On the regularity of axisymmetric olutions to the Navier–Stokes equations

Abstract

We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations
with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in
terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these
criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations.

Full text

Digital Object Identifier (DOI) 10.1007/s002090100317
Math. Z. 239, 645–671 (2002)
On the regularity of the axisymmetric solutions
of the Navier-Stokes equations
Dongho Chae, Jihoon Lee
Department of Mathematics, Seoul National University, Seoul 151-747, Korea
(e-mail : {dhchae,zhlee}@math.snu.ac.kr)
Received: 11 April 2000; in final firm: 26 November 2000 /
Published online: 28 February 2002 –
c
 Springer-Verlag 2002
Abstract. Weobtainimprovedregularitycriteriafortheaxisymmetricweak
solutions of the three dimensional Navier-Stokes equations with nonzero
swirl. In particular we prove that the integrability of single component of
vorticity or velocity fields, in terms of norms with zero scaling dimension
give sufficient conditions for the regularity of weak solutions. To obtain
these criteria we derive new a priori estimates for the axisymmetric smooth
solutions of the Navier-Stokes equations.
1 Introduction
In this paper, we are concerned with the initial value problem of the Navier-
Stokes equations in R
3
× [0,T).
∂u
∂t
+(u ·∇)u − ∆u = −∇p, (1)
div u =0, (2)
u(x, 0) = u
0
(x), (3)
where u =(u
1
,u
2
,u
3
) with u = u(x, t), and p = p(x, t) denote unknown
fluid velocity and scalar pressure, respectively, while u
0
is a given initial
velocity satisfying
div u
0
=0. In the above we denote
(u ·∇)u =
3

j=1
u
j
∂u
∂x
j
,∆u=
3

j=1
∂
2
u
∂x
2
j
.

646 D. Chae, J. Lee
For simplicity we assume that there is no external force term in the right
hand side of (1). Inclusion of the external force does not provide essential
difficulty in our works of this paper. We recall that the Leray-Hopf weak
solution of the Navier-Stokes equations is defined as a vector field u ∈
L
∞
(0,T; L
2
(R
3
)) ∩L
2
(0,T; H
1
(R
3
)) satisfying div u =0in the sense of
distribution, the energy inequality
u(t)
2
2
+2

t
0
∇u(s)
2
2
ds ≤u
0

2
2
a.e. t ∈ [0,T]
and

T
0

R
3
(u ·φ
t
+(u ·∇)φ ·u + u ·∆φ)dxdt +

R
3
u
0
(x) ·φ(x, 0)dx =0,
for all φ ∈ C
∞
0
(R
3
× [0,T))
3
with div φ =0.
For given u
0
∈ L
2
(R
3
) with div u
0
=0in the sense of distribution, a
global weak solution u to the Navier-Stokes equations was constructed by
Leray[14] and Hopf[10], which we call the Leray-Hopf weak solution. For
review on the theory of the Leray-Hopf weak solution see [7] and [25].
RegularityofsuchLeray-Hopf weaksolutionsisoneofthemostoutstanding
open problems in the mathematical fluid mechanics. For further discussion
below we introduce the Banach space L
α,γ
T
, equipped with the norm
u
L
α,γ
T
=


T
0
u(t)
α
γ
dt

1
α
,
where
u(t)
γ
=



R
3
|u(x, t)|
γ
dx

1
γ
if 1 ≤ γ<∞
esssup
x∈R
3
|u(x, t)| if γ = ∞
.
In particular, we use the same norm for scalar function u = u(x) and the
vector function u(x)=(u
1
(x),u
2
(x),u
3
(x)). Taking the curl operation of
(1), we obtain the evolution equation of the vorticity ω = curl u,
∂ω
∂t
− ∆ω +(u ·∇)ω − (ω ·∇)u =0. (4)
As an approach to the regularity problem Serrin[22](See also[20]) studied
the regularity criterion of the Leray-Hopf weak solutions, and obtained that
ifaweaksolutionu belongstoL
α,γ
T
where
2
α
+
3
γ
< 1,2 <α≤∞,3 <γ≤
∞,thenu isregular,andbecomesasmoothinspacevariableson(0,T].After
Serrin’s work, there are many improvements and developments regarding
the study of regularity criterion.(See e.g. [8], [9], [11], and [24].) It is found,
in particular, the Leray-Hopf weak solution becomes smooth in (x, t) if u ∈
L
α,γ
T
with
2
α
+
3
γ
≤ 1, 2 ≤ α ≤∞, 3 <γ≤∞. On the other hand, Beir
˜
ao

Axisymmetric solutions of the Navier-Stokes equations 647
da Veiga[1] obtained the regularity criterion by imposing the integrability
of the gradient of the velocity. Recently, one of the authors[3] improved
Beir
˜
ao da Veiga’s result by imposing the same integrability condition on
two componentsof the vorticity. Very Recently, Neustupa et al[18] obtained
regularitycriterionbyimposingintegrabilityofsinglecomponentofvelocity
field. The integrability condition here, however, is stronger than the Serrin’s
one, and is not optimal in the sense of scaling considerations. Moreover, the
weak solution concerned in [18] is not the Leray-Hopf weak solutions, but
the so called suitable weak solutions introduced first in [2].
We are concerned here with the regularity criteria of axisymmetric weak
solutions of the Navier-Stokes equations. By an axisymmetric solution of
the Navier-Stokesequations we mean a solutionof theequations of the form
u(x, t)=u
r
(r, x
3
,t)e
r
+ u
θ
(r, x
3
,t)e
θ
+ u
3
(r, x
3
,t)e
3
in the cylindrical coordinate system, where we used the basis
e
r
=(
x
1
r
,
x
2
r
, 0),e
θ
=(−
x
2
r
,
x
1
r
, 0),e
3
=(0, 0, 1),r=

x
2
1
+ x
2
2
.
In the above u
θ
is called the swirl component of the velocity u. For the
axisymmetric solutions, we can rewrite the equation (1) and (2) as follows.
˜
Du
r
Dt
− (∂
2
r
+ ∂
2
3
+
1
r
∂
r
)u
r
+
1
r
2
u
r
−
1
r
u
θ
u
θ
+ ∂
r
p =0,
˜
Du
θ
Dt
− (∂
2
r
+ ∂
2
3
+
1
r
∂
r
)u
θ
+
1
r
2
u
θ
+
1
r
u
θ
u
r
=0,
˜
Du
3
Dt
− (∂
2
r
+ ∂
2
3
+
1
r
∂
r
)u
3
+ ∂
3
p =0,
∂
r
(ru
r
)+∂
3
(ru
3
)=0,
where we denote
˜
D
Dt
=
∂
∂t
+ u
r
∂
r
+ u
3
∂
3
.
In the following, we will also use the notation for the axisymmetric vector
field u
˜u = u
r
e
r
+ u
3
e
3
,
and
˜
∇ =(∂
r
,∂
3
).
Fortheaxisymmetricvectorfieldu,wecancomputethevorticityω = curl u
as follows.
ω = ω
r
e
r
+ ω
θ
e
θ
+ ω
3
e
3
,

648 D. Chae, J. Lee
where
ω
r
= −∂
3
u
θ
,ω
3
= ∂
r
u
θ
+
u
θ
r
,ω
θ
= −∂
r
u
3
+ ∂
3
u
r
.
ForthestudyofaxisymmetricsolutionsoftheNavier-Stokesequationswith-
out swirl, Ukhovskii and Yudovich [26] ,and independently Ladyzhenskaya
[12] proved the existence of generalized solutions, uniqueness and the reg-
ularity. Recently, Leonardi, M
´
alek, Ne
ˇ
cas and Pokorn
´
y [13] gave a refined
proof. In the related case of helical symmetry, Mahalov, Titi and Leibovich
[15]provedtheglobal existenceof thestrongsolution. Forthe axisymmetric
Navier-Stokes equations with nonzero swirl component, however, the regu-
larityproblem isstill open.Forthe axisymmetricEulerequations withswirl,
theazimuthal componentof the vorticityin thecylindricalcoordinates alone
controls the blow-up of the velocity( See [5] and for the other studies on the
axisymmetric solutions of the Euler equations, see [4], [16], [17], [21], [26]
and references therein.). Our main results in this paper are the followings.
Theorem 1 Let u be an axisymmetric weak solution of the Navier-Stokes
equations with u
0
∈ H
2
(Ω) div u
0
=0and Q
T
= Ω × [0,T), where Ω
is bounded domain or R
3
.Ifω
θ
is in L
α,γ
T
(Q
T
) where α and γ satisfies
3
2
<γ<∞, 1 <α≤∞and
2
α
+
3
γ
≤ 2, then the weak solution u is
smooth in Q

T
= Ω

× (0,T) where Ω

⊂⊂ Ω.
Remark 1. Comparing with the result in [3], we find that Theorem 1 is an
obvious improvement of the corresponding criterion theorem in [3], which
is, in turn, improvement of [1] for the axisymmetric case.
The following theorem provides us the available a priori estimate for ω
θ
,
whichisnewfortheaxisymmetric solutionstotheknowledgeoftheauthors.
Theorem 2 If u is an axisymmetric smooth solution of the Navier-Stokes
equations with initial data u
0
∈ L
2
(R
3
) with div u
0
=0satisfying r
3
ω
θ
0
∈
L
2
(R
3
) and ru
θ
0
∈ L
4
(R
3
), then r
3
ω
θ
∈ L
∞,2
T
∩ L
2
(0,T; H
1
(R
3
)).
For a given axisymmetric function f on R
3
, we define the
˜
L
γ
norm by
f
˜
L
γ
=


∞
−∞

∞
0
|f(r, x
3
)|
γ
drdx
3

1
γ
.
Theorem 3 Let u be an axisymmetric weak solution of the Navier-Stokes
equations with the initial data u
0
satisfying ru
θ
0
∈ L
4
(R
3
), r
3
ω
θ
0
∈ L
2
(R
3
)
and u
0
∈ H
2
(R
3
).
(i) If u
r
and u
θ
is in L
α
(0,T; L
γ
(drdx
3
)) where 2 <γ<∞, 2 <α≤∞
and
1
α
+
1
γ
≤
1
2
, then the solution is smooth in R
3
× (0,T).

Axisymmetric solutions of the Navier-Stokes equations 649
(ii) If ω
θ
is in L
α
(0,T; L
γ
(drdx
3
)) where 1 <γ<∞, 1 <α≤∞and
1
α
+
1
γ
≤ 1, then the solution is smooth in R
3
× (0,T).
Remark 2. One of the interesting consequence of Theorem 3 (ii) is that
if ω
θ
∈ L
2
(0,T; L
2
(drdx
3
)), then the solution becomes smooth. By the
energy inequality for weak solutions, however, we know that ω ∈ L
2
(0,T;
L
2
(rdrdx
3
)) for the Leray-Hopf weak solutions. Thus there is a discrep-
ancy between the availableestimate and the criterion in the region only near
the axis of symmetry.
We note thatthe normu
L
α
(0,T ;L
γ
(drdx
3
))
haszero scalingdimension if
1
α
+
1
γ
=
1
2
, andω
L
α
(0,T ;L
γ
(drdx
3
))
has zeroscaling dimension if
1
α
+
1
γ
=
1. The above criteria are optimal in this sense.
As an immediate corollary of Theorem 3 (ii), we can reprove the following
result on the regularity of axisymmetric weak solutions in the case without
swirl, which was obtained previously by Ukhovski and Yudovich [26] and
Leonardi et al. [13] using different argument.
Corollary 1 Suppose u
0
is an axisymmetric initial data without swirl, sat-
isfying the hypothesis of Theorem 3. Then there exists a smooth solution on
R
3
× (0,T).
Next,we obtain regularitycriteriain termsof a singlecomponent ofvelocity
field.
Theorem 4 Let δ>0 be given, and set Γ
δ
= {x ∈ R
3
| r<δ}. Suppose
that u is an axisymmetric weak solution, and satisfies the hypothesis of
Theorem 3 and one of the following conditions.
Either
(i)
u
r
r
∈ L
α
(0,T; L
γ
(Γ
δ
)) where α and γ satisfy
3
2
<γ<∞, 1 <α≤
∞, and
2
α
+
3
γ
≤ 2,or
(ii) u
r
∈ L
α
(0,T; L
γ
(Γ
δ
)) where α and γ satisfy 3 <γ<∞, 2 <α≤
∞, and
2
α
+
3
γ
≤ 1.
Then u is smooth in R
3
× (0,T).
Remark 3. Comparing with the result in [18] on the regularity criterion
in terms of single component of velocity for the general case(without as-
sumption of any symmetry of solutions), we find that in Theorem 4 the
integrability assumption for the single component is much weaker, and op-
timal in the sense of scaling dimension of the norms.
The organization of this paper is the following:
In Sect.2 we establish the Calderon-Zygmund type of estimates for use in
the later sections. In Sect.3 we prove two a priori estimates for smooth

650 D. Chae, J. Lee
axisymmetric solutions of the Navier-Stokes equations. Theorem 2 is one
of them. In Sect.4, based on the previous estimates in Sects. 2 and 3, we
prove Theorem 1, Theorem 3 and Theorem 4 as well as Corollary 1.
A result similar to Theorem 4 (ii) was obtained independently at about the
same time by Neustupa et al., see [19].
2 Kinematic estimates
We recall thefollowing definition of the A
p
class and the weighted inequali-
ties for the singular integral operator of the convolution type. (See Stein[23]
pp. 194–217 for details.)
Definition 5 Let p ∈ (1, ∞). A real valued function w(x) is said to be in
A
p
class if it satisfies
sup
B⊂R
3

1
|B|

B
w(x)dx

1
|B|

B
w(x)
−
p

p
dx

p
p

< ∞,
where the supremum is taken over all balls B in R
3
. Here p

is the H
¨
older
conjugate of p, i.e.
1
p
+
1
p

=1.
For function w(x) ∈ A
p
we can extend the Calderon-Zygmund inequality
for the singular integral operator with the integral having weight function
w(x).
Theorem 6 ([23] p.205) Let p ∈ (1, ∞). Suppose T is a singular integral
operator of the convolution type, and w(x) ∈ A
p
. Then for f ∈ L
p
(R
3
),

R
3
|Tf(x)|
p
w(x)dx ≤ C

R
3
|f(x)|
p
w(x)dx.
Now we begin with the elementary lemma.
Lemma 1 For any p ∈ (1, ∞) the function w(x)=
1
√
x
2
1
+x
2
2
, where x =
(x
1
,x
2
,x
3
),isinA
p
class.
Proof. Let p

be the H
¨
older conjugate of p. We need to show that
sup
B⊂R
3

1
|B|

B
w(x)dx

1
|B|

B
w(x)
−
p

p
dx

p
p

< ∞.
Let x
0
∈ R
3
be given. We set B = {x ∈ R
3
||x − x
0
| <r}, and d is the
distance between x
0
and the x
3
axis. If d ≥ 2r , then
1
d+r
≤ w(x) ≤
1
d−r

Axisymmetric solutions of the Navier-Stokes equations 651
for all x in B.
Thus,
L :=

1
|B|

B
w(x)dx

1
|B|

B
w(x)
−
p

p
dx

p
p

≤

3
4πr
3

B
1
d − r
dx

3
4πr
3

B
(d + r)
p

p
dx

p
p

=
d + r
d − r
≤ 3.
If d<2r, then the cylinder {(x
1
,x
2
,x
3
) ∈ R
3
|x
2
1
+ x
2
2
< (d + r)
2
, |x
3
−
x
0
3
| <r} contains the ball B.
Thus it is easily seen that
L ≤

3
4πr
3
2π

x
0
3
+r
x
0
3
−r

d+r
0
1
ρ
· ρdρdx
3

×

3
4πr
3
2π

x
0
3
+r
x
0
3
−r

d+r
0
ρ
p

p
+1
dρdx
3

p
p

≤
3
2r
3
(d + r)2r

3
2r
3
p
p

+2p
(d + r)
p

p
+2
2r

p
p

≤ 27

27p
p

+2p

p
p

.
The lemma is proved. 
Combining Lemma 1 and Theorem 6, we immediately have the following.
Corollary 2 If T is a singular integral operator of the convolution type,
then

R
3
|Tf|
p
1
r
dx ≤ C

R
3
|f|
p
1
r
dx,
where r =

x
2
1
+ x
2
2
.
Lemma 2 If u is an axisymmetric vector field with div u =0, and ω =
curl u vanishes sufficiently fast near infinity in R
3
, then the gradients of ˜u
and u
θ
e
θ
can be represented as the singular integral form.
∇˜u(x)=Cω
θ
(x)e
θ
(x)+[K ∗ (ω
θ
e
θ
)](x),
∇(u
θ
e
θ
(x)) =
˜
C ˜ω(x)+[H ∗(˜ω)](x),
wherethekernels(K(x)) and(H(x)) arethematrixvaluedfunctionshomo-
geneous of degree −3, defining a singular integral operator by convolution,
and f ∗ g(x)=

R
3
f(x − y)g(y)dy denotes the standard convolution op-
erator. In the above, (C) and (
˜
C) are the constant matrices.

652 D. Chae, J. Lee
Proof. We observe that div ˜u =0, and curl ˜u = ω
θ
e
θ
. Similarly, div (u
θ
e
θ
)
=0and curl (u
θ
e
θ
)=ω
r
e
r
+ ω
3
e
3
. Then the conclusion is immediate.
(See [5] and [16].) 
The following is a localized version of the well-known Calderon-Zygmund
type of inequality for the velocity gradients and the vorticity.
Lemma 3 Supposep ∈ (1, ∞) isgiven andu(x) is adivergencefree vector
field in R
3
, which is in L
p
(B
2R
) and curl u = ω is in L
p
(B
2R
). Then, we
have the inequality
∇u
L
p
(B
R
)
≤ Cu
L
p
(B
2R
)
+ Cω
L
p
(B
2R
)
.
Proof. Choose cut off function ρ(x) ∈ C
∞
0
(R
3
) such that supp ρ ⊂ B
2R
,
ρ(x)=1if |x|≤R, and |∇ρ|≤
C
R
.
We have
div (ρu)=ρdiv u + ∇ρ · u = ∇ρ · u,
curl (ρu)=ρω + ∇ρ × u.
Notethatingeneral,foranyu ∈ W
1,p
(R
3
),∇u
L
p
(R
3
)
≤ C(div u
L
p
(R
3
)
+ curl u
L
p
(R
3
)
).
Fromtheabovestandardinequalityweobtaintheconclusionofthelemma.
Lemma 4 For any )>0,
lim
r→0

∞
−∞
∂
r

u
θ
r

u
θ
r

3
r

dx
3
=0.
Proof. We can prove this lemma similar to the proof of Corollary 1 of [13].
Since we have 
u
θ
r

6
≤ C∇
u
θ
r

2
≤ Cu
H
2
,
(u
θ
)
3
r
4−
and
1
r
1−
∂
r

u
θ
r

be-
long to L
2
(0,T; L
2
(R
3
)).(We can prove this fact similarly with the proof of
Lemma 4 of [13] if we set g =

u
θ
r

3
.) And we also get

∞
−∞
r
δ

u
θ
r

6
dx
3
and

∞
−∞
r
δ

∂
r

u
θ
r

2
dx
3
are bounded for small δ>0.Wehave

∞
−∞
∂
r

u
θ
r

u
θ
r

3
r

dx
3
≤


∞
−∞




∂
r

u
θ
r





2
r

2
dx
3

1
2


∞
−∞

u
θ
r

6
r

2
dx
3

1
2
r

2
.
Then the lemma follows immediately. 

Axisymmetric solutions of the Navier-Stokes equations 653
3 Estimates for smooth solutions
The following is rather well-known fact, and below we provide a proof of
it, which could not find in the literature.
Proposition 1 Suppose that u is a smooth axisymmetric solution of the
Navier-Stokes equations with initial data u
0
∈ L
2
(R
3
). Let p ∈ [2, ∞].If
ru
θ
0
∈ L
p
, then ru
θ
∈ L
∞
(0,T; L
p
) and (ru
θ
)
p
2
∈ L
2
(0,T; W
1,2
).
Proof. We observe that azimuthal component of the axisymmetric Navier-
Stokes equations reduces to
˜
D
Dt
(ru
θ
) − (∂
2
r
+ ∂
2
3
+
1
r
∂
r
)(ru
θ
)+
2
r
∂
r
(ru
θ
)=0. (5)
Multiplying the both sides of (5) by |ru
θ
|
p−2
(ru
θ
) and integrating over R
3
,
we obtain
1
p
d
dt

R
3
|ru
θ
|
p
dx +
4(p − 1)
p
2

R
3
|
˜
∇|ru
θ
|
p
2
|
2
dx
= −

R
3
2
r
∂
r
(ru
θ
)|ru
θ
|
p−2
(ru
θ
)dx := I. (6)
In (6), the integration by parts can be justified by use of the standard cut-off
argument, the details of which we omitted for simplicity. Before proceeding
further, we note that
∇f
2
= 
˜
∇f
2
,
for an arbitrary axisymmetric function f. Since ru
θ
is an axisymmetric
smooth function which vanishes at infinity and on the x
3
-axis, we obtain
I = −
4π
p

∞
−∞

∞
0
∂
r
|ru
θ
|
p
drdx
3
=0.
Thus we get
d
dt

R
3
|ru
θ
|
p
dx + C

R
3
|
˜
∇|ru
θ
|
p
2
|
2
dx
=0.
By Gronwall’s inequality, we have
sup
0≤t≤T

R
3
|ru
θ
|
p
dx + C

T
0

R
3
|
˜
∇|ru
θ
|
p
2
|
2
dxdt ≤ C(ru
θ
0

p
). (7)
The case p = ∞ is immediate if we let p →∞in the above. 
Using the Proposition 1, we now prove Theorem 2.

654 D. Chae, J. Lee
Proof of Theorem 2. Below we denote Θ = ru
θ
. Consider the e
r
,e
3
com-
ponents of the Navier-Stokes equations.
˜
Du
r
Dt
− (∂
2
r
+ ∂
2
3
)u
r
−
1
r
∂
r
u
r
+
1
r
2
u
r
= −∂
r
p +
Θ
2
r
3
,
˜
Du
3
Dt
− (∂
2
r
+ ∂
2
3
)u
3
−
1
r
∂
r
u
3
= −∂
3
p.
Applying the operator (∂
3
, −∂
r
) to the aboveequations, the azimuthal com-
ponent of the vorticity equation is obtained as the following.
˜
Dω
θ
Dt
+(∂
r
u
r
+∂
3
u
3
)ω
θ
−(∂
2
r
+∂
2
3
+
1
r
∂
r
)ω
θ
+
1
r
2
ω
θ
= ∂
3

Θ
2
r
3

. (8)
Suppose that r
3
ω
θ
∈ L
2
. Similar to the proof of Proposition 1, we first
assume ω
θ
decays sufficiently fast. Multiplying the both sides of (8) by
r
6
ω
θ
and integrating over R
3
,weget
1
2
d
dt

R
3
(r
3
ω
θ
)
2
dx +

R
3
[(u
r
∂
r
+ u
3
∂
3
)(r
3
ω
θ
)](r
3
ω
θ
)dx
−

R
3
u
r
∂
r
(r
3
)ω
θ
(r
3
ω
θ
)dx −

R
3
u
r
r
5
(ω
θ
)
2
dx
−

R
3
(r
3
ω
θ
)(∂
2
r
+ ∂
2
3
+
1
r
∂
r
)(r
3
ω
θ
)dx +

R
3
r
4
(ω
θ
)
2
dx
+

R
3
r
3
(ω
θ
)
2
(∂
2
r
+
1
r
∂
r
)(r
3
)dx +2

R
3
∂
r
ω
θ
∂
r
(r
3
)(r
3
ω
θ
)dx
=

R
3
(∂
3
Θ
2
)r
3
ω
θ
dx.
After integration by parts and more elementary computations, we have
1
2
d
dt

R
3
(r
3
ω
θ
)
2
dx +

R
3
|
˜
∇(r
3
ω
θ
)|
2
dx
=4

R
3
u
r
r
5
(ω
θ
)
2
dx +8

R
3
r
4
(ω
θ
)
2
dx
+6

R
3
∂
r
(r
3
ω
θ
)r
2
ω
θ
dx +

R
3
∂
3
Θ
2
r
3
ω
θ
dx
:= I
1
+ I
2
+ I
3
+ I
4
.
By use of H
¨
older’s inequality, Young’s inequality, and the Gagliardo-
Nirenberg inequality, we estimate I
1
as follows.

Axisymmetric solutions of the Navier-Stokes equations 655
|I
1
|≤u
r

2


R
3
r
10
(ω
θ
)
4
dx

1
2
≤ C


R
3
(r
3
ω
θ
)
10
3
(ω
θ
)
2
3
dx

1
2
≤ C


R
3
(ω
θ
)
2
dx

1
6


R
3
(r
3
ω
θ
)
5
dx

1
3
≤

R
3
(ω
θ
)
2
dx + C


R
3
(r
3
ω
θ
)
5
dx

2
5
≤

R
3
(ω
θ
)
2
dx + Cr
3
ω
θ

1
5
2

˜
∇(r
3
ω
θ
)
9
5
2
≤

R
3
(ω
θ
)
2
dx + )
˜
∇(r
3
ω
θ
)
2
2
+ C

r
3
ω
θ

2
2
.
For I
2
and I
3
, we estimate
|I
2
|≤C

R
3
(r
3
ω
θ
)
4
3
(ω
θ
)
2
3
dx
≤ C


R
3
(ω
θ
)
2
dx

1
3


R
3
(r
3
ω
θ
)
2
dx

2
3
≤

R
3
(ω
θ
)
2
dx + C

R
3
(r
3
ω
θ
)
2
dx,
and
|I
3
|≤C


R
3
(∂
r
(r
3
ω
θ
))
2
dx

1
2


R
3
r
4
(ω
θ
)
2
dx

1
2
≤ )

R
3
(∂
r
(r
3
ω
θ
))
2
dx + C


R
3
r
4
(ω
θ
)
2
dx
≤ )

R
3
(∂
r
(r
3
ω
θ
))
2
dx +

R
3
(ω
θ
)
2
dx + C

R
3
(r
3
ω
θ
)
2
dx.
Finally, we obtain
|I
4
|≤

R
3
(∂
3
Θ
2
)
2
dx +

R
3
(r
3
ω
θ
)
2
dx.
Combining the above estimates altogether, and choosing ) to be sufficiently
small, we obtain
d
dt

R
3
(r
3
ω
θ
)
2
dx + C

R
3
|
˜
∇(r
3
ω
θ
)|
2
dx
≤ C

R
3
(r
3
ω
θ
)
2
dx + C

R
3
(ω
θ
)
2
dx + C

R
3
(∂
3
(Θ
2
))
2
dx.

656 D. Chae, J. Lee
Applying Gronwall’s inequality, we get
sup
0≤t≤T
r
3
ω
θ

2
2
+ C

T
0

R
3
|
˜
∇(r
3
ω
θ
)|
2
dxdt
≤ C(T )r
3
ω
θ
0

2
2
+ C

T
0
ω
θ

2
2
dt + C

T
0
∂
3
Θ
2

2
2
dt
≤ C(T,r
3
ω
θ
0

2
, u
0

2
, ru
θ
0

4
). (9)
The last inequality of (9) is from the energy inequality and the Proposition
1. Similarly to the proof of Proposition 1, we can justify the integration
by parts in the above computations the proof by using the standard cut-off
function technique. 
4 Proof of regularity criteria
We can construct weak solutions in various ways.( See [6], [13], and [26].)
Forexample([2]and[6]),itispossibletoconstructweaksolutionsbyconsid-
ering thefollowing regularized equations. Let ρ
δ
be thestandard mollifier in
R
3
, ρ
δ
(x)=
1
δ
3
ρ(
|x|
δ
), where ρ ∈ C
∞
0
(R
3
), ρ ≥ 0 and supp ρ ⊂{|x|≤1}.
We regularize the Navier-Stokes equations as follows.



(∂
t
+(ρ
δ
∗ u
δ
) ·∇−∆)u
δ
+ ∇p
δ
=0,
div u
δ
=0,
u
δ
0
= ρ
δ
∗ u
0
.
For each δ>0, the above equations have a global axisymmetric smooth
solution if u
0
is axisymmetric, and belongs to L
2
(R
3
).Asδ → 0, we obtain
an axisymmetric Leray-Hopf weak solution as defined in the introduction.
In the proof of Theorem 1 below, we provide only a priori estimates. The
corresponding estimates for weak solutions can be justified by the above
regularization procedure.
Proof of Theorem 1. Let u be an axisymmetric smooth solution of the
Navier-Stokes equations. Taking curl on the both sides of the Navier-Stokes
equations, then we obtain the following equations.
∂ω
∂t
− ∆ω +(u ·∇)ω − (ω ·∇)u =0.
We may assume Q
T
= B
3R
× [0,T)={(x, t)|0 ≤ t<T,|x| < 3R}
where R>0 and Q

T
= B
2R
× [0,T)={(x, t)|0 ≤ t<T,|x| < 2R} for
simplicity. For the general domain the proof is similar. Let η = η(r) be a
smooth cut off function which has a support in B
2R
, 0 ≤ η ≤ 1, and η =1

Axisymmetric solutions of the Navier-Stokes equations 657
on B
R
. By multiplying ωη
2k
on the both sides of the above equations, and
integrating over R
3
, we get the following equation.
1
2
d
dt

B
2R
|ωη
k
|
2
dx +

B
2R
|∇(ωη
k
)|
2
dx
+2k
3

i,j=1

B
2R
(∂
j
η)ω
i
(∂
j
ω
i
)η
2k−1
dx
+

B
2R
|ω|
2
η
k
(∆η
k
)dx +

B
2R
[u ·∇(ωη
k
)](ωη
k
)dx
−

B
2R
(u ·∇η
k
)ω · ωη
k
dx
−

B
2R
(η
k
ω ·∇)u · ωη
k
dx =0.
From which it follows
1
2
d
dt

B
2R
|ωη
k
|
2
dx +

B
2R
|∇(ωη
k
)|
2
dx
= −

B
2R
(∆η
k
)|ω|
2
η
2k−2
dx − 2k
3

j=1

B
2R
(∂
j
η)(ωη
k−1
)∂
j
(ωη
k
)dx
+k

B
2R
|ω|
2
η
2k−1
(u ·∇)ηdx +

B
2R
(η
k
ω ·∇)u · ωη
k
dx
:= {1} + {2} + {3} + {4}.
First, {1} is estimated easily.
{1}≤C

B
2R
|ω|
2
dx,
where C = C(η).
On the other hand, {2} and {3} can be estimated by virtue of the Young
inequality, the H
¨
older inequality and the Gagliardo-Nirenberg inequality.
{2}≤C

B
2R
|ωη
k−1
||∇(ωη
k
)|dx
≤ )

B
2R
|∇(ωη
k
)|
2
dx + C


B
2R
|ω|
2
η
2k−2
dx
≤ )

B
2R
|∇(ωη
k
)|
2
dx+ C



B
2R
|ω|
2
dx

1
k


B
2R
(|ω|η
k
)
2
dx

k−1
k
≤ )

B
2R
|∇(ωη
k
)|
2
dx + C


B
2R
|ω|
2
dx + C


B
2R
(|ω|η
k
)
2
dx,

658 D. Chae, J. Lee
and
{3}≤C

B
2R
|u||ω|
2
η
2k−1
dx
≤ C


B
2R
|u|
2
dx

1
2


B
2R
|ω|
4
η
2(2k−1)
dx

1
2
≤

B
2R
|ω|
2
dx + C


B
2R
(|ω|η
k
)
2(2k−1)
k−1
dx

k−1
2k−1
≤

B
2R
|ω|
2
dx + C


B
2R
(|ω|η
k
)
2
dx

k−2
2(2k−1)
×


B
2R
|∇|ω|η
k
)
2
dx

3k
2(2k−1)
≤ C

B
2R
|ω|
2
dx + )

B
2R
|∇|ω|η
k
|
2
dx + C


B
2R
(|ω|η
k
)
2
dx.
To estimate {4}, we compute
{4} =

B
2R
[(η
k
ω
r
∂
r
−
1
r
η
k
ω
θ
∂
θ
+ η
k
ω
3
∂
3
)(u
r
e
r
+ u
θ
e
θ
+ u
3
e
3
)]
× (η
k
ω
r
e
r
+ η
k
ω
θ
e
θ
+ η
k
ω
3
e
3
)dx
=

B
2R
η
k
ω
r
(∂
r
u
r
)η
k
ω
r
−
1
r
η
k
ω
θ
u
θ
η
k
ω
r
+η
k
ω
3
(∂
3
u
r
)η
k
ω
r
+ η
k
ω
r
(∂
r
u
θ
)η
k
ω
θ
+
1
r
η
k
ω
θ
u
r
η
k
ω
θ
+ η
k
ω
3
(∂
3
u
θ
)η
k
ω
θ
+η
k
ω
r
(∂
r
u
3
)η
k
ω
3
+ η
k
ω
3
(∂
3
u
3
)η
k
ω
3
dx
:= I
1
+ ... + I
8
.
Now we estimate I
1
, ..., I
8
.
|I
1
|≤Cω
θ

L
p
(B
3R
)
η
k
ω
2
L
2p
p−1
(B
2R
)
≤ Cω
θ

L
p
(B
3R
)
∇(η
k
ω)
3
p
2
η
k
ω
2p−3
p
2
≤ C

ω
θ

2p
2p−3
L
p
(B
3R
)
η
k
ω
2
2
+ )∇(η
k
ω)
2
2
.
Similarly, we get the following.
|I
3
|, |I
6
|, |I
7
|, |I
8
|≤C

ω
θ

2p
2p−3
L
p
(B
3R
)
η
k
ω
2
2
+ )∇(η
k
ω)
2
2
.

Axisymmetric solutions of the Navier-Stokes equations 659
Since −
u
θ
r
= ∂
r
u
θ
+ ω
3
,wehave
I
2
=

B
2R
η
k
ω
θ
(∂
r
u
θ
)η
k
ω
r
dx +

B
2R
η
k
ω
θ
ω
3
η
k
ω
r
dx
=

B
2R
ω
θ
(∂
r
(η
k
u
θ
))η
k
ω
r
dx − k

B
2R
(∂
r
η)u
θ
ω
θ
ω
r
η
2k−1
dx
+

B
2R
η
k
ω
θ
ω
3
η
k
ω
r
dx
:= I
1
2
+ I
2
2
+ I
3
2
.
In the proof of Lemma 3, we know that
∇(η
k
u)
2
≤ Cu
2
+ Cη
k
ω
2
.
Thus we get
|I
1
2
|, |I
2
2
|, |I
3
2
|≤C

ω
θ

2p
2p−3
L
p
(B
3R
)
η
k
ω
2
2
+ )∇(η
k
ω)
2
2
.
Similarly, |I
4
| and |I
5
| can be estimated. Putting together the above esti-
mates, we get
1
2
d
dt

B
2R
|ωη
k
|
2
dx + C

B
2R
|∇(ωη
k
)|
2
dx
≤ Cω
θ

2p
2p−3
L
p
(B
3R
)
η
k
ω
2
L
2
(Ω
2R
)
+ Cω
2
L
2
(B
2R
)
.
Using Gronwall’s inequality, we have
sup
0≤t≤T
ωη
k
(t)
2
L
2
(B
2R
)
+ C

T
0
∇(ωη
k
)
2
L
2
(B
2R
)
dt
≤ C

ω
0

2
2
+

T
0
ω
2
L
2
(B
2R
)
dt

exp

C

T
0
ω
θ

2p
2p−3
L
p
(B
3R
)
dt

.
Applying Lemma 2 and Lemma 3, sup
t∈[0,T )
∇u(t)
L
2
(B
R
)
<Ci.e. u ∈
L
∞
(0,T; L
6
(B
R
)). Thus we get the interior regularity by applying Serrin’s
criterion. 
Remark4. Forlater use,we note thatif we setη ≡ 1 intheproof ofTheorem
1 then we get the following a priori estimate for the whole domain R
3
.
sup
0≤t≤T
ω
2
2
+ C

T
0
∇ω
2
2
dt ≤ Cω
0

2
2
exp

C

T
0
ω
θ

2p
2p−3
p
dt

.

660 D. Chae, J. Lee
For the proof of Theorem 3 and Theorem 4, we will use the standard con-
tinuation principle for the local strong solution.
Proof ofTheorem3. Proof of(i): First notethat thereexistsmaximal time T
0
such that there is a unique classical solution u ∈ C((0,T
0
); L
s
(R
3
)), s>3
and u(τ )
s
≥
C
(T
0
−τ)
s−3
2s
with constant C, which is independent of T
0
and s.( See [9].) We provide a priori estimate for the smooth axisymmetric
solution. We write the e
r
and e
θ
components of the vorticity equation as
follows.
˜
Dω
r
Dt
−

∂
2
r
+
1
r
∂
r
+ ∂
2
3

ω
r
+
1
r
2
ω
r
− (ω
r
∂
r
+ ω
3
∂
3
)u
r
=0. (10)
˜
Dω
θ
Dt
−

∂
2
r
+
1
r
∂
r
+ ∂
2
3

ω
θ
+
1
r
2
ω
θ
+
1
r
u
θ
ω
r
−{(ω
r
∂
r
+ ω
3
∂
3
) u
θ
+
1
r
ω
θ
u
r
} =0. (11)
Multiplying the both sides of (10) and (11) by ω
r
and ω
θ
respectively, and
integrating over (0, ∞) × (−∞, ∞), we get the following equalities.
1
2
d
dt

(ω
r
)
2
+(ω
θ
)
2
drdx
3
+

|
˜
∇ω
r
|
2
+ |
˜
∇ω
θ
|
2
drdx
3
+

(ω
θ
)
2
r
2
drdx
3
+

(ω
r
)
2
r
2
drdx
3
=

1
r
(∂
r
ω
r
)ω
r
drdx
3
−

[(u
r
∂
r
+ u
3
∂
3
)ω
r
]ω
r
drdx
3
+

[(ω
r
∂
r
+ ω
3
∂
3
)u
r
]ω
r
drdx
3
+

1
r
(∂
r
ω
θ
)ω
θ
drdx
3
−

[(u
r
∂
r
+ u
3
∂
3
)ω
θ
]ω
θ
drdx
3
+

[(ω
r
∂
r
+ ω
3
∂
3
)u
θ
]ω
θ
drdx
3
−

1
r
ω
r
ω
θ
u
θ
drdx
3
+

1
r
ω
θ
u
r
ω
θ
drdx
3
=

1
r
∂
r
ω
r
ω
r
drdx
3
−
1
2

u
r
r
(ω
r
)
2
drdx
3
+

ω
r
∂
r
u
r
ω
r
drdx
3
+

1
r
∂
r
ω
θ
ω
θ
drdx
3
+
1
2

u
r
r
(ω
θ
)
2
drdx
3
−

u
θ
r
∂
3
u
r
ω
r
drdx
3

Axisymmetric solutions of the Navier-Stokes equations 661
−

∂
r
u
θ
∂
3
u
r
ω
r
drdx
3
+

ω
r
∂
r
u
θ
ω
θ
drdx
3
−

u
θ
r
∂
3
u
θ
ω
θ
drdx
3
−

∂
r
u
θ
∂
3
u
θ
ω
θ
drdx
3
−

1
r
u
θ
ω
r
ω
θ
drdx
3
=

1
r
∂
r
ω
r
ω
r
drdx
3
−
1
2

u
r
r
(ω
r
)
2
drdx
3
+

ω
r
∂
r
u
r
ω
r
drdx
3
+

1
r
∂
r
ω
θ
ω
θ
drdx
3
+
1
2

u
r
r
(ω
θ
)
2
drdx
3
−

u
θ
r
∂
3
u
r
ω
r
drdx
3
−2

u
θ
r
ω
r
ω
θ
drdx
3
−

∂
r
u
θ
∂
3
u
r
ω
r
drdx
3
:= I
1
+ I
2
+ I
3
+ I
4
+ I
5
+ I
6
+ I
7
+ I
8
. (12)
We will estimate I
1
,...,I
8
as follows. First, it is easily seen from Young’s
inequality that
|I
1
|≤
1
2

(ω
r
)
2
r
2
drdx
3
+
1
2

(∂
r
ω
r
)
2
drdx
3
,
and
|I
4
|≤
1
2

(ω
θ
)
2
r
2
drdx
3
+
1
2

(∂
r
ω
θ
)
2
drdx
3
.
By means of the H
¨
older inequality, the Young inequality and the Gagliardo-
Nirenberg inequality, we get
|I
2
|≤


(ω
r
)
2
r
2
drdx
3

1
2


(u
r
)
2
(ω
r
)
2
drdx
3

1
2
≤ )

(ω
r
)
2
r
2
drdx
3
+ C

u
r

2
˜
L
γ
ω
r

2
˜
L
2γ
γ−2
≤ )

(ω
r
)
2
r
2
drdx
3
+ C

u
r

2
˜
L
γ
ω
r

2(γ−2)
γ
˜
L
2

˜
∇ω
r

4
γ
˜
L
2
≤ )

(ω
r
)
2
r
2
drdx
3
+ C

u
r

2γ
γ−2
˜
L
γ
ω
r

2
˜
L
2
+ )
˜
∇ω
r

2
˜
L
2
,
|I
3
| =2





u
r
(∂
r
ω
r
)ω
r
drdx
3





662 D. Chae, J. Lee
≤ )

(∂
r
ω
r
)
2
drdx
3
+ )
˜
∇ω
r

2
˜
L
2
+ C

u
r

2γ
γ−2
˜
L
γ
ω
r

2
˜
L
2
,
and
|I
5
|≤)

(ω
θ
)
2
r
2
drdx
3
+ C


(u
r
)
2
(ω
θ
)
2
drdx
3
≤ )

(ω
θ
)
2
r
2
drdx
3
+ )
˜
∇ω
θ

2
˜
L
2
+ C

u
r

2γ
γ−2
˜
L
γ
ω
θ

2
˜
L
2
.
On the other hand, we note that 
˜
∇u
r

˜
L
p
≤ Cω
θ

˜
L
p
, which follows from
Corollary 2 and Lemma 2. By use of the above fact, the Young inequality,
the H
¨
older inequality, and the Gagliardo-Nirenberg inequality again, we are
lead to
|I
6
|≤C

u
θ
∂
3
u
r

2
˜
L
2
+ )

(ω
r
)
2
r
2
drdx
3
≤ C

u
θ

2
˜
L
γ
∂
3
u
r

˜
L
2γ
γ−2
+ )

(ω
r
)
2
r
2
drdx
3
≤ C

u
θ

2
˜
L
γ
ω
θ

˜
L
2γ
γ−2
+ )

(ω
r
)
2
r
2
drdx
3
≤ C

u
θ

2
˜
L
γ
ω
θ

2(γ−2)
γ
˜
L
2

˜
∇ω
θ

4
γ
˜
L
2
+ )

(ω
r
)
2
r
2
drdx
3
≤ C

u
θ

2γ
γ−2
˜
L
γ
ω
θ

2
˜
L
2
+ )
˜
∇ω
θ

2
˜
L
2
+ )

(ω
r
)
2
r
2
drdx
3
.
Similarly to the above, we estimate
|I
7
|≤C

u
θ

2γ
γ−2
˜
L
γ
ω
r

2
˜
L
2
+ )
˜
∇ω
r

2
˜
L
2
+ )

(ω
θ
)
2
r
2
drdx
3
.
Since it is not easy to estimate I
8
directly, we integrate by parts first.
I
8
=

∂
r
∂
3
u
θ
u
r
ω
r
drdx
3
+

∂
r
u
θ
u
r
∂
3
ω
r
drdx
3
=

∂
r
ω
r
u
r
ω
r
drdx
3
−

u
θ
∂
r
u
r
∂
3
ω
r
drdx
3
−

u
θ
u
r
∂
r
∂
3
ω
r
drdx
3
=2

∂
r
ω
r
u
r
ω
r
drdx
3
−

u
θ
∂
r
u
r
∂
3
ω
r
drdx
3
+

u
θ
∂
3
u
r
∂
r
ω
r
drdx
3
:= I
1
8
+ I
2
8
+ I
3
8
.

Axisymmetric solutions of the Navier-Stokes equations 663
As previously, thanks tothe Young inequality, the H
¨
older inequality, and the
Gagliardo-Nirenberg inequality, we obtain
|I
1
8
|≤)

(∂
r
ω
r
)
2
drdx
3
+ C


(u
r
)
2
(ω
r
)
2
drdx
3
≤ )

(∂
r
ω
r
)
2
drdx
3
+ )
˜
∇ω
r

2
˜
L
2
+ C

u
r

2γ
γ−2
˜
L
γ
ω
r

2
˜
L
2
,
|I
2
8
|≤)

(∂
3
ω
r
)
2
drdx
3
+ C


(u
θ
)
2
(∂
r
u
r
)
2
drdx
3
≤ )

(∂
3
ω
r
)
2
drdx
3
+ )
˜
∇ω
θ

2
˜
L
2
+ C

u
θ

2γ
γ−2
˜
L
γ
ω
θ

2
˜
L
2
,
and
|I
3
8
|≤)

(∂
r
ω
r
)
2
drdx
3
+ )
˜
∇ω
θ

2
˜
L
2
+ C

u
θ

2γ
γ−2
˜
L
γ
ω
θ

2
˜
L
2
.
Choosing ) sufficiently small, we have
d
dt

(ω
r
)
2
+(ω
θ
)
2
drdx
3
+ C

|
˜
∇ω
r
|
2
+ |
˜
∇ω
θ
|
2
drdx
3
+C

(ω
θ
)
2
r
2
+
(ω
r
)
2
r
2
drdx
3
≤ C(u
r

2γ
γ−2
˜
L
γ
+ u
θ

2γ
γ−2
˜
L
γ
)(ω
θ

2
˜
L
2
+ ω
r

2
˜
L
2
). (13)
Using Gronwall’s inequality, we get the following a priori estimates.
sup
t∈[0,T
0
)
(ω
r
(t)
2
˜
L
2
+ ω
θ
(t)
2
˜
L
2
)+C

T
0

˜
∇ω
r

2
˜
L
2
+ 
˜
∇ω
θ

2
˜
L
2
dt
+C

T
0





ω
r
r




2
˜
L
2
+




ω
θ
r




2
˜
L
2

dt (14)
≤ C(ω
r
0

2
˜
L
2
+ ω
θ
0

2
˜
L
2
)

1 + exp

C

T
0
u
r

2γ
γ−2
˜
L
γ
+ u
θ

2γ
γ−2
˜
L
γ
dt

.
Applying Theorem 2 and above a priori estimates, we get

R
3
|ω
θ
|
2
dx =

∞
−∞

1
0
|ω
θ
|
2
rdrdx
3
+

∞
−∞

∞
1
|ω
θ
|
2
rdrdx
3
≤

∞
−∞

1
0
|ω
θ
|
2
drdx
3
+

∞
−∞

∞
1
|ω
θ
|
2
r
7
drdx
3
.
In the above inequality, right hand side is bounded by some constant by
Theorem2and (15).ByRemark4 below theendoftheproof oftheTheorem

664 D. Chae, J. Lee
1 and the Sobolev embedding theorem, we find that u
L
∞
((0,T
0
);L
6
(R
3
))
is
bounded by some constant. Thus we can continue our local smooth solution
until T by the standard continuation argument.
Proofof(ii):Theθ-componentoftheaxisymmetricNavier-Stokesequations
is as follows.
∂
t
u
θ
+(u
r
∂
r
+ u
3
∂
3
)u
θ
−(∂
2
r
+ ∂
2
3
+
1
r
∂
r
)u
θ
+
1
r
2
u
θ
+
1
r
u
θ
u
r
=0. (15)
Multiplyingthebothsidesof(15)and(8)by2|u
θ
|
2
u
θ
andω
θ
,andintegrating
over (−∞, ∞) × (0, ∞),weget
1
2
d
dt

|u
θ
|
4
drdx
3
+
1
2
d
dt

|ω
θ
|
2
drdx
3
+
3
2

|
˜
∇(u
θ
)
2
|
2
drdx
3
+

|
˜
∇ω
θ
|
2
drdx
3
+2

|u
θ
|
4
r
2
drdx
3
+

(ω
θ
)
2
r
2
drdx
3
≤

1
r



∂
r
(u
θ
)
2



(u
θ
)
2
drdx
3
+
5
4

|u
θ
|
4




u
r
r




drdx
3
+





1
r
∂
r
ω
θ
ω
θ




drdx
3
+
1
2





u
r
r




(ω
θ
)
2
drdx
3
+

|∂
3
(u
θ
)
2
|




ω
θ
r




drdx
3
:= I
1
+ I
2
+ I
3
+ I
4
+ I
5
.
Using Young’s inequality, we estimate
|I
1
|≤
1
2




(u
θ
)
2
r




2
˜
L
2
+
1
2

˜
∇(u
θ
)
2

2
˜
L
2
,
|I
3
|≤
1
2




ω
θ
r




2
˜
L
2
+
1
2

˜
∇(ω
θ
)
2
˜
L
2
,
and
|I
5
|≤
1
2

˜
∇(u
θ
)
2

2
˜
L
2
+
1
2




ω
θ
r




2
˜
L
2
.
On theother hand, byvirtue of the Gagliardo inequalityand H
¨
older inequal-
ity,wehave
|I
2
|≤Cω
θ

γ
γ−1
˜
L
γ
(u
θ
)
2

2
˜
L
2
+ )
˜
∇(u
θ
)
2

2
˜
L
2
and
|I
4
|≤Cω
θ

γ
γ−1
˜
L
γ
ω
θ

2
˜
L
2
+ )
˜
∇ω
θ

2
˜
L
2
.

Axisymmetric solutions of the Navier-Stokes equations 665
Choosing ) sufficiently small and using Gronwall’s inequality, we obtain
that
sup
0≤t≤T
0
(u
θ

4
˜
L
4
+ω
θ

2
˜
L
2
)≤C(u
θ
0

4
˜
L
4
+ω
θ
0

2
˜
L
2
) exp(C

T
0
0
ω
θ

γ
γ−1
˜
L
γ
dt).
By the same argument below (15) in the proof of (i), we obtain (ii). 
Proof of Corollary 1. For the axisymmetric solution for initial data without
swirl, u
θ
≡ 0, and thus ω
3
≡ 0. In view of (ii) of Theorem 3, it suffices to
prove

T
0

|ω
θ
|
2
drdx
3
dt < ∞. (16)
The main difficulty of the global existence is that we do not know if
ω
θ
r
2
∈
L
2
(0,T; L
2
(R
3
))( See [13].). Choose
ω
θ
r
as a test function of the vorticity
equation and integrate over R
3
. Since ω
r
, ω
3
and u
θ
vanish in this case, we
get the following inequality by integrating by parts.
1
2
d
dt

|ω
θ
|
2
drdx
3
+
1
2

(ω
θ
)
2
r
2
drdx
3
+
1
2

|
˜
∇ω
θ
|
2
drdx
3
≤ C

(ω
θ
)
2
drdx
3
.
The Gronwall inequality gives us that
ω
θ
∈ L
∞
(0,T;
˜
L
2
(R
3
)).
Thus, (16) is now proved. 
ProofofTheorem4. (i)LetT
0
bethemaximaltimeasintheproofofTheorem
3. We wish to multiply the both sides of (8) and (15) by
ω
θ
r
2
and
(u
θ
)
3
r
4
,
respectively, and integrate over R
3
. Indicated in the proof of Corollary 1 (
Seealso[13].), however, wedonotknowiftheybelongtoL
2
(0,T
0
; L
2
(R
3
).
To justify the procedure, we multiply the both sides of (8) and (15) by
ω
θ
r
2−
and
(u
θ
)
3
r
4−
with sufficiently small )>0, respectively and integrate over R
3
.
Then we have the following by the integration by parts.
1
2
d
dt

R
3

ω
θ
r
1−

2

2
dx +
1
4
d
dt

R
3

u
θ
r
1−

4

4
dx
+

R
3




˜
∇

ω
θ
r
1−

2





2
dx +
1
2

R
3





˜
∇

u
θ
r
1−

4

2





2
dx

666 D. Chae, J. Lee
)

1 −
)
4


R
3

ω
θ
r
2−

2

2
dx +
)
4

3 −
)
2


R
3
(u
θ
)
4
r
6−
dx
≤ )

R
3




u
r
r





ω
θ
r
1−

2

2
dx +

R
3





∂
3

u
θ
r
1−

2

2









ω
θ
r
1−

2




dx
+(2 − ))

R
3




u
r
r





u
θ
r
1−

4

4
dx := I
1
+ I
2
+ I
3
.
Note that boundary terms vanish (see [13] and Lemma 4). By virtue of
the H
¨
older inequality, the Gagliardo-Nirenberg inequality, and the Young
inequality, we obtain
|I
1
|≤)




u
r
r




γ




ω
θ
r
1−

2




2
2γ
γ−1
≤ C




u
r
r




2γ
2γ−3
γ




ω
θ
r
1−

2




2
2
+
1
4




˜
∇

ω
θ
r
1−

2





2
2
,
|I
2
|≤





˜
∇

u
θ
r
1−

4

2





2




ω
θ
r
1−

2




2
≤ C




ω
θ
r
1−

2




2
2
+
1
8





˜
∇

u
θ
r
1−

4

2





2
2
,
and
|I
3
|≤(2 − ))




u
r
r




γ






u
θ
r
1−

4

2





2
2γ
γ−1
≤ C




u
r
r




2γ
2γ−3
γ






u
θ
r
1−

4

2





2
2
+
1
8





˜
∇

u
θ
r
1−

4

2





2
2
.
By the use of the Gronwall inequality, we have
1
2
sup
t∈[0,T
0
]




ω
θ
(t)
r
1−

2




2
2
+
1
4
sup
t∈[0,T
0
]




u
θ
(t)
r
1−

4




4
4
+C

T
0
0






˜
∇

ω
θ
r
1−

2





2
2
+





˜
∇

u
θ
r
1−

4

2





2
2


dt (17)
≤ C






ω
θ
0
r
1−

2




2
2
+






u
θ
0
r
1−

4

2





2
2



1 + exp


T
0
0




u
r
r




2γ
2γ−3
γ
dt


Axisymmetric solutions of the Navier-Stokes equations 667
If ) → 0, then we have the following by the Lebesgue dominated conver-
gence theorem.
1
2
sup
t∈[0,T
0
]




ω
θ
(t)
r




2
2
+
1
4
sup
t∈[0,T
0
]




u
θ
(t)
r




4
4
+C

T
0
0






˜
∇

ω
θ
r





2
2
+





˜
∇

u
θ
r

2





2
2


dt (18)
≤ C






ω
θ
0
r




2
2
+






u
θ
0
r

2





2
2



1 + exp


T
0
0




u
r
r




2γ
2γ−3
γ
dt

In order to complete the proof, we choose cut-off function η(r) such that
0 ≤ η ≤ 1 on r ≤
r
0
2
and supp η ∈{r ≤ r
0
}. Multiplying the both sides of
(8) and (15) by
ω
θ
r
1−

2
η
4
and

u
θ
r
1−

4
η

3
, respectively, and integrate over R
3
.
Since the procedure is rather standard, we omit the details here. then we can
deduce the following inequality.
1
2
sup
t∈[0,T
0
]




ω
θ
(t)
r
η
2




2
2
+
1
4
sup
t∈[0,T
0
]




u
θ
(t)
r
η




4
4
+C

T
0
0






˜
∇

ω
θ
r
η
2





2
2
+





˜
∇

u
θ
r
η

2





2
2


dt
≤ C






ω
θ
0
r
η
2




2
2
+






u
θ
0
r
η

2





2
2



1 + exp


T
0
0




u
r
r




2γ
2γ−3
L
γ
(Γ
r
0
)
dt

+C

T
0
0
ru
θ

4
4
dt + C

T
0
0
ω
θ

2
2
dt. (19)
The right hand side of (19) is controlled by the initial datum u
0

2
, ru
θ
0

4
and


u
r
r


L
α
(0,T ;L
γ
(Γ
r
0
))
,whichisfinitebyhypothesis.Similarlytotheproof
of Theorem 3, we consider the following estimate.

R
3
|ω
θ
|
2
dx =

∞
−∞

r
0
2
0
|ω
θ
|
2
rdrdx
3
+

∞
−∞

∞
r
0
2
|ω
θ
|
2
rdrdx
3
≤ C

∞
−∞

r
0
2
0
(ω
θ
)
2
r
drdx
3
+ C

∞
−∞

∞
r
0
2
|ω
θ
|
2
r
7
drdx
3
.
Thus it follows that u ∈ L
∞
(0,T
0
; L
6
(R
3
)), which implies that we can
continue our local smooth solution by the standard continuation argument.

668 D. Chae, J. Lee
Proof of (ii) is similar to that of (i). For simplicity, we do not present the
cut-off function technique. It will be shown that the integrability of u
θ
is
controlled by that of u
r
. First we multiply the both sides of the azimuthal
componentof theNavier-Stokes equationsby|u
θ
|
2
u
θ
andintegrateoverR
3
,
then we get
1
4
d
dt

R
3
|u
θ
|
4
dx +
3
4

R
3
|
˜
∇|u
θ
|
2
|
2
dx
+

R
3
|u
θ
|
4
r
2
dx = −

R
3
|u
θ
|
4
r
u
r
dx := I
1
. (20)
By meansof the Young inequality, the H
¨
older inequality, and the Gagliardo-
Nirenberg inequality, it is established that
|I
1
|≤)

R
3

|u
θ
|
2
r

2
dx + C

R
3
|u
θ
|
4
(u
r
)
2
dx
≤ )

R
3

|u
θ
|
2
r

2
dx + Cu
r

2
γ
(|u
θ
|)
2

2
2γ
γ−2
≤ )

R
3

|u
θ
|
2
r

2
dx + Cu
r

2
γ
|u
θ
|
2

2(1−
3
γ
)
2

˜
∇|u
θ
|
2

6
γ
2
≤ )

R
3

|u
θ
|
2
r

2
dx + )
˜
∇|u
θ
|
2

2
2
+ Cu
r

2γ
γ−3
γ
|u
θ
|
2

2
2
.
Thus from (20), we have the following inequality.
1
4
d
dt

R
3
|u
θ
|
4
dx + C

R
3
|
˜
∇|u
θ
|
2
|
2
dx + C

R
3

|u
θ
|
2
r

2
dx
≤ Cu
r

2γ
γ−3
γ
(u
θ
)
2

2
2
.
Using Gronwall’s inequality, we find that the following is immediate.
1
4
sup
0≤t≤T
0
u
θ

4
4
+ C

T
0
0

R
3
|
˜
∇(|u
θ
|)
2
|
2
dxdt (21)
+C

T
0
0

R
3

|u
θ
|
2
r

2
dxdt ≤ C(u
θ
0

4
4
) exp


T
0
0
u
r

2γ
γ−3
γ
dt

.
The right hand side of (22) is controlled by u
θ
0

4
and

T
0
u
r

2γ
γ−3
γ
dt.
Multiplying the both sides of (8) by ω
θ
and integrating over R
3
,weget
1
2
d
dt

R
3
(ω
θ
)
2
dx +

R
3
|
˜
∇ω
θ
|
2
dx +

R
3

ω
θ
r

2
dx
= −

R
3
(∂
3
(u
θ
)
2
)
ω
θ
r
dx +

R
3
u
r
(ω
θ
)
2
r
dx := I
2
+ I
3
. (22)

Axisymmetric solutions of the Navier-Stokes equations 669
As previously, by means of the H
¨
older inequality, the Young inequality, and
the Gagliardo-Nirenberg inequality, it can be obtained that
|I
2
|≤)

R
3

ω
θ
r

2
dx + C


R
3
(∂
3
|u
θ
|
2
)
2
dx.
Due to the hypothesis on the integrability of u
r
, it is derived that
|I
3
|≤)

R
3

ω
θ
r

2
dx + C


R
3
(u
r
)
2
(ω
θ
)
2
dx
≤ )

R
3

ω
θ
r

2
dx + )
˜
∇ω
θ

2
2
+ C

u
r

2γ
γ−3
γ
ω
θ

2
2
.
Choosing ) sufficiently small, the inequality (22) reduces to the following.
1
2
d
dt

R
3
(ω
θ
)
2
dx + C

R
3
|
˜
∇(ω
θ
)|
2
dx + C

R
3

ω
θ
r

2
dx
≤ C

R
3
(∂
3
|u
θ
|
2
)
2
dx + Cu
r

2γ
γ−3
γ
ω
θ

2
2
.
Forthecompletionoftheproofwechooseη asintheproofof(i).Multiplying
the both sides of (15) and of (i). Multiplying the both sides of (15) and (8)
by |u
θ
|u
θ
η
4
and ω
θ
η
4
, respectively, and proceeding similarly, we get
1
2
sup
0≤t≤T
0
ω
θ
η
2

2
2
+ C

T
0
0

R
3
|
˜
∇(ω
θ
η
2
)|
2
dx
+C

T
0
0

R
3

ω
θ
η
2
r

2
dxdt
≤ C

ω
θ
0
η
2
2
+

T
0
0

˜
∇(|u
θ
|η)
2

2
2
+ ω
θ

2
2
dt

×exp

C

T
0
0
u
r

2γ
γ−3
L
γ
(Γ
r
0
)
dt

. (23)
Theright handsideof (23)iscontrolled by theinitialdatum u
0

H
1
,ru
θ
0

4
and the norm u
r

L
α
(0,T ;L
γ
(Γ
r
0
))
, which is finite by hypothesis. Similarly
to the case (i), we conclude that u is regular. 
Acknowledgements. We deeply thank to the anonymous referee for very careful reading of
the paper, and many helpful and constructive criticism. This research is supported partially
by KOSEF(K97-07-02-02-01-3) and BSRI-MOE.

670 D. Chae, J. Lee
References
1. H. Beir
˜
ao da Veiga, A new regularity class for the Navier-Stokes equations in R
n
,
Chinese Ann. Math. 16, (1995) 407–412
2. L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions
of the Navier-Stokes equations, Comm. Pure Appl. Math. 35, (1982), 771–831
3. D.ChaeandH-J. Choe,Onthe regularitycriterion forthesolutionsoftheNavier-Stokes
equations, Electron. J. Differential Equations 1999, (1999), No. 05, 1–7
4. D. Chae and O. Y. Imanuvilov, Generic solvability of the axisymmetric 3-D Euler
equations and 2-D Boussinesq equations, J. Diff. Eq. 156, (1999), 1–17
5. D. Chae and N. Kim, On the breakdown of axisymmetric smooth solutions for the 3-D
Euler equation, Comm. Math. Phys. 178, (1996), 391–398
6. P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regu-
larity for the Navier-Stokes equations, Indiana Univ. Math. J. 42, (1993), 775–789
7. P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press,
Chicago, 1988
8. E. Fabes, B. Jones and Riviere, The initial value problem for the Navier-Stokes equa-
tionswith data in L
p
, Arch. Rat. Mech. Anal. 45, (1972), 222-248
9. Y. Giga, Solutions for semilinear parabolic equations in L
p
and regularity of weak
solutions of the Navier-Stokes system, J. Diff. Eq. 62, (1986), 186–212
10. E.Hopf,
¨
UberdieAnfangwertaufgabenf
¨
urdiehydromischenGrundgleichungen,Math.
Nach. 4, (1951), 213–321
11. T. Kato, Strong L
p
-solutions of the Navier-Stokes equation in R
m
, with applications
to weak solutions, Math. Z. 187, (1984), 471–480
12. O. A. Ladyzhenskaya, On the unique global solvability to the Cauchy problem for the
Navier-Stokes equations in the presence of the axial symmetry, Zapisky Nauc. Sem.
LOMI 7, (1969), 155–177
13. S. Leonardi, J. M
´
alek, J. Ne
ˇ
cas and M. Pokorn
´
y, On Axially Symmetric Flows in R
3
,
Z. Anal. Anwendungen 18, (1999), 639–649
14. J. Leray,
´
Etude de divers
´
equations int
´
egrales nonlinearies et de quelques problemes
que posent lhydrodinamique, J. Math. Pures. Appl. 12, (1931), pp.1–82.
15. A. Mahalov, E. S. Titi and S. Leibovich, Invariant helical subspaces for the Navier-
Stokes equations, Arch. Rational Mech. Anal. 112, (1990), 193–222
16. A. Majda, Lecture Notes, Princeton University graduate course, 1986–1987
17. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm.
Pure & Appl. Math. 39, (1986), S 187-S 220
18. J.Neustupa, A.Novotn
´
yand P.Penel, ARemarktoInteriorregularityof asuitableweak
solution to the Navier-Stokes equations, preprint, University of Toulon-Var, (1999)
19. J. Neustupa and M. Pokorn
´
y, Axisymmetric flow of the newtonian fluid in the whole
space with non-zero tangential velocity component, preprint, (2000)
20. T. Ohyama, Interior regularity of weak solutions to the Navier-Stokes equation, Proc.
Japan Acad. 36, (1960), 273-277
21. X. Saint Raymond, Remarks on axisymmtric solutions of the incompressible Euler
system, Comm. P.D.E. 19(1 & 2), (1994), 321–334
22. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,
Arch. Rat. Mech. Anal. 9, (1962), 187–191
23. E.M.Stein,HarmonicAnalysis:Real-VariableMethods,OrthogonalityandOscillatory
Integrals, Princeton University Press, Princeton, 1993
24. M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure
Appl. Math. 41, (1988), 437-458

Axisymmetric solutions of the Navier-Stokes equations 671
25. R. Temam, Navier-Stokes equations and Nonlinear Functional Analysis, SIAM,
Philadelphia, 1983
26. M. R. Ukhovskii and V. I. Yudovich, Axially symmetric flows of ideal and viscous
fluids filling the whole space, J. Appl. Math. Mech. 32, (1968), 52–62