Electronic Journal of Differential Equations, Vol. 1999(1999), No. 05, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
REGULARITY OF SOLUTIONS TO
THE NAVIER-STOKES EQUATION
Dongho Chae & Hi-Jun Choe
Abstract. Recently, Beirão da Veiga [15] obtained regularity for the Navier-Stokes
equation in R3 by imposing conditions on the vorticity rather than the velocity. In
this article, we obtain regularity by imposing conditions on only two components of
the vorticity vector.
1. Introduction
We are concerned with the initial value problem of the Navier-Stokes equation
in R3 × (0, T ),
∂v
+ (v · ∇)v = −∇p + ν∆ , ,
∂t
div v = 0 ,
v(x, 0) = v0 (x) ,
(1)
(2)
(3)
where v(x, t) = (v1 (x, t), v2 (x, t), v3 (x, t)), x ∈ R3 , and t ∈ (0, T ). For simplicity we
assume that the external force is zero, but it is easy to extend our results to the
nonzero-external-force case.
Recall that a weak solution to the Navier-Stokes equation, which is called the
Leray-Hopf weak solution, is defined as a vector field v ∈ L∞ (0, T ; L2 (R3 )) ∩
L2 (0, T ; H 1 (R3 )) satisfying div v = 0 in the distributional sense, and
Z
0
T
Z
R3
[v · φt + (v · ∇)φ · v + v · ∆φ]dx dt = 0
for all φ ∈ [C0∞ (R3 × (0, T ))]3 with div φ = 0.
For v0 ∈ L2 (R3 ) with div v0 = 0, the existence of weak solutions was established
by Leray[13] and
Hopf in [11].
solution ofthe Navier-Stokes equation that
A weak
∞
1
3
2
belongs to L 0, T ; H (R ) ∩ L 0, T ; H 2 (R3 ) is called a strong solution. It
is well-known that for v0 ∈ H 1 (R3 ) with div v0 = 0 there exists a local unique
strong solution v ∈ C([0, T ); H 1 (R3 )), and that the maximal time of existence T∗
1991 Mathematics Subject Classifications: 35Q35, 76C99.
Key words and phrases: Navier-Stokes equations, regularity conditions.
c
1999
Southwest Texas State University and University of North Texas.
Submitted February 8, 1999. Published February 28, 1999.
The first author was partially supported by GARC-KOSEF, BSRI-MOE, KOSEF(K95070),
and SNU Research Fund.
The second author was supported by KOSEF and BSRI-prg.
1
Dongho Chae & Hi-Jun Choe
2
EJDE–1999/05
depends on the initial data kv0 kH 1/2 (R3 ) . Moreover, the strong solution belongs to
the class C (0, T∗ ); C ∞ (R3 ) , i.e., the solution becomes regular in the space variable
immediately after the initial moment. The global-in time continuation of the local
strong solution is an outstanding open problem in mathematical fluid mechanics.
Another notion of solution, useful for the study of the Navier-Stokes equation, is
that of mild solution initiated by Fujita and Kato [9].
In this note we are concerned with obtaining a sufficient condition for the global
continuation of strong solutions. In this direction there is a classical result due to
Serrin[14], which states that if a weak solution belongs to Lα,γ
T , then v becomes the
α,γ
α
γ
strong solution in (0, T ]. Here Lα,γ
=
L
=
L
(0,
T
;
L
(R3 ) with α2 + γ3 < 1,
T
[0,T ]
and α < ∞. Later, Fabes-Jones-Riviere [8] extended the above criterion to the case
2
+ γ3 = 1. The problem of regularity and uniqueness for the marginal case α = ∞,
α
γ = 3 in Serrin’s condition has been extensively studied by many authors; see for
example [3], [10], and [12]. Recently, Beirão da Veiga [15] obtained a sufficient
condition for regularity using the vorticity rather than the velocity. His result says
that if the vorticity ω = curl v of a weak solution v belongs to the space Lα,γ
with
T
2
3
+ γ ≤ 2 with 1 < α < ∞, then v becomes the strong solution on (0, T ]. Here we
α
prove that it is sufficient to control only two components of the vorticity vector, or
the gradients of the two components of the velocity vector. See Theorem 1 below.
Given a velocity field v, the two-component vorticity field is denoted by ω̃ =
ω1 e1 + ω2 e2 , where e1 = (1, 0, 0), e2 = (0, 1, 0).
Theorem 1. Let v0 ∈ L2 (R3 ) with div v0 = 0 and ω0 = curl v0 ∈ L2 (R3 ). If a
Leray-Hopf weak solution v, satisfies ω̃ ∈ Lα,γ
with α2 + γ3 ≤ 2, 1 < α < ∞ and
T
3
< γ < ∞, or if kω̃k ∞, 23 is sufficiently small, then v becomes the classical solution
2
on (0, T ].
LT
Remark 1. As an immediate corollary of the above theorem, we find that if the
classical solution of the 3-D Navier-Stokes equations blows up at time T , then
kω̃kLα,γ = ∞, where ω̃ is any two component vector of ω, and (α, γ) is a pair of
T
real numbers satisfying α2 + γ3 ≤ 2, 1 < α < ∞. In other words, at finite blow-up
time at least two components of the vortices must simultaneously blow up.
A related result is studied by Beale-Kato-Majda [1]. They show that for 3-D
Euler equations, the blow up of full gradients of the velocity field is controlled by
only three components of the vorticity field.
Remark 2. As another immediate corollary we obtain the global regularity for the
2-D Navier-Stokes equations, since in this case ω̃(x, t) = 0, for all (x, t) ∈ R3 ×(0, T ).
Our second theorem concerns the regularity criterion in terms of gradients of the
two components of velocity.
Theorem 2. Let ṽ = v1 e1 + v2 e2 be the first two components of a Leray-Hopf
weak solution of the Navier-Stokes equation corresponding to v0 ∈ H 1 (R3 ) with
div v0 = 0. Suppose that Dṽ ∈ Lα,γ
with α2 + γ3 ≤ 1, where 2 ≤ α ≤ ∞, and
T
3 ≤ γ ≤ ∞, then v becomes a classical solution in (0, T ].
2. Proof of Main Theorems
The key idea in booth proofs is a careful observation of the structure of the
nonlinear terms of the vorticity equations for the Navier-Stokes system. The struc-
EJDE–1999/05
Regularity of solutions to the Navier-Stokes equation
3
ture of the nonlinear term has been emphasized in the works by Constantin and
Fefferman [4], [5], [6], and [7].
Below we use the notation
Z
1/p
p
kukp =
|u(x)| dx
, 1 ≤ p < ∞.
R3
We also use C for various constants in the estimates below.
Proof of Theorem 1. Taking the curl on (1), we obtain
ωt + (v · ∇)ω = (ω · ∇)v + ν∆ω.
Multiplying (4) by ω in L2 (R3 ), and integrating by parts, we obtain
Z
1 d
2
2
(ω · ∇)v · ω dx .
kω(t)k2 + νk∇ω(t)k2 =
2 dt
R3
Using the Biot-Savart law, v is written in terms of ω:
Z
1
(x − y) × ω(y, t)
v(x, t) = −
dy .
4π R3
|x − y|3
(4)
(5)
(6)
Substituting this into the right hand side of (5), we have
Z
ZZ
3
y
y
(ω · ∇)v · ω dx =
× ω(x + y, t) · ω(x, t) dy dx
· ω(x, t)
4π
|y|
|y|4
R3
We decompose ω = ω̃ + ω 0 , ω̃ = ω1 e1 + ω2 e2 , ω 0 = ω3 e3
for the vorticities in {·}
ZZ
3
y
y
0
=
× ω̃(x + y, t) · ω (x, t) dy dx
· ω(x, t)
4π
|y|
|y|4
ZZ
3
y
y
+
× ω̃(x + y, t) · ω̃(x, t) dy dx
· ω(x + y, t)
4π
|y|
|y|4
ZZ
3
y
y
0
+
× ω (x + y, t) · ω̃(x, t) dy dx ,
· ω(x, t)
4π
|y|
|y|4
(7)
where all the integrations with respect to y are in the sense of principal value.
We have thus
Z
Z
0
3 (ω · ∇)v · ωdx ≤ C 3 |ω(x, t)||P (ω̃)||ω (x, t)|dx
R
R
Z
+C
|ω(x, t)||P (ω̃)||ω̃(x, t)|dx
R3
Z
+C
|ω(x, t)||P (ω 0 )||ω̃(x, t)|dx
3
ZR
Z
2
≤C
|ω| |P (ω̃)|dx + C
|ω||P (ω 0 )||ω̃|dx.
R3
=: I1 + I2 ,
R3
Dongho Chae & Hi-Jun Choe
4
EJDE–1999/05
where P (·) denotes the singular integral operator defined by the integrals with
respect to y in (7). We first consider the case 32 < γ < ∞.
We have the following estimates
I1 ≤ kP (ω̃)kγ kωk22γ
γ−1
(by Hölder’s inequality)
2γ−3
γ
3
≤ Ckω̃kγ kωk2
k∇ωk2γ
2γ
ν
≤ Ckω̃kγ2γ−3 kωk22 + k∇ωk22
4
(8)
(by Young’s inequality),
where we used the Calderon-Zygmund and the Gagliardo-Nirenberg inequalities in
the second inequality. For the second term of the right hand side of (8) we have by
the Höder inequality and the Calderon-Zygmund inequality,
I2 ≤ kω̃kγ kP (ω 0 )k
≤ Ckω̃kγ kω 0 k
2γ
γ−1
2γ
γ−1
kωk
kωk
(by Hölder’s inequality)
2γ
γ−1
2γ
γ−1
(by the Calderon-Zygmund inequality)
(9)
≤ Ckω̃kγ kωk22γ
γ−1
2γ
≤ Ckω̃kγ2γ−3 kωk22 +
ν
k∇ωk22
4
(by the similar estimates to (8)).
Thus, combining (8)-(9) with (5), we obtain
2γ
d
kω(t)k22 + νk∇ω(t)k22 ≤ Ckω̃(t)kγ2γ−3 kω(t)k22 .
dt
Applying the standard Gronwall lemma, we have
kω(t)k22
Z t
2γ
2γ−3
+
exp C
kω̃(s)kγ
ds dτ
0
τ
Z t
2γ
2γ−3
2
≤ kω0 k2 exp C
kω̃(s)kγ
ds .
Z
t
k∇ω(τ )k22
0
Since 0 <
2γ
2γ−3
≤ α, by the Hölder inequality we obtain
sup kω(t)k22 + ν
0≤t≤T
Z
0
T
k∇ω(t)k22 dt ≤ kω0 k22 exp C
≤
kω0 k22
Z
T
0
kω̃(t)kγ
2γ
2γ−3
α,γ
LT
exp Ckω̃k
2γ
2γ−3
T
!
dt
2γ
2
3
2γ−3 (2− α − γ
)
.
Thus, in this case kω̃kLα,γ
< ∞ implies
T
ω ∈ L∞ 0, T : L2 (R3 ) ∩ L2 0, T : H 1 (R3 ) .
Using the regularity of the strong solution, we obtain the conclusion of the theorem
for 1 < α < ∞, 32 < γ < ∞.
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Regularity of solutions to the Navier-Stokes equation
5
Next, we consider the case α = ∞, γ = 3/2. In this case, instead of (8) and (9),
we estimate as follows:
{1} ≤ kP (ω̃)k 32 kωk26 ≤ Ckω̃k 32 k∇ωk22 ,
and
(10)
I2 ≤ kω̃k 32 kP (ω 0 )k6 kωk6
≤ Ckω̃k 32 kω 0 k6 kωk6
≤
Ckω̃k 32 kωk26
≤
(11)
Ckω̃k 32 k∇ωk22 .
Combining (10)-(11) with (5), integrating over [0, T ], we deduce
Z T
Z T
sup kω(t)k22 + ν
k∇ω(t)k22 dt ≤ Ckω̃k ∞, 23
k∇ω(t)k22 dt .
0≤t≤T
LT
0
Thus, if Ckω̃kL∞,3/2 <
0
ν
,
2
then we have again
ω ∈ L∞ 0, T : L2 (R3 ) ∩ L2 0, T : H 1 (R3 ) ,
T
which implies the regularity of v as previously.
Proof of Theorem 2. We set ṽ = (v1 , v2 , 0). Then, taking the first two components
of the vorticity equation (4), we obtain
ω̃t + (v · ∇)ω̃ = (ω · ∇)ṽ + ν∆ω̃ .
Taking L2 (R3 ) inner product (12) with ω̃, we have, after integration by part,
Z
1 d
2
2
(ω · ∇)ṽ · ω̃ dx .
(12)
kω̃(t)k2 + νk∇ω̃(t)k2 =
2 dt
R3
We first consider the case 2 ≤ α < ∞, 3 < γ ≤ ∞. Using the Hölder and the
Gagliardo-Nirenberg inequalities, we estimate
Z
(ω · ∇)ṽ · ω̃ dx ≤ kωk2 k∇ṽkγ kω̃k 2γ
γ−2
R3
γ−3
3
(13)
≤ Ckωk2 k∇ṽkγ kω̃k2 γ k∇ω̃k2γ
2γ
ν
k∇ω̃k22 ,
2
where the case γ = ∞ corresponds to the obvious limit γ → ∞ for the norms of
the estimates. The estimates (13), combined with (12), yield
≤ Ckωk22 + Ck∇ṽkγγ−3 kω̃k22 +
2γ
d
kω̃(t)k22 + νk∇ω̃(t)k22 ≤ Ckω(t)k22 + Ck∇ṽ(t)kγγ−3 kω̃(t)k22 .
dt
Using the Gronwall lemma similarly to the proof of Theorem 1, we obtain
Z T
2
sup kω̃(t)k2 + ν
k∇ω̃(t)k22 dt
0≤t≤T
0
≤
kω0 k22
≤
Z
T
+
kω0 k22
0
!
kω(t)k22 dt
Z
T
+
0
Z
exp C
!
kDv(t)k22 dt
0
T
2γ
γ−3
k∇ṽ(t)kγ
2γ
γ−3
Lα,γ
T
exp Ck∇ṽk
T
!
dt
2γ
2
3
γ−3 (1− α − γ
)
,
Dongho Chae & Hi-Jun Choe
6
EJDE–1999/05
2γ
where we used the fact γ−3
≤ α in the second inequality. Thus, if ∇ṽ ∈ Lα,γ
T , then
1
3
6
3
we have by the Sobolev embedding, H (R ) ,→ L (R ),
ω̃ ∈ L∞ 0, T : L2 (R3 ) ∩ L2 0, T : L6 (R3 ) .
Since α = 2 and γ = 6 satisfy α2 + γ3 ≤ 2, the conclusion of Theorem 2 for the case
2 ≤ α < ∞, 3 < γ ≤ ∞ follows from Theorem 1.
Next, we consider the case α = ∞, γ = 3. In this case we estimate
Z
(ω · ∇)ṽ · ω̃ dx ≤ kωk2 k∇ṽk3 kω̃k6
R3
≤ Ckωk2 k∇ṽk3 k∇ω̃k2
ν
≤ Ckωk22 k∇ṽk23 + k∇ω̃k22 .
2
This, together with (12), provide us with
sup
0≤t≤T
kω̃(t)k22
Z
+ν
T
0
k∇ω̃(t)k22 dt
≤
Ck∇ṽk2L∞,3
T
Z
T
0
kDv(t)k22 dt
after integrating over [0, T ]. This inequality, in turn, implies that if k∇ṽkL∞,3 < ∞,
T
then
ω̃ ∈ L∞ 0, T : L2 (R3 ) ∩ L2 0, T : L6 (R3 ) .
In a similar manner to the previous case, we conclude the present proof. Acknowledgement. The authors would like to thank Professor J. Serrin for his helpful comment on Remark 1.
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Dongho Chae
Department of Mathematics, Seoul National University
Seoul 151-742, Korea
E-mail address: dhchae@math.snu.ac.kr
Hi-Jun Choe
Department of Mathematics, KAIST
Taejon, Korea
E-mail address: ch@math.kaist.ac.kr