On the blow-up criterion of the periodic incompressible fluids
Jamel Benameur2013, Mathematical Methods in the Applied Sciences
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Communicated by T. Hishida Kato, Ponce, Beale and Majda prove the existence and uniqueness of maximal solution of Euler and Navier-Stokes equations and some blow-up criterion. In the periodic case, we establish that if the maximum time T is finite, then the growth of ku.t/k H m is at least of the order of .T t/ 2m=5 .
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Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.2577
MOS subject classification: 35-XX; 35Qxx
On the blow-up criterion of the periodic
incompressible fluids
Jamel Benameur* †
Communicated by T. Hishida
Kato, Ponce, Beale and Majda prove the existence and uniqueness of maximal solution of Euler and Navier–Stokes equations and some blow-up criterion. In the periodic case, we establish that if the maximum time T is finite, then the growth
of ku.t/kHm is at least of the order of .T t/2m=5 . Copyright © 2012 John Wiley & Sons, Ltd.
Keywords: periodic incompressible fluids; Euler equations; Navier–Stokes equations; Sobolev spaces; blow-up criterion
1. Introduction
The incompressible periodic Euler . D 0/ and Navier–Stokes. > 0/ systems in cartesian coordinates are given by
.NS /
8
< @t u u C .u.r/u D rp
div u D 0 in RC T 3
:
u.0/ D u0 in T 3
in RC T 3
,
where is the viscosity of the fluid, u D u.t, x/ D .u1 , u2 , u3 / and p D p.t, x/ denote respectively the unknown
velocity and the unknown
pressure of the fluid at the point .t, x/ 2 RC T 3 and .u.ru/ :D u1 @1 u C u2 @2 u C u3 @3 u, whereas u0RD u01 .x/, u02 .x/, u03 .x/ is a given
initial
velocity. If u0 is quite regular, the pressure p is determined modulo the average on torus, that is, T 3 p.t, x/dx. So, we can suppose
R
T 3 p.t, x/dx D 0.
The classical result of Leray (see [1, 2]) is as follows: if u is a non-regular solution of the system .NS /, . > 0/ and T is the first time
of non-regularity, then
c 3
kru.t/k4L2 , 8t 2 Œ0, T /.
t
(1)
T
Under the condition of smoothness on the initial data for . 0/, we have
ku.t/k2L2 C 2
Z
t
0
kru. /k2L2 d ku0 k2L2 .
(2)
In this paper, we treat only the regular solution, that is to say, u0 2 Hs .T 3 /. For the problem .NS /( 0), it is an open problem to
determine if solution with large smooth initial data of finite energy remains regular for all time. However, if u0 2 Hs .T 3 /, with s > 5=2,
Kato (see [3]) shows that there is existence and uniqueness for a local solution. Mainly,
Theorem 1.1
Let 0 and u0 2 Hs .T3 / be a divergence-free
vectors field,
where s > 5=2, then there exists a time T > 0 and a strong solution u of
.NS / that belongs to C Œ0, T; Hs T 3 \ C 1 Œ0, T; Hs1 T 3 .
For more information, Beale et al. [4] and Kato and Ponce [5] proved the following blow-up result:
Department of Mathematics College of Science, King Saud University , Riyadh 11451, Kingdom of Saudi Arabia
*Correspondence to: Jamel Benameur, Department of Mathematics College of Science, King Saud University , Riyadh 11451, Kingdom of Saudi Arabia.
† E-mail: [email protected]
Copyright © 2012 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
Theorem 1.2
Let 0. Let u 2 C.Œ0, T /; Hs .T 3 // be a maximal solution of .NS /. Suppose that T < 1, then
Z
T
kr u.t/kL1 .T 3 / dt D C1.
0
For s D m 2 N .m 3/, to prove Theorem 1.2, the authors used the following inequality
@t ku.t/k2Hm C.m/kru.t/kL1 ku.t/k2Hm .
(3)
Its proof was based on a general method that makes special use of some commutator estimates and energy.
Here, we deal with the case of non-smoothness overall. We are trying to find more information about the growth of solution, and we
give some properties of the maximum time of regularity. Our main result is the following:
Theorem 1.3
Let 0. Let u 2 C.Œ0, T /, Hm .T 3 //,(m 3 integer) be a maximal solution of .NS /. Suppose that T < 1, then for all t 2 Œ0, T /,
2m
C.m/
1
ku.t/kL25.T 3 / ku.t/kHm .T 3 / .
.T t/2m=5
(4)
Our second result is special for the Navier–Stokes equations. Precisely,
Theorem 1.4
Let u 2 C.Œ0, T /, Hs .T 3 //,.s > 5=2/ be a maximal solution of Navier–Stokes equations .NS /, > 0, given by Theorem 1.1. Suppose
that T < 1, then
(i) For all t 2 Œ0, T /,
p
X
=2
jF .u/.t, k/j.
p
T t
k
(5)
(ii) For all 2 .3=2, s,
1 2
3
C. /ku0 kL2
.T
=3
t/=3
ku.t/kH , 8t 2 Œ0, T /.
(6)
(iii) For all 2 Œ1, s and for all t 2 Œ0, T /,
c. /ku0 kL1
3=4
2
.T t/=4
ku.t/kH .
(7)
In the following theorem, we give a ‘stability’ result of maximum time of the solution of Navier–Stokes and Euler equations with
respect to the viscosity. Precisely,
Theorem 1.5
Let 0 0. Let u 2 C.Œ0, T0 /, Hs .T 3 //,s 4.s 3 if 0 > 0), be the maximal solution of .NS0 /, given by Theorem 1.1. Then,
T0 lim inf T .
(8)
!0
Finally, we give a ‘stability’ result with respect to the initial condition.
Theorem 1.6
If the maximal time of .NS / is denoted by T .u0 /, let 0, and the index of the regularity s 4(s 3 if > 0), then
T .u0 /
lim inf
kv 0 u0 kHs !0
T .v 0 /.
(9)
The proof of Theorem 1.3 is based on Lemma 3.1 and a frequency decomposition into high and low frequencies and is ended by an
application of the Gronwall lemma. To prove Theorem 1.4, we use the Fourier analysis, and technical inequalities are adapted to each
case. As in the preceding theorem, we conclude the proof by using the Gronwall inequality. For Theorems 1.5 and 1.6, the proofs are
based on classical analysis and the Gronwall lemma.
Copyright © 2012 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
2. Notations and preliminary results
In this section, we recall some notations and definitions that will be used later on.
We denote by Cb .X/ the space of bounded and continuous functions on the space X.
The Fourier transformation is normalized as
Z
F .f /.k/ D b
f .k/ D
exp.i x k/f .x/dx, k D .k1 , k2 , k3 / 2 Z3 ,
T3
where x k D x1 k1 C x2 k2 C x3 k3 .
The inverse Fourier formula is
F 1 ..ak /k2Z3 / .x/ D
X
k2Z3
exp.ik x/ak , x D .x1 , x2 , x3 / 2 T 3 .
For s 2 R, the non-homogeneous Sobolev space Hs .T 3 / is defined by
)
X
2 sb
2
.1 C jkj / jf .k/j < 1 ,
H .T / :D f 2 S .T /,
s
where the corresponding norm is
(
3
0
3
k
kf kHs :D
X
k
2 sb
.1 C jkj / jf .k/j
P s .T 3 / is defined by
For s 2 R, the homogeneous Sobolev space H
2
!1=2
.
)
(
X
jkj2s jb
f .k/j2 < 1 ,
f .0/ D 0,
HP s .T 3 / :D f 2 S 0 .T 3 /, b
k
where the corresponding norm is
0
kf kHP s :D @
X
k¤0
11=2
jkj2s jb
f .k/j2 A
.
For s 2 R, < .=. >Hs .T 3 / denotes the associated scalar product to the non-homogeneous Sobolev space, and < .=. >HP s .T 3 / denotes
the associated scalar product to the homogeneous Sobolev space.
For p 2 Œ1, 1/, we define the space Lp by
p
(
L D .ak /, .ak / 2 C
Z3
,
X
k
)
p
jak j < 1
endowed with the norm
X
k.ak /kLp :D
k
jak jp
!1=p
.
The convolution product of a suitable pair of functions f and g on T 3 is given by
.f g/.x/ :D
Z
T3
f .y/g.x y/dy.
The convolution product of a suitable pair of sequences a :D .ak / and b :D .bk / on Z3 is given by
.a b/.k/ :D
X
ap bkp .
p
p
For any Banach space .B, k.k/, any real number 1 p 1 and any time T > 0, we denote by LT .B/ the space of all
measurable functions
t 2 Œ0, T ! f .t/ 2 B,
Copyright © 2012 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
such that
.t ! kf .t/k/ 2 Lp .Œ0, T/.
If f D .f1 , f2 , f3 / and g D .g1 , g2 , g3 / are two vector fields, we set
f ˝ g :D .g1 f , g2 f , g3 f /
and
div .f ˝ g/ :D .div .g1 f /, div .g2 f /, div .g3 f //.
For any subset X of a set E, the symbol 1X denotes the characteristic function of X defined by
1X .x/ D 1 if x 2 X, and 1X .x/ D 0 elsewhere.
Let us recall the following product laws stated in [6]. It will be useful to handle the nonlinear part.
Lemma 2.1
Let s, s0 be two real numbers such that s < 3=2 and s C s0 > 0. There exists a positive constant C :D C.s, s0 /, such that for all f 2 Hs .T 3 /
0
and g 2 Hs .T 3 /,
kfgk sCs0 3 3 C kf kHs .T 3 / kgkHs0 .T 3 / C kf kHs0 .T 3 / kgkHs .T 3 / .
H
2
.T /
If s, s0 < 3=2 and s C s0 > 0, there exists a constant c D c.s, s0 /,
kfgk
0 3
2
HsCs
.T 3 /
ckf kHs .T 3 / kgkHs0 .T 3 / .
Remark 2.2
Without loss of generality, we can consider a zero mean initial conditions of zero mean. That is,
Z
u0 .x/dx D 0.
T3
3. Proof of Theorem 1.3
Let us first prove the following lemma.
Lemma 3.1
Let > 5=2. For any regular function f , we have
1
5
5
krf kL1 kf kL2 .T23 / kf kH2 .T 3 / .
Proof
For r 2 R and a :D .ak /k2Z3 , we define Xr .a/ by
Xr .a/ :D
X
2r
jkj jak j
k
2
!1=2
.
Suppose that
f :D
X
ak eikx .
k
Without loss of generality, we may assume that a0 D 0 and the function f not to be equal to zero.
For 1, we have
X
X
X
jkj.jak j
jkj.jak j C
jkj.jak j D
k
jkj>
jkj
0
@
X
jkj
11=2 0
jak j2 A
0
X0 .a/ @
X
jkj
Copyright © 2012 John Wiley & Sons, Ltd.
jkj
@
X
jkj
11=2
2A
11=2
jkj2 A
0
C X .a/ @
0
C@
X
jkj>
X
jkj>
jkj
11=2 0
jkj2 jak j2 A
11=2
@
X
jkj>
11=2
jkj2C2 A
.
2C2 A
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
As,
X
jkj
jkj2 c
Z
ŒC1
r4 dr
1
c.ΠC 1/5 =5
,
c5
and
X
jkj>
jkj2C2 c
Z
1
r2C4 dr
Œ
Œ2C5
2 5
2C5
c
2 5
,
c
we obtain
X
k
jkj.jak j c1 5=2 X0 .a/ C q
1
0
c2
5
5
2
. 2 / X .a/
.
c2 C 5=2
B
. 25 /
X
.a/
C
X
.a/
@c1 C q
A
0
52
We chose , such that
5
5=2 X0 .a/ D . 2 / X .a/,
that is
D .X .a/=X0 .a//1= 1.
This finishes the proof of the lemma.
Now, we return to the proof of Theorem 1.3.
By (3) and the Gronwall lemma, we have
ku.t/k2Hm ku.a/k2Hm eC.m/
Rt
a
kru. /kL1 d
.
(10)
Combining Lemma 3.1 and (10), we obtain
4m
5
kru.t/kL1
4m
5 2
L2
C.m/ku.t/k
ku.a/k2Hm eC.m/
Rt
a
kru. /kL1 d
.
Using the L2 -energy estimate, we have
kru.t/kL1 eC.m/
Rt
a
kru. /kL1 d
5
1 2m
C.m/ku0 kL2
5
ku.a/kH2mm .
We integrate on Œa, T to infer that
1 eC.m/
RT
a
kru.t/kL1
1
5
5
C.m/ku0 kL2 .T2m3 / ku.a/kH2mm .T a/.
Using Theorem 1.2 and by taking the limit as T ! T , we obtain
1
5
5
1 C.m/ku0 kL2 .T2m3 / ku.a/kH2mm .T a/.
R
For 0 t0 < T , we consider the system with the unknown functions .v, p1 / such that T 3 p1 .t, x/dx D 0,
8
< @t v v C .v.r/v D rp1 , in RC R3
t
.
.NS / 0
div v D 0 in RC R3
:
v.0/ D u.t0 / in R3
Copyright © 2012 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
The maximal solution given by Theorems 1.1 and 1.2 denoted by v belongs to C.Œ0, T /, Hs / where T D T t0 . Moreover, for
every 0 t < T , we have
5
1 2m
1 C.m/kv.0/kL2
5
kv.t/kH2mm c.m/.T t0 t/.
So,
5
1 2m
1 C.m/ku.t0 /kL2
5
ku.t C t0 /kH2mm .T t0 t/.
Taking t D 0, we obtain the desired result.
4. Proof of Theorem 1.4
From Remark 2.2, we can suppose ub0 .0/ D 0. Then, for all t 2 Œ0, T /,
b
u.t, 0/ D 0.
4.1. Proof of (i)
This proof is carried out in two steps.
Step 1. Taking the scalar product in Hs , we obtain
1
@t kuk2Hs C kruk2Hs j < u.t/.ru=u >Hs j
2
j < div.u ˝ u/=u >Hs j
.
j < u ˝ u=ru >Hs j
X
.1 C jkj2 /s jF .u ˝ u/.k/j.jF .ru/.k/j
k¤0
Cauchy–Schwarz inequality gives
0
11=2
X
1
@t kuk2Hs C kruk2Hs @ .1 C jkj2 /s jF .u ˝ u/.k/j2 A .krukHs ,
2
k¤0
and
X
.1 C jkj2 /s jF .u ˝ u/.k/j2 2s
k¤0
s
2
s
2
s
2
X
k¤0
X
k
X
k
X
k
jkj2s jF .u ˝ u/.k/j2
jkj2s jF .u/ F .u/.k/j2
.
jkj2s .jF .u/j jF .u/j.k//2
jkj
2s
X
p
jF .u/.p/jjF .u/.k p/j
!2
Using the elementary inequality jkjs 2s jpjs C 2s jk pjs and its symmetry regarding the elements .k p/ and p, we obtain the
estimates
!2
X X
X
s
2 s
2
3sC1
jpj jF .u/.p/jjF .u/.k p/j
.1 C jkj / jF .u ˝ u/.k/j 2
p
k
k¤0
23sC1
3sC1
2
The Young inequality yields
X
X X
p
k
jpjs jF .u/.p/jjF .u/.k p/j
k j.j jF .u/j jF .u/jk2L2
.
dk
s
.1 C jkj2 /s jF .u ˝ u/.k/j2 23sC1 kj.js jF .u/jk2L2 kF .u/k2L1
k¤0
Copyright © 2012 John Wiley & Sons, Ltd.
!2
.
23sC1 kuk2Hs kF .u/k2L1
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
It follows that
1
@t ku.t/k2Hs C kru.t/k2Hs 2.3sC1/=2 kukHs kF .u/kL1 krukHs .
2
The elementary inequality ˛ˇ 12 ˛ 2 C 12 ˇ 2 implies that
1
@t ku.t/k2Hs C kru.t/k2Hs 23sC1 1 kuk2Hs kF .u/k2L1 C kruk2Hs .
2
2
So, one infers that
@t ku.t/k2Hs C kru.t/k2Hs 23sC2 1 kuk2Hs kF .u/k2L1 .
By the Gronwall lemma, for every 0 a t < T , one obtains that
c.s/ 1
ku.t/k2Hs ku.a/k2Hs e
Rt
a
kF .u/. /k2 1 d
L
.
The fact that lim supt!T ku.t/k2Hs D 1 implies that
Z
T
a
kF .u/. /k2L1 d D 1, 80 a < T .
(11)
Step 2. Applying Fourier transformation to the first equation of .NS /, we obtain
1
@t jF .u/.t, k/j2 C jkj2 jF .u/.t, k/j2 C Re .F .u.ru/.t, k/.F .u/.t, k// D 0.
2
As, for any " > 0, we have
1
1
@t jF .u/.t, k/j2 D @t jF .u/.t, k/j2 C "
2
2
q
q
D jF .u/.t, k/j2 C "@t jF .u/.t, k/j2 C ",
it follows that
@t
Consequently, we have
!
q
F .u.ru/.t, k/.F .u/.t, k/
jF .u/.t, k/j2
C Re
jF .u/.t, k/j2 C " C jkj2 p
D 0.
p
jF .u/.t, k/j2 C "
jF .u/.t, k/j2 C "
@t
Integrating over Œa, t, we obtain
q
q
jF .u/.t, k/j2
jF .u.ru/.t, k/j.
jF .u/.t, k/j2 C " C jkj2 p
jF .u/.t, k/j2 C "
jF .u/.t, k/j2 C " C jkj2
Letting " ! 0, we have
jF .u/.t, k/j C jkj2
Z
a
Z
a
t
q
jF .u/.a, k/j2 C "
jF .u/.t, k/j2
p
jF .u/.t, k/j2 C "
Z
C
t
jF .u/.t, k/j jF .u/.a, k/j C
t
a
Z
a
jF .u.ru/. , k/jd .
t
jF .u.ru/. , k/jd .
Using Young and Cauchy–Schwarz inequalities, we deduce that
kF .u/.t/kL1 C
Z
a
t
kF .u/kL1 kF .u/.a/kL1 C
Z
a
t
3=2
1=2
kF .u/kL1 kF .u/kL1 .
Using the inequality
˛ˇ
Copyright © 2012 John Wiley & Sons, Ltd.
˛2
ˇ2
C
,
2
2
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
we obtained
kF .u/.t/kL1 C
Z
t
a
Z
kF .u/kL1 kF .u/.a/kL1 C 1
t
kF .u/k3L1 C
a
4
Z
t
a
kF .u/kL1 .
The Gronwall inequality implies that
Rt
1
a
kF .u/.t/kL1 kF .u/.a/kL1 e
kF .u/k2
L1
.
It follows that
2 1
kF .u/.t/k2L1 e
Rt
a
kF .u/k2
L1
kF .u/.a/k2L1 .
Integrating over the interval Œa, T / and taking into account the result given by Equation (11), we obtain
kF .u/.a/k2L1 .T a/.
2
This finishes the proof.
4.2. Proof of (ii)
We begin by proving the following lemma.
Lemma 4.1
For > 3=2 and a :D .ak /Z3 f0g , we have
X
k2Z3 f0g
0
X
jak j c. / @
k2Z3 f0g
jak j
11
3
2
2A
0
X
@
k2Z3 f0g
jkj
2
jak j
Remark 4.1
3
Using the above lemma, we can deduce that for > 3=2 and .ak / 2 C Z that
X
k2Z3
0
jak j c. / @
X
k2Z3
11
jak j2 A
Proof of Lemma 4.1.
Suppose that a ¤ 0. For 1, we have
X
X
X
jak j
jak j C
jak j D
k
jkj
jkj>
0
11=2 0
@
X
jkj
jak j
2A
@
0
By choosing such that
X
jkj
c1 3=2 X0 .a/ C q
1
c2
3
2
0
3
2
11=2
1A
@
X
k2Z3
0
C@
X
jkj>
1
1
jkj
jak j
3
. 2 / X .a/
11=2 0
2A
@
.
3
2
.1 C jkj2 / jak j2 A
2
3
2
2A
X
jkj>
.
jkj
11=2
2 A
c2 C 3=2
B
. 23 /
X .a/ .
@c1 C q
A X0 .a/ C
23
3
3=2 X0 .a/ D . 2 / X .a/,
that is
D .X .a/=X0 .a//1= 1,
we obtain the desired result.
Now, we return to the proof of Theorem 1.4-(ii). According to the first step of Section 4.1, we have
c.s/ 1
ku.t/k2H ku.a/k2H e
Copyright © 2012 John Wiley & Sons, Ltd.
Rt
a
kF .u/. /k2 1 d
L
, 80 a t < T .
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
Lemma 4.1 implies that
4
kF .u/.t/kL31
4
3
L2
c. /ku.t/k
c./ 1
ku.t/k2H ku.a/k2H e
2
Rt
kF .u/. /k2 1 d
a
L
.
So, it follows that
4
4
kF .u/.t/kL31 c. /ku.t/kL23
4
c. /ku0 kL23
2
2
c./ 1
ku.a/k2H e
c./
ku.a/k2Hs e
R
1 t
a
Rt
a
kF .u/. /k2 1 d
L
kF .u/. /k2 1 d
L
.
Then, the following holds
kF .u/.t/k2L1 .T 3 / e
Rt
RT
kF .u/. /k21
c./ 1
a
kF .u/. /k21 d
L
3
2 3
c.s/ku0 kL2 .T 3 / ku.a/kH .
We integrate on Œa, T to obtain
c./ 1
1e
a
L
2 3
3
c. / 1 ku0 kL2 .T 3 / ku.a/kH .T a/.
Taking the limit as T ! T , we deduce that
2 3
3
1 c. / 1 ku0 kL2
ku.a/kH .T a/.
This means that
1 2
3
c. /=3 =3 ku0 kL2
.T t/=3 ku.t/kH , 80 t < T .
4.3. Proof of (iii)
This proof is the generalization of the Leray property (1).
By taking the scalar product in HP , where 2 Œ1, s, we obtain
1
@t kuk2P C kruk2P ku ˝ ukHP krukHP .
H
H
2
By Lemma 2.1 and the interpolation results, we infer that
1
@t kuk2P C kruk2P c. /kukHP 1 kuk P C 1 krukHP
H
H
2
H
2
3=2
H
1=2
H
c. /kukHP 1 kuk P kruk P .
Inequality ab 14 a4 C 34 b4=3 yields
1
@t kuk2P C kruk2P c. / 3 kuk4P 1 kuk2P C kruk2P .
H
H
H
H
H
2
2
Then, it follows that
@t kuk2P C kruk2P c. / 1 kuk4P 1 kuk2P .
H
H
H
Using the Gronwall lemma, we obtain for every 0 a t < T that
c./ 1
ku.t/k2P ku.a/k2P e
H
H
Rt
a
H
ku. /k4P 1 d
H
(12)
.
In particular, the following holds
Z
T
a
ku. /k4P 1 d D 1, 8 a 2 Œ0, T /.
H
(13)
Using the interpolation result
1 1
ku.t/kHP 1 ku.t/kL2
Copyright © 2012 John Wiley & Sons, Ltd.
1=
H
ku.t/k P ,
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
we obtain that
c./ 3
ku.t/k4P 1 e
H
Rt
a
ku. /k4P 1 d
H
4 4
ku.t/kL2
4=
H
ku.a/k P .
The L2 -energy estimate (2) yields
c./ 3
ku.t/k4P 1 e
H
Rt
a
ku. /k4P 1 d
H
4 4
ku0 kL2
4=
H
ku.a/k P .
Integrating over Œa, T / and using the blow-up criterion (13), we deduce that
4 4
1 c. / 3 ku0 kL2
4=
H
ku.a/k P .T a/.
This completes the proof of Theorem 1.4.
5. Proof of Theorem 1.5
Suppose that a sequence of positive real numbers n exists and verifies both n ! 0 and Tn T < T0 . Let wn :D un u0 , and
we have
@t wn n wn D .wn .r/wn .wn .r/u0 .u0 .r/wn C .n 0 /u0 rpn .
Taking the scalar product in H3 , we obtain
@t kwn k2H3 C 2n krwn k2H3 ckwn k3H3 C ckru0 .t/kH3 kwn k2H3
C 2jn 0 jkru0 .t/kH3 krwn .t/kH3 .
Consequently, one obtains
@t kwn .t/k2H3 C n krwn .t/k2H3 ckwn .t/k3H3 C ckru0 .t/kH3 kwn k2H3
C jn 0 jkru0 .t/k2H3 .
By the Gronwall Lemma, we infer that
kwn .t/k2H3
jn 0 j
Z
t
kru0 .t/k2H3
0
Rt
c 0 kwn kH3
c
kwn .t/kH3 e 2
Rt
0
kwn kH3
Integrating over Œ0, Tn / and using Theorem 1.2, we deduce that
1
ec
Rt
0
kwn kH3 Cc
Rt
0
kru0 kH3
,
jn 0 jMT e
RT
R
T
where MT :D 0 kru0 .t/k2H3 ec 0 kru0 kH3 .
Then, it follows that
p
jn 0 jMT .
c p
T jn 0 jMT .
2
Taking the limit as n ! 1, we obtain a contradiction. This completes the proof of Theorem 1.3.
6. Proof of Theorem 1.6
Suppose that a sequence .u0n /n exists in Hs and verifies both ku0n u0 kHs ! 0 and T .u0n / T < T .u0 /. Let u (resp. un ) be the solution
of (NS ) with initial condition u0 (resp. u0n ).
Let Wn be the difference defined by Wn :D un u. We have
@t Wn Wn D .Wn .r/Wn .Wn .r/u .u.r/Wn rpn .
Taking the scalar product in H3 , we obtain
@t kWn k2H3 C krWn k2H3 ckWn k3H3 C ckru.t/kH3 kWn k2H3 .
Copyright © 2012 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2012
J. BENAMEUR
Applying the Gronwall lemma, one infers that
kWn .t/k2H3 ku0n u0 k2H3 ec
Rt
0
kWn kH3 Cc
Rt
0
krukH3
.
This implies that
c
kWn .t/kH3 e 2
Rt
0
kWn kH3
c
ku0n u0 kH3 e 2
RT
0
krukH3
.
Integrating over Œ0, T .u0n // and using Theorem 1.2, we deduce that
1
R
c T
c
Tku0n u0 kH3 e 2 0 krukH3 .
2
Taking the limit as n ! 1, we obtain a contradiction.
7. General remarks
In this section, we give some important remarks about the results in Theorems 1.3–1.5.
In (7), if we take D 1, we obtain the Leray property (1).
Combining the blow-up result (4), and the energy estimate (2), we can deduce the following lower bound:
1 2m
5
C.m/ku0 kL2
.T t/2m=5
ku.t/kHm , 8t 2 Œ0, T /.
(14)
Acknowledgements
The author extends its appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the
research group project no. RGP-VPP-117. The author thanks the referee for his/her careful reading of the manuscript and corrections.
References
1.
2.
3.
4.
Leray J. Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica 1933; 63:22–25.
Leray J. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica 1934; 63:193–248.
Kato T. Quasi-Linear equations of evolution, with applications to differential equations. Lecture Notes in Mathematics 1975; 448:25–70.
Beale J, Kato T, Majda A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Communications in Mathematical Physics 1984;
94:61–66.
5. Kato T, Ponce G. Commutator estimates and the Euler and Navier–Stokes equations. Communications on Pure and Applied Mathematics 1988;
41:891–907.
6. Chemin J-Y. About Navier–Stokes System. Publication du Laboratoire Jaques-Louis Lions: Université de Paris VI, 1996.
Copyright © 2012 John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci. 2012
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