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.
Trang 1This paper is available online at http://stdb.hnue.edu.vn
ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS
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-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
LetΩ be a (not necessarily bounded) domain in R2 with boundary Γ We consider the
following two-dimensional (2D) g-B´enard problem
∂u
∂t + (u · ∇)u − ν∆u + ∇p = ξθ + f1, x∈ Ω, t > 0,
∂θ
∂t + (u · ∇)θ − κ∆θ −2κ
g (∇g · ∇)θ − κ∆g
g θ= f2, x∈ Ω, t > 0,
(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, u0is the initial velocity and θ0is 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: thuylt@epu.edu.vn
Trang 2The 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]
For the 2D g-B´enard problem, in [8] Hitherto, M ¨Ozl¨uk and M Kaya considered Boussinesq equations in the bounded domainΩg = {(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 period1 in the x1 and x2directions
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 domainΩ 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
Z
Ω
φ2gdx≤ 1
λ1 Z
Ω
|∇φ|2gdx, for all φ∈ C∞
0 (Ω); (1.2)
(F) f1 ∈ L2(0, T ; Hg), f2∈ L2(0, T ; L2(Ω, g));
(G) g ∈ W1,∞(Ω) such that
0 < m0≤ g(x) ≤ M0for 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-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]
Trang 32 Preliminaries
Let L2(Ω, g) = (L2(Ω, g))2and H10(Ω, g) = (H01(Ω, g))2be endowed with the usual inner products and associated norms We define
V1= {u ∈ (C∞
0 (Ω, g))2 : ∇ · (gu) = 0},
Hg= the closure of V1in L2(Ω, g),
Vg= the closure of V1in H10(Ω, g),
V′
g = the dual space of Vg,
V2= {θ ∈ C∞
0 (Ω, g)},
Wg= the closure of V2in H01(Ω, g),
W′
g= the dual space of Wg,
V = Vg× Wg, H = Hg× L2(Ω, g)
The inner products and norms in Vg, Hgare given by
(u, v)g =
Z
Ω
u· vgdx, u, v ∈ Hg,
and
((u, v))g =
Z
Ω
2 X
i,j=1
∇uj · ∇vigdx, u, v ∈ Vg,
and norms|u|2
g = (u, u)g,kuk2
g = ((u, u))g The norms| · |gandk · kgare equivalent to the usual ones in L2(Ω, g) and H10(Ω, g) We also use k · k∗for the norm in V′
g, andh·, ·i for duality pairing
between Vgand V′
g The inclusions
Vg ⊂ Hg ≡ H′
g⊂ V′
g, Wg⊂ L2(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
hf, uig = (f, u)g,∀f ∈ Hg,∀u ∈ Vg
Also, we define the orthogonal projection Pg as Pg: Hg → Hg and ˜Pg as ˜Pg: 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 Agand Pgare self-adjoint, using integration by parts
we have
hAgu, uig= hPg[−1
g(∇ · g∇)u], uig =
Z
Ω (∇u · ∇u)gdx = (∇u, ∇u)g
Trang 4Therefore, for u∈ Vg, we can write|A1/2g u|g = |∇u|g= kukg.
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 ˜Agon W′
gsuch that
∀τ ∈ Wg,h ˜Agθ, τig = (∇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 X
i,j=1
Z
Ω
ui∂vj
∂xiwjgdx,
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
Also bgsatisfies the inequality
|bg(u, v, w)| ≤ c|u|1/2g kuk1/2g kvk2g|w|1/2g kwk1/2g (2.2) Similarly, for u∈ Vg and θ, τ ∈ Wgwe define ˜Bg(u, θ) = ˜Pg[(u · ∇)θ] and
˜bg(u, θ, τ ) =
n X
i,j=1
Z
Ω
ui(x)∂θ(x)
∂xj τ(x)gdx
Then, one obtains that ˜bg(u, θ, τ ) = −˜bg(u, τ, θ), which particularly implies that
And ˜bg satisfies the inequality
|˜bg(u, θ, τ )| ≤ c|u|1/2g kuk1/2g kθk2g|τ |1/2g kτ k1/2g (2.4)
We denote the operators Cgu= Pg 1
g(∇g · ∇)u and ˜Cgθ= ˜Pg 1
g(∇g · ∇)θ such that
hCgu, vig = bg(∇g
g , u, v), h ˜Cgθ, τig = ˜bg(∇g
g , θ, τ)
Finally, let ˜Dgθ= ˜Pg[∆g
g θ] such that
h ˜Dgθ, τig = −˜bg(∇g
g , θ, τ) − ˜bg(∇g
g , τ, θ)
Trang 5Using 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 + ˜Bg(u, θ) + κ ˜Agθ− κ ˜Cgθ− κ ˜Dgθ = f2,
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+ ˜bg(u, θ, τ ) + κ(∇θ, ∇τ )g+ κ˜bg(∇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 Vgis separable andV1is dense in
Vg, there exists a sequence{ui}i∈Nwhich forms a complete orthonormal system in Hgand a base for Vg Similarly, there exists a sequence{θi}i∈Nwhich 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 X
j=1
fj(m)(t)uj; θ(m)(t) =
m X
j=1
g(m)j (t)θj,
u(m)(0) = um0=
m X
j=1 (a0, uj)uj; θ(m)(0) = θm0=
m X
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+ ˜bg(u(m), θ(m), θk) + κ((θ(m), θk))g
+ κ˜bg(∇g
g , θk, θ
(m)) = (f2, θk)g
(3.3)
Trang 6This system forms a nonlinear first order system of ordinary differential equations for the functions
fj(m)(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 fj(m)(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 ˜bg(u(m), θ(m), θ(m)) = 0,
we get
(u′(m)(t), u(m)(t))g+ νku(m)(t)k2g+ 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+ κkθ(m)(t)k2g+κ˜bg(∇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+ νku(m)(t)k2g
≤ν|∇g|∞
m0λ1/21
ku(m)(t)k2g+ ǫνku(m)(t)k2g+ kξk
2
∞ 2ǫνλ2 1
kθ(m)(t)k2g+ 1
2ǫνλ1|f1|
2
g, d
2dt|θ
(m)(t)|2g+ κkθ(m)(t)k2g ≤ κ|∇g|∞
m0λ1/21
kθ(m)(t)k2g+ ǫκkθ(m)(t)k2g+ 1
4ǫκλ1|f2|
2
g,
so that for
ν′
= 2ν 1 − |∇g|∞
m0λ1/21
− ǫ
! , κ′
= 2κ 1 − |∇g|∞
m0λ1/21
− ǫ
! , c′
= kξk
2
∞
ǫλ21
we get
d
dt|u(m)(t)|2g+ ν′
ku(m)(t)k2g≤ c
′
νkθ(m)(t)k2g+ 1
ǫλ1ν|f1|2g, (3.6)
d
dt|θ(m)(t)|2g+ κ′
kθ(m)(t)k2g ≤ 1
2ǫλ1κ|f2|2g, (3.7)
where ǫ >0 is chosen such that 1 − |∇g|∞
m0λ1/21
− ǫ
!
>0
Integrating (3.7) and (3.6) from0 to t, we obtain
sup t∈[0,T ]
|θ(m)(t)|2g ≤ |θ0|2g+ T
2ǫλ1κ|f2|2g (3.8)
sup t∈[0,T ]
|u(m)(t)|2g ≤ |u0|2g+ c
′
νκ′|θ0|2g+ c
′T 2ǫλ1νκκ′|f2|2g+ T
ǫλ1ν|f1|2g (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′
Z T 0
kθ(m)(t)k2gdt≤ T
2ǫλ1κ|f2|2g, (3.10)
Trang 7|u(m)(T )|2g+ ν′
Z T 0
ku(m)(t)k2gdt≤ c
′T 2ǫλ1νκκ′|f2|2g+ T
ǫλ1ν|f1|2g, (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)}mand{θ(m)}msuch 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)}msuch 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 functionsΨ1(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,
−
Z T
0
(u(m),Ψ′
1uk)gdt+
Z T 0
bg(u(m), u(m),Ψ1uk)dt + ν
Z T 0 ((u(m),Ψ1uk))gdt+ ν
Z T 0
bg(∇g
g , u (m),Ψ1uk)dt
= (um0, uk)gΨ1(0) +
Z T 0 (ξθ(m),Ψ1uk)gdt+
Z T 0 (f1, uk)gdt,
−
Z T
0
(θ(m),Ψ′
2θk)gdt+
Z T 0
˜bg(u(m), θ(m),Ψ2θk)dt + κ
Z T 0 ((θ(m),Ψ2θk))gdt + κ
Z T 0
˜bg(∇g
g , θk,Ψ2θ(m))dt = (θm0, θk)gΨ2(0) +
Z T 0 (f2,Ψ2θk)gdt
Following the technique given in [15], as m→ ∞ we obtain
−
Z T 0
(u, Ψ′
1v)gdt+
Z T 0
bg(u, u, Ψ1v)dt + ν
Z T 0 ((u, Ψ1v))gdt +ν
Z T 0
bg(1
g∇g, u, Ψ1v)dt = (u0, v)gΨ1(0) +
Z T 0 (ξθ, Ψ1v)gdt +
Z T 0 (f1, v)gdt,
(3.12)
Trang 8Z T 0
(θ, Ψ′
2τ)gdt+
Z T 0
˜bg(u, θ, Ψ2τ)dt + κ
Z T 0 ((θ, Ψ2τ))gdt +κ
Z T 0
˜bg(∇g
g , τ,Ψ2θ)dt = (θ0, τ)gΨ2(0) +
Z 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) forΨ1(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) = u0and 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− u2andw˜ = 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 = (ξ ˜w, v)g, d
dt( ˜w, τ)g+ ˜bg(u1, θ1, τ) − ˜bg(u2, θ2, τ) + κ(∇ ˜w,∇τ )g+ κ˜bg(∇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+ νkwk2g ≤ |bg(w, u2, w)| + ν|bg(∇g
g , w, w)| + |(ξ ˜w, w)g|, 1
2
d
dt| ˜w|2g+ κk ˜wk2g+ ≤ |˜bg(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+ νkwk2g ≤ c
2
ǫν|w|2gku2k2g+ ν|∇g|∞
m0λ1/21 kwk2g+ ǫν
2 kwk
2
g+ kξk
2
∞ ǫνλ1| ˜w|2, (3.14)
1
2
d
dt| ˜w|2g+ κk ˜wk2g ≤ ǫν
2 kwk
2
g+ ǫκk ˜wk2g+ c
4|θ2|4g 16ǫ3ν2κλ21| ˜w|2g+κ|∇g|∞
m0λ1/21 k ˜wk2g (3.15)
We sum equations (3.14) and (3.15) to obtain
d
dt(|w|2g+ | ˜w|2g) + 2 1 − |∇g|∞
m0λ1/21 − ǫ
! (νkwk2g+ κk ˜wk2g)
≤ 2c
2ku2k2g
ǫν |w|2g+ 2kξk
2
∞ ǫνλ1 + c
4kθ2k4g 8ǫ3ν2κλ21
!
| ˜w|2g,
so that for
γ = max
( 2c2ku2k2
ǫν ;2kξk
2
∞ ǫνλ1 + c
2kθk4 8ǫ3ν2κλ21
) ,
we get
d
dt(|w|2g+ | ˜w|2g) ≤ γ(|w|2g+ | ˜w|2g)
Trang 9Thanks to the Gronwall inequality, we have
|w(t)|2g+ | ˜w(t)|2g ≤ |w(0)|2g + | ˜w(0)|2g 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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