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MULTISCALE MODEL. SIMUL.
c
2014 Society for Industrial and Applied Mathematics
Vol. 12, No. 4, pp. 1722–1776
TOWARD THE FINITE-TIME BLOWUP OF THE 3D
AXISYMMETRIC EULER EQUATIONS: A NUMERICAL
INVESTIGATION
∗
GUO LUO
†
AND THOMAS Y. HOU
‡
Abstract. Whether the three-dimensional incompressible Euler equations can develop a singu-
larity in finite time from smooth initial data is one of the most challenging problems in mathematical
fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open ques-
tion from a numerical point of view by presenting a class of potentially singular solutions to the Euler
equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condi-
tion along the axial direction and a no-flow boundary condition on the solid wall. The equations
are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on
specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions.
With a maximum effective resolution of over (3 × 10
12
)
2
near the point of the singularity, we are
able to advance the solution up to τ
2
=0.003505 and predict a singularity time of t
s
≈ 0.0035056,
while achieving a pointwise relative error of O(10
−4
) in the vorticity vector ω and observing a
(3 × 10
8
)-fold increase in the maximum vorticity ω
∞
. The numerical data are checked against
all major blowup/non-blowup criteria, including Beale–Kato–Majda, Constantin–Fefferman–Majda,
and Deng–Hou–Yu, to confirm the validity of the singularity. A local analysis near the point of the
singularity also suggests the existence of a self-similar blowup in the meridian plane.
Key words. 3D axisymmetric Euler equations, finite-time blowup
AMS subject classifications. 35Q31, 76B03, 65M60, 65M06, 65M20
DOI. 10.1137/140966411
1. Introduction. The celebrated three-dimensional (3D) incompressible Euler
equations in fluid dynamics describe the motion of ideal incompressible flows in the
absence of external forcing. First written down by Leonhard Euler in 1757, these
equations have the form
(1.1) u
t
+ u ·∇u = −∇p, ∇·u =0,
where u =(u
1
,u
2
,u
3
)
T
is the 3D velocity vector of the fluid and p is the scalar pres-
sure. The 3D Euler equations have a rich mathematical theory, for which the inter-
ested readers may consult the excellent surveys [2, 18, 24] and the references therein.
This paper primarily concerns the existence or nonexistence of globally regular solu-
tions to the 3D Euler equations, which is regarded as one of the most fundamental
yet most challenging problems in mathematical fluid dynamics.
The interest in the global regularity or finite-time blowup of (1.1) comes from
several directions. Mathematically, the question has remained open for over 250 years
and has a close connection to the Clay Millennium Prize Problem on the Navier–
Stokes equations. Physically, the formation of a singularity in inviscid (Euler) flows
may signify the onset of turbulence in viscous (Navier–Stokes) flows, and it may
∗
Received by the editors April 24, 2014; accepted for publication (in revised form) August 15,
2014; published electronically November 18, 2014.
http://www.siam.org/journals/mms/12-4/96641.html
†
Applied & Computational Mathematics, California Institute of Technology, MC 9-94, Pasadena,
CA 91125. Current address: Department of Mathematics, City University of Hong Kong, Kowloon
Tong, Hong Kong (gluo@caltech.edu).
‡
Applied & Computational Mathematics, California Institute of Technology, MC 9-94, Pasadena,
CA 91125 (hou@acm.caltech.edu).
1722
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FINITE-TIME SINGULARITY OF 3D EULER 1723
provide a mechanism for energy transfer to small scales. Numerically, the resolution
of nearly singular flows requires special numerical treatment, which presents a great
challenge to computational fluid dynamicists.
Considerable efforts have been devoted to the study of the regularity properties of
the 3D Euler equations. The main difficulty in the analysis lies in the presence of the
nonlinear vortex stretching term and the lack of a regularization mechanism, which
implies that even the local well-posedness of the equations can only be established
for sufficiently smooth initial data (see, for example, [37]). Despite these difficulties,
a few important partial results [3, 44, 22, 46, 19, 20, 25] have been obtained over the
years which have led to improved understanding of the regularity properties of the 3D
Euler. More specifically, the celebrated theorem of Beale, Kato, and Majda [3] and
its variants [22, 46] characterize the regularity of the 3D Euler equations in terms of
the maximum vorticity, asserting that a smooth solution u of (1.1) blows up at t = T
if and only if
T
0
ω(·,t)
L
∞
dt = ∞,
where ω = ∇×u is the vorticity vector of the fluid. The non-blowup criterion of
Constantin, Fefferman, and Majda [19] focuses on the geometric aspects of Euler flows
instead and asserts that there can be no blowup if the velocity field u is uniformly
bounded and the vorticity direction ξ = ω/|ω| is sufficiently “well behaved” near the
point of the maximum vorticity. The theorem of Deng, Hou, and Yu [20] is similar
in spirit to the Constantin–Fefferman–Majda criterion but confines the analysis to
localized vortex line segments.
Besides the analytical results mentioned above, there also exists a sizable litera-
ture focusing on the (numerical) search of a finite-time singularity for the 3D Euler
equations. Representative work in this direction include [27, 45], which studied Euler
flows with swirls in axisymmetric geometries, the famous computation of Kerr and his
collaborators [38, 8, 39], which studied Euler flows generated by a pair of perturbed
antiparallel vortex tubes, and the viscous simulations of [5], which studied the 3D
Navier–Stokes equations using Kida’s high-symmetry initial data. Other interesting
pieces of work are [10, 47], which studied axisymmetric Euler flows with complex ini-
tial data and reported singularities in the complex plane. A more comprehensive list
of interesting numerical results can be found in the review article [24].
Although finite-time singularities were frequently reported in numerical simula-
tions of the Euler equations, most such singularities turned out to be either numerical
artifacts or false predictions, as a result of either insufficient resolution or inadver-
tent data analysis procedure (more to follow on this topic in section 4.4). Indeed, by
exploiting the analogy between the two-dimensional (2D) Boussinesq equations and
the 3D axisymmetric Euler equations away from the symmetry axis, E and Shu [21]
studied the potential development of finite-time singularities in the 2D Boussinesq
equations, with initial data completely analogous to those of [27, 45]. They found no
evidence for singular solutions, indicating that the “blowups” reported by [27, 45],
which were located away from the axis, are likely to be numerical artifacts. Likewise,
Hou and Li [32] repeated the computation of [38] with higher resolutions, in an at-
tempt to reproduce the singularity observed in that study. Despite some ambiguity in
interpreting the initial data used by [38], they managed to advance the solution up to
t = 19, which is beyond the singularity time T =18.7 alleged by [38]. By using newly
developedanalytictoolsbasedonrescaledvorticitymoments,Kerralsoconfirmedin
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1724 GUO LUO AND THOMAS Y. HOU
a very recent study [39] that solutions computed from initial data analogous to that
used in [38] eventually converge to superexponential growth and hence are unlikely
to lead to a singularity. In a later work, Hou and Li [33] also repeated the computa-
tion of [5] and found that the singularity reported by [5] is likely an artifact due to
insufficient resolution. . . . To summarize, the existing numerical studies do not seem
to provide convincing evidence to support the existence of a finite-time singularity,
and the question of whether initially smooth solutions to (1.1) can blow up in finite
time remains open.
By focusing on solutions with axial symmetry and other special properties, we
have discovered, through careful numerical studies, a class of potentially singular
solutions to the 3D axisymmetric Euler equations in a radially bounded, axially pe-
riodic cylinder (see (2.1)–(2.2) below). The reduced computational complexity in the
cylindrical geometry greatly facilitates the computation of the singularity. With a
specially designed adaptive mesh, we are able to achieve a maximum mesh resolution
of over (3 × 10
12
)
2
near the point of the singularity. This allows us to compute the
vorticity vector with four digits of accuracy throughout the simulations and to ob-
serve a (3 × 10
8
)-fold amplification in maximum vorticity. The numerical data are
checked against all major blowup/non-blowup criteria, including Beale–Kato–Majda,
Constantin–Fefferman–Majda, and Deng–Hou–Yu, to confirm the validity of the sin-
gularity. A careful local analysis also suggests the existence of a self-similar blowup
in the meridian plane. Our numerical method makes explicit use of the special sym-
metries built in the blowing-up solutions, which eliminates symmetry-breaking per-
turbations and facilitates a stable computation of the singularity.
The main features of the potentially singular solutions are summarized as follows.
The point of the potential singularity, which is also the point of the maximum vorticity,
is always located at the intersection of the solid boundary r = 1 and the symmetry
plane z =0. Itisastagnation point of the flow, as a result of the special odd-even
symmetries along the axial direction and the no-flow boundary condition (see (2.3)).
The vanishing velocity field at this point could have positively contributed to the
formation of the singularity, given the potential regularizing effect of convection as
observed by [33, 31]. When viewed in the meridian plane, the point of the potential
singularity is a hyperbolic saddle of the flow, where the axial flow along the solid
boundary marches toward the symmetry plane z = 0 and the radial flow marches
toward the symmetry axis r = 0 (see Figure 16(a)). The axial flow brings together
vortex lines near the solid boundary r = 1 and destroys the geometric regularity
of the vorticity vector near the symmetry plane z = 0, violating the Constantin–
Fefferman-Majda and Deng–Hou–Yu geometric non-blowup criteria and leading to
the breakdown of the smooth vorticity field.
The asymptotic scalings of the various quantities involved in the potential finite-
time blowup are summarized as follows. Near the predicted singularity time t
s
,the
scalar pressure and the velocity field remain uniformly bounded while the maximum
vorticity blows up like O(t
s
− t)
−γ
,whereγ roughly equals
5
2
. Near the point of the
potential singularity, namely the point of the maximum vorticity, the radial and axial
components of the vorticity vector grow roughly like O(t
s
− t)
−5/2
while the angular
vorticity grows like O(t
s
−t)
−1
. The nearly singular solution has a locally self-similar
structure in the meridian plane near the point of blowup, with a rapidly collapsing
support scaling roughly like O(t
s
−t)
3
along both the radial and the axial directions.
When viewed in R
3
, this corresponds to a thin tube on the symmetry plane z =0
evolved around the ring r = 1, where the radius of the tube shrinks to zero as the
singularity forms.
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FINITE-TIME SINGULARITY OF 3D EULER 1725
We emphasize that the 3D axisymmetric Euler equations (2.1) are different from
their free-space counterpart (1.1) in that they have a constant of motion that is not
present in the nonsymmetric case [41]. In addition, it is well known that the choice
of the boundary conditions (periodic vs. no-flow) has a nontrivial impact on the
qualitative behavior of the solutions of the Euler equations, especially near the solid
boundaries [2, 18]. In view of these differences and the fact that the singularity we
discover lies right on the boundary, we stress that the work described in this paper
is not directly relevant to the Clay Millennium Prize Problem on the Navier–Stokes
equations, which is posed either in free space or on periodic domains.
1
Rather, it
should be viewed as an attempt at the understanding of the effect of solid bound-
aries in the creation of small scales and, in the case of zero viscosity, the creation of
singularities in incompressible flows.
The rest of this paper is devoted to the study of the potential finite-time singu-
larity and is organized as follows. Section 2 contains a brief review of the 3D Euler
equations in axisymmetric form and defines the problem to be studied. Section 3
gives a brief description of the numerical method that is used to track and resolve the
nearly singular solutions. Section 4 examines the numerical data in great detail and
presents evidence supporting the existence of a finite-time singularity. Finally section
5 concludes the paper with a brief discussion on future research directions.
2. Description of the problem. The 3D Euler equations (1.1) with axial sym-
metry can be conveniently described in the so-called stream-vorticity form. To derive
these equations, recall first that in cylindrical coordinates (r, θ, z), an axisymmetric
flow u can be described by the decomposition
u(r, z)=u
r
(r, z) e
r
+ u
θ
(r, z) e
θ
+ u
z
(r, z) e
z
,
where e
r
=(cosθ, sin θ, 0)
T
,e
θ
=(−sin θ, cos θ, 0)
T
,ande
z
=(0, 0, 1)
T
are coordi-
nate axes. The vorticity vector ω = ∇×u has a similar representation,
ω(r, z)=ω
r
(r, z) e
r
+ ω
θ
(r, z) e
θ
+ ω
z
(r, z) e
z
,
ω
r
= −u
θ
z
,ω
θ
= u
r
z
− u
z
r
,ω
z
=
1
r
(ru
θ
)
r
,
where for simplicity we have used subscripts to denote partial differentiations. The
incompressibility condition ∇·u = 0 implies the existence of a stream function
ψ(r, z)=ψ
r
(r, z) e
r
+ ψ
θ
(r, z) e
θ
+ ψ
z
(r, z) e
z
,
for which u = ∇×ψ and ω = −Δψ.Takingtheθ-components of the velocity equation
(1.1), the vorticity equation
ω
t
+ u ·∇ω = ω ·∇u,
and the Poisson equation −Δψ = ω gives an alternative formulation of the 3D Euler
equations
u
1,t
+ u
r
u
1,r
+ u
z
u
1,z
=2u
1
ψ
1,z
,(2.1a)
ω
1,t
+ u
r
ω
1,r
+ u
z
ω
1,z
=(u
2
1
)
z
,(2.1b)
−
∂
2
r
+(3/r)∂
r
+ ∂
2
z
ψ
1
= ω
1
,(2.1c)
1
Indeed, according to the partial regularity result of Caffarelli, Kohn, and Nirenberg [9], any
finite-time singularity of the 3D axisymmetric Navier–Stokes equations, if it exists, must lie on the
symmetry axis.
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1726 GUO LUO AND THOMAS Y. HOU
where u
1
= u
θ
/r, ω
1
= ω
θ
/r, and ψ
1
= ψ
θ
/r are transformed angular velocity,
vorticity, and stream functions, respectively.
2
The radial and axial components of the
velocity can be recovered from ψ
1
as
(2.1d) u
r
= −rψ
1,z
,u
z
=2ψ
1
+ rψ
1,r
,
for which the incompressibility condition
1
r
(ru
r
)
r
+ u
z
z
=0
is satisfied automatically. As shown by [40], (u
θ
,ω
θ
,ψ
θ
) must all vanish at r =0ifu is
a smooth velocity field. Thus (u
1
,ω
1
,ψ
1
) are well defined as long as the corresponding
solution to (1.1) remains smooth. The reason we choose to work with the transformed
variables (u
1
,ω
1
,ψ
1
) instead of the original variables (u
θ
,ω
θ
,ψ
θ
) is that the equations
satisfied by the latter,
u
θ
t
+ u
r
u
θ
r
+ u
z
u
θ
z
= −
1
r
u
r
u
θ
,
ω
θ
t
+ u
r
ω
θ
r
+ u
z
ω
θ
z
=
2
r
u
θ
u
θ
z
+
1
r
u
r
ω
θ
,
−
Δ −(1/r
2
)
ψ
θ
= ω
θ
,
have a formal singularity at r = 0, which is inconvenient to work with numerically.
We shall numerically solve the transformed equations (2.1) on the cylinder
D(1,L)=
(r, z): 0 ≤ r ≤ 1, 0 ≤ z ≤ L
,
with the initial condition
(2.2a) u
0
1
(r, z) = 100 e
−30(1−r
2
)
4
sin
2π
L
z
,ω
0
1
(r, z)=ψ
0
1
(r, z)=0.
The solution is subject to a periodic boundary condition in z,
(2.2b) u
1
(r, 0,t)=u
1
(r, L, t),ω
1
(r, 0,t)=ω
1
(r, L, t),ψ
1
(r, 0,t)=ψ
1
(r, L, t),
and a no-flow boundary condition on the solid boundary r =1:
(2.2c) ψ
1
(1,z,t)=0.
The pole condition
(2.2d) u
1,r
(0,z,t)=ω
1,r
(0,z,t)=ψ
1,r
(0,z,t)=0
is also enforced at the symmetry axis r = 0 to ensure the smoothness of the solution.
The initial condition (2.2a) describes a purely rotating eddy in a periodic cylinder,
and it satisfies special odd-even symmetries at the planes z
i
=
i
4
L, i =0, 1, 2, 3.
Specifically, u
0
1
is even at z
1
,z
3
,itisoddatz
0
,z
2
,andω
0
1
,ψ
0
1
are both odd at all
z
i
’s. These symmetry properties are preserved by the equations (2.1), so instead of
solving the problem (2.1)–(2.2) on the entire cylinder D(1,L), it suffices to consider
the problem on the quarter cylinder D(1,
1
4
L), with the periodic boundary condition
(2.2b) replaced by appropriate symmetry boundary conditions. It is also interesting
to notice that the boundaries of D(1,
1
4
L) behave like “impermeable walls”:
(2.3) u
r
= −rψ
1,z
=0 on r =1,u
z
=2ψ
1
+ rψ
1,r
=0 on z =0,
1
4
L,
which is a consequence of the no-flow boundary condition (2.2c) and the odd symmetry
of ψ
1
.
2
These variables should not be confused with the components of the velocity, vorticity, and
stream function vectors.
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