J. Differential Equations 248 (2010) 328–341
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Journal of Differential Equations
www.elsevier.com/locate/jde
Remarks on the blowup criteria for Oldroyd models
Zhen Lei a,∗ , Nader Masmoudi b , Yi Zhou a
a
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433,
China
b
Courant Institute of Mathematical Sciences, New York University, NY 10012, USA
a r t i c l e
i n f o
Article history:
Received 8 April 2009
Revised 16 June 2009
Available online 24 July 2009
Keywords:
Losing a priori estimate
Blowup criteria
Oldroyd model
a b s t r a c t
We provide a new method to prove and improve the Chemin–
Masmoudi criterion for viscoelastic systems of Oldroyd type in
[J.Y. Chemin, N. Masmoudi, About lifespan of regular solutions of
equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (1)
(2001) 84–112] in two space dimensions. Our method is much
easier than the one based on the well-known losing a priori estimate
and is expected to be easily adopted to other problems involving
the losing a priori estimate.
© 2009 Elsevier Inc. All rights reserved.
1. Introduction
In this paper, we are going to study the non-blowup criteria of solutions of a type of incompressible non-Newtonian fluid flows described by the Oldroyd-B model in the whole 2-D space:
⎧
⎨ ∂t v + v · ∇ v + ∇ p = ν v + μ1 ∇ · τ ,
∂ τ + v · ∇ τ + aτ = Q (τ , ∇ v ) + μ2 D ( v ),
⎩ t
∇ · v = 0,
(1.1)
where v is the velocity field, τ is the non-Newtonian part of the stress tensor and p is the pressure.
The constants ν (the viscosity of the fluid), a (the reciprocal of the relaxation time), μ1 and μ2
*
Corresponding author.
E-mail addresses: [email protected], [email protected] (Z. Lei), [email protected] (N. Masmoudi),
[email protected] (Y. Zhou).
0022-0396/$ – see front matter
doi:10.1016/j.jde.2009.07.011
© 2009 Elsevier Inc.
All rights reserved.
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
329
(determined by the dynamical viscosity of the fluid, the retardation time and a) are assumed to be
non-negative. The bilinear term Q has the following form:
Q (τ , ∇ v ) = W ( v )τ − τ W ( v ) + b D ( v )τ + τ D ( v ) .
∇ v +(∇ v )t
(1.2)
∇ v −(∇ v )t
Here b ∈ [−1, 1] is a constant, D ( v ) =
is the deformation tensor and W ( v ) =
is
2
2
the vorticity tensor. Fluids of this type have both elastic properties and viscous properties. More
discussions and the derivation of Oldroyd-B model (1.1) can be found in Oldroyd [22] or Chemin and
Masmoudi [5].
There has been a lot of work on the existence theory of Oldroyd model [5,8–10,14,17]. In particular,
the following theorem is established by Chemin and Masmoudi in [5]:
Theorem (Chemin and Masmoudi). In two space dimensions, the solutions to the Oldroyd model (1.1) with
smooth initial data do not develop singularities for t T provided that
T
τ (t , ·)
L∞
2
+ |b|τ (t , ·) L 2 dt < ∞.
(1.3)
0
To establish the blowup criterion (1.3), the authors in [5] use a losing a priori estimate for solutions
of transport equations which was developed by Bahouri and Chemin [1] and used later on by a lot
of authors (for example, see [5–7,16,18,19,21] and the references therein). Our purpose of this paper
is to provide a simple method which avoids using the complicated losing a priori estimate and to
improve the blowup criterion (1.3) for Oldroyd model (1.1) established by Chemin and Masmoudi [5].
To best illustrate our ideas and for simplicity, we will take a = 0 and ν = μ1 = μ2 = b = 1 throughout
this paper. More precisely, we study the following system
⎧
⎨ ∂t v + v · ∇ v + ∇ p = v + ∇ · τ ,
∂ τ + v · ∇ τ = ∇ v τ + τ (∇ v )t + D ( v ),
⎩ t
∇ · v = 0.
(1.4)
We point out here that the results in this paper are obviously true for general constants a, μ1 , μ2 0,
ν , b > 0 from our proofs.
Our main result concerning system (1.4) is:
Theorem 1.1. Assume that ( v , τ ) is a local smooth solution to the Oldroyd model (1.4) on [0, T ) and
v (0, ·) L 2 ∩Ċ 1+α (R2 ) + τ (0, ·) L 1 ∩Ċ α (R2 ) < ∞ for some α ∈ (0, 1). Then one has
v (t , ·)
Ċ 1+α
+ τ (t , ·)Ċ α < ∞
for all 0 t T provided that
τ (t , ·)
L 1T (BMO)
< ∞ and τ L ∞ ( L 1 ) < ∞.
T
(1.5)
Remark 1.2. This result is in the spirit of the Beale–Kato–Majda [2] non-blowup criterion for 3-D Euler
equations. There were many subsequent results improving the criterion, see for instance [12,13,23].
In particular our result still holds if we replace BMO with the Besov space B 0∞,∞ used in H. Kozono,
T. Ogawa, and Y. Taniuchi [12] or if we replace the condition by the one introduced in Planchon [23].
In other words, the first condition in (1.5) in the above theorem can be weakened to
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Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
T
T
τ B 0∞,∞ =
sup q τ (t , ·) L ∞ dt < ∞,
q
0
0
or to
T
q τ (t , ·)
lim sup
δ→0 q
for some sufficiently small
dt < L∞
T −δ
> 0. The second condition in (1.5) can be replaced by
τ L 2 ( L 2 ) < ∞,
T
which was used in [5].
Remark 1.3. It is easy to check that smooth solutions to (1.4) enjoy the following energy law:
v (t , ·)2 + tr τ (t , ·) dx +
t |∇ v |2 dx ds =
R2
0
R2
v (0, ·)2 + tr τ (0, ·) dx,
(1.6)
R2
which means that
2
∩ L 2T Ḣ 1
v ∈ L∞
T L
(1.7)
for all T > 0 under the second condition of (1.5). The a priori estimate (1.7) will be important to apply
Lemma 3.1.
Finally, it is well known that if A = 2τ + I is a positive definite symmetric matrix at t = 0 (which
is actually the physical case), then this property is conserved for later times. Indeed, A satisfies the
equation
∂t A + v · ∇ A = ∇ v A + A (∇ v )t .
Also, if at t = 0, we have det( A (0)) > 1 and A is positive definite, then this will also hold for later
times (see [11]). In particular this implies that tr(τ ) > 0 (or one has −1 < tr(τ ) 0, which contradicts
with det( A ) > 1). Hence, we have the following corollary where we also use the improved criterion
of Planchon.
Corollary 1.4. There exists an > 0, such that if ( v , τ ) is a local smooth solution to the Oldroyd model (1.4)
on [0, T ), v (0, ·) L 2 ∩Ċ 1+α (R2 ) + τ (0, ·) L 1 ∩Ċ α (R2 ) < ∞ for some α ∈ (0, 1) and that det( I + 2τ (0)) > 1,
A = I + 2τ (0) is positive definite symmetric, then one has
v (t , ·)
Ċ 1+α
+ τ (t , ·)Ċ α < ∞
for all 0 t T provided that
T
lim sup
δ→0 q
T −δ
q τ (t , ·)
L∞
dt < .
(1.8)
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
331
Our proof is based on careful Hölder estimates of heat and transport equations and the standard
Littlewood–Paley theory, which is much easier than the extensively used losing a priori estimates (for
example, see [1,5–7,16]). In fact, the main innovation of this paper is that our analysis may be viewed
as a replacement of the losing a priori estimate. Our method is expected to be easily adopted to other
problems via the losing a priori estimate. Moreover, our criterion slightly improves the one established
by Chemin and Masmoudi (see [5]).
Finally, let us make a remark on MHD:
⎧
⎨ ∂t v + v · ∇ v + ∇ p = ν v + ∇ · ( H × H ),
∂ H + v · ∇ H = H · ∇ v,
⎩ t
∇ · v = ∇ · H = 0,
(1.9)
where H denotes the magnetic field. A direct corollary of Theorem 1.1 for MHD (1.9) is the following:
Corollary 1.5. Assume that ( v , H ) is a local smooth solution to MHD (1.9) on [0, T ) and v (0, ·) L 2 ∩Ċ 1+α +
H (0, ·) L 2 ∩Ċ α < ∞ for some α ∈ (0, 1). Then one has
v (t , ·)
Ċ 1+α
+ H (t , ·)Ċ α < ∞
for all 0 t T provided that
T
( H × H )(t , ·)
BMO
dt < ∞.
(1.10)
0
The proof of this corollary is given in Section 4. Unfortunately, at present we are not able to
improve (1.10) as
T
H (t , ·)2
BMO
dt < ∞,
0
and this is still an open problem.
The paper is organized as follows: Section 2 is devoted to recalling some basic properties of
Littlewood–Paley theory and proving two interpolation inequalities. The proof of Theorem 1.1 is given
in Section 3. In the last section we sketch the proof of Corollary 1.5.
2. Preliminaries
Let S (R2 ) be the Schwartz class of rapidly decreasing functions. Given f ∈ S (R2 ), its Fourier
F f = f̂ (inverse
Fourier transform F −1 g = ğ, respectively) is defined by f̂ (ξ ) =
transform
−ix·ξ
e
f (x) dx ( ğ (x) = e ix·ξ g (ξ ) dξ , respectively). Now let us recall the Littlewood–Paley decomposition (see [3,4]). Choose two non-negative radial functions ψ, φ ∈ S (R2 ), supported respectively in
B = {ξ ∈ R2 : |ξ | 43 } and C = {ξ ∈ R2 : 34 |ξ | 83 } such that
ψ(ξ ) +
ξ φ j = 1 for ξ ∈ R 2 ,
j 0
2
−∞ j ∞
φ
ξ
2j
= 1 for ξ ∈ R 2 \ {0}.
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Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
The frequency localization operator is defined by
q f =
φ̌( y ) f x − 2−q y dy ,
Sq f =
R2
ψ̌( y ) f x − 2−q y dy .
(2.1)
R2
The following lemma is well known (for example, see [4]).
Lemma 2.1. For s ∈ R, 1 p ∞ and integer q, one has
⎧ qs
c2 q f L p ∇ s q f L p C 2qs q f L p ,
⎪
⎪
⎨
|∇|s S q f p C 2qs f L p ,
L
⎪
⎪
2q
⎩ −C 22q t
ce
q f L ∞ et q f L ∞ C e −c2 t q f L ∞ .
(2.2)
Here C and c are positive constants independent of s, p and q.
We also need the following lemma (see also [20,21] where similar estimates were used).
Lemma 2.2. Assume that β > 0. Then there exists a positive constant C > 0 such that
⎧
f L ∞ C 1 + f L 2 + f BMO ln e + f Ċ β ,
⎪
⎪
⎪
⎪
⎪
T
T
⎪
⎪
⎪
⎪
g (s, ·) 2 ds
⎪
g
(
s
,
·)
ds
C
1
+
∇
⎨
L∞
L
(2.3)
0
0
⎪
⎪
⎪
T
⎪
T
⎪
⎪
⎪
⎪
+ sup q ∇ g (s, ·) L ∞ ds ln e + ∇ g (s, ·)Ċ β ds .
⎪
⎪
⎩
q
0
0
Proof. The first inequality is well known. For example, see [2,13,15]. To prove the second inequality,
we use the Littlewood–Paley theory to compute that
T
∇ g (s, ·)
T ds
C
∇
g
(
s
,
·)
q
∞
L
0
q0
0
+
T C
L∞
ds + C N max
1q N
2−β q 2β q q ∇ g (s, ·) L ∞ ds
g (s, ·) 2 ds + sup
L
+ 2−β N
T
1q N
0
T
q ∇ g (s, ·)
0
0 q N +1
T
T
N q ∇ g (s, ·) L ∞ ds
0
∇ g (s, ·) β ds .
Ċ
0
Then the second inequality in Lemma 2.2 follows by choosing
N=
1
β
T
log2 e +
0
∇ g (s, ·) β
Ċ
T
C ln e +
0
∇ g (s, ·)
Ċ β
ds .
2
L∞
ds
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
333
3. Blowup criteria for Oldroyd-B model
This section is devoted to establishing the blowup criterion for the Oldroyd-B model (1.1) and
proving Theorem 1.1. Our analysis is based on careful Hölder estimates of heat and transport equations and the standard Littlewood–Paley theory, which is much easier than the extensively used losing
a priori estimates (for example, see [1,5–7,16]). Moreover, our criterion slightly improves the one established by Chemin and Masmoudi (see [5]). We divide our proof into two steps. The first step is
focused on establishing some a priori estimates for 2-D Navier–Stokes equations. Then we establish
Hölder estimates for the velocity field v and the stress tensor τ in the second step.
Step 1. The a priori estimates for 2-D Navier–Stokes equations. We need the following lemma which
is basically established by Chemin and Masmoudi in [5]. For completeness, the proof will be also
sketched here.
Lemma 3.1 (Chemin–Masmoudi). Let v be a solution of the Navier–Stokes equations with initial data in L 2
and an external force f ∈ L 1T (C −1 ) ∩ L 2T ( H −1 ):
⎧
⎨ ∂t v + v · ∇ v + ∇ p = v + f ,
∇ · v = 0,
⎩
v (0, x) = v 0 (x).
(3.1)
Then we have the following a priori estimate:
v L 1 (C 1 ) C sup q v 0 L 2 1 − exp −c22q T + v 0 L 2 + f L 2 ( Ḣ −1 ) ∇ v 2L 2 ( L 2 )
T
T
q
T
+ sup
q
−q
2 q f (s)
T
L∞
ds .
(3.2)
0
Proof. First of all, applying the operator q to the 2-D Navier–Stokes equations (3.1) and then using
Lemma 2.1 and the standard energy estimate, we deduce that
1 d
2 dt
q v 2L 2 + c22q q v 2L 2 2q q v L 2 2−q q f L 2 + q ( v ⊗ v )L 2
2
2 c22q q v 2L 2 + C 2−q q f L 2 + q ( v ⊗ v ) L 2 .
Integrating with respect to time and summing over q, we get
q
q v 2L ∞ ( L 2 ) v 0 2L 2 + C f 2L 2 ( Ḣ −1 ) + v ⊗ v 2L 2 ( L 2 )
T
T
T
v 0 2L 2 + C f 2L 2 ( Ḣ −1 ) + v 2L ∞ ( L 2 ) ∇ v 2L 2 ( L 2 ) ,
T
T
T
where we used the standard interpolation inequality v 2L 4 C v L 2 ∇ v L 2 . Recalling the basic energy estimate
v 2L ∞ ( L 2 ) + ∇ v 2L 2 ( L 2 ) v 0 2L 2 + f 2L 2 ( Ḣ −1 ) ,
T
T
T
(3.3)
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Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
one has
q v 2L ∞ ( L 2 ) C v 0 2L 2 + f 2L 2 ( Ḣ −1 ) 1 + v 0 2L 2 + f 2L 2 ( Ḣ −1 ) .
T
q
T
(3.4)
T
Next, let us apply q to (3.1) and use Lemma 2.1 to estimate
q v (t )
L∞
C q v 0 L ∞ e
−c22q t
t
+
q f (s)
L∞
2q
+ q ∇ · ( v ⊗ v )(s) L ∞ e −c2 (t −s) ds,
0
which yields
T
v L 1 (C 1 ) C sup
T
q
q v 0 L ∞ 2q e −c2
2q
t
dt
0
T t
+ C sup
q
0
0
T
t
0
0
+ C sup
q
q f (s) ∞ 2q e −c22q (t −s) ds dt
L
q ∇ · ( v ⊗ v )(s) ∞ 2q e −c22q (t −s) ds dt
L
2q C sup q v 0 L 2 1 − e −c2 T
q
T
+ C sup
q ( v ⊗ v )(s) ∞ ds + f 1 −1 .
L (C )
L
T
q
0
Using the Bony’s decomposition, one can write
q ( v ⊗ v )(s) ∞ =
q ( p v ⊗ r v )(s) ∞ +
q ( p v ⊗ r v )(s) ∞
L
L
L
| p −r |1
q ( p v ⊗ r v )(s) ∞ .
+
L
r − p 2
A straightforward computation gives
T q ( p v ⊗ r v )(s)
0 | p −r |1
C
T 0 | p −r |1
L∞
ds
2q q ( p v ⊗ r v )(s) L 2 ds
p −r 2
(3.5)
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
T
C
2q−
p +r
2
1
1 p
1
1
2 p v 2∞ r v 22 p v 2∞ 2r r v 22 ds
2q−
p +r
2
1 p
1
1 1
2 p v 2∞ r v 22 2 p p v 22 2r r v 22 ds
L
0 | p −r |1, p q−3
T
C
335
L
0 | p −r |1, p q−3
1
L
L
L
L
L
L
1
C v L2∞ ( L 2 ) ∇ v L 2 ( L 2 ) v 21
L T (C 1 )
T
T
.
Similarly, one has
T q ( p v ⊗ r v )(s)
0
p −r 2
T
C
q ( p v ⊗ r v )(s)
+
r − p 2
ds
L∞
p v L ∞ r v L ∞ ds
p −r 2, | p −q|2
0
T
C
0
L∞
p
1 1
p
2 p v 2∞ 2 p p v 22 2r − 2 r v 2 ds
L
L
L
p −r 2, | p −q|2
1
1
C v L2∞ ( L 2 ) ∇ v L 2 ( L 2 ) v 21
L T (C 1 )
T
T
.
Using the above two estimates, one can improve (3.5) as
2q v L 1 (C 1 ) C sup q v 0 L 2 1 − e −c2 T + v L ∞ ( L 2 ) ∇ v 2L 2 ( L 2 ) + f L 1 (C −1 ) .
T
T
q
T
T
Consequently, one can deduce (3.2) from the basic energy estimate (3.3) and the above inequality.
2
Now let us assume that f ∈ L 1T (Ċ −1 ) ∩ L 2T ( H −1 ). By Lemma 3.1, it is easy to see that
v L 1
[t 0 , T ]
2q
(C 1 ) C sup q v (t 0 ) L 2 1 − exp −c2 ( T − t 0 ) +
T
q
+ v (t 0 )
sup 2−q q f (s) L ∞ ds
q
t0
+ f L 2 ( Ḣ −1 ) ∇ v 2L 2 ( L 2 )
L2
[t 0 , T ]
[t 0 , T ]
holds for any t 0 ∈ [0, T ). By (3.4), one can choose some q0 such that
sup q v 2L ∞
q>q0
[t 0 , T ] ( L
2)
4C
.
Furthermore, by the basic energy estimate (3.3), one can choose some t 1 ∈ [t 0 , T ) such that
(3.6)
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Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
sup q v (t ) L 2 1 − exp −c22q ( T − t )
sup
t 1 t T qq0
sup v (t )L 2 2c22q0 ( T − t 1 )
t 1 t T
C 22q0 v 0 L 2 + f L 2
−1
[0, T ] ( Ḣ )
(T − t1 ) 4C
.
Consequently, one has
sup sup q v (t ) L 2 1 − exp −c22q ( T − t )
t 1 t T
q
2C
(3.7)
.
On the other hand, it is obvious that one can choose some t 2 ∈ [t 1 , T ) such that
T
sup v (t ) L 2 + f L 2
∇ v 2L 2 ( L 2 )
−1
[t 2 , T ] ( Ḣ )
[t 2 , T ]
t 2 t T
+
sup 2−q q f (s) L ∞ ds q
2C
.
(3.8)
t2
Combining (3.7) and (3.8) with (3.6), one arrives at
v L 1
[t 2 , T ] (C
1)
.
(3.9)
τ . First of all, by (3.9) and the assumption (1.5), one can choose
Step 2. Hölder estimate for v and
t ∈ [t 2 , T ) such that
v L 1
[t , T ] (C
1)
,
τ L 1
[t , T ] (BMO)
.
(3.10)
For 0 t < T , define
A (t ) = sup v (t , ·)Ċ 1+α ,
B (t ) = sup τ (t , ·)Ċ α .
0s<t
0s<t
We are about to estimate A (t ) and B (t ) for 0 t < T . For this purpose, let us apply q to the
Oldroyd-B system (1.4) to get
∂t q v − q v + ∇q p = ∇ · q (τ − v ⊗ v ),
∂t q τ + v · ∇q τ = q ∇ v τ + τ (∇ v )t + D ( v ) + [ v · ∇, q ]τ .
(3.11)
Let us first estimate v (t , ·)Ċ 1+α . By the first equation in (3.11) and Lemma 2.1, one has
q v L∞
Ce
−c22q t q v (0) L ∞ +
t
2q
e −c2 (t −s) ∇ · q (τ − v ⊗ v ) L ∞ (s) ds.
0
Multiplying 2q(1+α ) to both sides of (3.12), we have
(3.12)
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
q v (t , ·) 1+α C v (0, ·) 1+α + C
Ċ
Ċ
t
337
2q
22q e −c2 (t −s) q τ Ċ α ds
0
t
+C
3
2q
2 2 q e −c2 (t −s) ( v ⊗ v )(s, ·)Ċ 1/2+α ds
0
C v (0, ·)
Ċ 1+α
+ B (t ) + C
t
14
v 4L 4 v 4Ċ 1+α
ds
,
0
where we have used Hölder inequality and the fact that v ⊗ v Ċ 1/2+α C v L 4 v Ċ 1+α . Consequently,
there holds
v (t , ·)4 1+α C v (0, ·) 1+α + B (t ) 4 + C
Ċ
Ċ
t
v 4L 4 v 4Ċ 1+α ds
0
C v (0, ·)
Ċ 1+α
4
+ B (t̃ ) + C
t
v (s, ·)22 ∇ v (s, ·)22 v (s, ·)4 1+α ds
L
L
Ċ
0
for any fixed t̃: 0 t̃ < T and t t̃ < T . Here we used the fact that B (t ) is nondecreasing. Consequently, Gronwall’s inequality gives that
4
A (t̃ )4 = sup v (t , ·)Ċ 1+α
0t <t̃
C v (0, ·)
Ċ 1+α
t
2 2
4
+ B (t̃ ) exp C
v (s, ·) L 2 ∇ v (s, ·) L 2 ds .
0
Since t̃ ∈ [0, T ) is arbitrary, using the basic energy inequality (1.7), we in fact have
A (t ) C v (0, ·)Ċ 1+α + B (t ) ,
0 t < T.
(3.13)
Next, by the second equation in (3.11), we have
q τ (t , ·)
L∞
q τ (0, ·)
t
L∞
+
q
2 q v (s, ·)
L∞
+ q ∇ v τ + τ (∇ v )t (s, ·) L ∞
0
+ [ v · ∇, q ]τ (s, ·) L ∞ ds,
which implies that
q τ (t , ·) α C τ (0, ·) α +
Ċ
Ċ
t
0
v Ċ 1+α + ∇ v L ∞ τ Ċ α + τ L ∞ v Ċ 1+α
+ 2αq [ v · ∇, q ]τ (s, ·) L ∞ ds.
(3.14)
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Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
By Bony’s decomposition, one has
[ v · ∇, q ]τ =
| p −q
|1
[ p v · ∇, q ]q
τ
p q
−2
+
[ p v · ∇, q ]q
τ +
[q
v · ∇, q ] p τ
p q
−2
=
[ S q
−1 v · ∇, q ]q
τ + [q
v · ∇, q ] S q
−1 τ
|q
−q|2
+
[ p v · ∇, q ]q
τ .
| p −q
|1
Noting that
[
S q −1 v
h( y ) ( S q
−1 v )(x) − ( S q
−1 v ) x − 2−q y
, q ] f =
one has
[ S q
−1 v , q ] f L∞
C 2−q ∇ S q
−1 v L ∞ f L ∞ .
Consequently, we have
t
2αq [ S q
−1 v · ∇, q ]q
τ (s, ·) L ∞ ds
|q
−q|2 0
t
2αq 2−q ∇ S q
−1 v L ∞ q
∇ τ L ∞ (s, ·) ds
|q
−q|2 0
t
C
∇ S q
−1 v L ∞ 2αq q
τ L ∞ (s, ·) ds
|q
−q|2 0
t
C
∇ v L ∞ τ Ċ α ds.
0
Similarly, we have
t
2αq [q
v · ∇, q ] S q
−1 τ (s, ·) L ∞ ds
|q
−q|2 0
t
|q
−q|2
2αq 2−q q
∇ v L ∞ S q
−1 ∇ τ L ∞ (s, ·) ds
0
t
C
τ L ∞ v Ċ 1+α ds.
0
f x − 2−q y dy ,
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
339
At last, one computes that
t
| p −q
|1 0
αq 2 [ p v · ∇, q ]q
τ (s, ·) L ∞ ds
t
p ,q
q 0
t
+
2(1+α )q [ p v , q ]q
τ (s, ·) L ∞ ds
p ,q
q+2 0
2(1+α )q [ p v , q ]q
τ (s, ·) L ∞ ds
t
C
τ L ∞ v Ċ 1+α ds.
0
The above inequalities yield an improvement of (3.14):
τ (t , ·)
Ċ α
C τ (0, ·)Ċ α + C
t
∇ v L ∞ + τ L ∞ τ (s, ·)Ċ α + v (s, ·)Ċ 1+α ds.
0
Now let us insert (3.13) into (3.15) to get
B (t ) = sup τ (t , ·)Ċ α
0s<t
C τ (0, ·)
t
Ċ α
+C
∇ v L ∞ + τ L ∞ v (0, ·)Ċ 1+α + B (s) ds.
0
Noting that by the inequalities in Lemma 2.2, we can estimate
t
∇ v L ∞ + τ L ∞ ds
0
t
t
∇ v L ∞ + τ L ∞ ds + C
1 + v L 2 + τ L 2 ds
t
0
t
+ C sup
t
∇q v L ∞ ds ln e +
q
t
t
v Ċ 1+α ds + C
0
t
τ BMO ln e + τ Ċ α ds
C + C ln e + Ct v (0, ·)Ċ 1+α + B (t ) + C ln e + B (t )
C + C ln e + v (0, ·) 1+α + B (t ) .
Ċ
(3.15)
340
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
Here C is a positive constant depending on the solution ( v , τ ) on [0, t ]. Consequently, we have
B (t ) C 1 + τ (0, ·)Ċ α + v (0, ·)Ċ 1+α + C
t
∇ v L ∞ + τ L ∞ B (s) ds.
0
Then Gronwall’s inequality yields that
e + v (0, ·)Ċ 1+α + B (t )
t
∇ v L ∞ + τ L ∞ ds
C e + τ (0, ·) Ċ α + v (0, ·) Ċ 1+α exp C
t
C e + τ (0, ·)Ċ α + v (0, ·)Ċ 1+α e C +C ln(e+ v (0,·)Ċ 1+α + B (t ))
C C e + τ (0, ·)Ċ α + v (0, ·)Ċ 1+α e + v (0, ·)Ċ 1+α + B (t ) .
From the above inequalities and (3.13), we have
A (t ) + B (t ) C 1 + τ (0, ·)Ċ α + v (0, ·)Ċ 1+α
by choosing
=
1
2C
2
(3.16)
.
4. Proof of Corollary 1.5
In fact, it is easy to see that the tensor H ⊗ H satisfies the following transport equation:
∂t ( H ⊗ H ) + v · ∇( H ⊗ H ) = ∇ v ( H ⊗ H ) + ( H ⊗ H )(∇ v )t .
(4.1)
Hence, the tensor H ⊗ H − 12 I plays the role of τ . (However, it seems not being able to directly apply
Theorem 1.1 to get Corollary 1.5.) The rest part of the proof of Corollary 1.5 is similar as that of
Theorem 1.1.
In fact, by the assumption v (0, ·) L 2 ∩Ċ 1+α + H (0, ·) L 2 ∩Ċ α < ∞, one can easily derive that
v (0, ·)
L 2 ∩Ċ 1+α (R2 )
+ H ⊗ H (0, ·) L 1 ∩Ċ α (R2 ) < ∞.
(4.2)
Moreover, one has the following energy law
( v , H ) ∞ 2 + 2∇ v 2 2 = ( v , H )(0, ·) 2 ,
L (L )
L (L )
L
(4.3)
H ⊗ H L ∞ ( L 1 ) < ∞.
(4.4)
T
T
which gives that
T
Having (4.2), (4.3) and (4.4) in hand, and noticing the assumption (1.10) and the transport equation
(4.1) for H ⊗ H , one has
v (t , ·)
Ċ 1+α
+ H ⊗ H (t , ·)Ċ α < ∞
Z. Lei et al. / J. Differential Equations 248 (2010) 328–341
341
by exactly the same manner as in Section 3. Coming back to the transport equation for H in (1.9), we
have
H Ċ α < ∞
in a standard manner. This completes the proof of Corollary 1.5.
Acknowledgments
Zhen Lei was in part supported by NSFC (grant No. 10801029), a special Postdoc Science Foundation of China (grant No. 200801175) and SGST 09DZ2272900. Nader Masmoudi was in part supported
by NSF under Grant DMS-0703145. Yi Zhou is partially supported by the NSFC (grant No. 10728101),
the 973 project of the Ministry of Science and Technology of China, the Doctoral Program Foundation
of the Ministry of Education of China, the “111” project (B08018) and SGST 09DZ2272900.
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