Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 299, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
GLOBAL SOLUTIONS FOR 2D COUPLED
BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
HONGJUN GAO, LIN LIN, YAJUN CHU
Abstract. In this article, we study the periodic initial-value problem of the
2D coupled Burgers-complex-Ginzburg-Landau (Burgers-CGL) equations. Applying the Brezis-Gallout inequality which is available in 2D case and establishing some prior estimates, we obtain the existence and uniqueness of a global
solution under certain conditions.
1. Introduction
In this article, we are concerned with the periodic initial-value problem for the
following 2D coupled Burgers-complex-Ginzburg-Landau (Burgers-CGL) equations:
Pt = ξP + (1 + iu)∆P − (1 + iv)|P |2 P − ∇P · ∇Q − rP ∆Q + f (x),
(x, t) ∈ Ω × R+ ,
1
Qt = m∆Q − |∇Q|2 − ω|P |2 + g(x), (x, t) ∈ Ω × R+ ,
2
P (xi + 2L, t) = P (xi , t), Q(xi + 2L, t) = Q(xi , t), (x, t) ∈ Ω × R+ ,
P (x, 0) = P0 (x),
Q(x, 0) = Q0 (x),
x ∈ Ω,
(1.1)
(1.2)
(1.3)
(1.4)
where Ω = [−L, L]×[−L, L], 2L is the period, f (x) and g(x) are given real functions.
The complex function P (x, t) is the rescaled amplitude of the flame oscillations, the
real function Q(x, t) is the deformation of the first front. The Landau coefficients
u, v are real and the coefficients ξ, m are positive, while the coupling coefficients
ω > 0 and r = r1 + ir2 .
The coupled Burgers-CGL equations was derived from the nonlinear evolution of
the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction, having two
flame fronts corresponding to two reaction zones with a finite separation distance
between them (see [2, 9, 12, 16, 17, 18]). It describes the interaction of the excited
oscillatory mode and the damped monotonic mode, that is ξ > 0, m > 0.
Let us recall some previous works which are related to our results. If there
are no terms involved Q in (1.1), then (1.1) reduces to the well-known complex
Ginzburg-Landau equation (CGL) that describes the weakly nonlinear evolution of
2010 Mathematics Subject Classification. 35M13, 35D35.
Key words and phrases. Burgers-CGL equation; prior estimate; global solution;
Gagliardo-Nirenberg inequality; Hölder inequality.
c
2015
Texas State University.
Submitted June 2, 2015. Published 7, 2015.
1
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H. GAO, L. LIN, Y. CHU
EJDE-2015/299
a long-scale instability (see [15]). Many results on the well-posedness and global
attractors for the CGL equation have been established, one can find the details in
[6, 7, 8, 13, 14, 19]. And if taking the coupling coefficient ω = 0, (1.1) would be
the well-known Burgers equation (see [3]). There are also many works concerning
the Burgers equation, one can see [4, 20] and so on. So far, the mathematical analysis and physical study about the coupled Burgers-CGL have been done by a few
researchers. For the periodic initial-value problem of the 1-D Burgers-GinzburgLandau equation, well-posedness and global attractors are obtained in [10]. By
some prior estimates and so-called continuity method, the authors in [11] firstly
showed the global existence of solutions and attractors to 1-D coupled BurgersCGL equations for flames governed by sequential reaction, and then obtained the
existence of the global attractor A ⊂ H 2 (Ω) × H 3 (Ω) for the problem. In addition,
they established the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractors. However, to our knowledge, there is few work on the 2D
coupled Burgers-CGL equations. We will prove global well-posedness of problem
(1.1)-(1.4) as conjectured in [11]. The method used here is well established but
need to be applied skillfully to get the desired result.
The purpose of this article is to study the global solutions of (1.1)-(1.4). The
existence and uniqueness of the global solutions to (1.1)-(1.4) is obtained by virtue
of the Brezis-Gallout inequality, which is available in the 2D case. The main result
in our paper is as follows.
Theorem 1.1. Suppose that P0 (x) ∈ H 2 (Ω), Q0 (x) ∈ H 3 (Ω), f (x) ∈ H 1 (Ω) and
g(x) ∈ H 2 (Ω), for any given T > 0, then (1.1)-(1.4) has the unique solution P (x, t)
and Q(x, t) satisfying
P (x, t) ∈ L∞ 0, T ; H 2 (Ω) , Pt (x, t) ∈ L∞ 0, T ; L2 (Ω) ,
Q(x, t) ∈ L∞ 0, T ; H 3 (Ω) , Qt (x, t) ∈ L∞ 0, T ; H 1 (Ω) .
The rest of this article is organized as follows. In section 2, we briefly give some
preliminaries. Some prior estimates for the solutions to (1.1)-(1.2) are established
in section 3. Then, by the standard method, we can extend the local solutions to
global solutions and subsequently the proof of Theorem 1.1 is presented.
2. Preliminaries
In this section, We firstly recall some concepts and conventional notation. Let
W k,p (Ω) denote the usual kth order Sobolev space with its norm

1/p
R
 P
α p
|D
u|
dx
, 1 ≤ p < ∞,
kukW k,p (Ω) := P |α|≤k Ω

α
p = ∞.
|α|≤k ess supΩ |D u|,
When p = 2, we denote kukH k (Ω) = kukW k,2 (Ω) . The inner product in L2 (Ω) is
defined by
Z
(u, v) = Re
u(x)v̄(x)dx.
Ω
For simplicity, we write kuk = kukL2 . Without any ambiguity, we denote a generic
positive constant by C which may vary from line to line. The following inequalities
play an important role in the proof of the main results in R2 .
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3
Lemma 2.1 (Gagliardo-Nirenberg inequality [5]). Let Ω be a bounded domain with
∂Ω in C m , and let u be any function in W m,r (Ω) ∩ Lq (Ω), 1 ≤ q, r ≤ ∞. For any
integer j, 0 ≤ j ≤ m, and for any number a satisfying j/m ≤ a ≤ 1, set
1 m
1
1
j
+ (1 − a) .
= +a
−
p
n
r
n
q
If m − j − n/r is not a non-negative integer, then
kDj ukLp ≤ CkukaW m,r kuk1−a
Lq .
(2.1)
If m−j −n/r is a non-negative integer, then (2.1) holds for a = j/m. The constant
C depends only on Ω, r, q, j and a.
Lemma 2.2 (Brezis-Gallout inequality [1]). Let Ω be a bounded domain in R2 , and
let u be any function in H 2 (Ω). If kukH 1 ≤ 1, then there holds
p
(2.2)
kukL∞ ≤ C 1 + log(1 + kukH 2 ) .
3. Existence and uniqueness of solutions
In this section, we firstly establish some prior estimates for the solutions to
(1.1)-(1.4) which guarantee the existence of the global solutions.
Lemma 3.1. Assume that g(x) ∈ L2 (Ω), then for the problem (1.1)-(1.4), it holds
that
d
(kQk2 ) ≤ E0 1 + log(1 + kP kH 2 + kQkH 3 ) kP k2H 2 + kQk2H 3 + E1 ,
dt
where E0 , E1 are constants.
Proof. Multiplying (1.2) by Q and integrating with respect to x over Ω gives
Z
Z
Z
d
kQk2 = −2mk∇Qk2 −
|∇Q|2 Q dx − 2ω
|P |2 Q dx + 2
gQ dx. (3.1)
dt
Ω
Ω
Ω
Applying the Brezis-Gallout inequality (Lemma 2.2) and Hölder inequality, we deduce that
Z
−
|∇Q|2 Q dx ≤ kQkL∞ k∇Qk2
Ω
p
(3.2)
≤ C 1 + log(1 + kQkH 2 ) k∇Qk2
≤ C (1 + log(1 + kQkH 2 )) kQk2H 1
and
− 2ω
Z
|P |2 Q dx ≤ 2ωkQkL∞ kP k2
Ω
p
≤ C 1 + log(1 + kQkH 2 ) kP k2
(3.3)
≤ C (1 + log(1 + kQkH 2 )) kP k2 .
Since g(x) ∈ L2 (Ω), obviously
Z
2
gQ dx ≤ kQk2 + kgk2 ≤ kQk2 + C1 .
Ω
(3.4)
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Substituting (3.2)-(3.4) back into (3.1), we infer that
d
kQk2 ≤ −2mk∇Qk2 + C (1 + log(1 + kQkH 2 )) kQk2H 1
dt
+ C (1 + log(1 + kQkH 2 )) kP k2 + kQk2 + C1
(3.5)
≤ C(1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + C1
= E0 (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + E1 .
Thus, the proof is complete.
Lemma 3.2. Assume that f (x) ∈ L2 (Ω) and g(x) ∈ L2 (Ω), then for (1.1)-(1.4),
we have
d
kP k2 + k∇Qk2 ≤ E2 (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + E3 ,
dt
where E2 , E3 are constants.
Proof. By differentiating (1.2) with respect to x and setting
W = ∇Q,
(3.6)
Equations (1.1)-(1.2) become
Pt = ξP + (1 + iu)∆P − (1 + iv) | P |2 P − ∇P · W − rP div W + f,
1
Wt = m∇(div W ) − ∇(W 2 ) − ω∇(|P |2 ) + ∇g.
2
(3.7)
(3.8)
Multiplying (3.7) by P , integrating with respect to x over Ω and taking the real
part gives
Z
1 d
kP k2 = ξkP k2 − k∇P k2 − kP k4L4 − Re
∇P · W P dx
2 dt
Ω
Z
Z
(3.9)
− Re
r|P |2 div W dx + Re
f P dx,
Ω
Ω
where
Z
1
− Re
∇P · W P dx = −
2
Ω
Z
1
W · ∇(|P | ) dx =
2
Ω
2
Z
|P |2 div W dx.
(3.10)
Ω
While multiplying (3.8) by W and integrating over Ω yields
Z
Z
1 d
1
kW k2 = −mk div W k2 −
∇(W 2 ) · W dx +
∇g · W dx
2 dt
2 Ω
Ω
Z
−ω
∇(|P |2 ) · W dx,
(3.11)
Ω
where
Z
−ω
2
Z
∇(|P | ) · W dx = ω
Ω
Ω
|P |2 div W dx.
(3.12)
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BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
5
Combining (3.9) with (3.12), we arrive at
d
(kP k2 + kW k2 )
dt
= 2ξkP k2 − 2k∇P k2 − 2mk div W k2 − 2kP k4L4 dx
Z
Z
+ (2ω + 1 − 2r1 )
|P |2 div W dx −
∇(W 2 ) · W dx
Ω
Z
Z
∇g · W dx =:
f P dx + 2
+ 2 Re
Ω
Ω
Ω
5
X
(3.13)
Ii ,
i=1
where
I1 = 2ξ kP k2 − 2k∇P k2 − 2mk div W k2 − 2kP k4L4 ,
Z
Z
2
I2 = (2ω + 1 − 2r1 )
|P | div W dx, I3 = −
∇(W 2 ) · W dx,
Ω
Z
ZΩ
∇g · W dx.
I4 = 2 Re
f P dx, I5 = 2
(3.14)
Ω
Ω
We now estimate each term on the right-hand side of (3.13). By the Brezis-Gallout
inequality and Hölder inequality, I2 and I3 can be estimated as follows
|I2 | ≤ |2ω + 1 − 2r1 |kP k2L∞ |Ω|1/2 k div W k
m
≤ k div W k2 + CkP k4L∞
2
p
m
≤ k div W k2 + C(1 + log(1 + kP kH 2 ))4
2
m
≤ k div W k2 + C(1 + log2 (1 + kP kH 2 ))
2
m
≤ k div W k2 + CkP k2H 2 + C
2
≤ C(1 + log(1 + kP kH 2 + kW kH 2 ))(kP k2H 2 + kW k2H 2 )
m
+ k div W k2 + C
2
(3.15)
and
|I3 | ≤ kW kL∞ kW kk div W k ≤
≤
m
k div W k2 + CkW k2L∞ kW k2
2
m
k div W k2 + C(1 + log(1 + kW kH 2 ))kW k2
2
≤ C(1 + log(1 + kP kH 2 + kW kH 2 ))(kP k2H 2 + kW k2H 2 ) +
(3.16)
m
k div W k2 .
2
For I4 and I5 , one can deduce that
|I4 | ≤ kP k2 + kf k2 ≤ kP kH 2 + C2
≤ (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + C2
(3.17)
and
|I5 | ≤ 2kgkk div W k ≤ mk div W k2 + C3 ,
(3.18)
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H. GAO, L. LIN, Y. CHU
EJDE-2015/299
provided f (x) ∈ L2 (Ω) and g(x) ∈ L2 (Ω). Substituting the above estimates (3.15)–
(3.18) back into (3.13) leads to
d
kP k2 + kW k2
dt
Z
≤ 2ξkP k2 − 2k∇P k2 − 2
|P |4 dx + C 1 + log(1 + kP kH 2 + kW kH 2 ) (3.19)
Ω
2
2
× kP kH 2 + kW kH 2 + C + C2 + C3
= E2 (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + E3 .
Noting (3.6), we finally obtain
d
kP k2 + k∇Qk2
dt
≤ E2 (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + E3 ,
which is the desired result.
(3.20)
To obtain the higher order estimates with the complex amplitude of the flame
oscillations P (x, t) and the deformation of the first front Q(x, t), we can use the
transformed equations (3.7)-(3.8) and have the following lemmas.
Lemma 3.3. Assume that f (x) ∈ L2 (Ω and g(x) ∈ H 1 (Ω), then for the problem
(1.1)-(1.4), we have
d
k∇P k2 + k∆Qk2
dt
≤ E4 (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + E5 ,
where E4 , E5 are constants.
Proof. Multiplying (3.7) by (−∆P ), integrating with respect to x over Ω and taking
the real part, we obtain
Z
1 d
2
2
2
k∇P k = ξk∇P k − k∆P k + Re (1 + iv)|P |2 P ∆P dx
2 dt
Ω
Z
Z
+ Re
∇P · W ∆P dx + Re
rP div W ∆P dx
(3.21)
Ω
ZΩ
− Re
f ∆P dx.
Ω
On the other hand, multiplying (3.8) by (−∇(div W )) and integrating with respect
to x over Ω yields
Z
1 d
1
k div W k2 = −mk∇(div W )k2 +
∇(W 2 ) · ∇(div W ) dx
2 dt
2 Ω
Z
Z
(3.22)
2
+ω
∇(|P | ) · ∇(div W ) dx −
∇g · ∇(div W ) dx.
Ω
Ω
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BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
7
Adding (3.22) and (3.21), one has
d
k∇P k2 + k div W k2
dt
= 2ξk∇P k2 − 2k∆P k2 − 2mk∇(div W )k2
Z
Z
2
+ 2 Re (1 + iv)|P | P ∆P dx + 2 Re
∇P · W ∆P dx
Ω
Z
ZΩ
∇(W 2 ) · ∇(div W ) dx
+ 2 Re
rP div W ∆P dx +
Ω
Z Ω
Z
2
+ 2ω
∇(|P | ) · ∇(div W ) dx − 2 Re
f ∆P dx
Ω
Ω
Z
∇g · ∇(div W ) dx =
−2
(3.23)
Ω
8
X
Ji ,
i=1
where
J1 = 2ξk∇P k2 − 2k∆P k2 − 2mk∇(div W )k2 ,
Z
Z
J2 = 2 Re (1 + iv)|P |2 P ∆P dx, J3 = Re
∇P · W ∆P dx,
Ω
Ω
Z
Z
∇(W 2 ) · ∇(div W ) dx,
J4 = 2 Re
rP div W ∆P dx, J5 =
Ω
Ω
Z
Z
2
J6 = ω
∇(|P | ) · ∇(div W ) dx, J7 = −2 Re
f ∆P dx,
Ω
Ω
Z
J8 = −2
∇g · ∇(div W ) dx.
(3.24)
Ω
We now estimate Ji (i = 2, . . . , 8) in (3.23). From Lemma 2.2, it follows that
|J2 | ≤ 2|1 + iv|kP k2L∞ kP kk∆P k ≤ CkP k2L∞ kP k2H 2
≤ C (1 + log(1 + kP kH 2 )) kP k2H 2
≤ C (1 + log(1 + kP kH 2 + kW kH 2 ))
(3.25)
kP k2H 2
+
kW k2H 2
,
|J3 | ≤ 2kW kL∞ k∇P kk∆P k ≤ 2kW kL∞ kP k2H 2
≤ C(1 + log(1 + kW kH 2 ))kP k2H 2
(3.26)
≤ C(1 + log(1 + kP kH 2 + kW kH 2 ))(kP k2H 2 + kW k2H 2 )
and
|J4 | ≤ 2|r|kP kL∞ k div W kk∆P k ≤ CkP k2L∞ k∆P k2 + k div W k2
2
p
≤ C 1 + log(1 + kP kH 2 ) k∆P k2 + k div W k2
≤ C (1 + log(1 + kP kH 2 )) kP kH 2 + CkW kH 2
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 ,
(3.27)
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H. GAO, L. LIN, Y. CHU
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where we used the Brezis-Gallout and Hölder inequalities. By direct computations,
we have
|J5 |
1
≤ 2kW kL∞ k∇W kk∇(div W )k ≤ kW k2L∞ k∇W k2 + mk∇(div W )k2
m
(3.28)
≤ C (1 + log(1 + kW kH 2 )) kW k2H 2 + mk∇(div W )k2
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + mk∇(div W )k2
and
|J6 | ≤ 4ωkP kL∞ k∇P kk∇(div W )k ≤
+
m
k∇(div W )k2
2
8ω 2
kP k2L∞ k∇P k2
m
m
k∇(div W )k2
2
+ kW kH 2 )) kP k2H 2 + kW k2H 2
≤ C (1 + log(1 + kP kH 2 )) kP k2H 2 +
(3.29)
≤ C (1 + log(1 + kP kH 2
m
+ k∇(div W )k2 .
2
In addition, J7 and J8 can be estimated as follows
|J7 | ≤ k∆P k2 + kf k2 ≤ kP k2H 2 + C4
≤ (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + C4
and
|J8 | ≤ 2k∇gkk∇(div W )k ≤
2
m
k∇(div W )k2 + k∇gk2
2
m
m
k∇(div W )k2 + C5 .
2
Substituting the above estimate (3.25)-(3.31) into (3.23) leads to
(3.30)
(3.31)
≤
d
(k∇P k2 + k div W k2 )
dt
≤ 2ξk∇P k2 − 2k∆P k2 + (C + 1) 1 + log(1 + kP kH 2 + kW kH 2 )
(3.32)
× (kP k2H 2 + kW k2H 2 ) + C4 + C5
≤ E4 (1 + log(1 + kP kH 2 + kW kH 2 ))(kP k2H 2 + kW k2H 2 ) + E5 .
From the transformation (3.6), we finally obtain
d
(k∇P k2 + k∆Qk2 )
dt
≤ E4 (1 + log(1 + kP kH 2 + kQkH 3 ))(kP k2H 2 + kQk2H 3 ) + E5 ,
which completes the proof of this lemma.
(3.33)
Lemma 3.4. Assume that f (x) ∈ H 1 (Ω) and g(x) ∈ H 2 (Ω), then for the problem
(1.1)-(1.4), we have
d
k∆P k2 + k∇∆Qk2
dt
≤ E6 (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + E7 ,
where E6 and E7 are constants.
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BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
9
Proof. Multiplying (3.7) by ∆2 P , integrating with respect to x over Ω and taking
the real part gives
Z
1 d
2
2
2
k∆P k = ξk∆P k − k∇∆P k − Re (1 + iv)|P |2 P ∆2 P dx
2 dt
Ω
Z
Z
− Re
∇P · W ∆2 P dx − Re
rP div W ∆2 P dx
(3.34)
Ω
ZΩ
+ Re
f ∆2 P dx.
Ω
Taking the inner product of (3.8) with ∇∆(div W ) over Ω, one has
1 d
k∇(div W )k2
2 dt
Z
1
= −mk∆(div W )k −
∇(W 2 ) · ∇∆(div W ) dx
2 Ω
Z
Z
−ω
∇(|P |2 ) · ∇∆(div W ) dx +
∇g · ∇∆(div W ) dx.
2
Ω
(3.35)
Ω
Adding (3.35) and (3.34), we have
d
(k∆P k2 + k∇(div W )k2 )
dt
= 2ξk∆P k2 − 2k∇∆P k2 − 2mk∆(div W )k2
Z
Z
− 2 Re (1 + iv)|P |2 P ∆2 P dx − 2 Re
∇P · W ∆2 P d
Ω
Ω
Z
Z
2
− 2 Re
rP div W ∆ P dx −
∇(W 2 ) · ∇∆(div W ) dx
Ω
Ω
Z
Z
− 2ω
∇(|P |2 ) · ∇∆(div W ) dx + 2 Re
f ∆2 P dx
Ω
Ω
Z
∇g · ∇∆(div W ) dx =:
+2
(3.36)
Ω
8
X
Ki ,
i=1
where
K1 = 2ξk∆P k2 − 2k∇∆P k2 − 2mk∆(div W )k2
Z
Z
2
2
K2 = −2 Re (1 + iv)|P | P ∆ P dx, K3 = −2 Re
∇P · W ∆2 P d,
Ω
Ω
Z
Z
2
K4 = −2 Re
rP div W ∆ P dx, K5 = −
∇(W 2 ) · ∇∆(div W ) dx
Ω
Ω
Z
Z
K6 = −2ω
∇(|P |2 ) · ∇∆(div W ) dx, K7 = 2 Re
f ∆2 P dx,
Ω
Ω
Z
K8 = 2
∇g · ∇∆(div W ) dx.
(3.37)
Ω
Next we analyze the each integrand in (3.36). It follows from Lemmas 2.1 2.2 that
1
k∇∆P k2 + CkP k4L∞ k∇P k2
2
1
≤ k∇∆P k2 + C 1 + log2 (1 + kP kH 2 ) kP kk∆P k
2
|K2 | ≤ 6|1 + iv|kP k2L∞ k∇P kk∇∆P k ≤
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1
k∇∆P k2 + C(1 + kP kH 2 )kP kkP kH 2
2
1
1
≤ k∇∆P k2 + CkP k2H 2 + C|Ω| 2 kP kL∞ kP k2H 2
2
1
≤ k∇∆P k2 + C (1 + log(1 + kP kH 2 )) kP k2H 2
2
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2
1
+ k∇∆P k2 .
2
≤
(3.38)
With respect to the term K3 , by direct computation, one has
Z
|K3 | = 2 Re (D2 P · W · ∇∆P + ∇P · ∇W · ∇∆P ) dx
Ω
≤ 2kW kL∞ kD2 P kk∇∆P k + k∇P kL4 k∇W kL4 k∇∆P k
1
≤ k∇∆P k2 + 4kW k2L∞ kD2 P k2 + k∇P k2L4 k∇W k2L4
2
1
1
1
≤ k∇∆P k2 + 4kW k2L∞ kD2 P k2 + k∇P k4L4 + k∇W k4L4 .
2
2
2
(3.39)
Applying the Gagliardo-Nirenberg inequality, it is sufficient to see that
k∇P k4L4 ≤ kP k2 k∆P k2 ,
(3.40)
which enable us to estimate K3 as
|K3 |
1
1
1
≤ k∇∆P k2 + 4kW k2L∞ kD2 P k2 + kP k2H 2 kP k2L∞ + kW k2H 2 kW k2L∞
2
2
2
1
≤ k∇∆P k2 + C (1 + log(1 + kW kH 2 )) kP k2H 2
2
+ C (1 + log(1 + kP kH 2 )) kP k2H 2 + C (1 + log(1 + kW kH 2 )) kW k2H 2
1
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + k∇∆P k2 .
2
(3.41)
Similarly to (3.41), we have
|K4 |
Z
= 2 Re
r(div W ∇P · ∇∆P + P ∇(div W ) · ∇∆P ) dx
Ω
≤ 2|r|k div W kL4 k∇P kL4 k∇∆P k + 2|r|kP kL∞ k∇(div W )kk∇∆P k
1
≤ k∇∆P k2 + Ck∇W k2L4 k∇P k2L4 + CkP k2L∞ k∇(div W )k2
2
1
≤ k∇∆P k2 + Ck∇W k4L4 + Ck∇P k4L4 + CkP k2L∞ kW k2H 2
2
1
≤ k∇∆P k2 + CkW k2H 2 kW k2L∞ + CkP k2H 2 kP k2L∞ + CkP k2L∞ kW k2H 2
2
1
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + k∇∆P k2 .
2
(3.42)
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11
To estimate K5 , we denote W = (W 1 , W 2 ) for simplicity. Note that
Z
∆(W 2 )∆(div W ) dx
Ω
Z
=
(2Wx11 Wx11 + 2W 1 Wx11 x1 + 2Wx21 Wx21 + 2W 2 Wx21 x1
Ω
+ 2Wx12 Wx12 + 2W 1 Wx12 x2 + 2Wx22 Wx22 + 2W 2 Wx22 x2 )∆(div W ) dx
Z
(2Wx11 Wx11 + 2Wx21 Wx21 + 2Wx12 Wx12 + 2Wx22 Wx22 )∆(div W ) dx|
≤
Ω
Z
+ | (2W 1 Wx11 x1 + 2W 2 Wx21 x1 + 2W 1 Wx12 x2
(3.43)
Ω
+ 2W 2 Wx22 x2 )∆(div W ) dx|
≤ 8k∇W k2L4 k∆(div W )k + 8kW kL∞ kD2 W kk∆(div W )k.
By (3.43), we find that
|K5 | = Z
∆(W 2 )∆(div W ) dx
Ω
≤ 8k∇W k2L4 k∆(div W )k + 8kW kL∞ kD2 W kk∆(div W )k
96
96
m
≤ k∆(div W )k2 + kW k2H 2 kW k2L∞ + kW k2L∞ kW k2H 2
3
m
m
m
2
≤ C (1 + log(1 + kW kH 2 )) kW kH 2 + k∆(div W )k2
3
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2
m
+ k∆(div W )k2 .
3
(3.44)
For the last three terms of (3.36), we have
|K6 | = 2ω
Z
∆P P + 2∇P · ∇P + P ∆P ∆(div W ) dx
Ω
≤ 4ωkP kL∞ k∆P kk∆(div W )k + 4ωk∇P k2L4 k∆(div W )k
m
≤ k∆(div W )k2 + CkP k2L∞ kP k2H 2 + CkP k2H 2 kP k2L∞
3
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2
m
+ k∆(div W )k2 ,
3
|K7 | ≤ 2k∇f kk∇∆P k ≤
(3.45)
1
k∇∆P k2 + C6
2
(3.46)
m
k∆(div W )k2 + C7
3
(3.47)
and
|K8 | ≤ 2k∆gkk∆(div W )k ≤
12
H. GAO, L. LIN, Y. CHU
EJDE-2015/299
because f (x) ∈ H 1 (Ω) and g(x) ∈ H 2 (Ω). Substituting (3.38)-(3.47) into (3.36),
we obtain the following estimates
d
k∆P k2 + k∇(div W )k2
dt
≤ C (1 + log(1 + kP kH 2 + kW kH 2 )) kP k2H 2 + kW k2H 2 + 2ξk∆P k2
(3.48)
+ C6 + C7
≤ E6 1 + log(1 + kP kH 2 + kW kH 2 ))(kP k2H 2 + kW k2H 2 + E7 ,
where k∆P k ≤ kP kH 2 and E6 = 2ξ + C, E7 = C6 + C7 .
Under the conditions of Lemmas 3.1–3.4, we have the following result.
Lemma 3.5. Assume that f (x) ∈ H 1 (Ω) and g(x) ∈ H 2 (Ω), then for the problem
(1.1)-(1.4), there exist positive constants E, E 0 , E 00 , E 000 such that for any T > 0,
0
1
kP kH 2 ≤ E 00 exp eET +E (kP0 kH 2 +kQ0 kH 3 ) ,
2
0
1
000
kQkH 3 ≤ E exp eET +E (kP0 kH 2 +kQ0 kH 3 ) .
2
Proof. From Lemmas 3.1–3.4, it is easy to show that
d
kQk2 + kP k2 + k∇Qk2 + k∇P k2 + k∆Qk2 + k∆P k2 + k∇∆Qk2
dt
≤ (E0 + E2 + E4 + E6 ) (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 (3.49)
+ E3 + E5 + E7
= E8 (1 + log(1 + kP kH 2 + kQkH 3 )) kP k2H 2 + kQk2H 3 + E9 .
Denote
kQk2 + k∇Qk2 + k∆Qk2 + k∇∆Qk2 = [Q]2H 3 ,
kP k2 + k∇P k2 + k∆P k2 = [P ]2H 2 .
Inequality (3.49) becomes
d
([P ]2H 2 + [Q]2H 3 )
dt
(3.50)
≤ E8 (1 + log(1 + kP kH 2 + kQkH 3 ))(kP k2H 2 + kQk2H 3 ) + E9 .
1/2
P
α
, thus [·]H 2 and [·]H 3 are
We always use the notation k · kH s =
|α|≤s |∂ |L2
equivalent norms to k · kH 2 and k · kH 3 respectively. Furthermore, there exists
a positive constant C8 such that k · kH s ≤ C8 [·]H s , s = 2, 3. Applying Cauchy
inequality, from (3.50) it follows that
d
[P ]2H 2 + [Q]2H 3
dt
≤ E8 (1 + log(1 + C8 [P ]H 2 + C8 [Q]H 3 )) C82 [P ]2H 2 + C82 [Q]2H 3 + E9
≤ C82 E8 1 + log(1 + C82 + [P ]2H 2 + C82 + [Q]2H 3 ) [P ]2H 2 + [Q]2H 3 + E9
(3.51)
≤ C82 E8 1 + log(1 + 2C82 ) 1 + log(1 + [P ]2H 2 + [Q]2H 3 ) [P ]2H 2 + [Q]2H 3
+ E9
= E10 1 + log(1 + [P ]2H 2 + [Q]2H 3 ) [P ]2H 2 + [Q]2H 3 + E9 .
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13
If there exists a positive constant N such that [P ]2H 2 + [Q]2H 3 ≤ N for all T > 0,
then Lemma 3.5 holds. Otherwise, there exists a positive constant E such that
d
([P ]2H 2 + [Q]2H 3 ) ≤ E([P ]2H 2 + [Q]2H 3 ) log([P ]2H 2 + [Q]2H 3 ),
(3.52)
dt
which implies that
log log([P ]2H 2 + [Q]2H 3 ) ≤ ET + log log([P ]2H 2 (0) + [Q]2H 3 (0))
≤ ET + E 0 (kP0 kH 2 + kQ0 kH 3 ).
(3.53)
Therefore,
[P ]H 2 ≤ exp
[Q]H 3 ≤ exp
1
2
1
2
eET +E
0
(kP0 kH 2 +kQ0 kH 3 )
,
0
eET +E (kP0 kH 2 +kQ0 kH 3 ) .
Namely,
kQkH 3
1
0
eET +E (kP0 kH 2 +kQ0 kH 3 ) ,
2
1
0
≤ E 000 exp
eET +E (kP0 kH 2 +kQ0 kH 3 ) .
2
kP kH 2 ≤ E 00 exp
This proof is complete.
Based on the previous lemmas, we are ready to prove the main result.
Proof of Theorem 1.1. From Lemmas 3.1–3.4, when P0 (x) ∈ H 2 (Ω), Q0 (x) ∈ H 3 (Ω),
f (x) ∈ H 1 (Ω), g(x) ∈ H 2 (Ω) and for any T > 0, we obtain that
1
0
eET +E (kP0 kH 2 +kQ0 kH 3 ) ,
kP kH 2 ≤ E 00 exp
2
1
0
000
eET +E (kP0 kH 2 +kQ0 kH 3 ) ,
kQkH 3 ≤ E exp
2
where the positive constant C depends only on kP0 kH 2 , kQ0 kH 3 and T . Furthermore, by the standard method, we can extend the local solutions P (x, t) and Q(x, t)
of the periodic initial value problem (1.1)-(1.4) to global solutions. Thus, the proof
is complete.
Acknowledgments. The work wass supported in part by China NSF Grant Nos.
(11171158, 11271192), PAPD of Jiangsu Higher Education Institutions, the Jiangsu
Collaborative Innovation Center for Climate Change and the NSF of the Jiangsu
Higher Education Committee of China No. 14KJB110016.
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Hongjun Gao
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
E-mail address: gaohj@njnu.edu.cn
Lin Lin (corresponding author)
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
E-mail address: linlin@njnu.edu.cn
Yajun Chu
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
E-mail address: chuyjnjnu@163.com