515. We define (
R
1
(
t
)
,Z
1
(
t
)) to be the maximum location of
|
ω
1
|
.
As the solution approaches the potential blow-up, we observe that
R
1
(
t
) = 0, and
−
ω
1
becomes essentially one-dimensional. Notice that
Z
1
(
t
) decays very fast towards
zero, scaling roughly like (
T
−
t
)
c
with some terminal time
T
and an exponent
c >
1.
We also observe that the double logarithm curve log log
‖
ω
‖
L
∞
grows superlinear