THE CAUCHY PROBLEM AND DECAY RATES
FOR STRONG SOLUTIONS OF A BOUSSINESQ SYSTEM
FRANCISCO GUILLÉN GONZÁLEZ, MÁRCIO SANTOS DA ROCHA,
AND MARKO ROJAS MEDAR
Received 10 June 2004
The Boussinesq equations describe the motion of an incompressible viscous fluid subject
to convective heat transfer. Decay rates of derivatives of solutions of the three-dimensional Cauchy problem for a Boussinesq system are studied in this work.
1. Introduction
In this work we show some theoretical results about decay rates of strong solutions of the
three-dimensional Cauchy problem for Boussinesq equations, described by the following
partial differential equation problem (see [6]):
∂u
− νu + u · ∇u + ∇π = θf in (0,T) × R3 ,
∂t
div u = 0 in (0,T) × R3 ,
∂θ
− χ θ + u · ∇θ = 0 in (0,T) × R3 ,
∂t
u(0,x) = a(x) in R3 ,
θ(0,x) = b(x)
(1.1)
in R3 ,
where the unknown are u, θ, π which denote, respectively, the velocity field, the scalar
temperature and the scalar pressure. Data are the positive constants ν, χ, respectively, the
viscosity and the thermal conductivity coefficients and the function f the external force
field, and a(x), b(x), respectively, represent the initial velocity and initial temperature.
The main objective of this work is to obtain a decay rate of derivatives for the strong
solutions to the Cauchy problem (1.1). For this, we will consider the usual Lebesgue
p
∞
(R3 ) =
spaces L p (R3 ) with the usual norms | · | p . We will denote Lσ (R3 ) the closure of C0,σ
∞
p
3
P
q
3
{v ∈ C0 ;div v = 0} in L (R ). We will denote too by L (0,T;L (R )) the Banach space,
classes of functions defined a.e. in [0,T] on Lq (R3 ), that are L p -integrable in the sense of
Bochner. For more details see [1, 3].
We observe that this model of fluids includes as a particular case the classical NavierStokes equations, which has been thoroughly studied (see, e.g., [7, 8]). Rojas Medar and
Copyright © 2005 Hindawi Publishing Corporation
Abstract and Applied Analysis 2005:7 (2005) 757–766
DOI: 10.1155/AAA.2005.757
758
Boussinesq equations
Lorca obtained results of uniqueness and existence of the local solutions and regularity of
solutions for Boussinesq equations [9, 10].
Results of decay rates of strong solution were obtained by Cheng He and Ling Hsião
[4]. In this paper, we are interested to get similar results for Boussinesq equations.
2. Results of decay rates
The main objective of this work is to establish the decay rates of derivatives about time
variable and spaces variables for the strong solutions to the Cauchy problem of the Boussinesq equations (1.1). For this, we will consider a sequence of Cauchy problems for the
linearized Boussinesq equations
∂uk
− νuk + uk−1 · ∇ uk−1 + ∇π k = θ k−1 f
∂t
div uk = 0 on (0,T) × R3 ,
on (0, T) × R3 ,
∂θ k
− χ θ k + uk−1 · ∇θ k−1 = 0 on (0,T) × R3 ,
∂t
uk (0,x) = ak (x) on R3 ,
θ k (0,x) = bk (x)
(2.1)
on R3 ,
∞
(R3 ) and bk ∈ C0∞ (R3 ) such that
for k ≥ 1, where ak ∈ C0,σ
ak −→ a
in L3σ (R3 ) strongly,
bk −→ b
in L3 (R3 ) strongly,
(2.2)
with |ak |3 ≤ |a|3 and |bk |3 ≤ |b|3 . The first term, (u0 ,π 0 ,θ 0 ), of this sequence is solution
of the trivial Cauchy problem:
∂u0
− νu0 + ∇π 0 = 0 on (0, T) × R3 ,
∂t
div u0 = 0 on (0,T) × R3 ,
∂θ 0
− χ θ 0 = 0 on (0,T) × R3 ,
∂t
u0 (0,x) = a0 (x) on R3 ,
θ 0 (0,x) = b0 (x)
(2.3)
on R3 .
Let Γν (t,x;s, y) = (4νπt)−3/2 exp(−|x|2 /4νt) be a fundamental solution of the heat equation in R3 (with viscosity coefficient ν). Then, the solution of the linearized Boussinesq
Francisco Guillén González et al. 759
system (2.1) can be written as follows:
uki (t,x) =
R3
−
−
Γν (t,x;0, y)aki (y)d y
t
R3
0
R3
−
ukj −1 (s, y)
∂uik−1
(s, y)d y ds
∂x j
0
∂π k
Γν (t,x;s, y)
(s, y)d y ds
∂xi
R3
0
R3
+
k
3
j =1
t
t
θ (t,x) =
Γν (t,x;s, y)
(2.4)
Γν (t,x;s, y)θ k−1 (s, y) fi (s, y)d y ds,
Γχ (t,x;0, y)bk (y)d y
t
R3
0
Γχ (t,x;s, y)
3
j =1
ukj −1 (s, y)
∂θ k−1
(s, y)d y ds.
∂x j
(2.5)
The convergence for the above method can be seen in [2, 7].
Definition 2.1. A couple (u,θ) is called strong solution for the system (1.1), if
q
∩ L∞ 0, ∞;L3σ R3 ,
θ ∈ L p 0, ∞;Lq R3 ∩ L∞ 0, ∞;L3 R3
u ∈ L p 0, ∞;Lσ R3
(2.6)
for some p > 2 and q > 3, and satisfying
∞
0
R3
u·
=−
dϕ
+ u · ϕ + (u · ∇ϕ)u dx dt
dt
R3
a · ϕ(0,x)dx +
∞
0
R3
(2.7)
θf · ϕdx dt
∞
for all ϕ ∈ C0∞ (0, ∞;C0,σ
(R3 )) and
∞
0
R3
dψ
+ θ ψ + (u · ∇ψ)θ dx dt = −
θ
dt
R3
bψ(0,x)dx
(2.8)
for all ψ ∈ C0∞ (0, ∞;C0∞ (R3 )).
Lemma 2.2. For the pressure π k the following estimate holds:
∇ π k ≤ C u k −1 · ∇ u k −1 + θ k −1 f r
r
for 1 < r < ∞, k > 0.
(2.9)
760
Boussinesq equations
Lemma 2.3. Let a ∈ L3σ (R3 ), b ∈ L3 (R3 ), |f(t)|q ≤ C0 t −1+3/2q (|a|3 + |b|3 ), |∇f(t)|q ≤
C0 t −3/2+3/2q (|a|3 + |b|3 ), where the constant C0 is independent of t and q. If C ∗ C0 (|a|3 +
|b|3 ) ≤ 1 for some constant C ∗ , then
t 1/2−3/2q uk (t)q + θ k (t)q ≤ CC0 |a|3 + |b|3 ,
(2.10)
t 1−3/2q ∇uk (t)q + ∇θ k (t)q ≤ CC0 |a|3 + |b|3
for 3 ≤ q ≤ ∞, t ≥ 0 and k ≥ 0.
Proof. We put
Iqk = t 1/2−3/2q uk (t)q + θ k (t)q
(2.11)
Jqk = t 1−3/2q ∇uk (t)q + ∇θ k (t)q .
We will assume by inductive hypotheses that the estimates (2.10) are true for k − 1. By
the Young inequality for convolution, we can estimate the terms of (2.4) as follows:
R3
−3/2
Γν (t,x;0, y)aki (y)d y ≤ (4νπt)
q
≤ Ct
R3
−1/2+3/2q k ai
e−|x− y| /4νt d y
2
1/ p
k
a i
3
(2.12)
3,
where 1/ p + 1/3 = 1 + 1/q. Again, by the Young inequality we obtain
3
t
∂uik−1
k −1
Γ
(t,x;s,
y)
u
(s,
y)
(s,
y)d
y
ds
ν
j
0 R3
∂x j
≤C
t
j =1
q
(t − s)−3/2(1/2−1/q) uk−1 4 ∇uk−1 4 ds
(2.13)
0
2
≤ CC02 |a|3 + |b|3 t −1/2+3/2q ,
where 1/ p + 1/2 = 1 + 1/q. Now, using (2.9) we have
t 2 −1/2+3/2q
∂π k
2
Γν (t,x;s, y)
(s, y)d y ds
,
≤ CC0 |a|3 + |b|3 t
∂xi
0 R3
q
t 2
2 −1+3/2q
Γν (t,x;s, y)θ k−1 (s, y) fi (s, y)d y ds
|a|3 + |b|3 .
≤ CC0 t
3
0
R
(2.14)
q
Moreover
k u (t) ≤ Ct −1/2+3/2q C0 |a|3 + |b|3 + C 2 |a|3 + |b|3 2 .
0
q
(2.15)
Analogously, we can obtain the estimate for |θ k (t)|q . Now, differentiating (2.4) and using
the fact
∂
≤ C(t − s)−2 e−λ|x− y |2 /4ν(t−s)
Γ
(t,x;s,
y)
∂x ν
l
(2.16)
Francisco Guillén González et al. 761
for i = 1,2,3 and some constant λ > 0, follows
∂
∂x
l
−1+3/2q
Γν (t,x;0, y)aki (y)d y |a|3 .
≤ Ct
R3
(2.17)
q
Analogously, we obtain
3
∂ t
∂uik−1
k −1
Γ
(t,x;s,
y)
u
(s,
y)
(s,
y)d
y
ds
ν
j
∂xl 0 R3
∂x j
j =1
∂
∂x
≤ CC02
t
0
i
2
|a|3 + |b|3 t −1+3/2q ,
(2.18)
q
∂π k
Γν (t,x;s, y)
(s, y)d y ds
3
∂x
R
i
q
≤ CC02
2
|a|3 + |b|3 t −1+(3/2q)
1
0
(2.19)
(1 − w)
−1/2−3/2r
w
−3/2+3/2(1/r+1/q)
dw,
where r > 3 and 1/r + 1/q > 1/3, to obtain the convergence of the last integral in (2.19).
Finally, we obtain
∂ t
∂x
i
0
R3
Γν (t,x;s, y)θ
k −1
t
fi d y ds ≤ C (t − s)−1/2+3/2(1/q−1/l) θ k−1 fi l ds.
(2.20)
0
q
Without difficulty we can obtain for the equation of the temperature
Jqk ≤ C0 |a|3 + |b|3 + CC02 |a|3 + |b|3
2
≤ 2C0 |a|3 + |b|3 .
(2.21)
For q = ∞ we can obtain analogously as before
k
I∞ ≤ C |a|3 + |b|3 ) + Ct
+ Ct 1/2
t
0
1/2
t
0
(t − s)−3/4 uk−1 4 ∇uk−1 4 + ∇θ k−1 4 ds
2
(t − s)−3/4 θ k−1 4 |f |4 ds ≤ C |a|3 + |b|3 + CC02 |a|3 + |b|3 .
(2.22)
k.
Similarly, we obtain the estimative for J∞
Lemma 2.4. Let a ∈ L3σ (R3 ), b ∈ L3 (R3 ) and |f |q ≤ C0 t −1+3/2q (|a|3 + |b|3 ), |∇f |q ≤
C0 t −3/2+3/2q (|a|3 + |b|3 ). If C ∗ C0 (|a|3 + |b|3 ) ≤ 1 for some constant C ∗ , then for 3 ≤ q ≤ ∞,
the following estimate is true uniformly in k:
t 3/2−3/2q
3
l, j =1


∂ 2 uk ∂ 2 θ k 

+
≤ 2C0 |a|3 + |b|3 .
∂xl ∂x j ∂xl ∂x j q
q
(2.23)
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Boussinesq equations
Proof. The identity (2.4) can be written as follows:
uki (t,x) =
R
Γν (t,x;0, y)aki (y)d y −
3

×
−
t/2 R3
0
3
∂uk−1
ukj −1 (s, y) i (s, y) +
∂x j
j =1
t t/2 R3
Γν (t,x;s, y)
Γν (t,x;s, y)

∂π k
(s, y) − θ k−1 (s, y) fi (s, y) d y ds
∂xi
(2.24)
3
∂uk−1
ukj −1 (s, y) i (s, y)
∂x j
j =1
∂π k
(s, y) − θ k−1 (s, y) fi (s, y) d y ds
+
∂xi
analogously for temperature.
We will make the case l = j (for the case l = j the argument is analogous). By the Young
inequality we obtain
2 ∂
2
∂xl
R3
Γν (t,x;0, y)aki (y)d y q
2
xl − y l
1
k
k
≤C Γ
(t,x;0,
y)a
(y)
+
Γ
(t,x;0,
y)a
(y)
d
y
ν
ν
i
i
t R3
t2
(2.25)
q
≤ Ct −3/2+3/2q |a|3 .
By analogous computations, we have
3
∂2 t/2 ∂uki −1
k −1
2
Γ (t,x;s, y) u j (s, y)
(s, y)d y ∂x 0 R3 ν
∂x j
j =1
l
2
≤ CC02 |a|3 + |b|3 t −3/2+3/2q
The estimation for the terms that involve
t/2
0
1/2
0
(1 − w)
q
−7/4+3/2q
(2.26)
w
j =1
2
≤ CC02 |a|3 + |b|3 t −3/2+3/2q
+ CC0 |a|3 + |b|3
1
1/2
1
1/2
(1 − w)
∂xl
q
(1 − w)−1/2 w−2+3/2q dw
−1/2
−3/2+3/2q ∂2 uk−1 .
× sup s
(s)
2
s∈[t/2,t]
dw.
are obtained analogously. By other side
3
∂2 t ∂uik−1
k −1
2
Γ
(t,x;s,
y)
u
(s,
y)
(s,
y)d
y
ds
ν
j
∂x t/2 R3
∂x j
l
−3/4
q
w
−2+3/2q
dw
(2.27)
Francisco Guillén González et al. 763
Analogously
2 t ∂
2
∂xl
t/2 R3
≤C
Γν (t,x;s, y)
∂π
d y ds
∂xi
q
t
∂ k−1
∂θ k−1 f (t − s)−2 · ∇ u k −1 + ∂x ds
∂x u
t/2
l
l
q
q
(2.28)
and, finally,
2 t ∂
2
∂xl
t/2 R3
Γν (t,x;s, y)θ k−1 fi d y ds
2
≤ CC02 |a|3 + |b|3 t −3/2+3/2q
q
1
1/2
(2.29)
(1 − w)
−1/2
w
−2+3/2q
dw.
The proof of Lemma 2.4 is a consequence from the above estimative.
Lemma 2.5. Let a ∈ L3σ (R3 ), b ∈ L3 (R3 ) and |f |q ≤ C0 t −1+3/2q (|a|3 + |b|3 ), |∇f |q ≤
C0 t −3/2+3/2q (|a|3 + |b|3 ). If M(|a|3 + |b|3 ) ≤ 1 for some constant M, then for 3 ≤ q ≤ ∞,
the following estimative are verified uniformly in k:
2
t 3/2−3/2q ∇π k q ≤ C |a|3 + |b|3 ,
k k ∂u ∂θ 2
t 3/2−3/2q ∂t + ∂t ≤ C |a|3 + |b|3 + C |a|3 + |b|3 .
q
(2.30)
(2.31)
q
Proof. The proof is a consequence of Lemmas 2.2, 2.3, and 2.4, and the following facts
∂uk
= νuk − uk−1 · ∇ uk−1 − ∇π k + θ k−1 f,
∂t
∂θ k
= χ θ k − u k −1 · ∇ θ k −1 .
∂t
(2.32)
The main result in this paper is the following.
Theorem 2.6. Let a ∈ L3σ (R3 ), b ∈ L3 (R3 ) and |f |q ≤ C0 t −1+3/2q (|a|3 + |b|3 ), |∇f |q ≤
C0 t −3/2+3/2q (|a|3 + |b|3 ). Then, there exists a positive constant ε such that, if (|a|3 + |b|3 ) ≤
ε, there exists a unique solution (u,θ) for (1.1), which satisfy:
t 1/2−3/2q u ∈ BC [0, ∞);Lq R3 ,
t 1/2−3/2q θ ∈ BC [0, ∞);Lq R3 ,
t 1−3/2q |∇u| ∈ BC [0, ∞);Lq R3 ,
t 1/2−3/2q |∇θ | ∈ BC [0, ∞);Lq R3 ,
(2.33)
764
Boussinesq equations
for 3 ≤ q ≤ ∞ and moreover
t 3/2−3/2q
3 ∂2 u
∂x ∂x
j
i, j =1
t 3/2−3/2q
3
i, j =1
2
∂ θ
∂x ∂x
j
i
i
∈ BC [0, ∞);Lq R3 ,
∈ BC [0, ∞);Lq R3 ,
(2.34)
t 3/2−3/2q |∇π | ∈ BC [0, ∞);Lq R3 ,
t 3/2−3/2q
∂u
∈ BC [0, ∞);Lq R3 ,
∂t
for 3 ≤ q ≤ ∞.
Proof. Using Lemma 2.3 with q = 3, we obtain
k k
u + θ ≤ C |a|3 + |b|3 ,
3
3
∇uk + ∇θ k ≤ Ct −1/2 |a|3 + |b|3
3
(2.35)
3
then, for 1 < p < 2 it is easy to show
uk ∈ L∞ 0, ∞;L3 R3 ,
θ k ∈ L∞ 0, ∞;L3 R3 ,
p ∇uk ∈ Lloc 0, ∞;L3 R3 ,
p ∇θ k ∈ Lloc 0, ∞;L3 R3 .
(2.36)
Let q and q∗ such that 1/q + 1/q∗ = 1 and we consider the following estimative:
sup
|v|
1,q∗ =1
W0
uk ,v ≤
sup
|v|
1,q∗ =1
W0
sup
|v|
1,q∗ =1
W0
∇uk |∇v|q∗ ≤ ∇uk ,
q
q
k−1
2 u
· ∇ uk−1 ,v ≤ uk−1 ,
q
sup
|v|
1,q∗ =1
W0
(2.37)
∇π k ,v ≤ π k .
q
Now, using Hölder and Sobolev inequalities, we have
sup
|v|
1,q∗ =1
W0
k−1 θ f,v ≤ C
sup
|v|
1,q∗ =1
W0
k −1 θ f 3q/(q+3) |∇v |q∗
= C θ k−1 f 3q/(q+3) .
(2.38)
Thus, by using Lemma 2.4 together with Sobolev and Hölder inequalities, we obtain
k
∂u ∂t W −1,q
≤ ∇uk q + C uk−1 3 ∇uk−1 q + C θ k−1 3 |f |q ,
(2.39)
Francisco Guillén González et al. 765
consequently,
k
∂u ∂t W −1,q
≤ Ct −1+3/2q .
(2.40)
Analogously the following inequality can be proved for the temperature θ k
k
∂θ ∂t W −1,q (R3 )
≤ Ct −1+3/2q .
(2.41)
Therefore, for 1 < r < 2q/(2q − 3), we have
∂uk
∈ Lrloc 0, ∞;W −1,q R3 ,
∂t
∂θ k
∈ Lrloc 0, ∞;W −1,q R3 .
∂t
(2.42)
By using the compact embedding of W 1,3 (R3 ) on L3loc (R3 ) and the Compactness Theorem in [11, Cap. 3], we obtain that there exists (u,θ) such that
uk −→ u in L2loc 0, ∞;L3loc R3
k
θ −→ θ
in
L2loc
0, ∞;L3loc R3
strongly,
strongly.
(2.43)
Now, using the standard arguments, it is easily to show that (u,θ) is a unique solution
of (1.1) (see [5]).
Acknowledgments
The authors has been partially supported by D.G.E.S. and M.C. y T. (Spain), Project
BFM2003-06446. M.A. Rojas-Medar is partially supported by CNPq-Brazil, Grant
301354/03-0.
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Francisco Guillén González: Universidad de Sevilla, Facultad de Matemáticas, 41012 Sevilla, Spain
E-mail address: guillen@us.es
Márcio Santos da Rocha: DM-CCE-UEL, Londrina-PR, Brazil
E-mail address: marcio@uel.br
Marko Rojas Medar: DMA-IMECC-UNICAMP, CP 6065, 13083-970, Campinas-SP, Brazil
E-mail address: marko@ime.unicamp.br