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 Ω
g 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].
For the 2D g-B´enard problem, in [8] Hitherto, M. Ozl¨uk and M. Kaya considered ¨
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
for the number of determining modes. More recently, in [9] M. Ozl¨uk and M. Kaya investigated ¨
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.

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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-B ´ENARD PROBLEM IN UNBOUNDED DOMAINS
Tran Quang Thinh1 and Le Thi Thuy2
1Faculty of Basic Sciences, Nam Dinh University of Technology Education
2Faculty of Mathematics, Electric Power University
Abstract. We consider the 2D g-Be´nard problem in domains satisfying the Poincare´
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 Be´nard problem.
1. Introduction
Let Ω be a (not necessarily bounded) domain in R2 with boundary Γ. We consider the
following two-dimensional (2D) g-Be´nard problem
∂u
∂t
+ (u · ∇)u− ν∆u+∇p = ξθ + f1, x ∈ Ω, t > 0,
∇ · (gu) = 0, x ∈ Ω, t > 0,
∂θ
∂t
+ (u · ∇)θ − κ∆θ −
2κ
g
(∇g · ∇)θ −
κ∆g
g
θ = f2, x ∈ Ω, t > 0,
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,
(1.1)
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-Be´nard problem arises in a natural way when
we study the standard 3D Be´nard problem on the thin domain Ωg = Ω × (0, g). Here the
g-Be´nard 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 Be´nard problem, and when θ ≡ 0we 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: thuylt@epu.edu.vn
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 Be´nard 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].
For the 2D g-Be´nard problem, in [8] Hitherto, M. ¨Ozlu¨k and M. Kaya considered
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
for the number of determining modes. More recently, in [9] M. ¨Ozlu¨k and M. Kaya investigated
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-Be´nard
problem in domains that are not necessarily bounded but satisfy the Poincare´ 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 Poincare´ type
inequality ∫
Ω
φ2gdx ≤
1
λ1
∫
Ω
|∇φ|2gdx, for all φ ∈ C∞0 (Ω); (1.2)
(F) f1 ∈ L
2(0, T ;Hg), f2 ∈ L
2(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∞ < m20λ1, (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-Be´nard 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
Be´nard problem in [3] and 2D g-Navier-Stokes equations in [6].
24
On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains
2. Preliminaries
Let L2(Ω, g) = (L2(Ω, g))2 and H10(Ω, g) = (H10 (Ω, g))2 be endowed with the usual inner
products and associated norms. We define
V1 = {u ∈ (C
∞
0 (Ω, g))
2 : ∇ · (gu) = 0},
Hg = the closure of V1 in L2(Ω, g),
Vg = the closure of V1 in H10(Ω, g),
V ′g = the dual space of Vg,
V2 = {θ ∈ C
∞
0 (Ω, g)},
Wg = the closure of V2 in H10 (Ω, g),
W ′g = the dual space of Wg,
V = Vg ×Wg, H = Hg × L
2(Ω, g).
The inner products and norms in Vg, Hg are given by
(u, v)g =
∫
Ω
u · vgdx, u, v ∈ Hg,
and
((u, v))g =
∫
Ω
2∑
i,j=1
∇uj · ∇vigdx, u, v ∈ Vg,
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 V ′g , and 〈·, ·〉 for duality pairing
between Vg and V ′g .
The inclusions
Vg ⊂ Hg ≡ H
′
g ⊂ V
′
g , Wg ⊂ L
2(Ω, g) ⊂W ′g
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
g
(∇ · g∇u) = −∆u−
1
g
(∇g · ∇)u,
we define the g-Laplace operator and g-Stokes operator as −∆gu = −
1
g
(∇ · g∇u) and Agu =
Pg[−∆gu], respectively. Since the operators Ag and Pg are self-adjoint, using integration by parts
we have
〈Agu, u〉g = 〈Pg[−
1
g
(∇ · g∇)u], u〉g =
∫
Ω
(∇u · ∇u)gdx = (∇u,∇u)g.
25
Tran Quang Thinh and Le Thi Thuy
Therefore, for u ∈ Vg, we can write |A1/2g u|g = |∇u|g = ‖u‖g .
Next, since the functional
τ ∈Wg 7→ (∇θ,∇τ)g ∈ R
is a continuous linear mapping on Wg, we can define a continuous linear mapping A˜g on W ′g such
that
∀τ ∈Wg, 〈A˜gθ, τ〉g = (∇θ,∇τ)g, for all θ ∈Wg.
We denote the bilinear operator Bg(u, v) = Pg[(u · ∇)v] and the trilinear form
bg(u, v, w) =
2∑
i,j=1
∫
Ω
ui
∂vj
∂xi
wjgdx,
where u, v, w lie in appropriate subspaces of Vg. Then, one obtains that bg(u, v, w) =
−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
g ‖u‖
1/2
g ‖v‖
2
g|w|
1/2
g ‖w‖
1/2
g . (2.2)
Similarly, for u ∈ Vg and θ, τ ∈Wg we define B˜g(u, θ) = P˜g[(u · ∇)θ] and
b˜g(u, θ, τ) =
n∑
i,j=1
∫
Ω
ui(x)
∂θ(x)
∂xj
τ(x)gdx.
Then, one obtains that b˜g(u, θ, τ) = −b˜g(u, τ, θ), which particularly implies that
b˜g(u, θ, θ) = 0. (2.3)
And b˜g satisfies the inequality
|b˜g(u, θ, τ)| ≤ c|u|
1/2
g ‖u‖
1/2
g ‖θ‖
2
g|τ |
1/2
g ‖τ‖
1/2
g . (2.4)
We denote the operators Cgu = Pg
[1
g
(∇g · ∇)u
]
and C˜gθ = P˜g
[1
g
(∇g · ∇)θ
]
such that
〈Cgu, v〉g = bg(
∇g
g
, u, v), 〈C˜gθ, τ〉g = b˜g(
∇g
g
, θ, τ).
Finally, let D˜gθ = P˜g[
∆g
g
θ] such that
〈D˜gθ, τ〉g = −b˜g(
∇g
g
, θ, τ)− b˜g(
∇g
g
, τ, θ).
26
On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains
Using the above notations, we can rewrite the system (1.1) as abstract evolutionary
equations
du
dt
+Bg(u, u) + νAgu+ νCgu = ξθ + f1,
dθ
dt
+ B˜g(u, θ) + κA˜gθ − κC˜gθ − κD˜gθ = f2,
u(0) = u0,
θ(0) = θ0.
3. 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
d
dt
(u, v)g + bg(u, u, v) + ν(∇u,∇v)g + νbg(
∇g
g
, u, v) = (ξθ, v)g
+(f1, v)g ,
d
dt
(θ, τ)g + b˜g(u, θ, τ) + κ(∇θ,∇τ)g + κb˜g(
∇g
g
, τ, θ) = (f2, τ)g,
(3.1)
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), θm(t)) of (3.1) for 1 ≤ k ≤ m and t ∈ [0, T ] in the form,
u(m)(t) =
m∑
j=1
f
(m)
j (t)uj ; θ
(m)(t) =
m∑
j=1
g
(m)
j (t)θj,
u(m)(0) = um0 =
m∑
j=1
(a0, uj)uj ; θ
(m)(0) = θm0 =
m∑
j=1
(τ0, θj)θj,
d
dt
(u(m), uk)g + bg(u
(m), u(m), uk) + ν((u
(m), uk))g
+ νbg(
∇g
g
, u(m), uk) = (ξθ
(m), uk)g + (f1, uk)g,
(3.2)
d
dt
(θ(m), θk)g + b˜g(u
(m), θ(m), θk) + κ((θ
(m), θk))g
+ κb˜g(
∇g
g
, θk, θ
(m)) = (f2, θk)g.
(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
f
(m)
j (t) and g
(m)
j (t) and has a solution on some maximal interval of existence [0, Tm).
We multiply (3.2) and (3.3) by f (m)j (t) and g(m)j (t) respectively, then add these equations
for k = 1, . . . ,m. Taking into account bg(u(m), u(m), u(m)) = 0 and b˜g(u(m), θ(m), θ(m)) = 0,
we get
(u′(m)(t), u(m)(t))g + ν‖u
(m)(t)‖2g + νbg(
∇g
g
, u(m)(t), u(m)(t))
= (ξθ(m), u(m)(t))g + (f1, u
(m)(t)),
(3.4)
(θ′(m)(t), θ(m)(t))g + κ‖θ
(m)(t)‖2g+κb˜g(
∇g
g
, θ(m)(t), θ(m)(t))
= (f2, θ
(m)(t))g.
(3.5)
Using (2.2), (2.4), the Schwarz and Young inequalities in (3.4) and (3.5) we obtain
d
2dt
|u(m)(t)|2g + ν‖u
(m)(t)‖2g
≤
ν|∇g|∞
m0λ
1/2
1
‖u(m)(t)‖2g + ǫν‖u
(m)(t)‖2g +
‖ξ‖2
∞
2ǫνλ21
‖θ(m)(t)‖2g +
1
2ǫνλ1
|f1|
2
g,
d
2dt
|θ(m)(t)|2g + κ‖θ
(m)(t)‖2g ≤
κ|∇g|∞
m0λ
1/2
1
‖θ(m)(t)‖2g + ǫκ‖θ
(m)(t)‖2g +
1
4ǫκλ1
|f2|
2
g,
so that for
ν ′ = 2ν
(
1−
|∇g|∞
m0λ
1/2
1
− ǫ
)
, κ′ = 2κ
(
1−
|∇g|∞
m0λ
1/2
1
− ǫ
)
, c′ =
‖ξ‖2
∞
ǫλ21
we get
d
dt
|u(m)(t)|2g + ν
′‖u(m)(t)‖2g ≤
c′
ν
‖θ(m)(t)‖2g +
1
ǫλ1ν
|f1|
2
g, (3.6)
d
dt
|θ(m)(t)|2g + κ
′‖θ(m)(t)‖2g ≤
1
2ǫλ1κ
|f2|
2
g, (3.7)
where ǫ > 0 is chosen such that
(
1−
|∇g|∞
m0λ
1/2
1
− ǫ
)
> 0.
Integrating (3.7) and (3.6) from 0 to t, we obtain
sup
t∈[0,T ]
|θ(m)(t)|2g ≤ |θ0|
2
g +
T
2ǫλ1κ
|f2|
2
g. (3.8)
sup
t∈[0,T ]
|u(m)(t)|2g ≤ |u0|
2
g +
c′
νκ′
|θ0|
2
g +
c′T
2ǫλ1νκκ′
|f2|
2
g +
T
ǫλ1ν
|f1|
2
g. (3.9)
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
|θ(m)(T )|2g + κ
′
∫ T
0
‖θ(m)(t)‖2gdt ≤
T
2ǫλ1κ
|f2|
2
g, (3.10)
28
On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains
|u(m)(T )|2g + ν
′
∫ T
0
‖u(m)(t)‖2gdt ≤
c′T
2ǫλ1νκκ′
|f2|
2
g +
T
ǫλ1ν
|f1|
2
g, (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′1uk)gdt+
∫ T
0
bg(u
(m), u(m),Ψ1uk)dt
+ ν
∫ T
0
((u(m),Ψ1uk))gdt+ ν
∫ T
0
bg(
∇g
g
, u(m),Ψ1uk)dt
= (um0, uk)gΨ1(0) +
∫ T
0
(ξθ(m),Ψ1uk)gdt+
∫ T
0
(f1, uk)gdt,
−
∫ T
0
(θ(m),Ψ′2θk)gdt+
∫ T
0
b˜g(u
(m), θ(m),Ψ2θk)dt+ κ
∫ T
0
((θ(m),Ψ2θk))gdt
+ κ
∫ T
0
b˜g(
∇g
g
, θk,Ψ2θ
(m))dt = (θm0, θk)gΨ2(0) +
∫ T
0
(f2,Ψ2θk)gdt.
Following the technique given in [15], as m→∞ we obtain
−
∫ T
0
(u,Ψ′1v)gdt+
∫ T
0
bg(u, u,Ψ1v)dt + ν
∫ T
0
((u,Ψ1v))gdt
+ν
∫ T
0
bg(
1
g
∇g, u,Ψ1v)dt = (u0, v)gΨ1(0) +
∫ T
0
(ξθ,Ψ1v)gdt
+
∫ T
0
(f1, v)gdt,
(3.12)
29
Tran Quang Thinh and Le Thi Thuy
−
∫ T
0
(θ,Ψ′2τ)gdt+
∫ T
0
b˜g(u, θ,Ψ2τ)dt+ κ
∫ T
0
((θ,Ψ2τ))gdt
+κ
∫ T
0
b˜g(
∇g
g
, τ,Ψ2θ)dt = (θ0, τ)gΨ2(0) +
∫ T
0
(f2,Ψ2τ)gdt.
(3.13)
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) ∈ C∞0 (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
dt
(w, v)g + bg(u1, u1, v)− bg(u2, u2, v) + ν(∇w,∇v)g + ν(Cgw, v)g = (ξw˜, v)g,
d
dt
(w˜, τ)g + b˜g(u1, θ1, τ)− b˜g(u2, θ2, τ) + κ(∇w˜,∇τ)g + κb˜g(
∇g
g
, τ, w˜) = 0.
Taking v = w(t), τ = w˜(t) and (2.1), (2.3) we obtain
1
2
d
dt
|w|2g + ν‖w‖
2
g ≤ |bg(w, u2, w)| + ν|bg(
∇g
g
,w,w)| + |(ξw˜, w)g|,
1
2
d
dt
|w˜|2g + κ‖w˜‖
2
g+ ≤ |b˜g(w, θ2, w˜)|+ κ|b˜g(
∇g
g
, w˜, w˜)|.
By applying (2.2), (2.4) it then follows by the Cauchy-Schwarz inequality, we have
1
2
d
dt
|w|2g + ν‖w‖
2
g ≤
c2
ǫν
|w|2g‖u2‖
2
g +
ν|∇g|∞
m0λ
1/2
1
‖w‖2g +
ǫν
2
‖w‖2g +
‖ξ‖2
∞
ǫνλ1
|w˜|2, (3.14)
1
2
d
dt
|w˜|2g + κ‖w˜‖
2
g ≤
ǫν
2
‖w‖2g + ǫκ‖w˜‖
2
g +
c4|θ2|
4
g
16ǫ3ν2κλ21
|w˜|2g +
κ|∇g|∞
m0λ
1/2
1
‖w˜‖2g. (3.15)
We sum equations (3.14) and (3.15) to obtain
d
dt
(|w|2g + |w˜|
2
g) + 2
(
1−
|∇g|∞
m0λ
1/2
1
− ǫ
)
(ν‖w‖2g + κ‖w˜‖
2
g)
≤
2c2‖u2‖
2
g
ǫν
|w|2g +
(
2‖ξ‖2
∞
ǫνλ1
+
c4‖θ2‖
4
g
8ǫ3ν2κλ21
)
|w˜|2g,
so that for
γ = max
{
2c2‖u2‖
2
g
ǫν
;
2‖ξ‖2
∞
ǫνλ1
+
c2‖θ‖4g
8ǫ3ν2κλ21
}
,
we get
d
dt
(|w|2g + |w˜|
2
g) ≤ γ(|w|
2
g + |w˜|
2
g).
30
On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains
Thanks to the Gronwall inequality, we have
|w(t)|2g + |w˜(t)|
2
g ≤
(
|w(0)|2g + |w˜(0)|
2
g
)
eγt.
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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31