Electronic Journal of Differential Equations, Vol. 2010(2010), No. 08, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
DISSIPATIVE BOUSSINESQ EQUATIONS ON
NON-CYLINDRICAL DOMAINS IN Rn
HAROLDO R. CLARK, ALFREDO T. COUSIN, CÍCERO L. FROTA, JUAN LÍMACO
Abstract. This article concerns the initial-boundary value problem for the
nonlinear Boussinesq equations on time dependent domains in Rn with 1 ≤
n ≤ 4. Global solvability, uniqueness of solutions and the exponential decay
to the energy are established provided the initial data are bounded in some
sense.
1. Introduction
Let Ω ⊂ Rn be an open bounded set with smooth boundary Γ. By Q∞ =
Ω×(0, ∞) and Σ∞ = Γ×(0, ∞) we denote the cylindrical domain and its boundary,
respectively. Given k = k(t) a real function defined on [0, ∞), for each t ≥ 0 we
denote Ωt the transformed sets by the number k(t); that is,
Ωt = {x ∈ Rn such that x = k(t)y for all y ∈ Ω},
and Γt is the boundary of Ωt . Then the time dependent domain
b∞ = ∪t>0 Ωt × {t} ,
Q
is a subset of Rn+1 , with lateral boundary
b ∞ = ∪t>0 Γt × {t} .
Σ
In this article, we study the initial-boundary value problem for the dissipative
Boussinesq equation
b∞ ,
utt (x, t) − ∆ u(x, t) + ut (x, t) + u2 (x, t) + ∆2 u(x, t) = 0 in Q
(1.1)
∂u
b ∞,
= 0 on Σ
∂ν
u(x, 0) = u0 (x); ut (x, 0) = u1 (x) for x ∈ Ω0 .
u=
(1.2)
(1.3)
The theory of water waves for the case of shallow water and waves of small amplitude, idealized by Scott-Russell in 1834, had one of the first mathematical analysis
established in 1872 by Boussinesq [3]. His work derived a nonlinear dissipative wave
system which is now known as the Boussinesq equations. See also Boussinesq [4].
2000 Mathematics Subject Classification. 35L10, 35Q53, 35B40.
Key words and phrases. Boussinesq equation; time dependent domains; existence;
uniqueness; asymptotic behavior.
c
2010
Texas State University - San Marcos.
Submitted September 13, 2009. Published January 16, 2010.
1
2
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
EJDE-2010/08
A nice survey on the history of the derivation of models of Boussinesq type can be
found in Miles [10].
Initial-boundary value problem in a cylindrical domain with small initial data has
been considered by Varlamov [13, 14, 15] in both 1-dimensional and 2-dimensional
cases. As results, classical solutions were constructed, uniqueness of solutions and
the long-time asymptotic were obtained explicitly. For more information about
problems associated with Boussinesq equation, see Varlamov [16] and references
therein. Liu-Russell [9] studied the existence and uniqueness of solutions to initialboundary value problems on a 1-d periodic domain. There (1.1) have an internal
weak damping k1 ut and a linear feedback k2 (u − [u]).
For the one-dimensional case, we mention the works of Bona-Sachs [2] and
Tsutsumi-Matahshi [12]; where the authors studied the existence, uniqueness and
stability of solutions for Cauchy problems. Cauchy problem related to (1.1) in
an abstract framework on a Hilbert space H has also been studied by other authors; Biler [1] and Pereira [11] established results on existence, uniqueness and
asymptotic stability of solutions.
This article is motivated by the article [5] where a 1-d version of (1.1)-(1.3) is
investigated. Our proof is a slight modification of the one in [5]. However we had
b∞ .
to overcome some technically difficulties when considering this problem in Q
The paper is organized as follows: In section 2, we give some assumptions to be
used later, and state the main results. Subsequently, sections 3 and 4 are devoted
to prove the main results: Theorems 2.1 and 2.2.
2. Assumptions and main results
For the functional spaces we use standard notation as in Lions [7] and LionsMagenes [8]. The inner product and norm in L2 (Ω) and H01 (Ω) are, respectively,
denoted by
Z
Z
1/2
(f, g) =
f (ξ) g(ξ) dξ, |f | =
|f (ξ)|2 dξ
,
Ω
Ω
n Z
X
∂g
∂f
(ξ)
(ξ)dξ ,
((f, g)) =
∂ξi
∂ξi
i=1 Ω
kf k =
n Z
X
i=1
Ω
|
1/2
∂f
(ξ)|2 dξ
.
∂ξi
For the rest of this article we consider n ≤ 4, which implies that H01 (Ω) is continuously imbedded in L4 (Ω). Let C0 be such that k · kL4 (Ω) ≤ C0 k · k. Moreover, let
C1 and C2 be positive real constants satisfying the inequalities kf kH 2 (Ω) ≤ C1 |∆f |
and kf k ≤ C2 |∆f | for all f ∈ H02 (Ω). Since Ω is bounded there exists C3 such
that |yi | ≤ kykRn ≤ C3 , for all y = (y1 , . . . yn ) ∈ Ω. From Poincaré inequality
H01 (Ω) ,→ L2 (Ω) and we put C4 such that | · | ≤ C4 k · k. Henceforth we take for
simplicity
C = max Cj .
0≤j≤4
We now state some assumptions on the function k:
k ∈ C 2 [0, ∞) with k(0) = 1 ,
1
0 < k0 ≤ k(t) ≤ k1 < √
for all t ≥ 0 .
2C
(2.1)
(2.2)
(2.3)
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
3
Let 0 be a real number such that
0 >
4
,
1 − 4k12 C 2
(2.4)
and for each pair of functions (u0 , u1 ) ∈ H02 (Ω) × L2 (Ω) we denote
α(u0 , u1 ) =
3
1 + C 2 k12
1
|u1 |2 +
ku0 k2 + 4 |∆u0 |2 .
2
4
k0
2k0
(2.5)
We also introduce the following tree polynomials:
9
1
λ+
(2n + 1)2 + n4 C 8 k12 C 4 λ2
p(λ, η) = (2 + 9nC 2 k1 )C 2 k1 +
k0
4
9 2 4 4 9 2 4 2 2
+ n C λ + n C k1 η ;
2
8
3
q(λ) = λ;
k0
2
r(λ, η) =
+ C 3 k13 (2n + n2 C 2 ) λ + (2nC 4 k12 )λ2 + [nC 4 k13 ]η .
k0
(2.6)
(2.7)
(2.8)
Now that the notation and assumptions have been set, we state the main results.
Theorem 2.1 (Existence and exponential decay). Suppose n ≤ 4 and (2.2)–(2.3)
hold. If
p(|k 0 (t)|, |k 00 (t)|) <
1
,
4
q(|k 0 (t)|) <
1
,
4
r(|k 0 (t)|, |k 00 (t)|) <
1
,
4
(2.9)
for all t ≥ 0. Then for each (u0 , u1 ) ∈ H02 (Ω) × L2 (Ω) such that
p
1
20 C 8 k16 α(u0 , u1 ) + 8C 3 k12 α(u0 , u1 ) < ,
(2.10)
4
there exists at least one global weak solution, u, to the problem (1.1)-(1.3), such
that
2
u ∈ L∞
ut ∈ L2loc (0, ∞; H01 (Ωt )),
(2.11)
loc (0, ∞; H0 (Ωt )),
and it satisfies (1.1) in the sense of L2 (0, T ; H −2 (Ωt )). Moreover, there exist positive real constants κ0 , κ1 , κ2 , such that the energy
1 0 2
E(u, t) =
|u (t)|L2 (Ωt ) + |∇u(t)|2L2 (Ωt ) + |∆u(t)|2L2 (Ωt )
2
of system (1.1)-(1.3) satisfies
E(u, t) ≤
κ2 α(u0 , u1 ) −t/κ0
e
κ1
for all t ≥ 0,
(2.12)
where κ0 , κ1 , κ2 , are defined in (3.40), (3.43), (3.51), respectively.
Theorem 2.2 (Uniqueness of Solutions). Under the assumption of Theorem 2.1,
if k 0 and k 00 satisfy
1
1 |k 0 |L1 (0,+∞) + |k 00 |L1 (0,+∞) < min
,
,
(2.13)
4K1 4K2
where K1 , K2 are real constants defined by (4.18), then the global weak solution of
(1.1)-(1.3) is unique on [0, T ], for all T > 0.
4
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
EJDE-2010/08
3. Proof of Theorem 2.1
The idea is to transform the non-cylindrical mixed problem (1.1)-(1.3) into to a
problem on a cylindrical domain, by using a suitable change of variables. Whence
let us introduce the function F : Rn × [0, ∞) → Rn × [0, ∞) defined by
F (y, t) = (k(t)y, t) = (k(t)y1 , . . . , k(t)yn , t) ,
for y = (y1 , . . . , yn ) ∈ Rn .
(3.1)
It is not difficult to see that F is a diffeomorphism of class C 2 which satisfies:
b∞ ,
F (Q∞ ) = Q
F (Ω) = Ωt ,
b ∞,
F (Σ∞ ) = Σ
F −1 (x, t) =
x ,t .
k(t)
b∞ → R, using the diffeomorphism F , we define v = (u ◦ F ) :
Given a function u : Q
Q∞ → R; that is, v(y, t) = u(k(t)y, t). Then we get
u(x, t) = v(y, t)
where y =
∂u
1 ∂v
=
∂xi
k(t) ∂yi
0
∂u
k (t)
=−
∂t
k(t)
x
,
k(t)
for i = 1, . . . , n,
n
X
j=1
(3.2)
∂v
∂v
yj +
.
∂yj
∂t
For the second order derivatives we find
∂2u
1 ∂2v
= 2
for i = 1, . . . , n ,
2
∂xi
k (t) ∂yi2
n
n
n
k 0 (t) 2 X X ∂ 2 v
∂2v
k 0 (t) X ∂ 2 v
∂2u
=
−
2
y
+
yl y j
j
∂t2
∂t2
k(t) j=1 ∂t∂yj
k(t) j=1
∂yl ∂yj
l=1
n
2(k (t)) − k(t)k (t) X
∂v
+
yj ,
k 2 (t)
∂y
j
j=1
0
00
2
(3.3)
n
∂2u
k 0 (t) X ∂ 2 v
k 0 (t) ∂v
1 ∂2v
=− 2
yj − 2
+
,
∂xi ∂t
k (t) j=1 ∂yi ∂yj
k (t) ∂yi
k(t) ∂yi ∂t
n
k 0 (t) X ∂ 3 v
k 0 (t) ∂ 2 v
1
∂3v
∂ 2 ∂u
(
)
=
−
y
−
2
+
.
j
∂x2i ∂t
k 3 (t) j=1 ∂yi2 ∂yj
k 3 (t) ∂yi2
k 2 (t) ∂yi2 ∂t
Taking into account these computations, we have
1
1
∆v , ∆2 u = 4 ∆2 v ,
k 2 (t)
k (t)
n
∂u
k 0 (t) X
∂v
k 0 (t)
1
∂v
∆( ) = − 3
∆(
)yj − 2 3 ∆v + 2 ∆( );
∂t
k (t) j=1
∂yj
k (t)
k (t)
∂t
∆u =
∆(u2 ) =
1
∆(v 2 ).
k 2 (t)
(3.4)
(3.5)
(3.6)
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
5
From (3.3)-(3.6), a function u is a solution to the problem (1.1)-(1.3) if and only if
v is a solution to the problem
vtt (y, t) −
1
k 2 (t)
∆(v(y, t) + vt (y, t) + v 2 (y, t))
1
k 4 (t)
∆2 v(y, t)
n
n
+2
k 0 (t)
k 0 (t) X
∂v
k 0 (t) X ∂ 2 v
∆v(y, t) + 3
∆(
(y, t))yj − 2
(y, t)yj
3
k (t)
k (t) j=1
∂yj
k(t) j=1 ∂t∂yj
+
k 0 (t) 2 X X ∂ 2 v
(y, t)yl yj
k(t) j=1
∂yl ∂yj
n
(3.7)
n
l=1
n
2(k (t)) − k(t)k 00 (t) X
∂v
(y, t)yj = 0 in Q∞ ,
k 2 (t)
∂y
j
j=1
0
+
2
∂v
= 0 on Σ∞ ,
∂ν
v(y, 0) = v0 (y) = u0 (y), vt (y, 0) = v1 (y) = u1 (y)
v=
(3.8)
for y ∈ Ω.
(3.9)
According to the above statements, it suffices to prove that under the assumptions of Theorem 2.1 there exists at least a weak solution v to (3.7)-(3.9) satisfying
2
v ∈ L∞
loc (0, ∞; H0 (Ω)),
vt ∈ L2loc (0, ∞; H01 (Ω)).
(3.10)
H02 (Ω),
Let (wj )j∈N be a basis to the Sobolev space
and let Vm be the finite dimensional subspace of H02 (Ω) spanned by the vectors {w1 , w2 , . . . , wm }.
The
Pm theory of ordinary differential equations yields a local solution vm (x, t) =
j=1 gjm (t)wj (x) in Vm , defined in [0, Tm ] for each m ∈ N. This solution is a local
solution to the approximate initial value problem
1 1
0
2
00
(vm
(t), w) + 2
∇(vm (t) + vm
(t) + vm
(t)), ∇w + 4 (∆vm (t), ∆w)
k (t)
k (t)
n 0
0
X
k (t)
∂vm
k (t)
−2 3
∇(
∇vm (t), ∇w − 3
(t)), ∇(yj w)
k (t)
k (t) j=1
∂yj
−2
+
n
n
n
0
k 0 (t) X ∂vm
k 0 (t) 2 X X ∂ 2 vm
(t)yj , w + (
)
(t)yi yj , w
k(t) j=1 ∂yj
k(t) i=1 j=1 ∂yi ∂yj
n
2(k 0 (t))2 − k(t)k 00 (t) X
∂vm
(t)yj , w = 0
2
k (t)
∂yj
j=1
vm (0) = v0m → u0
in H02 (Ω)
and
(3.11)
for all w ∈ Vm ,
0
vm
(0) = v1m → u1
in L2 (Ω).
(3.12)
Now we need estimates independent of m and t which will allow us to extend the
solutions vm to the whole interval [0, ∞) and take to the limit in vm as m → ∞.
0
A priori estimates. First we take w = vm
in (3.11) to obtain
1 d 0
1 d
1
1
0
2
0
|v (t)|2 + 2
kvm (t)k2 + 2 kvm
(t)k2 + 2 (∇vm
(t), ∇vm
(t))
2 dt m
k (t) dt
k (t)
k (t)
1 d
2k 0 (t)
0
+ 4
|∆vm (t)|2 − 3 (∇vm (t), ∇vm
(t))
2k (t) dt
k (t)
6
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
−
n
k 0 (t) X ∂vm
0
∇(
(t)),
∇(y
v
(t))
j
m
k 3 (t) j=1
∂yj
−2
+
EJDE-2010/08
(3.13)
n
n
n
0
k 0 (t) 2 X X ∂ 2 vm
k 0 (t) X ∂vm
0
0
(t)yj , vm
(t) + (
)
(t)yi yj , vm
(t)
k(t) j=1 ∂yj
k(t) i=1 j=1 ∂yi ∂yj
n 2(k 0 (t))2 − k(t)k 00 (t) X
∂vm
0
(t)y
,
v
(t)
= 0.
j
m
k 2 (t)
∂yj
j=1
Now we study each term in (3.13):
d
d 1 kvm (t)k2
k 0 (t)
kvm (t)k2 = (
) + 3 kvm (t)k2 ;
2
dt
dt 2 k (t)
k (t)
1
k 2 (t)
(3.14)
1
2
0
(∇vm
(t), ∇vm
(t))R
2
k (t)
n
2 X
∂vm
∂v 0
≤ 2
kvm (t)kL4 (Ω) k
(t)kL4 (Ω) | m (t)|
k (t) i=1
∂yi
∂yi
≤
n
0
X
∂vm
2C 2
∂vm
kv
(t)k
k
(t)k
|
(t)|
m
k 2 (t)
∂yi
∂yi
i=1
(3.15)
2C 2
0
(t)k
≤ 2 kvm (t)k kvm (t)kH 2 (Ω) kvm
k (t)
2C 4
0
≤ 2 |∆vm (t)|2 kvm
(t)k
k (t)
0
|∆vm (t)|4
1 kvm
(t)k2
≤ 0 C 8
+
;
k 2 (t)
0 k 2 (t)
here we have used the imbedding H01 (Ω) ,→ L4 (Ω) and the constants C and 0
given in (2.1) and (2.4), respectively. We also find
2k 0 (t)
d 1
1 d
|∆vm (t)|2 =
|∆vm (t)|2 + 5 |∆vm (t)|2 ;
(3.16)
4
4
2k (t) dt
dt 2k (t)
k (t)
0
2k 0 (t)
(t)k2
|k 0 (t)| kvm (t)k2
|k 0 (t)| kvm
(∇vm (t), ∇vm (t))R ≤
+
.
(3.17)
3
2
2
k (t)
k0
k (t)
k0
k (t)
Taking δij = 0 if i = j and 1 if i 6= j, we have
n k 0 (t) X
∂vm
0
∇(
(t)),
∇(y
v
(t))
j
m
R
k 3 (t) j=1
∂yj
n 2
n 2
k 0 (t) h X
X
0
∂ vm
∂ vm
∂vm
0
= 3
(t),
v
(t)
+
(t),
y
(t)
i
m
k (t) i=1 ∂yi2
∂yi2
∂yi
i=1
+
n X
n
X
j=1 i=1
0
δij
∂2v
i
∂v 0
m
(t), yj m (t) ∂yi ∂yj
∂yi
R
|k (t)|
0
(t)k |∆vm (t)|
(2n + 1)C 2 kvm
k 3 (t)
0
9(2n + 1)2 C 4 0 2 kvm
(t)k2
1 |∆vm (t)|2
≤
|k (t)|
+
;
4
k 2 (t)
9 k 4 (t)
≤
(3.18)
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
7
n n
k 0 (t) X
0
0
|k 0 (t)| X ∂vm
∂vm
0
0
(t)yj , vm
(t) ≤ 2C
|
(t)| |vm
(t)|
2
k(t) j=1 ∂yj
k(t) j=1 ∂yj
R
|k 0 (t)| 0
0
|v (t)| kvm
(t)k
k(t) m
|k 0 (t)| 0
kvm (t)k2 ;
≤ 2C 2
k(t)
n X
n 2
k 0 (t) X
∂ vm
0
)2
(t)yi yj , vm
(t) (
k(t) i=1 j=1 ∂yi ∂yj
R
≤ 2C
≤ C2
n X
n
X
|k 0 (t)|2 0
∂ 2 vm
|v
(t)|
(t)|
|
m
2
k (t)
∂yi ∂yj
i=1 j=1
0
(3.19)
(3.20)
2
|k (t)|
0
kvm
(t)k |∆vm (t)|
k 2 (t)
1 |∆vm (t)|2
9n4 C 8 0 2 0
|k (t)| kvm (t)k2 +
;
≤
4
9 k 4 (t)
n 2(k 0 (t))2 − k(t)k 00 (t) X
∂vm
0
(t)y
,
v
(t)
j
m
2
k (t)
∂yj
R
j=1
≤ n2 C 4
kvm (t)k
|2(k 0 (t))2 − k(t)k 00 (t)| 0
kvm (t)k
k(t)
k(t)
2
0
9C 4 n2
(t)k
1 kvm (t)k2
kv
≤
|2(k 0 (t))2 − k(t)k 00 (t)|2 m2
+
;
4
k (t)
9 k 2 (t)
Inserting (3.14)-(3.21) in (3.13) we obtain
≤ C 2n
0 2
k
1 d 0
kvm (t)k2
|∆vm (t)|2 kvm
|vm (t)|2 +
+
+ 2
2
4
2 dt
k (t)
k (t)
k (t)
1
kvm (t)k2 h 1
2
1
≤
+ |k 0 (t)|
+
+ (2C 2 k1 + )|k 0 (t)|
9 k0
k 2 (t)
0
k0
9C 4 (2n + 1)2 + C 8 n4 k12 0 2 9C 4 n2 0 4
+
|k (t)| +
|k (t)|
4
2
i
0
9C 4 n2 k12 00 2 kvm
(t)k2
2
2 0 |∆vm (t)|2
+
|k (t)|
+
+
|k (t)|
8
k 2 (t)
9 k0
k 4 (t)
|∆vm (t)|4
+ 0 C 8
.
k 2 (t)
Now we go back to (3.11) and take w = vm (t). Hence
d 0
kvm (t)k2
1 d
0
(vm (t), vm (t)) − |vm
(t)|2 +
+ 2
kvm (t)k2
2
dt
k (t)
2k (t) dt
|∆v (t)|2
1 k 0 (t)
m
+ 2
∇(vm (t))2 , ∇vm (t) +
−
2
kvm (t)k2
k (t)
k 4 (t)
k 3 (t)
n
n 2k 0 (t) X
0
∂vm
k 0 (t) X ∂vm
∇(
(t)), ∇(yj vm (t)) −
(t)yj , vm (t)
− 3
k (t) j=1
∂yj
k(t) j=1 ∂yj
(3.21)
(3.22)
8
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
n
n
k 0 (t) X X ∂ 2 vm
(t)yi yj , vm (t)
k(t) i=1 j=1 ∂yi ∂yj
+
+
EJDE-2010/08
n 2(k 0 (t))2 − k(t)k 00 (t) X
∂v
k 2 (t)
j=1
m
∂yj
(3.23)
(t)yj , vm (t) = 0 .
We work with each term of (3.23):
1 d
1
d
k 0 (t)
2
2
kv
(t)k
kv
(t)k
kvm (t)k2 ;
=
+
m
m
2k 2 (t) dt
dt 2k 2 (t)
k 3 (t)
1 ∇(vm (t))2 , ∇vm (t) 2
k (t)
R
Z
n
2 X
vm (x, t) ∂vm (x, t) ∂vm (x, t) dy
≤ 2
R
R
R
k (t) i=1 Ω
∂yi
∂yi
(3.24)
n
≤
∂vm
∂vm
2 X
kvm (t)kL4 (Ω) k
(t)kL4 (Ω) |
(t)|
2
k (t) i=1
∂yi
∂yi
(3.25)
2C 2 n
kvm (t)k2 kvm (t)kH 2 (Ω)
k 2 (t)
2nC 3
≤ 2
kvm (t)k |∆vm (t)|2 ;
k (t)
n k 0 (t) X
∂vm
∇(
(t)),
∇(y
v
(t))
j
m
R
k 3 (t) j=1
∂yj
≤
≤
n
n X
n
i
X
X
∂ 2 vm
∂ 2 vm
|k 0 (t)| h
∂vm
|
|
|v
(t)|
(t)|
+
C
(t)||
(t)|
m
2
k 3 (t)
∂yi
∂yi ∂yj
∂yi
i=1
j=1 i=1
(3.26)
|k 0 (t)|
≤ 3
[n|vm (t)|kvm (t)kH 2 (Ω) + nCkvm (t)kkvm (t)kH 2 (Ω) ]
k (t)
|k 0 (t)|
≤ 2nC 3 3
|∆vm (t)|2 ;
k (t)
n n
2k 0 (t) X
0
0
∂vm
2C|k 0 (t)| X ∂vm
(t)y
,
v
(t)
≤
|
||vm (t)|
j m
k(t) j=1 ∂yj
k(t) j=1 ∂yj
R
kvm (t)k
0
nC 2 |k 0 (t)|kvm
(t)k
k(t)
1 kvm (t)k2
0
≤ 9nC 4 |k 0 (t)|2 kvm
(t)k2 +
;
9 k 2 (t)
≤2
(3.27)
n 2(k 0 (t))2 − k(t)k 00 (t) X
∂vm
(t)y
,
v
(t)
j
m
k 2 (t)
∂y
R
j
j=1
≤
≤
2|k 0 (t)|2 + k(t)|k 00 (t)| k 2 (t)
2|k 0 (t)|2 + k(t)|k 00 (t)| k 2 (t)
C|vm (t)|
n
X
∂vm
|
(t)|
∂yj
j=1
nC 4 |∆vm (t)|2 .
(3.28)
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
9
Since 0 < k0 ≤ k(t) ≤ k1 , taking into account (3.23)-(3.28) we find
d 0
1 kvm (t)k2 8 kvm (t)k2
|∆vm (t)|2
(vm (t), vm (t)) +
+
+
dt
2 k 2 (t)
9 k 2 (t)
k 4 (t)
2
0
1
kvm (t)k
kvm (t)k2
2 2
4 2 0
≤ |k 0 (t)|
+
k
C
+
9nC
k
|k
(t)|
1
1
k0
k 2 (t)
k 2 (t)
h
2
|∆vm (t)|
+ (2n + n2 C 2 )C 3 k13 |k 0 (t)|
+ 2nC 3 k12 kvm (t)k
k 4 (t)
i |∆v (t)|2
m
+ nC 4 k12 2|k 0 (t)|2 + k1 |k 00 (t)|
.
k 4 (t)
(3.29)
This inequality and (3.22) yields
0
(t)k2
1 kvm
7 kvm (t)k2
7 |∆vm (t)|2
dH
(t) + 1 − k12 C 2 −
+
+
2
2
dt
0
k (t)
9 k (t)
9 k 4 (t)
kv 0 (t)k2
kvm (t)k2
≤ p(|k 0 (t)|, |k 00 (t)|) m2
+ q(|k 0 (t)|)
k (t)
k 2 (t)
|∆vm (t)|2
|∆vm (t)|4
+ r(|k 0 (t)|, |k 00 (t)|)
+ 0 C 8
4
k (t)
k 2 (t)
|∆vm (t)|2
+ 2nC 3 k12 kvm (t)k
,
k 4 (t)
(3.30)
where
H(t) =
kvm (t)k2
1 |∆vm (t)|2
1 0
0
|vm (t)|2 +
+
+ (vm
(t), vm (t)).
2
2
k (t)
2 k 4 (t)
From (2.3) and (2.4) we can see that (1 − k12 C 2 −
(3.30) as
1
0 )
(3.31)
> 3/4. Therefore, we rewrite
0
kvm
dH
(t)k2
3
(t) +
− p(|k 0 (t)|, |k 00 (t)|)
2
dt
4
k (t)
2
3
kv
(t)k
3
|∆vm (t)|2
m
+
− q(|k 0 (t)|)
+ ( − r(|k 0 (t)|, |k 00 (t)|))
2
4
k (t)
4
k 4 (t)
|∆vm (t)|4
|∆vm (t)|2
− 0 C 8
− 2nC 3 k12 kvm (t)k
≤ 0.
2
k (t)
k 4 (t)
(3.32)
This inequality and (2.9) yield
|∆v (t)|2
0
dH
1 kvm
(t)k2 1 kvm (t)k2 1 |∆vm (t)|2 1
m
(t)+
+
+
+
−γ(t)
≤ 0 , (3.33)
dt
2 k 2 (t)
2 k 2 (t)
4 k 4 (t)
4
k 4 (t)
where
γ(t) = 0 C 8 k12 |∆vm (t)|2 + 2nC 3 k1 kvm (t)k .
On the other hand,
0
|(vm
(t), vm (t))| ≤
1 0
kvm (t)k2
|vm (t)|2 + C 2 k12
.
4
k 2 (t)
(3.34)
10
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
EJDE-2010/08
Taking into account the definition of H(t), (3.33) and (3.34), for all t ≥ 0 we find
1 0
1 kvm (t)k2
1 |∆vm (t)|2
|vm (t)|2 +
+
4
2 k 2 (t)
2 k 4 (t)
≤ H(t)
≤
3 0
(1
|vm (t)|2 +
4
+ C 2 k12 )
kvm (t)k2
k02
+
(3.35)
1
|∆vm (t)|2 ,
2k04
which in particular for t = 0 gives H(0) ≤ α(u0 , u1 ). Simple computations then
lead to
γ(t) ≤ 20 C 8 k16 H(t) + 2nC 3 k12 H 1/2 (t) ∀t ≥ 0 ,
(3.36)
p
8 6
3 2
and from (2.10), we obtain γ(0) ≤ 20 C k1 α(u0 , u1 ) + 2nC k1 α(u0 , u1 ) < 1/4.
Now we claim that
1
for all t ≥ 0.
(3.37)
γ(t) <
4
By contradiction let us suppose that (3.37) does not hold. The continuity gives
t∗ > 0 such that
1
1
γ(t) <
for all t ∈ [0, t∗ ) and γ(t∗ ) = .
(3.38)
4
4
Integrating (3.32) from 0 to t∗ we come to H(t∗ ) ≤ H(0) ≤ α(u0 , u1 ). This inequality, (3.36) and (2.10) yield γ(t∗ ) < 1/4, which contradicts (3.38) and our claim is
proved.
Since we have (3.32), (3.35) and (3.37) one can easily gets a constant A > 0 such
that
Z
t
0
|vm
(t)|2 + kvm (t)k2 + |∆vm (t)|2 +
0
kvm
(s)k2 ds ≤ A .
(3.39)
0
0
Hence for all T > 0 we have (vm )m∈N bounded in L∞ (0, T ; H02 (Ω)) and (vm
)m∈N
∞
2
2
1
bounded in L (0, T ; L (Ω)) ∩ L (0, T ; H0 (Ω)). From standard compactness arguments we are able to get the existence of global solutions.
To complete the proof of Theorem 2.1, we must to establish a rate decay estimate
to the total energy of the problem (1.1)-(1.3). In fact, from (3.33) and (3.37), we
get
0
(t)k2
1 kvm
1 kvm (t)k2
1 |∆vm (t)|2
dH
(t) +
+
+
≤ 0.
dt
2 k 2 (t)
2 k 2 (t)
4 k 4 (t)
From this inequality, (2.1) and (2.3), we obtain
0
dH
1 |vm
(t)|2
1 kvm (t)k2
1 |∆vm (t)|2
(t) +
+
+
≤ 0.
dt
2C
k12
2
k12
4
k14
From this inequality there exists a positive real constant κ0 such that
3
dH
(1 + C 2 k12 )
1
2
2
0
(t) + κ0 |vm
(t)|2 +
kv
(t)k
+
|∆v
(t)|
≤ 0,
m
m
dt
4
k02
2k04
where
2
k02
k04 ,
,
.
3Ck12 2k12 (1 + C 2 k12 ) 2k14
Therefore, by using (3.35)2 in this inequality we get
κ0 = min
dH
1
(t) + H(t) ≤ 0
dt
κ0
for all t ≥ 0,
(3.40)
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
11
which gives
H(t) ≤ H(0)e−t/κ0
for all t ≥ 0.
(3.41)
The total energy of the approximate system (3.11)-(3.12) comes from identity (3.13);
that is,
1 0
kvm (t)k2
|∆vm (t)|2 E(vm , t) =
|vm (t)|2 +
+
.
(3.42)
2
2
k (t)
k 4 (t)
From (3.42) and (3.35)1 there exists 0 < κ1 ≤ 1/2 such that κ1 Em (t) ≤ H(t). Also
we have that H(0) ≤ α(u0 , u1 ). Therefore, from (3.41) we get
E(vm , t) ≤
α(u0 , u1 ) −t/κ0
e
κ1
for all t ≥ 0.
(3.43)
The estimate (3.39) gives enough convergence to take to the limit m → ∞ in Em ,
via Banach-Steinhauss theorem, which implies
E(v, t) ≤
α(u0 , u1 ) −t/κ0
e
κ1
for all t ≥ 0,
(3.44)
where
E(v, t) =
1 0 2 kv(t)k2
|∆v(t)|2 |v (t)| + 2
+
,
2
k (t)
k 4 (t)
(3.45)
is the total energy associated with the system (3.7)-(3.9). Finally, we have to
compare the terms of E(u, t) with those of E(v, t). In fact, from (3.2)-(3.4) we have
the identities
1 ∂v
∂u
=
for i = 1, . . . , n;
∂xi
k(t) ∂yi
n
(3.46)
∂u
k 0 (t) X ∂v
1
∂v
=−
, ∆u = 2 ∆v.
yj +
∂t
k(t) j=1 ∂yj
∂t
k (t)
From the first identity above, we have
k∇u(x, t)k2Rn =
1
k∇v(y, t)k2Rn .
k 2 (t)
Integrating this over Ωt , using x = k(t)y and dx = k n (t)dy we get, thanks to
hypothesis (2.3), that
|∇u(t)|2L2 (Ωt ) ≤
k1n
|∇v(t)|2L2 (Ω) .
k02
(3.47)
From the second identity of (3.46), we obtain
∂u
(x, t) ≤ k2 k∇v(y, t)kRn kyk2Rn + ∂v (y, t) .
R
R
∂t
k0
∂t
Squaring both sides, yields
2 2
∂u
(x, t)2 ≤ 2k2 C k∇v(y, t)k2Rn + 2 ∂v (y, t)2 .
2
R
R
∂t
k0
∂t
Integrating this over Ωt and observing that dx = k n (t)dy, we get
2
|u0 (t)|L2 (Ωt ) ≤
2k22 C 2 k1n
2
2
|∇v(t)|L2 (Ω) + 2k1n |v 0 (t)|L2 (Ω) .
k02
(3.48)
12
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
EJDE-2010/08
Repeating the same arguments as above in the third identity of (3.46), we find
|∆u(t)|2L2 (Ωt ) ≤
k1n
|∆v(t)|2L2 (Ω) .
k04
(3.49)
Now, to compare the function E(u, t) with the function E(v, t) it will be used the
equivalences of the norms: kzk and |∇z| in H01 (Ω). Thus, from (3.47)-(3.49) we get
1 0 2
|u (t)|L2 (Ωt ) + |∇u(t)|2L2 (Ωt ) + |∆u(t)|2L2 (Ωt )
2
2k 2 C 2 k n kn
kn
≤ 2k1n |v 0 (t)|2L2 (Ω) + 12 + 2 2 1 |∇v(t)|2L2 (Ω) + 14 |∆v(t)|2L2 (Ω) .
k0
k0
k0
(3.50)
Thus, choosing
E(u, t) =
kn
2k 2 C 2 k n 2k n κ2 = max 4k1n , 2k1 12 + 2 2 1 , 41 ,
k0
k0
k0
(3.51)
we get from (3.45) and (3.50) that E(u, t) ≤ κ2 E(v, t) for all t ≥ 0. Therefore, from
(3.44) we obtain the desired estimate (2.12) and consequently the proof of Theorem
2.1 is finished
4. Proof of Theorem 2.2
Problems (1.1)-(1.3) and (3.7)-(3.9) are equivalent, then it is sufficient to show
the uniqueness of solutions to (3.7)-(3.9). Suppose v and vb two solutions of (3.7)(3.9). Thus, φ = v − vb satisfies
1
1
k 0 (t)
2
∆(φ(y,
t)
+
φ
(y,
t))
+
∆
φ(y,
t)
+
2
∆φ(y, t)
t
k 2 (t)
k 4 (t)
k 3 (t)
n
n
k 0 (t) X
∂φ
k 0 (t) X ∂ 2 φ
+ 3
∆(
(y, t)) yj − 2
(y, t)yj
k (t) j=1
∂yj
k(t) j=1 ∂t∂yj
φtt (y, t) −
n
n
2(k 0 (t))2 − k(t)k 00 (t) X
∂φ
k 0 (t) 2 X ∂ 2 φ
+
(y, t)yl yj +
(y, t)yj
2
k(t)
∂yl ∂yj
k (t)
∂yj
j=1
(4.1)
j, l=1
1
v 2 (y, t))
= − 2 (∆v 2 (y, t) − ∆b
k (t)
in Q∞ ,
∂φ
= 0 on Σ∞ ,
∂ν
φ(y, 0) = φt (y, 0) = 0 for y ∈ Ω.
φ=
(4.2)
(4.3)
Equation (4.1) is given
in the sense of L2 (0, T ; H −2 (Ω)) and φt ∈ L2 (0, T ; H01 (Ω)),
then the duality φtt (t), φt (t) H −2 (Ω)×H 1 (Ω) does not make sense. To overcome
0
this difficulty, the uniqueness will be obtained following the argument contained in
Ladyzhenskaya-Visik [6]. In fact, for each s ∈ (0, T ) let ψ(t), be a real function
defined for all t, in ]0, T [, by
( Rs
− t φ(y, r)dr if 0 < t ≤ s,
ψ(y, t) =
(4.4)
0
if s < t < T,
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
13
∞
2
where φ, is a solution of (4.1)-(4.3).
Since
φ is in L (0, T ; H0 (Ω)) and ψ in
∞
2
L (0, T ; H0 (Ω)), the duality φtt (t), ψ(t) H −2 (Ω)×H 2 (Ω) makes sense. Moreover,
0
ψt (y, t) = φ(y, t)
Setting ψ1 (y, t) =
Rt
0
and ψ(y, s) = 0.
(4.5)
φ(y, r)dr, we have
ψ(y, t) = ψ1 (y, t) − ψ1 (y, s)
and ψ(y, 0) = −ψ1 (y, s).
(4.6)
Taking the scalar product on L2 (Ω) of ψ with both sides of (4.1) and integrating
from 0, to s, yields
Z s
Z s
Z s
1
1
hφtt (t), ψ(t)idt −
h∆(φ(t) + φt (t)), ψ(t)idt +
h∆2 φ(t), ψ(t)idt
2 (t)
4 (t)
k
k
0
0
0
Z s 0
Z s 0
n
∂φ
k (t)
k (t) X
+2
∆(
h∆φ(t),
ψ(t)idt
+
(t)) yj , ψ(t) dt
3
3
∂yj
0 k (t)
0 k (t) j=1
Z s 0
Z
n
n
s
k (t) X ∂ 2 φ
k 0 (t) 2 X ∂ 2 φ
−2
(t)yj , ψ(t) dt +
(t)yl yj , ψ(t) dt
k(t)
∂yl ∂yj
0 k(t) j=1 ∂t∂yj
0
j, l=1
Z s
n
2(k 0 (t))2 − k(t)k 00 (t) X ∂φ
(t)yj , ψ(t) dt
(4.7)
+
2
k (t)
∂yj
0
j=1
Z s
1
=−
h(∆v 2 (t) − ∆b
v 2 (t)), ψ(t)idt.
2 (t)
k
0
Now, we modify each terms of (4.7) by using several times integration by parts,
the Green formula, the identities (4.5), (4.6), the null initial condition (4.3), the
hypotheses (2.2), (2.3), (2.12) and usual inequalities like: Cauchy-Schwartz, Young
so on. In fact, The first term can be changed as
Z s
s Z s
1
hφtt (t), ψ(t)idt = (φt (t), ψ(t)) −
(4.8)
(φt (t), φ(t))dt = − |φ(s)|2 .
2
0
0
0
The second term of (4.7) is modified as
Z s
Z s
1
1
h∆φ(t),
ψ(t)idt
=
(∇ψt (t), ∇ψ(t))dt
−
2
2
0 k (t)
0 k (t)
Z
1 s d 1
k 0 (t)
2
=
|∇ψ(t)|
−
2
|∇ψ(t)|2 dt
2
3
2 0 dt k (t)
k (t)
Z s 0
1 1
k
(t)
= − 2 |∇ψ1 (s)|2 −
|∇ψ1 (s)|2 dt.
3
2 k (0)
0 k (t)
The last integral above is bounded from above as follows:
Z s 0
Z s
Z s 0
|k (t)|
2
k (t)
2 2
2
−
|∇ψ1 (s)| dt R ≤ 2
|∇ψ1 (t)| dt + 3 |∇ψ1 (s)|
|k 0 (t)|dt
3
3
k0
0 k (t)
0 k (t)
0
Z s 0
|k (t)|
2
=2
|∇ψ1 (t)|2 dt + 3 |∇ψ1 (s)|2 |k 0 |L1 (0,∞) .
3 (t)
k
k
0
0
14
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
EJDE-2010/08
Therefore, as k(0) = 1, see (2.2), we obtain
Z
s
1
h∆φ(t), ψ(t)idt
2 (t)
k
0
Z s 0
|k (t)|
2
1
|∇ψ1 (t)|2 dt − 3 |k 0 |L1 (0,∞) |∇ψ1 (s)|2
≥ − |∇ψ1 (s)|2 − 2
3 (t)
2
k
k
0
0
−
(4.9)
The third term of (4.7) is modified as
s
Z
1
h∆φt (t), ψ(t)idt = −
2
k (t)
−
0
s
Z
1
0
∇ψ(t) )
2
k (t)
Z s 0
k (t)
1
2
dt
−
2
|∇φ(t)|
(φ(t), ∆ψ(t))dt.
2
3
k (t)
0 k (t)
(∇φ(t),
Z0 s
=−
0
The last integral above is the same that the fifth term of (4.7) with positive sign.
Thus, we have
s
Z
1
h∆φt (t), ψ(t)idt + 2
k 2 (t)
−
0
Z
0
s
k 0 (t)
h∆φ(t), ψ(t)idt = −
k 3 (t)
Z
0
s
1
|∇φ(t)|2 dt.
k 2 (t)
(4.10)
Now, we estimate the fourth term of (4.7):
s
Z
0
Z s 0
1
1
k (t)
2
2
h∆
φ(t),
ψ(t)idt
=
−
|∆ψ
(s)|
+
2
|∆ψ(t)|2 dt
1
5 (t)
k 4 (t)
2
k
Z s0 0
1
|k (t)|
2
2
≥ − |∆ψ1 (s)| −
5 |∆ψ1 (t)| dt
2
k
0
0
1
2 0
− 5 |∆ψ1 (s)| |k |L1 (0,∞) .
k0
(4.11)
The sixth term of (4.7) is estimated as
n
k 0 (t) X
∂φ
h
∆(
(t)) yj , ψ(t)idtR
3
∂yj
0 k (t) j=1
Z s 0
∂ψ
k (t) ∂φ
=
(
(t), 2
(t) + yj ∆ψ(t))dtR
3
∂yj
0 k (t) ∂yj
Z s 0
Z s 0
|k (t)|
|k (t)|
≤2
|∇φ(t)||∇ψ
(s)|dt
+
|∇φ(t)|kykRn |∆ψ(t)|dt
1
3 (t)
3
k
0 k (t)
0
Z s
31
k2
k2
2 0
1
≤
|∇φ(t)|2 + 2 4 |∇ψ1 (t)|2 dt +
4 |∇ψ1 (s)| |k |L (0,∞)
2
2k
(t)
k
k
1 0
1 0
0
Z
k22 C 2 s
C 2 k2
|∆ψ1 (t)|2 dt +
+
|∆ψ1 (s)|2 |k 0 |L1 (0,∞) ,
4
21 k0 0
21 k04
Z
s
(4.12)
where k2 is a constant that comes from the hypothesis (2.2). That is, |k 0 (t)| ≤ k2 .
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
The seventh term of (4.7) is estimated as
Z s 0
n
k (t) X ∂ 2 φ
−2
(t)yj , ψ(t) dtR
k(t)
∂t∂y
j
0
j=1
Z s 0 X
n
k (t)
∂φ
= 2
(t)yj , φ(t) dtR
∂yj
0 k(t)
j=1
Z
n
s k 0 (t) X
∂
=
(
[φ(t)]2 , yj )dtR
0 k(t) j=1 ∂yj
Z s 0
nk (t)
|φ(t)|2 dtR
=
k(t)
0
Z
nk2 s
≤
|φ(t)|2 dt.
k0 0
The eighth term of (4.7) is estimated as
n
Z s k 0 (t) X
∂2φ
2
(t)yl yj , ψ(t) dt
k(t)
∂yl ∂yj
R
0
j, l=1
n Z s k 0 (t) X
∂φ
∂ψ 2
(t), [δl j yj + yl ]ψ(t) + yl yj
(t) dt
=−
k(t)
∂yl
∂yj
R
0
j, l=1
Z s 0
n
∂φ ∂ψ k (t) 2 X ∂φ ≤
2
(t) d(Ω)|ψ(t)| + (t)[d(Ω)]2 (t) dt
k(t)
∂yl
∂yl
∂yj
0
j, l=1
Z s 0
k (t) 2 √
2n nC|∇φ(t)||ψ(t)| + nC 2 |∇φ(t)||∇ψ(t)| dt
≤
k(t)
0
Z s
Z
2n3 C 2 k24 s 1
1
2
≤ 2
|∇φ(t)|
dt
+
|ψ1 (t)|2 dt
2
2
2
0 k (t)
0 k (t)
Z
2n2 C 2 k23
n2 C 4 k24 s 1
2 0
1
+
|∇ψ1 (t)|2 dt
|ψ
(s)|
|k
|
+
1
L (0,∞)
2
2 k02
2
0 k (t)
n2 C 4 k23
+
|∇ψ1 (s)|2 |k 0 |L1 (0,∞) .
22 k02
15
(4.13)
(4.14)
The ninth term of (4.7) is estimated as
n
Z s 2(k 0 (t))2 − k(t)k 00 (t) X
∂φ
]
(t)y
,
ψ(t)
dt
[
j
k 2 (t)
∂yj
R
0
j=1
Z
n s 2(k 0 (t))2 − k(t)k 00 (t) X
∂ψ φ(t),
ψ(t)
+
y
=
[−
]
(t) dt
j
k 2 (t)
∂yj
R
0
j=1
Z s
(2k2 + k1 ) 0
(|k (t)| + |k 00 (t)|) [n|φ(t)||ψ(t)| + d(Ω)|φ(t)||∇ψ(t)|]dt
≤
2 (t)
k
0
Z
n2 (2k2 + k1 )2
C 2 (2k2 + k1 )2 s
≤
+
|φ(t)|2 dt
(4.15)
k04
4k04
0
Z s
+ 2(C 2 + 1)
(|k 0 (t)| + |k 00 (t)|)2 |∇ψ1 (t)|2 dt
0
16
H. R. CLARK, A. T. COUSIN, C. L. FROTA, J. LÍMACO
EJDE-2010/08
+ 4(C 2 + 1)(k2 + k3 )|∇ψ1 (s)|2 (|k 0 |L1 (0,∞) + |k 00 |L1 (0,∞) ).
where k3 is a constant due to (2.2) defined by |k 00 (t)| ≤ k3 .
The last term of (4.7) is estimated as
Z s 1
2
2
h(∆v
(t)
−
∆b
v
(t)),
ψ(t)idt
−
2
R
0 k (t)
Z
s
1
([v(t)
+
v
b
(t)]φ(t),
∆ψ(t))dt
=−
2 (t)
k
R
0
Z s 1
2
1
1
≤ C kvkL∞ (0,T ;H0 (Ω)) + kb
v kL∞ (0,T ;H0 (Ω))
|∇φ(t)||∆ψ(t)|dt
2
0 k (t)
Z s
1
≤ 3
|∇φ(t)|2 dt
2 (t)
k
0
2
C 4 kvkL∞ (0,T ;H01 (Ω)) + kb
v kL∞ (0,T ;H01 (Ω)) Z s
|∆ψ1 (t)|2 dt
+
23 k02
0
2
C 4 kvkL∞ (0,T ;H01 (Ω)) + kb
v kL∞ (0,T ;H01 (Ω)) s
+
|∆ψ1 (s)|2 .
(4.16)
23 k02
Inserting (4.8)-(4.16) in (4.7), we have
1
1
|φ(s)|2 +
− K1 [|k 0 |L1 (0,∞) + |k 00 |L1 (0,∞) ] |∇ψ1 (s)|2
2
2
2k1
1
+ [ 4 − K2 |k 0 |L1 (0,∞) − K3 s]|∆ψ1 (s)|2
2k1
Z s
1
|∇φ(t)|2 dt
+ [1 − (21 + 22 + 3 )]
2 (t)
k
0
Z s
2
≤ K4
[|φ(t)| + |∇ψ1 (t)|2 + |∆ψ1 (t)|2 ]dt,
(4.17)
0
where
k2
2n2 C 3 k23
n2 C 4 k23
2
+
+
+
+ 4(C 2 + 1)(k2 + k3 );
k03
1 k04
2 k02
22 k02
1
C 2 k2
;
K2 = 5 +
k0
21 k04
2
C4 1
1
K3 =
kvk
+
kb
v
k
;
∞
∞
L (0,T ;H0 (Ω))
L (0,T ;H0 (Ω))
23 k02
4k2
k2
31
k2
k2 C 2
nk2
2
2n3 C 3 k24
n2 C 4 k24
K4 = 3 + 5 + 2 + 2 4 + 2 4 +
+ 2+
+
k0
k0
2k0
1 k0
21 k0
k0
k0
2
2 k02
n2 (2k2 + k1 )2
C 2 (2k2 + k1 )2
3
+ 2(C 2 + 1) +
+
+ 2 + K3 .
4
4
k0
4k0
k0
(4.18)
Thus, if 1 , 2 , 3 , are taking such that 21 + 22 + 3 ≤ 1/2, and k 0 , k 00 , satisfying
the hypothesis (2.13), we get
Z
1
1
1
1 s 1
2
2
2
|φ(s)| + |∇ψ1 (s)| + ( − K3 s)|∆ψ1 (s)| +
|∇φ(t)|2 dt
2
4
4
2 0 k 2 (t)
K1 =
EJDE-2010/08
DISSIPATIVE BOUSSINESQ EQUATIONS
Z
≤ K4
s
h
17
i
|φ(s)|2 + |∇ψ1 (s)|2 + |∆ψ1 (s)|2 dt.
0
Now, if s ≤ T0 = 1/8K3 , then
|φ(s)|2 + |∇ψ1 (s)|2 + |∆ψ1 (s)|2 ≤ 8K4
s
Z
h
i
|φ(s)|2 + |∇ψ1 (s)|2 + |∆ψ1 (s)|2 dt.
0
From this and Gronwall’s inequality we find φ(x, s) = 0 a. e. for all s ∈ [0, T0 ].
This gives the uniqueness of solutions over the interval [0, T0 ].
The task now is to show the uniqueness of the solutions over the interval [T0 , 2T0 ].
In fact, being the solutions unique on [0, T0 ] we obtain from (4.3) that
φ(y, T0 ) = φt (y, T0 ) = 0
for y ∈ Ω.
Now, we consider in (4.4) the variable s ∈ (T0 , T ) and take ψ1 (y, t) =
Thus, we get
ψ(y, t) = ψ1 (y, t) − ψ1 (y, s)
Rt
T0
φ(y, r)dr.
and ψ(y, T0 ) = −ψ1 (y, s).
Repeating the steps (4.7)-(4.17) on the whole interval (T0 , T ) × Ω we get
Z
1
1
1
1 s 1
|φ(s)|2 + |∇ψ1 (s)|2 +
− K3 (s − T0 ) |∆ψ1 (s)|2 +
|∇φ(t)|2 dt
2
4
4
2 T0 k 2 (t)
Z sh
i
≤ K4
|φ(s)|2 + |∇ψ1 (s)|2 + |∆ψ1 (s)|2 dt.
T0
Now, if s ≤ 2T0 = 1/4K3 then we will have
Z sh
i
2
2
2
|φ(s)| + |∇ψ1 (s)| + |∆ψ1 (s)| ≤ 8K4
|φ(s)|2 + |∇ψ1 (s)|2 + |∆ψ1 (s)|2 dt.
0
From this and Gronwall’s inequality we have φ(x, s) = 0 a. e. for all s ∈ [T0 , 2T0 ].
This gives the uniqueness of solutions over the interval [0, 2T0 ]. Repeating the
precedent process N times until that N T0 ≥ T we will obtain the uniqueness of
solutions over the interval [0, T ], and thus the proof of the Theorem 2.2 is complete
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[11] Pereira, D. C.; Existence, uniqueness and asymptotic behavior for solutions of the nonlinear
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Haroldo R. Clark
Universidade Federal Fluminense, IM-GAN, RJ, Brasil
E-mail address: hclark@vm.uff.br
Alfredo T. Cousin
Universidade Estadual de Maringá, DMA, PR, Brasil
E-mail address: atcousin@uem.br
Cı́cero Lopes Frota
Universidade Estadual de Maringá, DMA, PR, Brasil
E-mail address: clfrota@uem.br
Juan Lı́maco
Universidade Federal Fluminense, IM-GMA, RJ, Brasil
E-mail address: jlimaco@vm.uff.br