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.
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 http://stdb.hnue.edu.vn ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS ´ TO 2D G-BENARD PROBLEM IN UNBOUNDED DOMAINS Tran Quang Thinh1 and Le Thi Thuy2 Faculty of Basic Sciences, Nam Dinh University of Technology Education 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 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 > is the kinematic viscosity coefficient, ξ is a constant vector, κ > 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 ≡ 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 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] ă uk and M Kaya considered For the 2D g-Benard problem, in [8] Hitherto, M Ozlă Boussinesq equations in the bounded domain Ωg = {(y1 , y2 , y3 ) ∈ R3 : (y1 , y2 ) ∈ Ω2 , < 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 ă uk and M Kaya investigated for the number of determining modes More recently, in [9] M Ozlă 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 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 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 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 < m0 ≤ g(x) ≤ M0 for all x = (x1 , x2 ) ∈ Ω, and |∇g|2∞ < m20 λ1 , (1.3) where λ1 > 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 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 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 ((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 − (∇ · g∇u) = −∆u − (∇g · ∇)u, g g 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 = 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 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) = (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) = (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 (∇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) (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 ≤ 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) ) = 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) |u (t)|g + ν u(m) (t) 2dt ν|∇g|∞ (m) ≤ u (t) 1/2 m λ1 d (m) |θ (t)|g + κ θ (m) (t) 2dt g g + ǫν u(m) (t) g ≤ κ|∇g|∞ 1/2 m λ1 g + ξ 2∞ (m) θ (t) 2ǫνλ21 θ (m) (t) g g + ǫκ θ (m) (t) + g |f1 |2g , 2ǫνλ1 + |f2 |2g , 4ǫκλ1 so that for ν ′ = 2ν 1− |∇g|∞ 1/2 m λ1 − ǫ , κ′ = 2κ − |∇g|∞ 1/2 m0 λ1 − ǫ , c′ = ξ 2∞ ǫλ21 we get d (m) c′ (m) |u (t)|g + ν ′ u(m) (t) 2g ≤ θ (t) 2g + |f1 |2g , dt ν ǫλ1 ν d (m) |θ (t)|g + κ′ θ (m) (t) 2g ≤ |f2 |2g , dt 2ǫλ1 κ where ǫ > is chosen such that 1− |∇g|∞ −ǫ 1/2 m0 λ1 Integrating (3.7) and (3.6) from to t, we obtain sup |u(m) (t)|2g ≤ |u0 |2g + t∈[0,T ] (3.7) > T |f2 |2g 2ǫλ1 κ (3.8) c′ c′ T T |θ | + |f2 |2g + |f1 |2g 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 to T to get T T |f2 |2g , (3.10) |θ (m) (T )|2g + κ′ θ (m) (t) 2g dt ≤ 2ǫλ κ 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 ≤ 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 ) = and Ψ2 (T ) = We multiply (3.2) and (3.3) by Ψ1 (t) and Ψ2 (t) respectively and then integrate by parts, T − (u(m) , Ψ′1 uk )g dt+ T bg (u(m) , u(m) , Ψ1 uk )dt T +ν T ((u(m) , Ψ1 uk ))g dt + ν bg ( 0 T = (um0 , uk )g Ψ1 (0) + ∇g (m) , u , Ψ1 uk )dt g T (θ (m) , Ψ′2 θk )g dt + T +κ T (f1 , uk )g dt, 0 − T (ξθ (m) , Ψ1 uk )g dt + T ˜bg (u(m) , θ (m) , Ψ2 θk )dt + κ ((θ (m) , Ψ2 θk ))g dt ˜bg ( ∇g , θk , Ψ2 θ (m) )dt = (θm0 , θk )g Ψ2 (0) + g T (f2 , Ψ2 θk )g dt Following the technique given in [15], as m → ∞ we obtain T +ν T T T − (u, Ψ′1 v)g dt + ((u, Ψ1 v))g dt bg (u, u, Ψ1 v)dt + ν T bg ( ∇g, u, Ψ1 v)dt = (u0 , v)g Ψ1 (0) + g (ξθ, Ψ1 v)g dt (3.12) T (f1 , v)g dt, + 29 Tran Quang Thinh and Le Thi Thuy T T − T +κ (θ, Ψ′2 τ )g dt + T ˜bg (u, θ, Ψ2 τ )dt + κ ˜bg ( ∇g , τ, Ψ2 θ)dt = (θ0 , τ )g Ψ2 (0) + g ((θ, Ψ2 τ ))g dt T (3.13) (f2 , Ψ2 τ )g dt 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) ˜ = dt g Taking v = w(t), τ = w(t) ˜ and (2.1), (2.3) we obtain d ∇g |w|2 + ν w 2g ≤ |bg (w, u2 , w)| + ν|bg ( , w, w)| + |(ξ w, ˜ w)g |, dt g g ∇g d |w| ˜ 2g + κ w˜ 2g + ≤ |˜bg (w, θ2 , w)| ˜ + κ|˜bg ( , w, ˜ w)| ˜ dt g By applying (2.2), (2.4) it then follows by the Cauchy-Schwarz inequality, we have d |w|2 + ν w dt g g ≤ d |w| ˜ + κ w˜ dt g g ≤ c2 |w| u2 ǫν g ǫν w 2 g g + ξ 2∞ |w| ˜ , ǫνλ1 (3.14) c4 |θ2 |4g κ|∇g|∞ |w| ˜ 2g + w ˜ 2g 1/2 16ǫ ν κλ1 m λ1 (3.15) ν|∇g|∞ 1/2 m λ1 + ǫκ w ˜ g + w g + ǫν w 2 g + We sum equations (3.14) and (3.15) to obtain |∇g|∞ d − ǫ (ν w (|w|2g + |w| ˜ 2g ) + − 1/2 dt m λ1 ≤ 2c2 u2 ǫν g |w|2g + g + κ w˜ 2g ) c4 θ2 4g ξ 2∞ + 2 ǫνλ1 8ǫ ν κλ1 |w| ˜ 2g , so that for γ = max 2c2 u2 ǫν g ; c2 θ 4g ξ 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 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 ≥ In particular, the solution is unique REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] C.T Anh and D.T Son, 2013 Pullback attractors for nonautonomous 2D B´enard problem in some unbounded domains Math Methods Appl Sci., 36, pp 1664-1684 H Bae and J Roh, 2004 Existence of solutions of the g-Navier-Stokes equations, Taiwanese J Math., 8, pp 85-102 M Cabral, R Rosa and R Temam, 2004 Existence and dimension of the attractor for the B´enard problem on channel-like domains Discrete Contin Dyn Syst., 10, pp 89-116 L Friz, M.A Rojas-Medar and M.D Rojas-Medar, 2016 Reproductive solutions for the g-Navier-Stokes and g-Kelvin-Voight equations Elect J Diff Equations, No 37, 12 p J Jiang, Y Hou and X Wang, 2011 Pullback attractor of 2D nonautonomous g-Navier-Stokes equations with linear dampness Appl Math Mech -Engl Ed., 32, pp 151-166 M Kwak, H Kwean and J Roh, 2006 The dimension of attractor of the 2D g-Navier-Stokes equations J Math Anal Appl., 315, pp 436-461 H Kwean and J Roh, 2005 The global attractor of the 2D g-Navier-Stokes equations on some unbounded domains Commun Korean Math Soc., 20, pp 731-749 ă uk and M Kaya, 2018 On the weak solutions and determining modes of the g-B´enard problem M Ozlă Hacet J Math Stat., 47, pp 1453-1466 ¨ uk and M Kaya, 2018 On the strong solutions and the structural stability of the g-Benard M Ozlă problem Numer Funct Anal Optim., 39, pp 383-397 D.T Quyet, 2014 Asymptotic behavior of strong solutions to 2D g-Navier-Stokes equations Commun Korean Math Soc., 29, pp 505-518 J.C Robinson, 2001 Infinite-Dimensional Dynamical Systems Cambridge University Press, Cambridge J Roh, 2001 G-Navier-Stokes Equations PhD thesis, University of Minnesota J Roh, 2005 Dynamics of the g-Navier-Stokes equations J Differential Equations, 211, pp 452-484 J Roh, 2009 Convergence of the g-Navier-Stokes equations Taiwanese J Math., 13, pp 189-210 R Temam, 1984 Navier-Stokes Equations, Theory and Numerical Analysis North-Holland, Amsterdam, The Netherlands, 3rd edition 31 ... 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. .. setting of the 2D g-B´enard problem Section 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. .. number of determining modes More recently, in [9] M Ozlă the existence, uniqueness of strong solutions, and the continuous dependence of the solutions on the viscosity parameter for problem (1.1) in