Electronic Journal of Differential Equations, Vol. 2002(2002), No. 29, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Regularity for solutions to the Navier-Stokes
equations with one velocity component regular
∗
Cheng He
Abstract
In this paper, we establish a regularity criterion for solutions to the
Navier-stokes equations, which is only related to one component of the
velocity field. Let (u, p) be a weak solution to the Navier-Stokes equations.
We show that if any one component of the velocity field u, for example
u3 , satisfies either u3 ∈ L∞ (R3 × (0, T )) or ∇u3 ∈ Lp (0, T ; Lq (R3 )) with
1/p + 3/2q = 1/2 and q ≥ 3 for some T > 0, then u is regular on [0, T ].
1
Introduction
We consider the initial-value problem of the Navier-Stokes equations in R3 as
follows:
∂u
− ν∆u + (u · ∇)u = −∇p,
∂t
(1.1)
div u = 0, (x, t = 0) = a(x)
in which u = u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)) and p = p(x, t) denote the
unknown velocity field and the scalar pressure function of the fluid at the point
(x, t) ∈ R3 × (0, +∞), while a is a given initial velocity field, and ν is the
viscosity coefficient of the fluid. We will assume that ν ≡ 1 for simplicity.
For any given a ∈ L2 (R3 ) and div a = 0 in the sense of the distribution, a
global weak solution u ∈ L∞ (0, ∞; L2 (R3 )) to (1.1) with ∇u ∈ L2 (R3 × (0, ∞))
was constructed by Leary [6] for a long time ago. But the regularity and the
uniqueness of his weak solutions remain yet open, in the general case, till now, in
spite of the great efforts made. Moreover, the uniqueness of the weak solutions
follows if the regularity for weak solutions can be obtained. Cf. [10]. Therefore
many regularity criteria have been obtained for weak solutions. In a space-time
cylinder R3 × (0, T ), the regularity class was showed for weak solutions which
belong to the class Lp (0, T ; Lq (R3 )) by Serrin [8] in the case of 1/p + 3/2q < 1
and q > 3; by Fabes, Jones and Riviere [4], Sohr and von Wahl [11] in the case
of 1/p + 3/2q = 1 and q > 3. Also see Giga [5] and Takahashi [9]. For the
critical case as p = ∞ and q = 3, W.von Wahl showed that C([0, T ]; L3 (R3 )) is
∗ Mathematics Subject Classifications: 35Q30, 76D05.
Key words: Navier-Stokes equations, weak solutions, regularity.
c
2002
Southwest Texas State University.
Submitted November 7, 2001. Published March 17, 2002.
1
2
Regularity for solutions to the Navier-Stokes equations
EJDE–2002/29
a regularity class for weak solutions. Also see [2]. Another kind of regularity
class was obtained by da Veiga [1], in which he showed that u is regular if
∇u ∈ Lα (0, T ; Lβ (R3 )) with 1/α + 3/2β = 1/2 and 1 < α ≤ 2. It is obvious
that, for the assumptions of all regularity criteria, it need that every components
of the velocity field must satisfies the same assumptions, and it don’t make
any difference between the components of the velocity field. Thus, it don’t
exploit the relation between the regularity of velocity field and that of any one
component of the velocity field, which is hard to be used in some applications.
As pointed out by Neustupa and Penel [7], it is interesting to know how to effect
the regularity of the velocity field by the regularity of only one component of
the velocity field, when one try to construct an counter-example of solutions to
(1.1), which develops blowup at finite time, i.e., construct an irregular solution.
In this respect, the first result was obtained by Neustupa and Penel [7]. They
showed that, for the suitable weak solutions which essentially differ from the
usual weak solutions in that they should verify a generalized energy inequality,
if u3 is essentially bounded in one subdomain D, then u has no singular points
in D. Their arguments depend heavily on the partial regularity of the suitable
weak solutions developed by Caffarelli, Kohn and Nirenberg [3].
In this paper, we are also interested in establishing the regularity criteria which is only related to one component of the velocity field, and which is
valid for any weak solutions satisfying the energy inequality. Actually, let u be
any weak solution to the Navier-Stokes equations (1.1) in L∞ (0, ∞; L2 (R3 )) ∩
L2loc (0, ∞; H 1 (R3 )) which verifies the energy inequality, if any one component,
e.g., u3 of the velocity field u satisfying either u3 ∈ L∞ (R3 × (0, T )) or ∇u3 ∈
Lp (0, T ; Lq (R3 )) with 1/p + 3/2q = 1/2 and q ≥ 3, then u is regular. Our
analysis is motivated by the argument of Neustupa and Penel [7], and here we
improve their arguments.
This paper is organized as follows: in section 2, we state our main results
after introducing some notations, in section 3, we given the proof of the main
result.
2
Main Result
In this section, we first introduce some notations and the definition of weak
solutions, then state our main result.
Let Lp (R3 ), 1 ≤ p ≤ +∞, represent the usual Lesbegue space of scalar
functions as well as that of vector-valued functions with norm denoted by k ·
∞
kp . Let C0,σ
(R3 ) denote the set of all C ∞ real vector-valued functions φ =
(φ1 , φ2 , φ3 ) with compact support in R3 , such that div φ = 0. Lpσ (R3 ), 1 ≤ p <
∞
(R3 ) with respect to k · kp . H m (R3 ) denotes the usual
∞, is the closure of C0,σ
Sobolev Space. Finally, given a Banach space X with norm k · kX , we denote by
Lp (0, T ; X), 1 ≤ p ≤ +∞, the set of function f (t) defined on (0, T ) with values
RT
in X such that 0 kf (t)kpX dt < +∞. Let QT = R3 × (0, T ). At last, by symbol
C, we denote a generic constant whose value is unessential to our aims, and it
may change from line to line.
EJDE–2002/29
Cheng He
3
Definition. A measurable vector u on QT is called a weak solution to the
Navier-Stokes equations (1.1), if
1. u ∈ L∞ (0, T ; L2σ (R3 )) ∩ L2 (0, T ; H 1 (R3 ));
2. u verifies the Navier-Stokes equations in the sense of the distribution, i.e.,
Z ∞Z
∂φ
u(x, t) ·
− ∇u(x, t) · ∇φ + (u(x, t) · ∇)φ · u(x, t) dxdt
∂t
0
R3
Z
+
a(x)φ(x, 0)dx = 0
R3
for every φ ∈ C0∞ (R3 × R) with div φ = 0.
Our main result can be stated as follows.
Theorem 2.1 Let the initial velocity a ∈ L2σ (R3 ) ∩ H 1 (R3 ), and u is the weak
solution to (1.1) which satisfies the energy inequality. If any one component
of the velocity field u, e.g., u3 satisfies either u3 ∈ L∞ (R3 × (0, T )) or ∇u3 ∈
Lp (0, T ; Lq (R3 )) with 1/p + 3/2q = 1/2 for q ≥ 3, then u is regular on [0, T ],
and for t ∈ [0, T ] the following two estimates are satisfied.
ku(t)k22 + 2
t
Z
k∇u(s)k22 ds ≤ kak22 ,
(2.1)
0
k∇u(t)k22
+
Z
t
k∆u(s)k22 ds ≤ A
(2.2)
0
In the various cases, A depends on the initial data as follows:
A1 = k∇ak22 + CM12 kak22 1 + kak42 M12 + Ckak22 kak2 M1 + k∇ak2
× k∇ak22 + kak22 M12 1 + kak42 M12 eCkak2 kak2 M1 +k∇ak2
where M1 := k∇u3 kL∞ (0,T ;L3 (R3 )) < ∞,
A2
4q
1/2
= k∇ak22 + CA3 kak2q−3 M2 + kak2 kakH 1 (R3 ) (1 + M2 eCM2 ) + M2
q−3
+Ckak22 M2 q ,
where M2 :=
A3
A4
RT
0
k∇u3 kpLp (0,T ;Lq (R3 )) ds < ∞,
q−3 k∇ak22 + Ckak22 M2 q
4q
× exp M2 kak2q−3 + kak2 kakH 1 (R3 ) 1 + M 1/2 eCM2 + M2 ,
8/3
4/3 = k∇ak22 + C M02 + M0 kak2 kak22
4/3
5/3 +A5 kak2 k∇ak2 + M0 kak2 + M0 kak2 ,
=
4
Regularity for solutions to the Navier-Stokes equations
EJDE–2002/29
where M0 := ku3 kL∞ (R3 ×(0,T )) < ∞, and
A5
=
4/3
8/3
k∇ak22 + C(M02 + kak2 M0 )kak22
4/3
5/3 × exp Ckak2 k∇ak2 + M0 kak2 + M0 kak2
.
The constant C, above, is independent of T . Thus T can approach +∞.
Remarks. 1. The weak solution u ∈ L∞ (0, T ; L2σ (R3 )) ∩ L2 (0, T ; H 1 (R3 ))
was constructed by Leray [6] as the initial velocity field a ∈ L2σ (R3 ), such that
u satisfies the energy inequality.
2. Neustupa and Penel [7] showed that, for the suitable weak solution u, if u3 is
essentially bounded in a subdomain D of a time-space cylinder Ω × (0, T ), then
u has no singular points in D.
3. For solutions to the Navier-Stokes equations, either the class Lp (0, T ; Lq ) for
1/p + 3/2q ≤ 1/2 and q > 3, or C([0, T ]; L3 (Ω)) is a regularity class, Cf. [4], [5],
[8], [9],[10] and [2]. Further, if ∇u ∈ Lα (0, T ; Lβ (R3 )) with 1/α + 3/2β = 1 for
1 < α ≤ 2, then u also is regular, see [1].
3
Proof of the main theorem
In this section, we present the proof of Theorem 2.1, preceded by the following
Lemmas. The first lemma follows from direct calculations.
Lemma 3.1 Let a ∈ L2σ (R3 ). Then for any t ≥ 0,
ku(t)k22
+2
Z
t
k∇u(s)k22 ds ≤ kak22 .
(3.1)
0
Let ω = curl u(x, t). Then ω satisfies weakly the equations
∂ω
− ∆ω + (u · ∇)ω = (ω · ∇)u.
∂t
(3.2)
For the third component ω3 , of ω, one obtain the following estimate.
Lemma 3.2 Let a ∈ L2σ (R3 ) ∩ H 1 (R3 ). If u3 ∈ L∞ (R3 × (0, T )), then
kω3 (t)k22
+
Z
t
k∇ω3 (s)k22 ds ≤ kω3 (0)k22 + M02 kak22
(3.3)
0
for any 0 ≤ t ≤ T with M0 = ku3 kL∞ (R3 ×(0,T )) .
Proof. Multiplying both sides of the third equation of (3.2) by ω3 and integrating for x ∈ R3 , we obtain
Z
1 d
2
2
kω3 k2 + k∇ω3 k2 =
(ω · ∇)u3 · ω3 dx.
(3.4)
2 dt
R3
EJDE–2002/29
Cheng He
5
Using integration by parts and the Cauchy inequality, the right term of (3.4)
can be estimated as
Z
Z
(ω · ∇)u3 · ω3 dx = −
(ω · ∇)ω3 · u3 dx
R3
R3
≤ k∇ω3 k2 ku3 k∞ kωk2
1
1
k∇ω3 k22 + ku3 k2∞ k∇uk22 ,
≤
2
2
where we used the fact that kωk2 ≤ k∇uk2 . Thus
d
kω3 k22 + k∇ω3 k22 ≤ M02 k∇uk22 .
dt
(3.5)
Integrating (3.5) over (0, t) gives (3.3).
Lemma 3.3 Let a ∈ L2σ (R3 ) ∩ H 1 (R3 ). If ∇u3 ∈ Lp (0, T ; Lq (R3 )) with 1/p +
3/2q = 1/2 for q ≥ 3, then
(
Z t
C kak2H 1 (R3 ) + kak22 M12
for q = 3,
2
2
kω3 (t)k2 +
k∇ω3 (s)k2 ds ≤
(3.6)
CM
2
CkakH 1 (R3 ) 1 + M2 e
for q > 3
0
for 0 < t ≤ T . Here M1 := k∇u3 kL∞ (0,T ;L3 (R3 )) and M2 :=
q > 3.
RT
0
k∇u3 (s)kpq ds for
Proof. If q = 3, by the Hölder inequality and the Sobolev inequality, the right
term of (3.4) can be estimated as
Z
|
(ω · ∇)u3 · ω3 dx| ≤ kωk2 k∇u3 k3 kω3 k6
R3
≤ Ckωk2 k∇u3 k3 k∇ω3 k2
1
≤
k∇ω3 k22 + Ck∇u3 k23 k∇uk22 .
2
Then
d
kω3 k22 + k∇ω3 k22 ≤ Ck∇u3 k22 k∇uk22
dt
which implies (3.6) as q = 3.
If q > 3, by the Hölder inequality and the Gagliardo-Nirenberg inequality,
we estimate the right term of (3.4) as
Z
|
(ω · ∇)u3 · ω3 dx| ≤ kωk2 k∇u3 kq kω3 k2q/(q−2)
R3
1−3/q
3/q
≤ Ckωk2 k∇u3 kq kω3 k2
k∇ω3 k2
1
≤
k∇ω3 k22 + Ck∇u3 kpq kω3 k22 + Ck∇uk22 .
2
6
Regularity for solutions to the Navier-Stokes equations
EJDE–2002/29
Then
d
kω3 k22 + k∇ω3 k22 ≤ Ck∇u3 kpq kω3 k22 + Ck∇uk22
dt
which implies that
Rt
p
d −C R t k∇u3 kpq ds
0
e
kω3 k22 ≤ Ce−C 0 k∇u3 kq ds k∇uk22 .
dt
Thus using Lemma 3.1,
kω3 k22 ≤ CeCM2 kω3 (0)k22 + kak22 .
(3.7)
(3.8)
Substituting (3.8) into (3.7), (3.6) follows for q > 3.
Lemma 3.4 Let a ∈ L2σ (R3 ) ∩ H 1 (R3 ). If ∇u3 ∈ Lp (0, T ; Lq (R3 )) with 1/p +
3/2q = 1/2 for q ≥ 3, then for any t ≥ 0,
Z t
2
k∇u(t)k2 +
k∆u(s)k22 ds ≤ A1 ,
(3.9)
0
where, for q = 3,
A1
= k∇ak22 + CM12 kak22 1 + kak42 M12 + Ckak22 kak2 M1 + k∇ak2
× k∇ak22 + kak22 M12 1 + kak42 M12 eCkak2 kak2 M1 +k∇ak2 .
When q > 3, then for any t ≥ 0,
k∇u(t)k22 +
Z
t
k∆u(s)k22 ds ≤ A2
(3.10)
0
with
A2
A3
4q
1/2
= k∇ak22 + CA3 kak2q−3 M2 + kak2 kakH 1 (R3 ) (1 + M2 eCM2 )
q−3
+M2 + Ckak22 M2 q
q−3 4q
=
k∇ak22 + Ckak22 M2 q exp M2 kak2q−3
+kak2 kakH 1 (R3 ) 1 + M 1/2 eCM2 + M2 .
The constant C, above, is independent of T . Thus T can approach +∞.
Proof.
The Navier-Stokes equations (1.1) can be rewritten as
∂u
1
− ∆u + ω × u = −∇(p + |u|2 ).
(3.11)
∂t
2
Multiplying both sides of (3.11) by ∆u, it follows that
Z
1 d
k∇uk22 + k∆uk22 =
ω × u · ∆udx
2 dt
R3
Z
Z
=
(ω2 u3 − ω3 u2 )∆u1 dx +
(ω3 u1 − ω1 u3 )∆u2 dx
R3
R3
Z
+
(ω1 u2 − ω2 u1 )∆u3 dx.
(3.12)
R3
EJDE–2002/29
Cheng He
7
In the following, we estimate the each terms at the right hand side of (3.12)
by the Hölder inequality, Young inequality and Gagliardo- Nirenberg inequality.
Let δ be one small parameter determined later. One has that
Z
I1 =
ω2 u3 ∆u1 dx ≤ k∆u1 k2 kω2 k4 ku3 k4
R3
1/2
1/2
1/4
3/4
≤ Ck∆uk2 ku3 k2 k∇u3 k3 kω2 k2 k∇ω2 k2
7
1/2
1/2
1/4
≤ Ck∆uk24 kuk2 k∇u3 k3 k∇uk2
≤ δk∆uk22 + C(δ)kak42 k∇u3 k43 k∇uk22 ,
for q = 3, and
I1
= |
Z
ω2 u3 ∆u1 dx| ≤ k∆u1 k2 kω2 k2r/(r−2) ku3 kr
R3
2rq+6q−6r
3q(r−2)
r−3
3
≤ Ck∆uk2 ku3 k2 r(5q−6) k∇u3 kqr(5q−6) kω2 k2 r k∇ω2 k2r
3q(r−2)
2rq+6q−6r
r+3
r−3
≤ Ck∆uk2 r kuk2 r(5q−6) k∇u3 kqr(5q−6) k∇uk2 r
4q
≤ δk∆uk22 + C(δ)kak2q−3 k∇u3 kpq k∇uk22 ,
for q > 3 with r = 9q/(2q + 3) > 3.
Z
I2 = |
ω3 u2 ∆u1 dx| ≤ k∆u1 k2 kω3 k3 ku2 k6
R3
≤ δk∆uk22 + C(δ)k∇u2 k22 kω3 k2 k∇ω3 k2
≤ δk∆uk22 + C(δ)k∇ω3 k2 k∇uk32 .
Similar to the estimates as I2 ,
Z
I3 = |
ω3 u1 ∆u2 dx| ≤ δk∆uk22 + C(δ)k∇ω3 k2 k∇uk32 .
R3
Similar to the estimates as I1 , one can obtain that
Z
I4 =
ω1 u3 ∆u2 dx
R3
(
δk∆uk22 + C(δ)kak42 k∇u3 k43 k∇uk22 ,
if q = 3,
4q
≤
q−3
p
2
2
δk∆uk2 + C(δ)kak2 k∇u3 kq k∇uk2 , if q > 3.
Let
I5 = |
Z
ω1 u2 ∆u3 dx| ≤ |
R3
Z
∂2 u3 · u2 · ∆u3 dx| + |
R3
Z
∂3 u2 · u2 · ∆u3 dx|.
R3
Then
I51
= |
Z
∂2 u3 · u2 · ∆u3 dx| ≤ k∆u3 k2 k∇u3 kq ku2 k2q/(q−2)
R3
q−3
3
≤ Ck∆uk2 k∇u3 kq kuk2 q k∇uk2q
2(q−3)
q
≤ δk∆uk22 + C(δ)kak2
6
k∇u3 k2q k∇ukqq
8
Regularity for solutions to the Navier-Stokes equations
EJDE–2002/29
and
I52
Z
1
∂2 u2 · u2 · ∆u3 dx| = |
|u2 |2 ∆∂u3 dx|
2
3
3
R
ZR
Z
|
u2 ∆u2 · ∂u3 dx| + |
|∇u2 |2 ∂3 u3 dx|
= |
≤
Z
R3
≤
≤
R3
δ
k∆u2 k22 + C(δ)k∇u3 k2q ku2 k22q/(q−2) + k∇u3 kq k∇u2 k22q/(q−2)
2
2(q−3)
2q−3
6
3
δ
k∆uk22 + C(δ)k∇u3 k2q kuk2 q k∇uk2q + Ck∇u3 kq k∇uk2 q k∆uk2q
2
2(q−3)
q
≤ δk∆uk22 + C(δ)k∇u3 k2q kak2
6
k∇uk2q + C(δ)k∇u3 kpq k∇uk22 .
Similarly,
I6
I61
= |
Z
= |
ZR
ω2 u1 ∆u3 dx| ≤ |
3
Z
∂3 u1 · u1 · ∆u3 dx| + |
R3
Z
∂1 u3 · u1 · ∆u3 dx|
R3
∂3 u1 · u1 · ∆u3 dx|
R3
2(q−3)
q
I62
≤ δk∆uk22 + C(δ)k∇u3 k2q kak2
Z
= |
∂1 u3 · u1 · ∆u3 dx|
6
k∇uk2q + C(δ)k∇u3 kpq k∇uk22
R3
2(q−3)
q
≤ δk∆uk22 + C(δ)kak2
6
k∇u3 k2q k∇ukqq .
Let δ = 1/16. Substituting above estimates into (3.12), it follows that
d
k∇uk22 + k∆uk22 ≤ CM12 1 + kak42 M12 k∇uk22 + Ck∇ω3 k2 k∇uk32
dt
for q = 3, and
d
k∇uk22 + k∆uk22
dt
(3.13)
4q
≤ C kak2q−3 k∇u3 kpq + k∇ω3 k2 k∇uk2 + k∇u3 kpq k∇uk22
2(q−3)
q
+Ckak2
6/q
k∇u3 k2q k∇uk2
(3.14)
for q > 3. (3.13) implies that
Z t
d
exp − C
k∇ω3 k2 k∇uk2 ds k∇uk22
dt
0
Z t
≤ CM12 1 + kak42 M12 exp − C
k∇ω3 k2 k∇uk2 ds k∇uk22 ,
0
which implies
k∇u(t)k22
Z t
≤ k∇ak22 + CM12 kak22 1 + kak42 M12 exp C
k∇ω3 k2 k∇uk2 ds
0
≤ k∇ak22 + CM12 kak22 1 + kak42 M12 exp Ckak2 kak2 M1 + k∇uk2 ,
EJDE–2002/29
Cheng He
9
This inequality and (3.13) imply (3.9). From (3.14), it follows that
Z t
4q
d
exp C
kak2q−3 k∇u3 kpq + k∇ω3 k2 k∇uk2 + k∇u3 kpq ds k∇uk22
dt
0
2(q−3)
q
6
k∇u3 k2q k∇uk2q
Z t
4q
× exp C
kak2q−3 k∇u3 kpq + k∇ω3 k2 k∇uk2 + k∇u3 kpq ds ,
≤ Ckak2
0
which implies
k∇u(t)k22
exp C
≤
t
Z
0
4q
kak2q−3 k∇u3 kpq + k∇ω3 k2 k∇uk2 + k∇u3 kpq ds
Z
2(q−3)
× k∇ak22 + Ckak2 q
≤
k∇ak22
+
q−3
q
Ckak22 M2
+kak2 kakH 1 (R3 ) 1 +
T
k∇u3 kpq dt
0
q−3
q
Z
T
k∇uk2 dt
0
q3 4q
q−3
exp M2 kak2
∆
1/2
M2 eCM2 =
A3 ,
This inequality and (3.14) imply (3.10).
Lemma 3.5 Let a ∈ L2σ (R3 ) ∩ H 1 (R3 ). If u3 ∈ L∞ (R3 × (0, T )), then for any
t ≥ 0,
Z
t
k∇u(t)k22 +
k∆u(s)k22 ds ≤ A4
(3.15)
0
with
A4
A5
4
8/3
= k∇ak22 + C(M02 + M0 kak23 kak22
4/3
5/3 +A5 kak2 k∇ak2 + M0 kak2 + M0 kak2 ,
4/3
8/3
= k∇ak22 + C(M02 + kak2 M0 )kak22
4/3
5/3 × exp Ckak2 k∇ak2 + M0 kak2 + M0 kak2
.
In above, constant C is a absolute constant and independent of T . Thus T can
be taken as +∞.
Proof. We need to re-estimate the terms at the right hand side of (3.12)
by the assumption that u3 ∈ L∞ (QT ) to replace the assumption that ∇u3 ∈
Lp (0, T ; Lq (R3 )). In view of the above procedure, we don’t use the assumption
as estimating the I2 and I3 , except the estimates about k∇ω3 k2 obtained in
Lemmas 3.2 and 3.3. So there are same as u3 ∈ L∞ (QT ), i.e.,
Z
I2 = |
ω3 u2 ∆u1 dx| ≤ δk∆uk22 + C(δ)k∇ω3 k2 k∇uk32
R3
Z
I3 = |
ω3 u1 ∆u2 dx| ≤ δk∆uk22 + C(δ)k∇ω3 k2 k∇uk32 .
R3
10
Regularity for solutions to the Navier-Stokes equations
EJDE–2002/29
Now we estimate other terms. By Hölder inequality, the estimate of I1 is easy.
In fact,
I1 = |
Z
R3
ω2 u3 ∆u1 dx| ≤ k∆uk2 ku3 k∞ kω2 k2
≤ δk∆uk22 + C(δ)ku3 k2∞ k∇uk22 .
Similarly,
I4 = |
Z
R3
ω1 u3 ∆u2 dx| ≤ δk∆uk22 + C(δ)ku3 k2∞ k∇uk22 .
By Hölder inequality again, one has
I51 = |
Z
∂2 u3 u2 ∆u3 dx| ≤
R3
δk∆uk22
+ C0
Z
|u2 |2 |∂2 u3 |2 dx.
R3
Integrating by parts, we have
Z
|u2 |2 |∂2 u3 |2 dx
Z
Z
2
≤ |2
u2 ∂2 u2 ∂2 u3 u3 dx| + |
|u2 |2 |∂22
u3 u3 dx|
R3
R3
Z
Z
2
≤ |
|∂2 u2 |2 |u3 |2 dx| + |
u2 ∂22
u2 |u3 |2 dx|
R3
R3
R3
1/3
2
+k∆u3 k2 ku2 k6 ku3 k2 ku3 k2/3
∞
≤
2/3
ku3 k2∞ k∇u2 k22 + k∆u2 kku2 k6 ku3 k2 ku3 k4/3
∞
1/3
+k∆u3 k2 ku2 k26 ku3 k2 ku3 k2/3
∞
δ
2/3
4
k∆uk22 + C(δ)kak2 ku3 k4/3
≤
∞ k∇uk2
C0
4/3
2
+C ku3 k2∞ + kak2 ku3 k8/3
∞ k∇uk2 .
For I52 , by integration by part, we can treat as
I52
=
≤
≤
≤
Z
Z
1
|
∂3 u2 u2 ∆u3 dx| = |
|u2 |2 ∆∂3 u3 dx|
2 R3
R3
Z
Z
|
(u2 ∆u2 )∂3 u3 dx| + |
|∇u2 |2 ∂3 u3 dx|
3
3
R
ZR
Z
|
(u2 ∆u2 )∂3 u3 dx| + |2
(∇u2 · ∇∂3 u2 )u3 dx|
R3
R3
Z
δk∆uk22 + C1 (δ)
|u2 |2 |∂3 u3 |2 dx + C(δ)ku3 k2∞ k∇u2 k22 .
R3
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Cheng He
11
Integrating by parts again,
Z
|
|u2 |2 |∂3 u3 |2 dx|
R3
Z
Z
2
≤ 2|
u2 · ∂3 u2 · ∂3 u3 · u3 dx| + |
|u2 |2 ∂33
u3 · u3 dx|
3
3
R
R
Z
Z
2
≤ |
|∂3 u2 |2 |u3 |2 dx| + |
u2 ∂33
u2 · |u3 |2 dx|
R3
R3
1/3
2
+k∆u3 k2 ku2 k6 ku3 k2 ku3 k2/3
∞
2
≤
ku3 k2∞ k∇uk22 + k∆u2 k2 ku2 k6 ku3 k23 ku3 k4/3
∞
1/3
+k∆u3 k2 ku2 k26 ku3 k2 ku3 k2/3
∞
δ
2/3
4
≤
k∆uk22 + Ckak2 ku3 k4/3
∞ k∇uk2
C1
4/3
2
+C ku3 k2∞ + kak2 ku3 k8/3
∞ k∇uk2 .
Therefore,
Z
I5 = |
ω1 u2 ∆u3 dx|
R3
2/3
4/3
4
2
8/3
2
≤ 4δk∆uk22 + Ckak2 ku3 k4/3
∞ k∇uk2 + C ku3 k∞ + kak2 ku3 k∞ k∇uk2 .
Similarly,
Z
I6 = |
ω2 u1 ∆u3 dx|
R3
2/3
4/3
8/3
2
4
2
≤ 4δk∆uk22 + Ckak2 ku3 k4/3
∞ k∇uk2 + C ku3 k∞ + kak2 ku3 k∞ k∇uk2 .
Substituting above estimates into (3.12) and taking δ = 1/16, one obtain that
d
k∇uk22 + k∆uk22
dt
2/3
2
2
≤ C k∇ω3 k2 k∇uk2 + kak2 ku3 k4/3
∞ k∇uk2 k∇uk2
4/3
2
+C ku3 k2∞ + kak2 ku3 k8/3
(3.16)
∞ k∇uk2 ,
which implies
k∇u(t)k22
≤
4/3
8/3 k∇ak22 + C M02 + kak2 M0 kak22
5/3
4/3 ,
× exp Ckak2 k∇ak2 + M0 kak2 + kak2 M0
here we used Lemmas 3.1 and 3.2. The above inequality and (3.16) implies
(3.15).
Proof of Theorem 2.1 Since the initial velocity field a is in L2σ (R3 )∩H 1 (R3 ),
it is well-known that there is a T0 > 0, such that there is a unique strong solution
u ∈ L∞ (0, T0 ; H 1 (R3 )) ∩ L2 (0, T0 ; H 2 (R3 )) to the Navier-Stokes equations (1.1).
12
Regularity for solutions to the Navier-Stokes equations
EJDE–2002/29
See [11]. According the result about uniqueness [11], our weak solution identity
with the strong solution in (0, T0 ). If u3 ∈ L∞ (R3 × (0, T ))(t0 < T ≤ +∞) or
∇u3 ∈ Ls (0, T ; Lq (R3 )) for 1/s + 3/2q = 1/2 with q ≥ 3, then Lemmas 3.1-3.5
imply
Z T
sup ku(t)kH 1 (R3 ) +
k∇u(s)k22 + k∆u(s)k22 ds ≤ C(kakH 1 (R3 ) ).
0≤t≤T
0
Thus the local strong solution u can be extended to time T , and also identity
with the weak solution. Moreover the classical regular criteria [11] implies that
u is a regular solution on [0, T ].
Remark. The referee has kindly pointed out the recent works [12] -[14]. In
[12], the authors consider the interior regularity of the suitable weak solutions
under the assumption that one component of the velocity is assumed to belong to
the anisotropic Lebesgue space Lp,q (p in time and q in space) with 2/p + 3/q ≤
1/2 for p ≥ 4 and q > 6; In [13], the authors obtained the regular criteria
under the different assumptions on three components of velocity; While in [14],
the authors defined many regularity classes by means of derivatives of some
components of velocity. For example, they obtained the regularity of the weak
solutions which satisfy the energy inequality and ∂3 u belong to Lp (0, T ; Lq (R3 ))
with 1/p + 3/2q ≤ 3/4 for q ≥ 2, or ∂3 u3 ∈ L∞ (0, T ; L∞ (R)), etc. But our
assumptions are different from theirs. Through our arguments also based on
some estimates of vorticity as do in [12]-[14], the particular technique is different.
Acknowledgement. The author would like to express his sincere gratitude
to researchers: Professor Tetsuro Miyakawa for his heuristic discussion about
this problem, the anonymous referee for his helpful comments and for pointing
out the existence of related works [12]-[14], Professor M. Pokorny for sending me
his preprint [14], and for pointing out that the weak solution here should satisfy
the energy inequality. The present work was completed while the author was
visiting the Department of Mathematics at Kobe University as a reseach fellow
of the Japan Society for the Promotion of Sciences (JSPS). The financial support
and the warm hospitality at these institutions are gratefully acknowledged here.
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Cheng He
13
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http://www.karlin.mff.cuni.cz/ms-preprints/.
Cheng He
Department of Mathematics, Faculty of Science
Kobe University
Rokko, Kobe, 657-8501, Japan
e-mail: chenghe@math.kobe-u.ac.jp
On leave from:
Academy of Mathematics and System Sciences,
Academia Sinica, Beijing, 100080, China.