Potential Singularity of the axisymmetric Euler equations

POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER

EQUATIONS WITH

INITIAL VORTICITY FOR A LARGE

RANGE OF

. PART I: THE

-DIMENSIONAL CASE

THOMAS Y. HOU

∗

AND

SHUMAO ZHANG

†

Abstract.

In Part I of our sequence of 2 papers, we provide numerical evidence for a potential

finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with

initial vorticity for a large range of

. We employ an adaptive mesh method using a highly effective

mesh to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution

study shows that our numerical method is at least second-order accurate. Scaling analysis and

the dynamic rescaling formulation are presented to quantitatively study the scaling properties of

the potential singularity. We demonstrate that this potential blow-up is stable with respect to the

perturbation of initial data. Our study shows that the 3D Euler equations with our initial data

develop finite-time blow-up when the H ̈older exponent

is smaller than some critical value

∗

By properly rescaling the initial data in the

-axis, this upper bound for potential blow-up

∗

can

asymptotically approach 1

3. Compared with Elgindi’s blow-up result in a similar setting [15], our

potential blow-up scenario has a different H ̈older continuity property in the initial data and the

scaling properties of the two initial data are also quite different.

Key words.

3D axisymmetric Euler equations, finite-time blow-up

AMS subject classifications.

35Q31, 76B03, 65M60, 65M06, 65M20

1. Introduction.

The three-dimensional (3D) incompressible Euler equations

in fluid dynamics describe the motion of inviscid incompressible flows and are one

of the most fundamental equations in fluid dynamics. Despite their wide range of

applications, the question regarding the global regularity of the Euler equations has

been widely recognized as a major open problem in partial differential equations

(PDEs) and is closely related to the Millennium Prize Problem on the Navier-Stokes

equations listed by the Clay Mathematics Institute [16]. In 2014, Luo and Hou [35,

36] considered the 3D axisyemmtric Euler equations with smooth initial data and

boundary, and presented strong numerical evidences that they can develop potential

finite time singularity. The presence of the boundary, the symmetry properties and

the direction of the flow in the initial data collaborate with each other in the formation

of a sustainable finite time singularity. Recently, Chen and Hou [6] provided a rigorous

justification of the Luo-Hou blow-up scenario.

In 2021, Elgindi [15] showed that given appropriate

initial vorticity with

α >

sufficiently small, the 3D axisymmetric Euler equations with no swirl can develop

finite-time singularity. In Elgindi’s work, the initial data for the vorticity

have

H ̈older continuity near

= 0 and

= 0. When

is small enough, Elgindi

approximated the 3D axisymmetric Euler equations by a fundamental model that

develops a self-similar finite-time singularity. The blow-up result obtained in [15] has

infinite energy. In a subsequent paper [13], the authors improved the result obtained

in [15] to have finite energy blow-up.

In this work we study potential finite time singularity of the 3D axisymmetric

Euler equations with no swirl and

initial vorticity for a large range of

. Define

∇×

as the vorticity vector, and then the 3D incompressible Euler equations

∗

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).

arXiv:2212.11912v1 [math.AP] 22 Dec 2022

THOMAS Y. HOU AND SHUMAO ZHANG

can be written in the vorticity stream function formulation:

(1.1)

·∇

−∇

ω,

∇×

ψ,

where

is the vector-valued stream function. Let us use

= (

) to denote a

point in

R

, and let

be the unit vectors of the cylindrical coordinate system

(

, e

(

−

, e

= (0

where

√

and

. We say a vector field

is axisymmetric if it admits

the decomposition

(

r,z

)

(

r,z

)

(

r,z

)

namely,

and

are independent of the angular variable

. Denote by

, and

the angular velocity, vorticity and stream function, respectively. The axisymmetric

condition can then simplify the 3D Euler equations (1.1) to [38]:

−

(1.2a)

(1.2b)

−

(1.2c)

−

, u

(1.2d)

In the case of no swirl, i.e.

≡

0, the axisymmetric Euler equations are further

simplified into:

(1.3a)

−

(1.3b)

−

, u

(1.3c)

When the initial condition for the angular vorticity

is smooth, it is well known that

the 3D axisymmetric Euler equations with no swirl (1.3) will not develop finite-time

blow-up [47]. Therefore, we consider (1.3) when the initial condition for the angular

vorticity

H ̈older continuous for a large range of

. By using an effective

adaptive mesh method, we will provide convincing numerical evidences that the 3D

axisymmetric Euler equations with no swirl and

initial voriticity with 0

< α <

develop potential finite-time self-similar blow-up.

We perform scaling analysis and use the dynamic rescaling formulation [21, 5, 8] to

study the behavior of the potential self-similar blow-up. An operator splitting method

is proposed to solve the dynamic rescaling formulation and the late time solution from

the adaptive mesh method is used as our initial condition for the dynamic rescaling

formulation. We observe rapid convergence to a steady state, which implies that this

potential singularity is self-similar. We will demonstrate that this potential blow-up is

POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER EQUATIONS, PART I

stable with respect to the perturbation of initial data, suggesting that the underlying

blow-up mechanism is generic and insensitive to the initial data. By introducing a

parameter

to control the stretching of the physical domain and the initial data in

the

-axis, we find that the 3D Euler equations with

initial vorticity can develop

potential finite-time blow-up when the H ̈older exponent

is smaller than some

∗

This upper bound

∗

can asymptotically approach 1

3 as

→

0. This result supports

Conjecture 8 of [13].

We choose the following

initial data for

with a stretching parameter

δ >

−

12000

(

−

)

sin(2

πδz

)

1 + 12

5 cos

(

πδz

)

Note that the initial condition is a smooth and periodic function in

and is

The velocity field

becomes

,α

continuous. We further introduce the new variables:

(

r,z

) =

(

r,z

)

, ψ

(

r,z

) =

(

r,z

)

(1.4)

to remove the formal singularity in (1.3) near

= 0. In terms of the new variables

(

,ψ

), the 3D axisymmetric Euler equations with no swirl have the following equiv-

alent form

−

)

(1.5a)

−

,rr

−

,zz

−

(1.5b)

−

rψ

, u

= 2

rψ

(1.5c)

The above reformulation is crucial for us to perform accurate numerical computation

of the potential singular solution and allow us to push the computation sufficiently

close to the singularity time.

It is important to note that the initial condition for the rescaled vorticity field

is a smooth function of

and

. Using the above reformualtion enables us to resolve

the potential singular solution sufficiently close to the potential singularity time. If

we solve the original 3D Euler equations (1.3a)–(1.3c), it is extremely difficult to

resolve the H ̈older continuous vorticity even with an adaptive mesh, especially for

small

. For this reason, we have not been able to compute the finite time singularity

in Elgindi’s work [15] since such a reformulation is not available for his initial data.

Compared with Elgindi’s blow-up result [15], our potential blow-up scenario has

very different scaling properties. The scaling factor

in our scenario increases with

and tends to infinity as

approaches

∗

. In contrast, the scaling factor

in Elgindi’s

scenario is 1

/α

, which decreases with

and tends to infinity as

approaches 0.

Another difference is that Elgindi’s initial vorticity is

in both

√

and

, while our initial vorticity is

,α

continuous in

, but smooth in

There has been a number of theoretical analysis of the 3D Euler equations. The

Beale-Kato-Majda (BKM) blow-up criterion [2, 17] gives a necessary and sufficient

condition for the finite-time singularity for the smooth solutions of the 3D Euler

equations at time

if and only if

∫

‖

(

)

‖

∞

= +

∞

. In [10], Constantin,

Fefferman and Majda asserted that there will be no finite-time blow-up if the velocity

is uniformly bounded and the direction of vorticity

ω/

is sufficiently regular

(Lipschitz continuous) in an

(1) domain containing the location of the maximum

vorticity. Inspired by the work of [10], Deng-Hou-Yu developed a more localized

non-blow-up criterion using a Lagrangian approach in [12].

THOMAS Y. HOU AND SHUMAO ZHANG

There have been a number of numerical attempts in search of the potential finite-

time blow-up. The finite-time blow-up in the numerical study was first reported by

Grauer and Sideris [19] and Pumir and Siggia [41] for the 3D axisymmetric Euler

equations. However, the later work of E and Shu [14] suggested that the finite-time

blow-up in [19, 41] could be caused by numerical artifact. Kerr and his collaborators

[24, 3] presented finite-time singularity formation in the Euler flows generated by a

pair of perturbed anti-parallel vortex tubes. In [22], Hou and Li reproduced Kerr’s

computation using a similar initial condition with much higher resolutions and did

not observe finite time blow-up. The maximum vorticity grows slightly slower than

double exponential in time. Later on, Kerr confirmed in [25] that the solutions from

[24] eventually converge to a super-exponential growth and are unlikely to lead to a

finite-time singularity.

In [4, 45], Caflisch and his collaborators studied axisymmetric Euler flows with

complex initial data and reported singularity formation in the complex plane. The

review paper [18] lists a more comprehensive collection of interesting numerical results

with more detailed discussions.

Due to the lack of stable structure in the potentially singular solutions, the pre-

viously mentioned numerical results remain inconclusive. In [35, 36], Luo and Hou

reported that the 3D axisymmetric Euler equations with a smooth initial condition

developed a self-similar finite time blow-up in the meridian plane on the boundary

= 1, see also [37]. The Hou-Luo blow-up scenario has generated a great deal of

interests in both the mathematics and fluid dynamics communities, and inspired a

number of subsequent developments [28, 27, 26, 9, 7, 5, 8, 6].

The rest of this paper is organized as follows. In Section 2, we briefly introduce

the numerical method. We present the evidence of the potential self-similar blow-

up in Section 3, and provide the resolution study and scaling analysis. In Section

4 we use the dynamic rescaling method to provide further evidence of the potential

blow-up for the case of

= 0

1. In Section 5, we consider the potential finite-time

blow-up in the general case of the H ̈older exponent

, and introduce the anisotropic

scaling parameter

. The sensitivity of the potential blow-up to the initial data is

considered in Section 6, and the comparison of our potential blow-up scenario with

Elgindi’s scenario in [15] is discussed in Section 7. Some concluding remarks are made

in Section 8.

2. Problem set up and numerical method.

In this section, we give details

about the setup of the problem, the initial data, the boundary conditions, and some

basic properties of the equations, and our numerical method.

2.1. Boundary conditions and symmetries.

We consider (1.5) in a cylinder

region

cyl

{

(

r,z

) : 0

≤

}

We impose a periodic boundary condition in

with period 1:

(

r,z

) =

(

r,z

+ 1)

, ψ

(

r,z

) =

(

r,z

+ 1)

(2.1)

In addition, we enforce that (

,ψ

) are odd in

= 0:

(

r,z

) =

−

(

−

)

, ψ

(

r,z

) =

−

(

−

)

(2.2)

And this symmetry will be preserved dynamically by the 3D Euler equations.

= 0, it is easy to see that

) = 0, so there is no need for the boundary

condition for

= 0. Since

rψ

will at least be

-continuous, according

POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER EQUATIONS, PART I

to [32, 33],

must be an odd function of

. Therefore, we impose the following pole

condition for

) = 0

(2.3)

We impose the no-flow boundary condition at the boundary

= 1:

) = 0

(2.4)

This implies that

) = 0. So there is no need to introduce a boundary condition

for

= 1.

Due to the periodicity and the odd symmetry along the

direction, the equations

(1.3) only need to be solved on the half-periodic cylinder

{

(

r,z

) : 0

≤

}

The above boundary conditions of

show that there is no transport of the flow across

its boundaries. Indeed, we have

= 0

= 0 or 1

and

= 0

= 0 or 1

Thus, the boundaries of

behave like “impermeable walls”.

Fig. 1

3D profiles of the initial value

−

◦

1

and

−

◦

1

Fig. 2

The initial data for the angular vorticity

2.2. Initial data.

Inspired by the potential blow-up scenario in [20], we propose

the following initial data for

◦

−

12000

(

−

)

sin(2

πz

)

1 + 12

5 cos

(

πz

)

(2.5)

THOMAS Y. HOU AND SHUMAO ZHANG

Later we will see in Section 6 that the self-similar singularity formation has some

robustness to the choice of initial data. We solve the Poisson equation (1.5b) to get

the initial value

◦

Fig. 3

Initial velocity fields

and

Fig. 4

A heuristic diagram of the hyperbolic flow.

The 3D profiles of (

◦

,ψ

◦

) can be found in Figure 1. Since most parts of

◦

and

◦

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

= 0. We can see that there is a sharp drop to zero of

−

near

= 0, which is due

to the H ̈older continuous of

= 0.

We plot the initial velocity field

and

in Figure 3. We can see that

is primarily positive near

= 0 and negative near

= 1

2 when

is small, and

is mainly negative when

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

2.3. Self-similar solution.

For nonlinear PDEs, people are particularly inter-

ested in studying self-similar blow-up solutions. 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 (1.5), the

POTENTIAL SINGULARITY OF THE AXISYMMETRIC EULER EQUATIONS, PART I

self-similar profile is the ansatz

(2.6)

(

x,t

)

≈

(

−

)

Ω

(

−

(

−

)

(

x,t

)

≈

(

−

)

(

−

(

−

)

for some parameters

and

. Here

is considered as the blow-up time,

and

is the location of the self-similar blow-up. The parameters

are called

scaling factors.

It is also important to notice that the 3D Euler equations (1.1) enjoy the following

scaling invariant property: if (

u,ω,ψ

) is a solution to (1.1), then (

λ,τ

,ω

λ,τ

,ψ

λ,τ

) is

also a solution, where

λ,τ

(

x,t

) =

(

)

, ω

λ,τ

(

x,t

) =

(

)

, ψ

λ,τ

(

x,t

) =

(

)

and

λ >

τ >

0 are two constant scaling factors. In the case of the axisymmetric 3D

Euler equations with no swirl (1.5), the scaling invariant property can be equivalently

translated to: if (

,ψ

) is a solution of (1.5), then

{

(

)

(

)}

(2.7)

is also a solution.

If we assume the existence of the self-similar solution (2.6), then the new solutions

in (2.7) should also admit the same ansatz, regardless of the values of

and

. As a

result, we must have

= 1 +

αc

, c

= 1

−

(2.8)

Therefore, the self-similar profile (2.7) of (1.5) only has one degree of freedom, for

example

, in the scaling factors. In fact,

cannot be determined by straightforward

dimensional analysis.

As a consequence of the ansatz (2.6) and the scaling relation (2.8), we have

‖

(

x,t

)

‖

∞

∼

−

‖

(

x,t

)

‖

∞

∼

−

(2.9)

which should always hold true regardless of the value of

2.4. Numerical method.

Although the initial data are very smooth, the so-

lutions of Euler equations quickly become very singular and concentrate in a rapidly

shrinking region. Therefore, we use the adaptive mesh method to resolve the singular

profile of the solutions. A detailed description of the adaptive mesh method can be

found in [21, 37, 48]. Here we briefly introduce the idea behind the adaptive mesh

method. The specific parameter setting used for the experiments in this work can be

found in the appendix of [48].

The Euler equations (1.5) are originally posted as an initial-boundary value prob-

lem on the computational domain (

r,z

)

∈

2]. To capture the singular

part of the solution, we introduce two variables (

ρ,η

)

∈

1], and the maps

(

)

, z

(

)