On the existence and uniqueness of solutions to 2D G-benard problem in unbounded domains

HNUE JOURNAL OF SCIENCE
DOI: 10.18173/2354-1059.2020-0025
Natural Science, 2020, Volume 65, Issue 6, pp. 23-31
This paper is available online at

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS
´
TO 2D G-BENARD
PROBLEM IN UNBOUNDED DOMAINS
Tran Quang Thinh1 and Le Thi Thuy2
1 Faculty

of Basic Sciences, Nam Dinh University of Technology Education
2 Faculty of Mathematics, Electric Power University

Abstract. We consider the 2D g-B´enard problem in domains satisfying the Poincar´e
inequality with homogeneous Dirichlet boundary conditions. We prove the existence and
uniqueness of global weak solutions. The obtained results particularly extend previous
results for 2D g-Navier-Stokes equations and 2D B´enard problem.

1.

Introduction

Let Ω be a (not necessarily bounded) domain in R2 with boundary Γ. We consider the
following two-dimensional (2D) g-B´enard problem

∂u

+ (u · ∇)u − ν∆u + ∇p = ξθ + f1 ,
x ∈ Ω, t > 0,




∂t



∇ · (gu) = 0,
x ∈ Ω, t > 0,




2κ
κ∆g
∂θ


+ (u · ∇)θ − κ∆θ −
(∇g · ∇)θ −
θ = f2 , x ∈ Ω, t > 0,

∂t
g
g
(1.1)

u = 0,
x ∈ Γ, t > 0,





θ = 0,
x ∈ Γ, t > 0,





u(x, 0) = u0 (x),
x ∈ Ω,



θ(x, 0) = θ0 (x),
x ∈ Ω,

where u ≡ u(x, t) = (u1 , u2 ) is the unknown velocity vector, θ ≡ θ(x, t) is the temperature,
p ≡ p(x, t) is the unknown pressure, f1 is the external force function, f2 is the heat source
function, ν > 0 is the kinematic viscosity coefficient, ξ is a constant vector, κ > 0 is thermal
diffusivity, u0 is the initial velocity and θ0 is the initial temperature.
As derived and mentioned in [8], 2D g-B´enard problem arises in a natural way when
we study the standard 3D B´enard problem on the thin domain Ωg = Ω × (0, g). Here the
g-B´enard problem is a couple system which consists of g-Navier-Stokes equations and the
advection-diffusion heat equation in order to model convection in a fluid. Moreover, when
g ≡ const we get the usual B´enard problem, and when θ ≡ 0 we get the g-Navier-Stokes equations.
In what follows, we will list some related results.
Received June 5, 2020. Revised June 19, 2020. Accepted June 26, 2020
Contact Le Thi Thuy, e-mail address:

23


Tran Quang Thinh and Le Thi Thuy

The existence and long-time behavior of solutions in terms of existence of an attractor
for the 2D B´enard problem have been studied in [3] in the autonomous case and in [1] in the
non-autonomous case.
The 2D g-Navier-Stokes equations and its relationship with the 3D Navier-Stokes equations
in the thin domain Ωg was introduced by Roh in [12]. Since then there have been many
works devoted to studying mathematical questions related to these equations. In particular, the
existence and long-time behavior of solutions to 2D g-Navier-Stokes equations have been studied
extensively, in the both autonomous and non-autonomous cases, see e.g. [2, 5, 6, 7, 10, 13, 14].
The existence of time-periodic solutions to g-Navier-Stokes and g-Kelvin-Voight equations was
also studied more recently in [4].
¨ uk and M. Kaya considered
For the 2D g-B´enard problem, in [8] Hitherto, M. Ozl¨
Boussinesq equations in the bounded domain Ωg = {(y1 , y2 , y3 ) ∈ R3 : (y1 , y2 ) ∈ Ω2 , 0 < y3 <
g}, where Ω2 is a bounded region in the plane and g = g(y1 , y2 ) is a smooth function defined
on Ω2 . They proved the existence and uniqueness of weak solutions and derived upper bounds
¨ uk and M. Kaya investigated
for the number of determining modes. More recently, in [9] M. Ozl¨
the existence, uniqueness of strong solutions, and the continuous dependence of the solutions on
the viscosity parameter for problem (1.1) in the non-autonomous case and the function g to be
periodic with period 1 in the x1 and x2 directions.
In this paper we will study the existence and uniqueness of weak solutions to 2D g-B´enard
problem in domains that are not necessarily bounded but satisfy the Poincar´e inequality. To do
this, we assume that the domain Ω and functions f1 , f2 , g satisfy the following hypotheses:
(Ω) Ω is an arbitrary (not necessarily bounded) domain in R2 satisfying the Poincar´e type

inequality
φ2 gdx ≤
Ω

1
λ1

|∇φ|2 gdx,

for all φ ∈ C0∞ (Ω);

(1.2)

Ω

(F) f1 ∈ L2 (0, T ; Hg ), f2 ∈ L2 (0, T ; L2 (Ω, g));
(G) g ∈ W 1,∞ (Ω) such that
0 < m0 ≤ g(x) ≤ M0 for all x = (x1 , x2 ) ∈ Ω, and |∇g|2∞ < m20 λ1 ,

(1.3)

where λ1 > 0 is the constant in the inequality (1.2).
The paper is organized as follows. In Section 2, for convenience of the reader, we recall the
functional setting of the 2D g-B´enard problem. Section 3 is devoted to proving the existence and
uniqueness of global weak solutions to the problem by combining the Galerkin method and the
compactness lemma. The results obtained here extend and improve some previous results for 2D
B´enard problem in [3] and 2D g-Navier-Stokes equations in [6].
24

On the existence and uniqueness of solutions to 2D g-B´enard problem in unbounded domains

2.

Preliminaries

Let L2 (Ω, g) = (L2 (Ω, g))2 and H10 (Ω, g) = (H01 (Ω, g))2 be endowed with the usual inner
products and associated norms. We define
V1 = {u ∈ (C0∞ (Ω, g))2 : ∇ · (gu) = 0},
Hg = the closure of V1 in L2 (Ω, g),
Vg = the closure of V1 in H10 (Ω, g),
Vg′ = the dual space of Vg ,
V2 = {θ ∈ C0∞ (Ω, g)},
Wg = the closure of V2 in H01 (Ω, g),
Wg′ = the dual space of Wg ,
V = Vg × Wg , H = Hg × L2 (Ω, g).
The inner products and norms in Vg , Hg are given by
(u, v)g =

u · vgdx, u, v ∈ Hg ,
Ω

and

2

((u, v))g =

∇uj · ∇vi gdx, u, v ∈ Vg ,
Ω i,j=1

and norms |u|2g = (u, u)g , u 2g = ((u, u))g . The norms | · |g and · g are equivalent to the usual
ones in L2 (Ω, g) and H10 (Ω, g). We also use · ∗ for the norm in Vg′ , and ·, · for duality pairing
between Vg and Vg′ .
The inclusions
Vg ⊂ Hg ≡ Hg′ ⊂ Vg′ , Wg ⊂ L2 (Ω, g) ⊂ Wg′
are valid where each space is dense in the following one and the injections are continuous. By the
Riesz representation theorem, it is possible to write
f, u

g

= (f, u)g , ∀f ∈ Hg , ∀u ∈ Vg .

Also, we define the orthogonal projection Pg as Pg : Hg → Hg and P˜g as P˜g : L2 (Ω, g) →
Wg . By taking into account the following equality
1
1
− (∇ · g∇u) = −∆u − (∇g · ∇)u,
g
g
1
we define the g-Laplace operator and g-Stokes operator as −∆g u = − (∇ · g∇u) and Ag u =
g
Pg [−∆g u], respectively. Since the operators Ag and Pg are self-adjoint, using integration by parts
we have
Ag u, u

g

1
= Pg [− (∇ · g∇)u], u
g

g

(∇u · ∇u)gdx = (∇u, ∇u)g .

=
Ω

25


Tran Quang Thinh and Le Thi Thuy
1/2

Therefore, for u ∈ Vg , we can write |Ag u|g = |∇u|g = u g .
Next, since the functional
τ ∈ Wg → (∇θ, ∇τ )g ∈ R
is a continuous linear mapping on Wg , we can define a continuous linear mapping A˜g on Wg′ such
that
∀τ ∈ Wg , A˜g θ, τ g = (∇θ, ∇τ )g , for all θ ∈ Wg .
We denote the bilinear operator Bg (u, v) = Pg [(u · ∇)v] and the trilinear form
2

bg (u, v, w) =

ui
i,j=1 Ω

∂vj
wj gdx,
∂xi
Then, one obtains that bg (u, v, w) =

where u, v, w lie in appropriate subspaces of Vg .
−bg (u, w, v), which particularly implies that

bg (u, v, v) = 0.

(2.1)

Also bg satisfies the inequality
|bg (u, v, w)| ≤ c|u|1/2
u
g

1/2
g

v 2g |w|1/2
w
g

1/2
g .

(2.2)

˜g (u, θ) = P˜g [(u · ∇)θ] and
Similarly, for u ∈ Vg and θ, τ ∈ Wg we define B
n

˜bg (u, θ, τ ) =

ui (x)
i,j=1 Ω

∂θ(x)
τ (x)gdx.
∂xj

Then, one obtains that ˜bg (u, θ, τ ) = −˜bg (u, τ, θ), which particularly implies that
˜bg (u, θ, θ) = 0.

(2.3)

And ˜bg satisfies the inequality
|˜bg (u, θ, τ )| ≤ c|u|1/2
u
g

g

θ 2g |τ |1/2
τ
g

1/2

g .

1
1
(∇g · ∇)u and C˜g θ = P˜g (∇g · ∇)θ such that
g
g

We denote the operators Cg u = Pg
Cg u, v

1/2
g

= bg (

∇g
, u, v), C˜g θ, τ
g

g

= ˜bg (

∇g
, θ, τ ).
g

˜ g θ = P˜g [ ∆g θ] such that
Finally, let D

g
˜ g θ, τ
D
26

g

= −˜bg (

∇g
∇g
, θ, τ ) − ˜bg (
, τ, θ).
g
g

(2.4)


On the existence and uniqueness of solutions to 2D g-B´enard problem in unbounded domains

Using the above notations, we can rewrite the system
equations

du


+ Bg (u, u) + νAg u + νCg u



 dt

 dθ
˜g (u, θ) + κA˜g θ − κC˜g θ − κD
˜gθ
+B
dt



u(0)



θ(0)

3.

(1.1) as abstract evolutionary
= ξθ + f1 ,
= f2 ,
= u0 ,
= θ0 .

Existence and uniqueness of weak solutions

Definition 3.1. A pair of functions (u, θ) is called a weak solution of problem (1.1) on the interval
(0, T ) if u ∈ L2 (0, T ; Vg ) and θ ∈ L2 (0, T ; Wg ) satisfy

∇g

d


(u, v)g + bg (u, u, v) + ν(∇u, ∇v)g + νbg (
, u, v) = (ξθ, v)g


g
 dt
(3.1)
+(f1 , v)g ,


∇g
d


 (θ, τ )g + ˜bg (u, θ, τ ) + κ(∇θ, ∇τ )g + κ˜bg (
, τ, θ) = (f2 , τ )g ,
dt
g
for all test functions v ∈ Vg and τ ∈ Wg .

The following theorem is our main result.
Theorem 3.1. Let the initial datum (u0 , θ0 ) ∈ H be given, let the external forces f1 , f2 satisfy
hypothesis (F) and the function g satisfy hypothesis (G). Then there exists a unique weak solution
(u, θ) of problem (1.1) on the interval (0, T ).
Proof. Existence. We use the standard Galerkin method. Since Vg is separable and V1 is dense in
Vg , there exists a sequence {ui }i∈N which forms a complete orthonormal system in Hg and a base
for Vg . Similarly, there exists a sequence {θi }i∈N which forms a complete orthonormal system in

L2 (Ω, g) and a base for Wg .
Let m be an arbitrary but fixed positive integer. For each m we define an approximate
solution (um (t), θ m (t)) of (3.1) for 1 ≤ k ≤ m and t ∈ [0, T ] in the form,
m
(m)

u

(t) =

(m)
fj (t)uj ;

m

θ

(m)

(t) =

(m)

gj

m

m

u(m) (0) = um0 =

(a0 , uj )uj ;
j=1

(t)θj ,

j=1

j=1

θ (m) (0) = θm0 =

(τ0 , θj )θj ,
j=1

d (m)
(u , uk )g + bg (u(m) , u(m) , uk ) + ν((u(m) , uk ))g
dt
∇g (m)
, u , uk ) = (ξθ (m) , uk )g + (f1 , uk )g ,
+ νbg (
g
d (m)
(θ , θk )g + ˜bg (u(m) , θ (m) , θk ) + κ((θ (m) , θk ))g
dt
∇g
, θk , θ (m) ) = (f2 , θk )g .
+ κ˜bg (
g

(3.2)

(3.3)

27


Tran Quang Thinh and Le Thi Thuy

This system forms a nonlinear first order system of ordinary differential equations for the functions
(m)
(m)
fj (t) and gj (t) and has a solution on some maximal interval of existence [0, Tm ).
(m)

(m)

We multiply (3.2) and (3.3) by fj (t) and gj (t) respectively, then add these equations
for k = 1, . . . , m. Taking into account bg (u(m) , u(m) , u(m) ) = 0 and ˜bg (u(m) , θ (m) , θ (m) ) = 0,
we get
∇g (m)
, u (t), u(m) (t))
(u′(m) (t), u(m) (t))g + ν u(m) (t) 2g + νbg (
g
(3.4)
= (ξθ (m) , u(m) (t))g + (f1 , u(m) (t)),
∇g (m)
, θ (t), θ (m) (t))
(θ ′(m) (t), θ (m) (t))g + κ θ (m) (t) 2g +κ˜bg (
g

(3.5)

= (f2 , θ (m) (t))g .
Using (2.2), (2.4), the Schwarz and Young inequalities in (3.4) and (3.5) we obtain
d (m) 2
|u (t)|g + ν u(m) (t)
2dt
ν|∇g|∞ (m)
≤
u (t)
1/2
m 0 λ1
d (m) 2
|θ (t)|g + κ θ (m) (t)
2dt

2
g
2
g

+ ǫν u(m) (t)

2
g

≤

κ|∇g|∞

1/2
m 0 λ1

2
g

+

ξ 2∞ (m)
θ (t)
2ǫνλ21

θ (m) (t)

2
g

2
g

+ ǫκ θ (m) (t)

+
2
g

1
|f1 |2g ,
2ǫνλ1

+

1
|f2 |2g ,
4ǫκλ1

so that for
ν ′ = 2ν

1−

|∇g|∞
1/2

m 0 λ1

− ǫ , κ′ = 2κ 1 −

|∇g|∞
1/2

m0 λ1

− ǫ , c′ =

ξ 2∞
ǫλ21

we get
d (m) 2

c′ (m)
1
|u (t)|g + ν ′ u(m) (t) 2g ≤
θ (t) 2g +
|f1 |2g ,
dt
ν
ǫλ1 ν
1
d (m) 2
|θ (t)|g + κ′ θ (m) (t) 2g ≤
|f2 |2g ,
dt
2ǫλ1 κ
where ǫ > 0 is chosen such that

1−

|∇g|∞

−ǫ
1/2
m0 λ1
Integrating (3.7) and (3.6) from 0 to t, we obtain

sup |u(m) (t)|2g ≤ |u0 |2g +

t∈[0,T ]

(3.7)

> 0.

T
|f2 |2g .
2ǫλ1 κ

(3.8)

c′
c′ T
T
2
|θ
|
+
|f2 |2g +
|f1 |2g .
0
g
′
′
νκ
2ǫλ1 νκκ
ǫλ1 ν

(3.9)

sup |θ (m) (t)|2g ≤ |θ0 |2g +

t∈[0,T ]

(3.6)

These inequalities imply that the sequences {u(m) }m and {θ (m) }m remain in a bounded set of
L∞ (0, T ; Hg ) and L∞ (0, T ; L2 (Ω, g)), respectively. We then integrate (3.6) and (3.7) from 0 to
T to get
T
T
|f2 |2g ,
(3.10)
|θ (m) (T )|2g + κ′
θ (m) (t) 2g dt ≤
2ǫλ
κ
1
0
28


On the existence and uniqueness of solutions to 2D g-B´enard problem in unbounded domains

|u(m) (T )|2g + ν ′

T

u(m) (t) 2g dt ≤

0

c′ T
T
|f1 |2g ,
|f2 |2g +
′
2ǫλ1 νκκ
ǫλ1 ν

(3.11)

which shows that the sequences {u(m) }m and {θ (m) }m are bounded in L2 (0, T ; Vg ) and
L2 (0, T ; Wg ), respectively. Due to the estimates (3.8)-(3.11), we assert the existence of elements
u ∈ L2 (0, T ; Vg ) ∩ L∞ (0, T ; Hg ),
θ ∈ L2 (0, T ; Wg ) ∩ L∞ (0, T ; L2 (Ω, g)),
and the subsequences {u(m) }m and {θ (m) }m such that
u(m) ⇀ u in L2 (0, T ; Vg ),
θ (m) ⇀ θ in L2 (0, T ; Wg ),
and
u(m) ⇀ u weakly-star in L∞ (0, T ; Hg ),
θ (m) ⇀ θ weakly-star in L∞ (0, T ; L2 (Ω, g)).
Applying the Aubin-Lions lemma, we have subsequences {u(m) }m and {θ (m) }m such that
u(m) → u in L2 (0, T ; Hg ),
θ (m) → θ in L2 (0, T ; L2 (Ω, g)).
In order to pass to the limit, we consider the scalar functions Ψ1 (t) and Ψ2 (t) continuously
differentiable on [0, T ] and such that Ψ1 (T ) = 0 and Ψ2 (T ) = 0. We multiply (3.2) and (3.3) by
Ψ1 (t) and Ψ2 (t) respectively and then integrate by parts,
T

−
0

(u(m) , Ψ′1 uk )g dt+

T

bg (u(m) , u(m) , Ψ1 uk )dt

0
T

+ν

T

((u(m) , Ψ1 uk ))g dt + ν

bg (
0

0
T

= (um0 , uk )g Ψ1 (0) +

∇g (m)
, u , Ψ1 uk )dt
g

T
0

(θ (m) , Ψ′2 θk )g dt +
T

+κ
0

T

(f1 , uk )g dt,
0

0

−

T

(ξθ (m) , Ψ1 uk )g dt +
T

˜bg (u(m) , θ (m) , Ψ2 θk )dt + κ

0

((θ (m) , Ψ2 θk ))g dt

0

˜bg ( ∇g , θk , Ψ2 θ (m) )dt = (θm0 , θk )g Ψ2 (0) +

g

T

(f2 , Ψ2 θk )g dt.
0

Following the technique given in [15], as m → ∞ we obtain

0
T

+ν
0

T

T

T

−

(u, Ψ′1 v)g dt +

((u, Ψ1 v))g dt

bg (u, u, Ψ1 v)dt + ν
0
T

0

1
bg ( ∇g, u, Ψ1 v)dt = (u0 , v)g Ψ1 (0) +
g

(ξθ, Ψ1 v)g dt

(3.12)

0
T

(f1 , v)g dt,

+
0

29


Tran Quang Thinh and Le Thi Thuy
T

T

−
0
T

+κ
0

(θ, Ψ′2 τ )g dt +

T

˜bg (u, θ, Ψ2 τ )dt + κ

0

˜bg ( ∇g , τ, Ψ2 θ)dt = (θ0 , τ )g Ψ2 (0) +
g

((θ, Ψ2 τ ))g dt
0
T

(3.13)

(f2 , Ψ2 τ )g dt.
0

The equations (3.12) and (3.13) hold for v and τ which are finite linear combinations of the uk
and θk for k = 1, . . . , m and by continuity (3.12) and (3.13) hold for v ∈ Vg and τ ∈ Hg
respectively. Rewriting (3.12) and (3.13) for Ψ1 (t), Ψ2 (t) ∈ C0∞ (0, T ) we see that (u, θ) satisfy
(3.1). Furthermore, applying similar techniques given in [13, 15] it is easy to show that (u, θ)
satisfies the initial conditions u(0) = u0 and θ(0) = θ0 .
Uniqueness. For the uniqueness of weak solutions, let (u1 , θ1 ) and (u2 , θ2 ) be two weak

solutions with the same initial conditions. Putting w = u1 − u2 and w
˜ = θ1 − θ2 . Then we have
d
(w, v)g + bg (u1 , u1 , v) − bg (u2 , u2 , v) + ν(∇w, ∇v)g + ν(Cg w, v)g = (ξ w,
˜ v)g ,
dt
∇g
d
(w,
˜ τ )g + ˜bg (u1 , θ1 , τ ) − ˜bg (u2 , θ2 , τ ) + κ(∇w,
˜ ∇τ )g + κ˜bg (
, τ, w)
˜ = 0.
dt
g
Taking v = w(t), τ = w(t)
˜ and (2.1), (2.3) we obtain
1 d
∇g
|w|2 + ν w 2g ≤ |bg (w, u2 , w)| + ν|bg (
, w, w)| + |(ξ w,
˜ w)g |,
2 dt g
g
∇g
1 d
|w|
˜ 2g + κ w˜ 2g + ≤ |˜bg (w, θ2 , w)|
˜ + κ|˜bg (
, w,

˜ w)|.
˜
2 dt
g
By applying (2.2), (2.4) it then follows by the Cauchy-Schwarz inequality, we have
1 d
|w|2 + ν w
2 dt g

2
g

≤

1 d
|w|
˜ 2 + κ w˜
2 dt g

2
g

≤

c2 2
|w| u2
ǫν g
ǫν
w
2

2
g

2
g

+

ξ 2∞ 2
|w|
˜ ,
ǫνλ1

(3.14)

c4 |θ2 |4g
κ|∇g|∞
|w|
˜ 2g +
w
˜ 2g .
2
3
2
1/2
16ǫ ν κλ1
m 0 λ1

(3.15)

ν|∇g|∞
1/2

m 0 λ1

+ ǫκ w
˜

2
g

+

w

2
g

+

ǫν
w
2

2
g

+

We sum equations (3.14) and (3.15) to obtain
|∇g|∞
d
− ǫ (ν w
(|w|2g + |w|
˜ 2g ) + 2 1 −
1/2
dt
m 0 λ1
≤

2c2 u2
ǫν

2
g

|w|2g +

2
g

+ κ w˜ 2g )

c4 θ2 4g
2 ξ 2∞
+ 3 2 2
ǫνλ1
8ǫ ν κλ1

|w|
˜ 2g ,

so that for
γ = max

2c2 u2
ǫν

2
g

;

c2 θ 4g
2 ξ 2∞
+ 3 2 2
ǫνλ1
8ǫ ν κλ1

we get
d
(|w|2g + |w|
˜ 2g ) ≤ γ(|w|2g + |w|
˜ 2g ).
dt
30

,

On the existence and uniqueness of solutions to 2D g-B´enard problem in unbounded domains

Thanks to the Gronwall inequality, we have
2
γt
|w(t)|2g + |w(t)|
˜ 2g ≤ |w(0)|2g + |w(0)|
˜
g e .

Hence, the continuous dependence of the weak solution on the initial data in any bounded interval
for all t ≥ 0. In particular, the solution is unique.
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