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On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating

  • Angela Iannelli EMAIL logo
Published/Copyright: August 26, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

This article surveys some results in the study of Iannelli [Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for t → ∞ of the time-dependent solutions is proved.

Keywords: Oberbeck-Boussinesq model; viscous fluid; heating
MSC 2010: 35Q30; 35Q35

1 Introduction

Oberbeck-Boussinesq equations have been considered in the literature since long time, and several problems are still under investigation, e.g., the long-time behavior of solutions (see [1–8] and references quoted therein). In this study, we provide a survey of our works [9–11], which we believe are of interest but have been somewhat forgotten since they appeared in Italian. Thus, in this work, we provide a (partial) translation, reformulation, and summary of the main results. In particular, we consider the case of a viscous and compressible fluid, contained in a bounded domain Ω , with a wall subject to heating. It is known that a density gradient is produced, which, in turn, in the presence of mass forces, causes the movement of the fluid. This motion is governed by the Navier-Stokes equations and by the energy conservation equation, i.e., by the system

(1.1) ϱ ∂ u ̲ ∂ t − μ Δ u ̲ − ξ ∇ ( ∇ ⋅ u ̲ ) + ϱ ( u ̲ ⋅ ∇ ) u ̲ + ∇ p = f ̲ ,

(1.2) ∂ ϱ ∂ t + ∇ ⋅ ( ϱ u ̲ ) = 0 ,

(1.3) ϱ ∂ T ∂ t − K T C P Δ T + ϱ u ̲ ⋅ ∇ T + p ∇ ⋅ u ̲ + μ 2 ∑ i , k = 1 2 ( D k u i + D i u k ) 2 − ξ ( ∇ ⋅ u ̲ ) 2 = 0 ,

where u ̲ represents the velocity of the fluid, ϱ is the density, p is the pressure, T is the temperature, μ and ξ are the viscosity coefficients, K T is the thermal conductivity coefficient, C p is the specific heat, and f ̲ is the force acting on the fluid. In the sequel, for the sake of simplicity and given that this is the most common case in applications, it will be assumed that f ̲ reduces to the force of gravity alone. In the event that there are no heat sources in Ω and some simplifying hypotheses are verified (such as independence of density from pressure, density seen as a linear function of temperature and always considered constant except in terms of the force in the momentum equation, see [12]), the motion of the fluid can be described with the following set of Oberbeck-Boussinesq equations:

(1.4) ϱ 0 ∂ u ̲ ∂ t − μ Δ u ̲ + ϱ 0 ( u ̲ ⋅ ∇ ) u ̲ + ∇ p = ϱ 0 { 1 − α T ( T − T 0 ) } g ̲ ,

(1.5) ∇ ⋅ u ̲ = 0 ,

(1.6) ∂ T ∂ t − γ T Δ T + u ̲ ⋅ ∇ T = 0 ,

where ϱ 0 and T 0 are the reference (constant) values for density and temperature, respectively, g ̲ is the gravitational acceleration, γ T = K T ϱ 0 C p is the thermal diffusivity, and α T = R T P R g , in which R T represents the thermal Rayleigh number, P R represents the Prandtl number, and g = ∣ g ̲ ∣ . We will also assume that Ω is the square ( − 1 , 1 ) 2 with boundary Γ = ∂ Ω and sides Γ 1 , Γ 2 , Γ 3 , Γ 4 (to be understood as arranged consecutively, in an anti-clockwise order starting from the side Γ 1 parallel to the abscissa axis and located in the lower half-plane). The boundary conditions are associated with system (1.4)–(1.6):

(1.7) u ̲ ∣ Γ × ( 0 ; T ) = 0 , T ∣ Γ 2 × ( 0 ; T ) = T 1 , T ∣ Γ 4 × ( 0 ; T ) = T 2 , and ∂ T ∂ ν ̲ ( Γ 1 ∪ Γ 3 ) × ( 0 ; T ) = 0 ,

where (0, T ) is the time interval considered , ν ̲ = ( ν 1 , ν 2 ) is the external normal to Γ , T 1 and T 2 are known functions of time with T 2 > T 1 .

This article is organized as follows. In Section 1, we study problem (1.4)–(1.7), and we prove a global existence and uniqueness theorem for the solutions (understood in the appropriate sense) of (1.4)–(1.7), satisfying the initial conditions

(1.8) u ̲ ( 0 ) = u ˜ ̲ , T ( 0 ) = T ˜ , on Ω ,

where u ˜ ̲ and T ˜ are the known functions of space, defined on Γ . The theorems obtained essentially correspond to the classical results for the Navier-Stokes equations in the two-dimensional case. Section 2 deals with the stationary solutions of (1.4)–(1.7) (in this case, obviously, T 1 and T 2 are the constants); precisely, the proof of an existence and uniqueness result of solutions is given in the case in which the ∣ T 1 − T 2 ∣ is small enough and the viscosity coefficient is large enough, in analogy with the case of stationary solutions of the Navier-Stokes system. Finally, in Section 3, we study the asymptotic behavior of solutions, and we prove that these solutions converge, for t → + ∞ , to appropriate stationary solutions.

Other aspects of the model presented in this work have been studied in notes [10] and [11]. Precisely,

  • in [11], the periodic solutions of the Oberbeck-Boussinesq model are studied, and it is demonstrated that if the data are sufficiently small, the problem admits one and only one periodic solution.

  • in [10], the Oberbeck-Boussinesq model is analyzed with a collocation-Legendre-type spectral method.

2 Time-dependent solutions

We deal here with the study of the evolutionary mathematical problem represented by system (1.4)–(1.6) with Conditions (1.7) and (1.8).

In order to define the solution of the problem considered, we introduce the following functional spaces:

(2.1) V = { s ̲ ∈ D ( Ω ) 2 ∣ ∇ ⋅ s ̲ = 0 } , V = { s ̲ ∈ H 0 1 ( Ω ) 2 ∣ ∇ ⋅ s ̲ = 0 } = V ¯ H 1 , H = { s ̲ ∈ L 2 ( Ω ) 2 ∣ ∇ ⋅ s ̲ = 0 , γ ν ̲ s ̲ = 0 } = V ¯ L 2 , H 0 1 ˜ ( Ω ) = { ϕ ∈ H 1 ( Ω ) , ∣ ϕ ∣ Γ 2 ∪ Γ 4 = 0 } , H 2 ˜ ( Ω ) = ϕ ∈ H 2 ( Ω ) , ∂ ϕ ∂ v Γ 1 ∪ Γ 3 = 0 , H 0 2 ˜ ( Ω ) = { ϕ ∈ H ˜ 2 ( Ω ) , ∣ ϕ ∣ Γ 2 ∪ Γ 4 = 0 } ,

where γ ν ̲ s ̲ indicates the restriction of ν ̲ ⋅ s ̲ to Γ . From now on, we will also denote the cylinder Γ × ( 0 , T ) by Q T . We will then say that the pair ( u ̲ , T ) is a “solution of system (1.4)–(1.6) in the time interval (0, T ), satisfying (1.7) and (1.8)” if:

  1. u ̲ ( t ) ∈ L 2 ( 0 , T ; V ) ∪ L ∞ ( 0 , T ; H ) , T ( t ) ∈ L 2 ( 0 , T ; H ˜ 2 ( Ω ) ) ∩ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) , T ( 0 ) = T ˜ a.e. on Ω , T ∣ Γ 4 × ( 0 ; T ) = T 2 , T ∣ Γ 2 × ( 0 ; T ) = T 1 ,

  2. the following identities are satisfied:

(2.2) ( u ̲ ( T ) , s ̲ ( T ) ) L 2 ( Ω ) − ( u ˜ ̲ , s ̲ ( 0 ) ) L 2 ( Ω ) + ∫ 0 T − u ̲ ( η ) , ∂ s ̲ ∂ η ( η ) L 2 ( Ω ) + μ ϱ 0 ( u ̲ ( η ) , s ̲ ( η ) ) H 0 1 ( Ω ) + ( ( u ̲ ( η ) ⋅ ∇ ) u ̲ ( η ) , s ̲ ( η ) ) L 2 ( Ω ) } d η = − ∫ 0 T α T ( T ( η ) g ̲ , s ̲ ( η ) ) L 2 ( Ω ) d η , ∀ s ̲ ∈ L 2 ( 0 , T ; V ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) .

(2.3) ∫ 0 T ∂ T ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) + ( u ̲ ( η ) ⋅ ∇ T ( η ) , ϕ ( η ) ) L 2 ( Ω ) + − γ T ( Δ T ( η ) , ϕ ( η ) ) L 2 ( Ω ) d η = 0 , ∀ ϕ ∈ L 2 ( 0 , T ; L 2 ( Ω ) ) .

Remark 1

The term ∫ 0 T { ( u ̲ ( η ) ⋅ ∇ T ( η ) , ϕ ( η ) ) L 2 ( Ω ) } d η is well defined. As a matter of fact,

∫ 0 T { ( u ̲ ( η ) ⋅ ∇ T ( η ) , ϕ ( η ) ) L 2 ( Ω ) } d η = ∑ j = 1 2 ∫ Q T u j ∂ T ∂ x j ϕ d Q T ;

now, observe that from ( a 1 ) , it follows that (by interpolation)

u ̲ ∈ L 4 ( 0 , T ; H 1 ⁄ 2 ( Ω ) ) ⊂ L 4 ( Q T ) and ∂ T ∂ x j ∈ L 2 ( 0 , T ; H 1 ( Ω ) ) ∩ L ∞ ( 0 , T ; L 2 ( Ω ) ) and so ∂ T ∂ x j ∈ L 4 ( Q T ) .

Since ϕ belongs to L 2 ( Q T ) and from the Cauchy-Schwarz inequality, it follows that u j ∂ T ∂ x j ϕ ∈ L 1 ( Q T ) . The well definedness of the stated integral term can also be established using directly the Gagliardo-Nirenberg and Hölder inequalities.

Besides, we comment that the apparently missing term proportional to ( g ̲ , s ̲ ) is equal to zero, in fact given any conservative field z ̲ (which is, of course, the case if z ̲ = g ̲ ), there exists V such that z ̲ = ∇ V , and hence, ( z ̲ , s ̲ ) = ( ∇ V , s ̲ ) = − ( V , ∇ ⋅ s ̲ ) = 0 by assumption, where ( ⋅ , ⋅ ) denotes the scalar product in L 2 ( Ω ) .

This section deals with the following topics: first, we transform the original problem into an equivalent homogeneous problem; then, we study some auxiliary theorems that are quite useful in showing existence and uniqueness of the solution, which, in turn, are separately analyzed in the final part.

2.1 Equivalent homogeneous problem

Problems (1.4)–(1.8) can be reduced to a problem with homogeneous boundary conditions by means of an appropriate change of variables, precisely:

d ( x , t ) = T 1 ( t ) − T 2 ( t ) 2 x + T 1 ( t ) + T 2 ( t ) 2 , θ ( x , y , t ) = T ( x , y , t ) − d ( x , t ) .

In this case, from (1.7), it follows that:

θ ( − 1 , y , t ) = θ ( 1 , y , t ) = 0 ∀ y ∈ [ − 1 , 1 ] .

Then, passing to the new variable θ in problem (1.4)–(1.8), we obtain the following homogeneous problem equivalent to the original one:

(2.4) ϱ 0 ∂ u ̲ ∂ t + ϱ 0 ( u ̲ ⋅ ∇ ) u ̲ − μ Δ u ̲ + ∇ p = ϱ 0 { 1 − α T ( θ + d − T 0 ) } g ̲ ,

(2.5) ∇ ⋅ u ̲ = 0 ,

(2.6) ∂ θ ∂ t − γ T Δ θ + u ̲ ⋅ ∇ θ = T 1 − T 2 2 u 1 − ∂ d ∂ t ,

with the initial conditions

(2.7) u ̲ ( 0 ) = u ˜ ̲ , θ ( 0 ) = θ ˜ ,

θ ˜ ( x , y ) = T ˜ ( x , y ) − d ( x , 0 ) = T ˜ ( x , y ) − T 1 ( 0 ) 2 ( 1 + x ) − T 2 ( 0 ) 2 ( 1 − x ) ,

and boundary conditions

(2.8) u ̲ ∣ Γ × ( 0 ; T ) = 0 , θ ∣ Γ 2 ∪ Γ 4 × ( 0 ; T ) = 0 , ∂ θ ∂ ν ( Γ 1 ∪ Γ 3 ) × ( 0 ; T ) = 0 .

We will say that a pair ( u ̲ , θ ) is “solution of problem (2.4)–(2.8) in the time interval (0, T )” if:

  1. u ̲ ( t ) ∈ L 2 ( 0 , T ; V ) ∩ L ∞ ( 0 , T ; H ) ,

    θ ( t ) ∈ L 2 ( 0 , T ; H ˜ 0 2 ( Ω ) ) ∩ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) , θ ( 0 ) = θ ˜ a.e., on Ω ,

  2. the following conditions are satisfied:

(2.9) ( u ̲ ( T ) , s ̲ ( T ) ) L 2 ( Ω ) − ( u ˜ ̲ , s ̲ ( 0 ) ) L 2 ( Ω ) + ∫ 0 T − u ̲ ( η ) , ∂ s ̲ ∂ η ( η ) L 2 ( Ω ) + μ ϱ 0 ( u ̲ ( η ) , s ̲ ( η ) ) H 0 1 ( Ω ) + ( ( u ̲ ( η ) ⋅ ∇ ) u ̲ ( η ) , s ̲ ( η ) ) L 2 ( Ω ) d η = − ∫ 0 T α T ( ( θ ( η ) + d ( θ ) ) g ̲ , s ̲ ( η ) ) L 2 ( Ω ) d η , ∀ s ̲ ∈ L 2 ( 0 , T ; V ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) )

(2.10) ∫ 0 T ∂ θ ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) − γ T ( Δ θ ( η ) , ϕ ( η ) ) L 2 ( Ω ) + ( u ̲ ( η ) ⋅ ∇ θ ( η ) , ϕ ( η ) ) L 2 ( Ω ) d η = ∫ 0 T T 2 ( η ) − T 1 ( η ) 2 u 1 ( η ) , ϕ ( η ) L 2 ( Ω ) d η − ∫ 0 T ∂ d ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) d η , ∀ ϕ ∈ L 2 ( Q T ) .

2.2 Auxiliary theorems

Some results regarding the individual equations of the problem under consideration are collected here. The proofs of the following two lemmas can be found in [9] and are easy consequences of Sobolev and Gagliardo-Nirenberg inequalities.

Lemma 1

If f ∈ H 0 2 ˜ ( Ω ) , then the two norms ∣ ∇ f ∣ H 1 ( Ω ) and ∣ Δ f ∣ L 2 ( Ω ) are equivalent.

Lemma 2

If u ̲ ∈ L 4 ( Ω ) and v ∈ H 1 ( Ω ) , then the following inequality holds:

∣ u ̲ ∣ L 4 ( Ω ) ∣ s ∣ H 1 ( Ω ) ∣ s ∣ L 4 ( Ω ) ⩽ ε + ε ̲ c ˆ 2 4 ε ∣ s ∣ 2 H 1 ( Ω ) + c ˆ 2 16 ε ε ¯ ∣ s ∣ 2 L 2 ( Ω ) ∣ u ̲ ∣ L 4 ( Ω ) 4 ,

where ε and ε ˜ are the arbitrary positive constants and c ˆ is the embedding constant of H 1 ⁄ 2 ( Ω ) in L 4 ( Ω ) .

Theorem 1

Let u ̲ = ( u 1 , u 2 ) ∈ L 2 ( 0 , T ; V ) ∩ L ∞ ( 0 , T ; H ) , θ ˜ ∈ H 1 ( Ω ) , K ∈ L 2 ( 0 , T ) , K ˜ ∈ L 2 ( 0 , T ; L 2 ( Ω ) ) . Then, there exists a unique function θ ( t ) such that

  1. θ ( t ) ∈ L 2 ( 0 , T ; H ˜ 0 2 ( Ω ) ) ∪ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) , θ ( 0 ) = θ ˜ ,

  2. (2.11) ∫ 0 T ∂ θ ∂ η ( η ) − γ T Δ θ ( η ) + u ̲ ( η ) ⋅ ∇ θ ( η ) , φ ( η ) L 2 ( Ω ) d η = ∫ 0 T ( K ( η ) u 1 ( η ) + K ˜ ( η ) , φ ( η ) ) L 2 ( Ω ) d η , ∀ ϕ ∈ L 2 ( Q T ) .

Proof – existence

We prove the existence of the function θ ( t ) with the Faedo-Galerkin method, just sketching the main lines, which are by now standard, in order to arrive to the estimates necessary to finalize the proof. Let { g j } be { λ j } , respectively, a basis of eigenfunctions of the operator − Δ and the corresponding sequence of eigenvalues, for which we have

− Δ g j = λ j g j , j = 1 , 2 , … .

For the assumptions made on Γ , it follows (see, e.g., [13]) g j ∈ H ˜ 0 2 ( Ω ) . Then, setting θ n ( t ) = ∑ j = 1 n α j ( t ) g j , the approximation of

∂ θ ∂ t − γ T Δ θ + u ̲ ⋅ ∇ θ = K u 1 + K ˜ ,

with the condition θ ( 0 ) = θ ˜ , has the following form:

(2.12) ∂ θ n ∂ t − γ T Δ θ n + u ̲ ⋅ ∇ θ n = K u 1 + K ˜ ,

with the initial condition θ n ( 0 ) = Π n θ ˜ , where Π n θ ˜ is the projection of θ ˜ onto the space generated by g 1 , … , g n . Multiplying (2.12) by g j and integrating over Ω follows:

(2.13) ∂ θ n ∂ t − γ T Δ θ n + u ̲ ⋅ ∇ θ n , g j L 2 ( Ω ) = ( K u 1 + K ˜ , g j ) L 2 ( Ω ) , j = 1 … , n .

Standard calculations, which make use of Green’s formula, yield

(2.14) 1 2 ∂ ∂ t ∣ ∇ θ n ∣ L 2 ( Ω ) 2 + γ T ∣ Δ θ n ∣ L 2 ( Ω ) 2 = ( u ̲ ⋅ ∇ θ n , Δ θ n ) L 2 ( Ω ) − ( K u 1 + K ˜ , Δ θ n ) L 2 ( Ω ) .

By the integrability properties of u ̲ , δ θ n , ∇ θ n , Cauchy-Schwarz inequality, and the fact that the two norms ∣ Δ θ n ∣ L 2 ( Ω ) and ∣ ∇ θ n ∣ H 1 ( Ω ) are equivalent (Lemma 1), one obtains

(2.15) 1 2 ∂ ∂ t ∣ ∇ θ n ∣ L 2 ( Ω ) 2 + γ T ∣ Δ θ n ∣ H 1 ( Ω ) 2 ⩽ ∣ u ̲ ∣ L 4 ( Ω ) ∣ ∇ θ n ∣ L 4 ( Ω ) ∣ ∇ θ n ∣ H 1 ( Ω ) + ∣ K ∣ ∣ u ̲ ∣ L 2 ( Ω ) ∣ ∇ θ n ∣ H 1 ( Ω ) + ∣ K ˜ ∣ L 2 ( Ω ) ∣ ∇ θ n ∣ H 1 ( Ω ) .

On the other hand, recalling the result proved in Lemma 2, with arbitrary ε and ε ˜ , we have:

(2.16) 1 2 ∂ ∂ t ∣ ∇ θ n ∣ L 2 ( Ω ) 2 + H ∣ ∇ θ n ∣ H 1 ( Ω ) 2 ⩽ 1 4 ε 1 ∣ K ∣ 2 ∣ u ̲ ∣ L 2 ( Ω ) + 1 4 ε 2 ∣ K ˜ ∣ L 2 ( Ω ) 2 + c ˆ 2 16 ε ε ˜ ∣ u ̲ ∣ 4 L 4 ( Ω ) ∣ ∇ θ n ∣ L 2 ( Ω ) 2 .

where H = ( γ T − ε − ε ˜ c ˆ 2 4 ε − ε 1 − ε 2 ) can be taken to be positive for appropriate ε , ε ˜ , ε 1 , ε 2 > 0 . Integrating in time, one has

(2.17) 1 2 ∣ ∇ θ n ∣ L 2 ( Ω ) 2 ⩽ 1 4 ε 1 ∫ 0 T ∣ K ( η ) ∣ 2 ∣ u ̲ ( η ) ∣ L 2 ( Ω ) 2 d η + 1 4 ε 2 ∫ 0 T ∣ K ˜ ( η ) ∣ L 2 ( Ω ) 2 d η + c ˆ 2 16 ε ε ˜ ∫ 0 t ∣ u ̲ ( η ) ∣ L 4 ( Ω ) 4 ∣ ∇ θ n ( η ) ∣ L 2 ( Ω ) 2 d η + 1 2 ∣ ∇ θ ˜ ∣ L 2 ( Ω ) 2 ,

which, upon setting ψ ( η ) = c ˆ 2 8 ε ε ˜ ∣ u ̲ ( η ) ∣ L 4 ( Ω ) 4 gives, for an appropriate c ˜ :

∣ ∇ θ n ( η ) ∣ L 2 ( Ω ) 2 ⩽ c ˜ + ∫ 0 t ψ ( η ) ∣ ∇ θ n ( η ) ∣ L 2 ( Ω ) 2 d η , ∀ t ∈ [ 0 , T ] ,

which immediately implies, by a Gronwall-type argument,

(2.18) ∣ ∇ θ n ( η ) ∣ L 2 ( Ω ) 2 ⩽ M , ∀ t ∈ [ 0 , T ] ,

namely, θ n ∈ L ∞ ( 0 , T , H 1 ( Ω ) ) . Integrating in time (2.15) also yields

(2.19) H ∫ 0 T ∣ ∇ θ n ( η ) ∣ H 1 ( Ω ) 2 d η ⩽ 1 4 ε 1 ∫ 0 T ∣ K ( η ) ∣ 2 ∣ u ̲ ( η ) ∣ L 2 ( Ω ) 2 d η + 1 2 ∣ ∇ θ ˜ ∣ L 2 ( Ω ) 2 + 1 4 ε 2 ∫ 0 T ∣ K ˜ ( η ) ∣ L 2 ( Ω ) 2 d η + c ˆ 2 16 ε ε ˜ ∫ 0 T ∣ u ̲ ( η ) ∣ L 4 ( Ω ) 4 ∣ ∇ θ n ( η ) ∣ L 2 ( Ω ) 2 d η ,

which, since u ̲ ∈ L 4 ( Q T ) and by (2.18), yields

(2.20) ∫ 0 T ∣ ∇ θ n ( η ) ∣ H 1 ( Ω ) 2 d η 1 2 ⩽ M ′ .

Then, θ n ∈ L 2 ( 0 , T ; H 2 ( Ω ) ) , with ∣ θ n ∣ L 2 ( 0 , T ; H 2 ( Ω ) ) ⩽ M ′ , and being θ n = ∑ j = 1 n α j g j with g j ∈ H 0 2 ˜ ( Ω ) , we obtain that θ n ∈ L 2 ( 0 , T ; H ˜ 0 2 ( Ω ) ) .

Finally, to prove that θ n belongs to the space H 1 ( 0 , T ; L 2 ( Ω ) ) , we note that from (2.13), one has

∂ θ n ∂ t − γ T Δ θ n + u ̲ ⋅ ∇ θ n , ∂ θ n ∂ t L 2 ( Ω ) = K u 1 + K ˜ , ∂ θ n ∂ t L 2 ( Ω ) , j = 1 … , n .

From this, it is easy to obtain that

∂ θ n ∂ t L 2 ( Q T ) ⩽ γ T ∣ Δ θ n ∣ L 2 ( Q T ) + ∣ u ̲ ⋅ ∇ θ n ∣ L 2 ( Q T ) + ∣ K u 1 + K ˜ ∣ L 2 ( Q T ) .

Therefore, taking into account (2.20) and the fact that ∇ θ n ∈ L 4 ( Q T ) , we obtain:

(2.21) ∂ θ n ∂ t L 2 ( Q T ) ⩽ 2 γ T M ′ + ∣ u ̲ ∣ L 4 ( Q T ) ∣ ∇ θ n ∣ L 4 ( Q T ) + ∣ K u 1 + K ˜ ∣ L 2 ( Q T ) ,

which allows us to conclude that ∂ θ n ∂ t ∈ L 2 ( Q T ) or, equivalently, that θ n ∈ H 1 ( 0 , T ; L 2 ( Ω ) ) . Therefore, we have

θ n ∈ L 2 ( 0 , T ; H ˜ 0 2 ( Ω ) ) ∩ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) ,

with θ n ( 0 ) = Π n θ ˜ . By the aforementioned bounds, we can extract a subsequence (again indicated with { θ n } n ∈ N ) such that

θ n ⟶ θ in the weak topology * of L ∞ ( 0 , T ; H 1 ( Ω ) ) ,

θ n ⟶ θ in the weak topology of L 2 ( 0 , T ; H ˜ 0 2 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) .

The point ( i 1 ) is thus proven.

Moving now to the proof of the point ( i i 1 ) , if v ∈ L 2 ( Q T ) , then v ( t ) = ∑ j = 1 + ∞ β j ( t ) g j for a suitable sequence of functions { β j } , with β j = β j ( t ) ; given v k ( t ) = ∑ j = 1 k β j ( t ) g j , multiply (2.13) by { β j } and sum, to obtain

∂ θ n ∂ η − γ T Δ θ n + u ̲ ⋅ ∇ θ n , v k L 2 ( Ω ) = ( K u 1 + K ˜ , v k ) L 2 ( Ω ) .

Integrating in time and taking the limit for n ⟶ + ∞ , we obtain:

(2.22) ∫ 0 T ∂ θ ∂ η ( η ) − γ T Δ θ ( η ) + u ̲ ( η ) ⋅ ∇ θ ( η ) , v k ( η ) L 2 ( Ω ) d η = ∫ 0 T ( K ( η ) u 1 ( η ) + K ˜ ( η ) , v k ( η ) ) L 2 ( Ω ) d η .

From (2.25), for k ⟶ + ∞ , ( i i 1 ) follows.

Uniqueness

Let θ 1 and θ 2 be two functions verifying ( i 1 ) and ( i i 1 ) . Setting δ = θ 1 − θ 2 , it follows that

δ ∈ L 2 ( 0 , T ; H ˜ 0 2 ( Ω ) ) ∩ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) ,

δ ( 0 ) = 0 a.e., on Ω ,

∫ Q T ∂ δ ∂ t − γ T Δ δ + u ̲ ⋅ ∇ δ ϕ d Q T = 0 , ∀ ϕ ∈ L 2 ( Q T ) .

If one takes

ϕ = δ , on ( 0 , t ) , 0 , on ( t , T ) ,

for a generic time t of the interval [ 0 , T ] , we then obtain:

(2.23) ∫ 0 t ∂ δ ∂ η ( η ) − γ T Δ δ ( η ) + u ̲ ( η ) ⋅ ∇ δ ( η ) , δ ( η ) L 2 ( Ω ) d η = 0 .

Now observe that since ( u ̲ ⋅ ∇ δ , δ ) L 2 ( Ω ) = 0 (see Lemma 4 of Section 2), then (2.26) reduces to

1 2 ∫ 0 t ∂ ∂ η ∣ δ ( η ) ∣ L 2 ( Ω ) 2 d η + ∫ 0 t γ T ∣ ∇ δ ( η ) ∣ L 2 ( Ω ) 2 d η = 0 , ∀ t ∈ [ 0 , T ] ,

from which it follows

δ ( t ) L 2 ( Ω ) 2 − δ ( 0 ) L 2 ( Ω ) 2 ⩽ 0 , ∀ t ∈ [ 0 , T ] ,

and since ∣ δ ( 0 ) ∣ L 2 ( Ω ) = 0 , we obtain that ∣ δ ( t ) ∣ L 2 ( Ω ) = 0 and therefore uniqueness.□

From Theorem 1, the following result follows easily.

Theorem 2

Let u ̲ = ( u 1 , u 2 ) ∈ L 2 ( 0 , T ; V ) ∩ L ∞ ( 0 , T ; H ) , T ˜ ∈ H 1 ( Ω ) , T 1 , T 2 ∈ H 1 ( 0 , T ) , T ˜ ∣ Γ 2 = T 1 ( 0 ) , T ˜ ∣ Γ 4 = T 2 ( 0 ) . Then, there exists a unique function T ( t ) such that

  1. T ( t ) ∈ L 2 ( 0 , T ; H ˜ 2 ( Ω ) ) ∩ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) , T ( 0 ) = T ˜ , T ∣ Γ 2 × ( 0 ; T ) = T 1 , T ∣ Γ 4 × ( 0 ; T ) = T 2 ,

  2. (2.24) ∫ 0 T ∂ T ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) + ( u ̲ ( η ) ⋅ ∇ T ( η ) , ϕ ( η ) ) L 2 ( Ω ) − γ T ( Δ T ( η ) , ϕ ( η ) ) L 2 ( Ω ) d η = 0 , ∀ ϕ ∈ L 2 ( Q T ) .

Remark 2

It is interesting to note that assuming that the hypotheses of Theorem 2 hold and that T ˜ ∈ L ∞ ( Ω ) , we have T ∈ L ∞ ( Q T ) , thanks to an application of the maximum principle (see, e.g., [14]); therefore, there exists a constant E such that ∣ T ∣ L ∞ ( Q T ) ⩽ E .

Theorem 3

Let f ̲ ∈ L 2 ( 0 , T ; V ′ ) , u ˜ ̲ ∈ H , ξ constant. There is then one and only one function u ̲ such that

  1. u ̲ ∈ L 2 ( 0 , T ; V ) ∩ L ∞ ( 0 , T ; H ) ,

  2. (2.25) ( u ̲ ( T ) , s ̲ ( T ) ) L 2 ( Ω ) − ( u ˜ ̲ , s ̲ ( 0 ) ) L 2 ( Ω ) + ∫ 0 T − u ̲ ( η ) , ∂ s ̲ ∂ η ( η ) L 2 ( Ω ) + ξ ( u ̲ ( η ) , s ̲ ( η ) ) H 0 1 ( Ω ) + ( ( u ̲ ( η ) ⋅ ∇ ) u ̲ ( η ) , s ̲ ( η ) ) L 2 ( Ω ) d η = ∫ 0 T < f ̲ ( η ) , s ̲ ( η ) > d η , ∀ s ̲ ∈ L 2 ( 0 , T ; V ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) .

    The function u ̲ is also weakly continuous from ( 0 , T ) → H .

Proof

See [15] or [16] or even [17]. In fact, since u ̲ ∈ L 2 ( 0 , T ; V ) ∩ H 1 ( 0 , T ; V ′ ) , classical interpolation theory implies u ∈ C ( [ 0 , T ] ; H ) ⊂ L ∞ ( 0 , T ; H ) .□

2.3 Existence of a solution of the problem

Now, to prove the existence of a solution ( u ̲ , T ) of the problem under examination in the sense specified in points ( a 1 ) and ( b 1 ) : in the proof, the Leray-Schauder fixed point theorem is fundamental.

Theorem 4

Let u ̲ ∈ H , T ˜ ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) , T 1 , T 2 ∈ H 1 ( 0 , T ) with T ˜ ∣ Γ 2 = T 1 ( 0 ) , T ˜ ∣ Γ 4 = T 2 ( 0 ) . Then, there exists a couple ( u ̲ , T ) , solution of problem (1.4)–(1.8), in the sense specified in ( a 1 ) and ( b 1 ) .

Proof

Consider the transformation:

F : L 2 ( Q T ) ⟶ L 2 ( Q T )

such that F ( T ˜ ) = T , where T is obtained starting from T ˜ with the following procedure: given T ˜ ∈ L 2 ( Q T ) , replace it in equation (2.2) instead of T and observe that (2.2) coincides with ( i i 3 ) as long as we take ξ = μ ϱ 0 and f = − α T T ˜ g ̲ ∈ L 2 ( Q T ) . By Theorem 3, there exists a unique u ˜ ̲ ∈ L 2 ( 0 , T ; V ) ∩ L ∞ ( 0 , T ; H ) verifying (2.2). Substitute this u ˜ ̲ for the term u ̲ into (2.3). Then, by Theorem 2, we obtain, in a unique way, T satisfying the conditions relating to it, listed in points ( a 1 ) and ( b 1 ) . Observe that T ∈ L 2 ( Q T ) , and the transformation F is thus completely defined. By the same Theorem 2, it follows that T ∈ L 2 ( 0 , T ; H 2 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) . But being L 2 ( 0 , T ; H 2 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) ⊂ [ L 2 ( 0 , T ; H 2 ( Ω ) ) , H 1 ( 0 , T ; L 2 ( Ω ) ) ] β = H 1 − β ( 0 , T ; H 2 β ( Ω ) ) (where the notation [ A , B ] β means the interpolated space between A and B with index β ), for β = 1 2 , it follows that T ∈ H 1 2 ( 0 , T ; H 1 ( Ω ) ) ⊂ L 2 ( 0 , T ; L 2 ( Ω ) ) with completely continuous immersion. It is thus proven that the transformation F is completely continuous. Now, let { F λ } be a family of transformations dependent on the parameter λ ∈ [ 0 , 1 ] and defined as follows: consider the system

(2.26) ∂ u ̲ ∂ t + ( u ̲ ⋅ ∇ ) u ̲ − μ ϱ 0 Δ u ̲ + 1 ϱ 0 ∇ p = { 1 − α T ( T − T 0 ) } g ̲ ,

(2.27) ∇ ⋅ u ̲ = 0 ,

(2.28) ∂ T δ t + u ̲ ⋅ ∇ T − γ T Δ T = 0 ,

with the associated initial conditions

(2.29) u ̲ ( 0 ) = λ u ̲ and T ( 0 ) = λ T ˜ ,

and the boundary conditions

(2.30) u ̲ ∣ Γ × ( 0 ; T ) = 0 , T ∣ Γ 4 × ( 0 ; T ) = λ T 2 , T ∣ Γ 2 × ( 0 ; T ) = λ T 1 , ∂ T ∂ ν ∣ ( Γ 1 ∪ Γ 3 ) × ( 0 ; T ) = 0 ,

with the usual meaning of the symbols μ , ϱ 0 , T 0 , g ̲ , … .

In analogy with what was explained in the introductory part of this section, it will be said that ( u ̲ , T ) is a “solution in the time interval ( 0 , T ) of system (2.26)–(2.28) satisfying (2.29), (2.30),” if:

  1. u ̲ ( t ) ∈ L 2 ( 0 , T ; V ) ∩ L ∞ ( 0 , T ; H ) ,

    T ( t ) ∈ L 2 ( 0 , T ; H ˜ 2 ( Ω ) ) ∩ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ H 1 ( 0 , T ; L 2 ( Ω ) ) ,

    T ( 0 ) = λ T ˜ a.e. on Ω , T ∣ Γ 2 × ( 0 ; T ) = λ T 1 , T ∣ Γ 4 × ( 0 ; T ) = λ T 2 ,

  2. (2.2) and (2.3) are satisfied.

Now, define the transformation F λ : L 2 ( Q T ) ⟶ L 2 ( Q T ) such that F λ is associates with T ˜ ∈ L 2 ( Q T ) the function T ∈ L 2 ( Q T ) obtained with the same procedure used to describe F, applied now under Conditions ( a 3 ) and ( b 3 ) .

Note that F 1 = F and F 0 = 0 . Finally, let T λ be a fixed point of the transformation F λ ; following Remark 2, it turns out that

∣ T λ ∣ L ∞ ( Q T ) ⩽ λ E ,

from which it follows

∣ T λ ∣ L 2 ( Q T ) ⩽ 2 E T 1 2 .

Therefore, applying the Leray-Schauder theorem, it follows that there exists a fixed point T of the transformation F, which together with u ̲ (obtainable from T with the procedure previously described) provides a solution ( u ̲ , T ) of the original problem.□

We comment that it is likely that a Galerkin method, and an application of parabolic regularity theory to deduce the additional smoothness of the temperature, can also be used as an alternative proof earlier.

2.4 Uniqueness of the solution

Our next result involves, for completeness, a statement on uniqueness, a fact that was already known.

Theorem 5

Under the assumptions of Theorem 4, the solution ( u ̲ , T ) of problem (1.4)–(1.8) in the sense specified in ( a 1 ) and ( b 1 ) is unique.

Proof

For this proof, which uses a Friedrichs regularization operator (mollifier [16]), see [9].□

3 Stationary solutions

In this section, we look for stationary solutions of system (1.4)–(1.6) satisfying (1.7), i.e., solutions of

(3.1) − μ Δ u ̲ + ϱ 0 ( u ̲ ⋅ ∇ ) u ̲ + ∇ p = ϱ 0 { 1 − α T ( T − T 0 ) } g ̲ ,

(3.2) ∇ ⋅ u ̲ = 0 ,

(3.3) − γ T Δ T + u ̲ ⋅ ∇ T = 0 ,

with associated boundary conditions

(3.4) u ̲ ∣ Γ = 0 , T ∣ Γ 2 = T 1 , T ∣ Γ 4 = T 2 , and ∂ T ∂ ν Γ 1 ∪ Γ 3 = 0 ,

where T 1 and T 2 are the constant values, with T 2 > T 1 .

The “solution of problem (3.1)–(3.4)” is defined as a pair ( u ̲ , T ) such that

  1. u ̲ ∈ V , T ∈ H 1 ( Ω ) , γ 0 T = T 1 on Γ 2 , γ 0 T = T 2 on Γ 4 ,

  2. the following conditions are satisfied:

(3.5) μ ( u ̲ , s ̲ ) H 0 1 ( Ω ) + ϱ 0 ( ( u ̲ ⋅ ∇ ) u ̲ , s ̲ ) L 2 ( Ω ) = − α T ϱ 0 ( T g ̲ , s ̲ ) L 2 ( Ω ) , ∀ s ̲ ∈ V ,

(3.6) γ T ( ∇ T , ∇ ϕ ) L 2 ( Ω ) + ( u ̲ ⋅ ∇ T , ϕ ) L 2 ( Ω ) = 0 , ∀ ϕ ∈ H ˜ 0 1 ( Ω ) ,

where γ 0 denotes the restriction operator on the Γ boundary. It should be noted that, with a specific change of variables, namely,

T ( x , y ) = θ ( x , y ) + d ˜ ( x ) , d ˜ ( x ) = T 1 − T 2 2 x + T 1 − T 2 2

(in this regard, see paragraph 1.1), problem (3.1)–(3.4) are transformed into an equivalent stationary homogeneous problem expressed by

(3.7) − μ Δ u ̲ + ϱ 0 ( u ̲ ⋅ ∇ ) u ̲ + ∇ p = ϱ 0 g ̲ { 1 − α T ( θ + d ˜ − T 0 ) } ,

(3.8) ∇ ⋅ u ̲ = 0 ,

(3.9) − γ T Δ θ + u ̲ ⋅ ∇ θ + T 1 − T 2 2 u 1 = 0 ,

with the boundary conditions

(3.10) u ̲ ∣ Γ = 0 , θ ∣ Γ 2 ∪ Γ 4 = 0 , ∂ θ ∂ ν ( Γ 1 ∪ Γ 3 ) = 0

We then call “solution of problem (3.7)–(3.10)” or also, for convenience, “homogeneous stationary solution,” a pair ( u ̲ , θ ) such that

  1. u ̲ ∈ V , θ ∈ H ˜ 0 1 ( Ω ) ;

  2. the following conditions are satisfied:

(3.11) μ ( u ̲ , s ̲ ) H 0 1 ( Ω ) + ϱ 0 ( ( u ̲ ⋅ ∇ ) u ̲ , s ̲ ) L 2 ( Ω ) = − ϱ 0 α T ( g ̲ ( θ + d ˜ ) , s ̲ ) L 2 ( Ω ) , ∀ s ̲ ∈ V ,

(3.12) γ T ( ∇ θ , ∇ ϕ ) L 2 ( Ω ) + ( u ̲ ⋅ ∇ θ , ϕ ) L 2 ( Ω ) + T 1 − T 2 2 ( u 1 , ϕ ) L 2 ( Ω ) = 0 , ∀ ϕ ∈ H ˜ 0 1 ( Ω ) ,

We now proceed to the discussion of problem (3.1)–(3.4) and, precisely, in the next paragraph, we prove some auxiliary theorems, while the second and third paragraphs are dedicated, respectively to the existence and uniqueness of the solution.

3.1 Auxiliary theorems

Lemma 3

Let f ∈ H − s ( Ω ) , 0 < s < 1 . Then,

  1. there exists a unique χ ∈ H 2 − s ( Ω ) ∩ H 0 1 ( Ω ) solution of

    (3.13) Δ χ = f , χ ∣ Γ = 0 ,

  2. there exists a unique χ ∈ H 2 − s ( Ω ) ∩ H ˜ 0 1 ( Ω ) solution of:

    (3.14) Δ χ = f , χ ∣ Γ 2 ∪ Γ 4 = 0 , ∂ χ ∂ ν Γ 1 ∪ Γ 3 = 0 .

Proof

See [9].□

Lemma 4

Let u ̲ ∈ V and θ ∈ H 1 ( Ω ) . Then,

( u ̲ ⋅ ∇ θ , θ ) L 2 ( Ω ) = 0 .

Proof

See [9].□

Theorem 6

Let u ̲ = ( u 1 , u 2 ) ∈ V . There is then one and only one function θ such that

  1. θ ∈ H ˜ 0 1 ( Ω ) ,

  2. h ( ∇ θ , ∇ ϕ ) L 2 ( Ω ) + ( u ̲ ⋅ ∇ θ , ϕ ) L 2 ( Ω ) = K ( u 1 , ϕ ) L 2 ( Ω ) , ∀ ϕ ∈ H ˜ 0 1 ( Ω ) ,

with K and h being the real constants, h > 0. The function θ also belongs to C 0 ( Ω ¯ ) .

Proof

Observing that H ˜ 0 1 ( Ω ) is a Hilbert space, set

a ( θ , ϕ ) = h ( ∇ θ , ∇ ϕ ) L 2 ( Ω ) + ( u ̲ ⋅ ∇ θ , ϕ ) L 2 ( Ω ) , ∀ θ , ϕ ∈ H ˜ 0 1 ( Ω ) .

Therefore,

∣ a ( θ , ϕ ) ∣ ⩽ ∣ h ( ∇ θ , ∇ ϕ ) L 2 ( Ω ) ∣ + ∣ ( u ̲ ⋅ ∇ θ , ϕ ) L 2 ( Ω ) ∣ ⩽ h ∣ θ ∣ H 0 1 ( Ω ) ∣ ϕ ∣ H 0 1 ( Ω ) + + c ˆ 2 ∣ u ̲ ∣ H 0 1 ( Ω ) ∣ θ ∣ H 0 1 ( Ω ) ∣ ϕ ∣ H 0 1 ( Ω ) = ( h + c ˆ 2 ⋅ ∣ u ̲ ∣ H 0 1 ( Ω ) ) ∣ ϕ ∣ H 0 1 ( Ω ) ∣ θ ∣ H 0 1 ( Ω ) ,

with c ˆ being the embedding constant of H 0 1 ( Ω ) in L 4 ( Ω ) , from which we deduce that the form a: H ˜ 0 1 ( Ω ) × H ˜ 0 1 ( Ω ) ⟶ R is linear and continuous. Now, observe that by Lemma 4, the form a is also coercive; therefore, by the Lax-Milgram theorem, θ ∈ H ˜ 0 1 satisfying ( i i 5 ) exists and is unique. As regards the continuity of θ , it verifies (in a weak sense) the following equation:

h Δ θ = u ̲ ⋅ ∇ θ + K u 1 .

From u ̲ ∈ V and θ ∈ H ˜ 0 1 ( Ω ) , it follows that θ satisfies an equation of the type Δ θ = f with f ∈ L p ( Ω ) , p < 2 . Since L p ( Ω ) ⊂ H − s ( Ω ) for 0 < s < 1 , we deduce from Lemma 3 that θ ∈ H 2 − s ( Ω ) ∩ H ˜ 0 1 ( Ω ) . On the other hand, for s < 1 , H 2 − s ( Ω ) ⊂ C 0 ( Ω ¯ ) , and thus, the continuity of θ is obtained.□

Remark 3

In the hypotheses of Theorem 6, it also holds

∣ ∇ θ ∣ L 2 ( Ω ) ⩽ c ˆ K h ∣ u ̲ ∣ L 2 ( Ω ) ,

where c ˆ is here and in the following a positive constant providing the equivalence of the norms of H 1 ( Ω ) and H 0 1 ( Ω ) on H ˜ 0 1 ( Ω ) .

Corollary 1

Let u ̲ = ( u 1 , u 2 ) ∈ V . There is then one and only one function T such that

  1. T ∈ H 1 ( Ω ) ∩ C 0 ( Ω ¯ ) , γ 0 T = T 1 , on Γ 2 , γ 0 T = T 2 , on Γ 4 ,

  2. γ T ( ∇ T , ∇ ϕ ) L 2 ( Ω ) + ( u ̲ ⋅ ∇ T , ϕ ) L 2 ( Ω ) = 0 , ∀ ϕ ∈ H ˜ 0 1 ( Ω ) .

Proof

See [9], or the proof of Theorem 6.□

Remark 4

From an application of the maximum principle (see [14]), it follows that T 1 ⩽ T ( x , y ) ⩽ T 2 , ∀ ( x , y ) ∈ Ω .

Remark 5

From the definition of θ and from Remark 3, it follows that

∣ ∇ T ∣ L 2 ( Ω ) ⩽ T 2 − T 1 2 c ˆ γ T ∣ u ̲ ∣ L 2 ( Ω ) + 2 .

3.2 Existence of the solution

In order to prove the existence of the solution ( u , T ) of problem (3.1)–(3.4) in the sense specified in ( c 1 ) and ( d 1 ) , consider the following new problem, very similar to the initial one but for the presence of the projection Π T defined as follows:

(3.15) − μ Δ u ̲ + ϱ 0 ( u ̲ ⋅ ∇ ) u ̲ + ∇ p = ϱ 0 { 1 − α T Π T ( T − T 0 ) } g ̲ ,

(3.16) ∇ ⋅ u ̲ = 0 ,

(3.17) − γ T Δ T + u ̲ ⋅ ∇ θ = 0 ,

with associated boundary conditions

(3.18) u ̲ ∣ Γ = 0 , T ∣ Γ 2 = T 1 , T ∣ Γ 4 = T 2 , and ∂ T ∂ ν Γ 1 ∪ Γ 3 = 0 ,

with the usual meaning of the symbols μ , ϱ 0 , T 0 , g ̲ , … and with Π T the projection operator defined in the following way:

Π T ( T ) = T 1 , if T < T 1 , T , if T 1 ⩽ T ⩽ T 2 , T 2 , if T > T 2 .

Furthermore, the “solution of problem (3.15)–(3.18)” is defined as a pair ( u ̲ , T ) such that

  1. the conditions listed in ( c 1 ) are satisfied,

  2. the following equations hold:

    (3.19) μ ϱ 0 ( u ̲ , s ̲ ) H 0 1 ( Ω ) + ϱ 0 ( ( u ̲ ⋅ ∇ ) u ̲ , s ̲ ) L 2 ( Ω ) = ( − α T g ̲ ( Π T T − T 0 ) , s ̲ ) L 2 ( Ω ) , ∀ s ̲ ∈ V ,

    (3.20) γ T ( ∇ T , ∇ ϕ ) L 2 ( Ω ) + ( u ̲ ⋅ ∇ T , ϕ ) L 2 ( Ω ) = 0 , ∀ ϕ ∈ H ˜ 0 1 ( Ω ) .

Theorem 7

If ( u ̲ , T ) is a solution to problem (3.15)–(3.18), then it is a solution to the original problem (3.1)–(3.4).

Proof

The conditions ( c 1 ) are valid as a consequence of ( c 3 ) ; by the maximum principle, T 1 ⩽ T ⩽ T 2 (that implies Π T T = T ), and therefore, ( u ̲ , T ) verifies ( d 1 ) .□

Theorem 8

If the boundary data T 1 and T 2 satisfy T 1 ≤ T 0 ≤ T 2 and

(3.21) α T ∣ T 2 − T 1 ∣ < μ 2 2 c ˆ 2 ϱ 0 2 g ,

where c ˆ is the embedding constant of H 0 1 ( Ω ) in L 4 ( Ω ) , then there exists a solution to problem (3.15)–(3.18).

Proof

The existence of this solution can be proved by the Leray-Schauder theorem, applied here to a suitable transformation F that we now define. For this purpose, let T ˜ be a function in L 2 ( Ω ) : substituting T ˜ for T in (3.19), we obtain a Navier-Stokes equation (in variational form) with the known term

f ̲ = − α T g ̲ ( Π T T ˜ − T 0 ) ∈ L ∞ ( Ω ) .

It is interesting to observe that, since T 0 can be arbitrarily chosen, which is only a constant reference value, one can certainly assume T 1 ⩽ T 0 ⩽ T 2 , from which ∣ Π T T ˜ − T 0 ∣ ⩽ ∣ T 2 − T 1 ∣ ; this last relation then allows us to obtain

(3.22) ∣ f ̲ ∣ L 2 ( Ω ) = g α T ( Π T T ˜ − T 0 ) L 2 ( Ω ) ⩽ 2 g α T ∣ T 2 − T 1 ∣ ,

and by (3.21),

(3.23) ∣ f ̲ ∣ L 2 ( Ω ) < μ 2 c ˆ 2 ϱ 0 2 .

For known theorems on the Navier-Stokes equations (stationary case, see, for example, [15], or even [17]), it follows that u ˜ ̲ ∈ V exists and is unique, solution of (3.19).

Now, substituting u ˜ ̲ in place of u ̲ into (3.20), thanks to Corollary 1, there exists T endowed with all the properties highlighted in the corollary itself so (remember also Remark 2), in particular, T ∈ L ∞ ( Ω ) .

Let F : L 2 ( Ω ) ⟶ L 2 ( Ω ) be the transformation that is associated with the function T obtained with the procedure just described to the function T ˜ . From the previous theorems, it also follows that T ∈ H 2 − s ( Ω ) with arbitrarily small positive s , and therefore, F is completely continuous from L 2 ( Ω ) in itself. Now, let { F λ } λ ∈ [ 0 , 1 ] be a family of transformations of L 2 ( Ω ) defined in the following way: let system (3.15)–(3.17) with associated boundary conditions (for fixed λ )

(3.24) u ̲ ∣ Γ = 0 , T ∣ Γ 2 = λ T 1 , T ∣ Γ 4 = λ T 2 , and ∂ T ∂ ν Γ 1 ∪ Γ 3 = 0 .

A pair ( u ̲ , T ) will be called “solution of problem (3.15)–(3.17), (3.23)” if

  1. u ̲ ∈ V , T ∈ H 1 ( Ω ) , γ 0 T = λ T 1 , on Γ 2 , γ 0 T = λ T 2 , on Γ 4 , (3.19)

  2. and (3.20) are verified.

Let us then define F λ : L 2 ( Ω ) ⟶ L 2 ( Ω ) as the transformation, which is associated with T ˜ the function T determined with the procedure described previously applied here to problem (3.15)–(3.17) and (3.23). We observe that F 1 = F and F 0 = 0 . Finally, let T ˜ λ be a fixed point of the transformation F λ : in this case,

(3.25) λ T 1 ⩽ T ˜ λ ⩽ λ T 2 .

Hence, being λ ∈ [ 0 , 1 ] and setting

(3.26) M T = max { ∣ T 1 ∣ , ∣ T 2 ∣ } ,

we obtain

(3.27) ∣ T λ ˜ ∣ L 2 ( Ω ) ⩽ 2 M T .

The statement follows from the Leray-Schauder theorem.□

Corollary 2

In the hypothesis of Theorem 8, there exists at least one solution to the original problem (3.1)–(3.4).

Proof

This follows from Theorems 7 and 8.□

Remark 6

If ( u ̲ , T ) is a solution to problem (3.1)–(3.4), remembering Remark 5 and (2.22) and keeping in mind that

(3.28) ∣ u ̲ ∣ H 0 1 ( Ω ) ⩽ c ϱ 0 μ ∣ f ̲ ∣ L 2 ( Ω ) ,

it follows that

(3.29) ∣ ∇ T ∣ L 2 ( Ω ) ⩽ T 2 − T 1 2 c ˆ c 2 μ c ˆ 2 γ T ϱ 0 + 2 ,

where c is the embedding constant of H 0 1 ( Ω ) in L 2 ( Ω ) .

3.3 Uniqueness of the solution

We next state and prove a uniqueness result.

Theorem 9

There exist positive constants μ 0 and τ 0 such that, if the viscosity coefficient μ ≥ μ 0 and ∣ T 2 − T 1 ∣ ≤ τ 0 , then the solution of the original problems (3.1)–(3.4), whose existence has been proved in Theorem 8, is unique.

Proof

Thanks to Corollary 2, we can assume that there exist two solutions ( u ̲ , T ) and ( u ˜ ̲ , T ˜ ) of problem (3.1)–(3.4), in the sense specified in points ( c 1 ) and ( d 1 ) . Then, setting δ ̲ u = u ̲ − u ˜ ̲ and δ t = T − T ˜ , the pair ( δ ̲ u , δ T ) ∈ V × H ˜ 0 1 ( Ω ) satisfies

(3.30) μ ( δ ̲ u , s ̲ ) H 0 1 ( Ω ) + ϱ 0 ( ( u ̲ ⋅ ∇ ) δ ̲ u + ( δ ̲ u ⋅ ∇ ) u ̲ − ( δ ̲ u ⋅ ∇ ) δ ̲ u , s ̲ ) L 2 ( Ω ) = − α T ϱ 0 ( δ T g ̲ , s ̲ ) L 2 ( Ω ) , ∀ s ̲ ∈ V ,

(3.31) γ T ( ∇ δ T , ∇ ϕ ) L 2 ( Ω ) + ( δ ̲ u ⋅ ∇ δ T , ϕ ) L 2 ( Ω ) − ( δ ̲ u ⋅ ∇ T , ϕ ) L 2 ( Ω ) + − ( u ̲ ⋅ ∇ δ T , ϕ ) L 2 ( Ω ) = 0 , ∀ ϕ ∈ H ˜ 0 1 ( Ω ) .

Now, taking s ̲ = δ ̲ u , ϕ = δ T and remembering Lemma 4, we obtain

(3.32) μ ∣ δ ̲ u ∣ H 0 1 ( Ω ) 2 + ϱ 0 ( ( δ ̲ u ⋅ ∇ ) u ̲ , δ ̲ u ) L 2 ( Ω ) = − ϱ 0 α T ( δ T g ̲ , δ ̲ u ) L 2 ( Ω ) ,

(3.33) γ T ∣ δ T ∣ H 0 1 ( Ω ) 2 − ( δ ̲ u ⋅ ∇ T , δ T ) L 2 ( Ω ) = 0 .

By adding (3.30) and (3.31), we obtain

(3.34) μ ∣ δ ̲ u ∣ H 0 1 ( Ω ) 2 + ϱ 0 ( ( δ ̲ u ⋅ ∇ ) u ̲ , δ ̲ u ) L 2 ( Ω ) + γ T ∣ δ T ∣ H 0 1 ( Ω ) 2 − ( δ ̲ u ⋅ ∇ T , δ T ) L 2 ( Ω ) = − ϱ 0 α T ( δ T g ̲ , δ ̲ u ) L 2 ( Ω ) .

Besides, thanks to (3.22), it can be proved that

(3.35) ∣ ( ϱ 0 ( δ ̲ u ⋅ ∇ ) u ̲ , δ ̲ u ) L 2 ( Ω ) ∣ ⩽ c β μ ∣ δ ̲ u ∣ H 0 1 ( Ω ) 2 ,

where β is the constant, 0 < β < 1 ; it also holds

(3.36) ∣ ( δ ̲ u ⋅ ∇ T , δ T ) L 2 ( Ω ) ∣ ⩽ c ˆ 2 ∣ δ ̲ u ∣ H 0 1 ( Ω ) ∣ ∇ T ∣ L 2 ( Ω ) ∣ δ T ∣ H 0 1 ( Ω ) .

Following Remark 6, we also obtain

(3.37) ∣ ( δ ̲ u ⋅ ∇ T , δ T ) L 2 ( Ω ) ∣ ⩽ c ˆ 2 M 1 ∣ δ ̲ u ∣ H 0 1 ( Ω ) ∣ δ T ∣ H 0 1 ( Ω ) ⩽ c ˆ 2 M 1 ε 1 ∣ δ ̲ u ∣ H 0 1 ( Ω ) 2 + c ˆ 2 M 1 4 ε 1 ∣ δ T ∣ H 0 1 ( Ω ) 2 ,

where ε 1 is the positive constant to be chosen appropriately, and M 1 is a bound on ∣ ∇ T ∣ L 2 ( Ω ) as given by the right-hand side of (3.29). In conclusion, from (3.32), (3.33), and (3.35), it follows:

(3.38) 1 − c β c 2 μ − c ˆ 2 M 1 c 2 ε 1 − ϱ 0 α T g 4 ε 2 ∣ δ ̲ u ∣ L 2 ( Ω ) 2 + γ T c 2 − c ˆ 2 M 1 4 c 2 ε 1 − ε 2 ϱ 0 α T g ⋅ ∣ δ T ∣ L 2 ( Ω ) 2 ⩽ 0 ,

where ε 2 is the arbitrary positive constant and c ⩽ 1 .

From (3.36), it is clear that recalling the expression of M 1 (Remark 6), if the difference ∣ T 2 − T 1 ∣ and the viscosity μ are, respectively, small enough and large enough so that, with an appropriate choice of ε 1 and ε 2 , the coefficients of ∣ δ ̲ u ∣ L 2 ( Ω ) 2 and ∣ δ T ∣ L 2 ( Ω ) 2 are positive, then the solution is unique.□

4 Asymptotic behavior

In Section 1, we examinated the problem constituted by the system of Oberbeck-Boussinesq equations (1.4)–(1.6) with associated boundary conditions (1.7) and initial conditions (1.8), and the existence and uniqueness of the “evolutionary solution” ( u ̲ , T ) of this problem have been proven, to be understood in the sense specified in ( a 1 ) and ( b 1 ) and in the hypothesis that it is: u ˜ ̲ ∈ H , T ˜ ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) , T 1 and T 2 ∈ H 1 ( 0 , T ) .

Now observe that, due to the arbitrariness of T and thanks to the possibility of repeating the same procedure to prove the existence and uniqueness of the solution on any time interval contained in ( 0 , + ∞ ) , (for this purpose, it would be sufficient to move the origin of times), it can be stated that the “homogeneous evolutionary solution” ( u ̲ , θ ) of problem (3.9)–(3.13), defined in ( a 2 ) and b 2 , verifies the conditions:

  1. u ̲ ( t ) ∈ L loc 2 ( 0 , + ∞ ; V ) ∩ L ∞ ( 0 , + ∞ ; H ) ,

    θ ( t ) ∈ L loc 2 ( 0 , + ∞ ; H ˜ 0 2 ( Ω ) ) ∩ L ∞ ( 0 , + ∞ ; H 1 ( Ω ) ) ∩ H loc 1 ( 0 , + ∞ ; L 2 ( Ω ) ) ,

    with u ̲ ( 0 ) = u ˜ ̲ a.e. on Ω , θ ( 0 ) = θ ˜ a.e. on Ω , where θ ˜ ( x , y ) = T ˜ ( x , y ) − d ( x , 0 ) , with

    d ( x , t ) = T 1 ( t ) − T 2 ( t ) 2 x + T 1 ( t ) + T 2 ( t ) 2

    (see paragraph 1.1);

  2. for an arbitrary T > 0 , the weak formulations (2.14) and (2.15), are satisfied, with test functions in L loc 2 ( 0 , + ∞ ; L 2 ( Ω ) ) .

It seems appropriate to observe that these considerations are valid for T 1 and T 2 ∈ H loc 1 ( 0 , + ∞ ) ; if we also assume that there exist the finite limits T 1 ˜ and T 2 ˜ of the functions T 1 and T 2 for t ⟶ + ∞

T 1 ˜ = lim t → + ∞ T 1 ( t ) , and T 2 ˜ = lim t → + ∞ T 2 ( t ) ,

we can then consider the “stationary solution” ( u ˜ ̲ , T ˜ ) of the system (3.1)–(3.3) with associated boundary conditions

(4.1) u ˜ ̲ ∣ Γ = 0 , T ˜ ∣ Γ 2 = T 1 ˜ , T ˜ ∣ Γ 4 = T 2 ˜ , and ∂ T ˜ ∂ ν Γ 1 ∪ Γ 3 = 0 ,

in the sense specified in ( c 1 ) and ( d 1 ) , and the “homogeneous stationary solution” ( u ˜ ̲ , θ ˜ ) of system (3.7)–(3.10) with T 1 = T 1 ˜ , T 2 = T 2 ˜ , and finally, d ˜ ( x ) = T 1 ˜ − T 2 ˜ 2 x + T 1 ˜ + T 2 ˜ 2 , characterized by ( c 2 ) and ( d 2 ) .

The aim of this section is to prove that, in the circumstances described so far, the evolutionary solution ( u ̲ , T ) converges to the stationary one ( u ˜ ̲ , T ˜ ) for t ⟶ + ∞ , which is equivalent to proving the convergence of ( u ̲ , θ ) to ( u ˜ ̲ , θ ˜ ) . This argument is developed in two paragraphs; the first presents some auxiliary theorems, and the second obtains the conclusive result.

4.1 Auxiliary theorems

Let w ̲ = ( w 1 , w 2 ) = u ̲ − u ˜ ̲ , δ θ = θ − θ ˜ . Note that the weak formulations stated below only for the convenience of the reader are identical to the ones given in (2.9) and (2.10), but for the functional space involved.

Lemma 5

The “difference” functions w ̲ and δ θ belong, respectively, to the functional spaces L loc 2 ( 0 , + ∞ ; V ) ∩ L ∞ ( 0 , + ∞ ; H ) and to

L loc 2 ( 0 , + ∞ ; H ˜ 0 1 ( Ω ) ) ∩ L ∞ ( 0 , + ∞ ; H 1 ( Ω ) ) ∩ H loc 1 ( 0 , + ∞ ; L 2 ( Ω ) )

and verify

(4.2) ( w ̲ ( T ) , s ̲ ( T ) ) L 2 ( Ω ) − ( w ̲ ( 0 ) , s ̲ ( 0 ) ) L 2 ( Ω ) + ∫ 0 T { ( − w ̲ ( η ) , s ̲ ′ ( η ) ) L 2 ( Ω ) + μ ϱ 0 ( ( w ̲ ( η ) , s ̲ ( η ) ) H 0 1 ( Ω ) + ( ( u ̲ ( η ) ⋅ ∇ ) w ̲ ( η ) , s ̲ ( η ) ) L 2 ( Ω ) + ( ( w ̲ ( η ) ⋅ ∇ ) u ˜ ̲ , s ̲ ( η ) ) L 2 ( Ω ) } d η = − ∫ 0 T α T ( g ̲ ( δ θ ( η ) + d ( θ ) ) − d ˜ ) , s ̲ ( η ) L 2 ( Ω ) d η ,

(4.3) ∫ 0 T ∂ δ θ ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) − γ T ( ∇ δ θ ( η ) , ϕ ( η ) ) L 2 ( Ω ) + ( u ̲ ( η ) ⋅ ∇ δ θ ( η ) , ϕ ( η ) ) + ( w ̲ ( η ) ⋅ ∇ θ ˜ , ϕ ( η ) ) L 2 ( Ω ) d η = ∫ 0 T ( T 2 ( η ) − T 2 ˜ 2 − T 1 ( η ) − T 1 ˜ 2 ) ( w 1 ( η ) , ϕ ( η ) ) L 2 ( Ω ) − ∂ d ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) d η , ∀ s ̲ ∈ L loc 2 ( 0 , + ∞ ; V ) ∩ H loc 1 ( 0 , + ∞ ; L 2 ( Ω ) ) , ∀ ϕ ∈ L loc 2 ( 0 , + ∞ ; H 0 1 ˜ ( Ω ) ) and ∀ T > 0 .

Proof

See [9]. Similar results have already been established in earlier sections of this article using a standard continuation argument.□

A further result can be proved.

Lemma 6

Let [ t 1 , t 2 ] ⊂ ( 0 , + ∞ ) ; in this case, the functions w ̲ and δ θ , previously defined, satisfy

(4.4) ( w ̲ ( t 2 ) , s ̲ ( t 2 ) ) L 2 ( Ω ) − ( w ̲ ( t 1 ) , s ̲ ( t 1 ) ) L 2 ( Ω ) + ∫ t 1 t 2 − w ̲ ( η ) , ∂ s ̲ ∂ η ( η ) L 2 ( Ω ) + μ ϱ 0 ( w ̲ ( η ) , s ̲ ( η ) ) H 0 1 ( Ω ) + ( ( w ̲ ( η ) ⋅ ∇ ) u ˜ ( η ) , s ̲ ( η ) ) L 2 ( Ω ) + ( ( u ̲ ( η ) ⋅ ∇ ) w ̲ ( η ) , s ̲ ( η ) ) L 2 ( Ω ) } d η = − ∫ t 1 t 2 α T ( g ̲ ( δ θ ( η ) + d ( θ ) − d ˜ ) , s ̲ ( η ) ) L 2 ( Ω ) d η ,

(4.5) ∫ t 1 t 2 ∂ δ θ ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) − γ T ( ∇ δ θ ( η ) , ∇ ϕ ( η ) ) H 0 1 ( Ω ) + ( u ̲ ( η ) ⋅ ∇ δ θ ( η ) , ϕ ( η ) ) L 2 ( Ω ) + ( w ̲ ( η ) ⋅ ∇ θ ˜ , ϕ ( η ) ) L 2 ( Ω ) d η = ∫ t 1 t 2 T 2 ( η ) − T 2 ˜ 2 − T 1 ( η ) − T 1 ˜ 2 w 1 ( η ) , ϕ ( η ) L 2 ( Ω ) − ∂ d ∂ η ( η ) , ϕ ( η ) L 2 ( Ω ) d η , ∀ s ̲ ∈ L loc 2 ( 0 , + ∞ ; V ) ∩ H loc 1 ( 0 , + ∞ ; H ) , ∀ ϕ ∈ L loc 2 ( 0 , + ∞ ; H 0 1 ˜ ( Ω ) ) .

Proof

See again [9]. The proofs follow again along lines similar to previous ones.□

4.2 On the convergence of the evolutionary solution to the stationary solution

From the results of the previous paragraph, it is possible to obtain a “energy relations” for the functions w ̲ and δ θ , as will be demonstrated in the following theorem:

Theorem 10

The functions w ̲ and δ θ , previously defined, verify

(4.6) 1 2 ∣ w ̲ ( t 2 ) ∣ L 2 ( Ω ) 2 − 1 2 ∣ w ̲ ( t 1 ) ∣ L 2 ( Ω ) 2 + ∫ t 1 t 2 μ ϱ 0 ∣ w ̲ ( η ) ∣ H 0 1 ( Ω ) 2 + ( ( w ̲ ( η ) ⋅ ∇ ) u ˜ , w ̲ ( η ) ) L 2 ( Ω ) d η = ∫ t 1 t 2 − ( α T g ̲ ( δ θ ( η ) + d ( θ ) ) − d ˜ ) , w ̲ ( η ) L 2 ( Ω ) d η ,

(4.7) 1 2 ∣ δ θ ( t 2 ) ∣ L 2 ( Ω ) 2 − 1 2 ∣ δ θ ( t 1 ) ∣ L 2 ( Ω ) 2 + ∫ t 1 t 2 { γ T ∣ ∇ δ θ ( η ) ∣ L 2 ( Ω ) 2 + ( w ̲ ( η ) ⋅ ∇ θ ˜ , δ θ ( η ) ) L 2 ( Ω ) } d η = ∫ t 1 t 2 T 2 ( η ) − T 2 ˜ 2 − T 1 ( η ) − T 1 ˜ 2 ( w 1 ( η ) , δ θ ( η ) ) L 2 ( Ω ) − ∂ d ∂ η ( η ) , δ θ ( η ) L 2 ( Ω ) d η .

Proof

Note that, by Lemma 6, the functions w ̲ and δ θ verify (4.4) and (4.5); to obtain (4.7), it will be sufficient to take ϕ = δ θ , in (4.5). The deduction of (4.6) is, however, more complicated. Define the following two functions on the interval [ t 1 , t 2 ] : w ̲ 1 is linear with respect to time and such that w ̲ 1 ( t 1 ) = w ̲ ( t 1 ) and w ̲ 1 ( t 2 ) = w ̲ ( t 2 ) , and w ̲ 2 = w ̲ − w ̲ 1 , from which it immediately follows that w ̲ 2 ( t 1 ) = w ̲ 2 ( t 2 ) = 0 . Let w ̲ * be the function obtained from w ̲ 2 by extension outside the interval [ t 1 , t 2 ] with zero values, and let J ε (real ε > 0 ) be a Friedrichs regularization operator (mollifier), operating with respect to the time variable only. For a precise definition of J ε , we refer the reader to [16]. Then, take (4.7) s ̲ = w ̲ 1 + J ε 2 w ̲ * ; in this case, we obtain

(4.8) ( w ̲ ( t 2 ) , w ̲ 1 ( t 2 ) + J ε 2 w ̲ * ( t 2 ) ) L 2 ( Ω ) − ( w ̲ ( t 1 ) , w ̲ 1 ( t 1 ) + J ε 2 w ̲ * ( t 1 ) ) L 2 ( Ω ) + ∫ t 1 t 2 − w ̲ ( η ) , ∂ w 1 ̲ ∂ η ( η ) + ∂ J ε 2 ∂ η w ̲ * ( η ) L 2 ( Ω ) + μ ϱ 0 ( w ̲ ( η ) , w ̲ 1 ( η ) + J ε 2 w ̲ * ( η ) ) H 0 1 ( Ω ) + ( w ̲ ( η ) ⋅ ∇ ) u ˜ ̲ + ( u ̲ ( η ) ⋅ ∇ ) w ̲ ( η ) , w ̲ 1 ( η ) + J ε 2 w ̲ * ( η ) L 2 ( Ω ) } d η = − ∫ t 1 t 2 ( α T g ̲ ( δ θ ( μ ) + d ( η ) − d ˜ ) , w ̲ 1 ( η ) + J ε 2 w ̲ * ( η ) ) L 2 ( Ω ) d η .

About the limit behavior of the terms of (4.8), for ε ⟶ 0 , it can be proved that (see again [16]):

(4.9) ( w ̲ ( t 2 ) , w ̲ 1 ( t 2 ) + J ε 2 w ̲ * ( t 2 ) ) L 2 ( Ω ) − ( w ̲ ( t 1 ) , w 1 ( t 1 ) + J ε 2 w ̲ * ( t 1 ) ) L 2 ( Ω ) ⟶ ∣ w ̲ ( t 2 ) ∣ L 2 ( Ω ) 2 − ∣ w ̲ ( t 1 ) ∣ L 2 ( Ω ) 2 ;

(4.10) ∫ t 1 t 2 − w ̲ ( η ) , ∂ w 1 ̲ ∂ η ( η ) + ∂ J ε 2 ∂ η w ̲ * ( η ) L 2 ( Ω ) d η ⟶ − 1 2 ∣ w ̲ ( t 2 ) ∣ L 2 ( Ω ) 2 + 1 2 ∣ w ̲ ( t 1 ) ∣ L 2 ( Ω ) 2 ;

(4.11) ∫ t 1 t 2 μ ϱ 0 ( ( w ̲ ( η ) , w ̲ 1 ( η ) ) + J ε 2 w ̲ * ( η ) ) H 0 1 ( Ω ) d η ⟶ ∫ t 1 t 2 μ ϱ 0 ∣ w ̲ ( η ) 2 ∣ H 0 1 ( Ω ) d η ;

(4.12) ∫ t 1 t 2 ( ( u ̲ ( η ) ⋅ ∇ ) w ̲ ( η ) , w ̲ 1 ( η ) + J ε 2 w ̲ * ( η ) ) L 2 ( Ω ) d η ⟶ ∫ t 1 t 2 ( ( u ̲ ( η ) ⋅ ∇ ) w ̲ ( η ) , w ̲ ( η ) ) L 2 ( Ω ) d η = 0 .

Therefore, passing to the limit for ε ⟶ 0 in (4.8), taking the results (4.9), (4.19), (4.11), and (4.12), (4.6) follows.□

We can now proceed with the fundamental result of this section.

Theorem 11

Let u ˜ ̲ ∈ H , T ˜ ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) , T 1 and T 2 ∈ H 1 ( 0 , T ) , with

(4.13) lim t → + ∞ ∂ T 1 ∂ t ( t ) = 0 , lim t → + ∞ ∂ T 2 ∂ t ( t ) = 0 ,

(4.14) lim t → + ∞ T 1 ( t ) = T 1 ˜ , lim t → + ∞ T 2 ( t ) = T 2 ˜ , ( − ∞ < T 1 , T 2 < + ∞ ) ,

and with

T 1 < T 2 , T 1 ˜ < T 2 ˜ , T ˜ ∣ Γ 2 = T 1 ( 0 ) , T ˜ ∣ Γ 4 = T 2 ( 0 ) .

If the viscosity μ is “large enough” and the difference ∣ T 2 ˜ − T 1 ˜ ∣ is “small enough,” then the “evolutionary solution” ( u ̲ , T ) relative to the data u ˜ ̲ , T ˜ , T 1 , and T 2 , is asymptotic, for t ⟶ + ∞ , to the “stationary solution” relative to the data T 1 ˜ and T 2 ˜ .

Proof

Setting again w ̲ = u ̲ − u ˜ ̲ and δ θ = θ − θ ˜ (observe that in the hypotheses of the present theorem, ( u ̲ , T ) and ( u ˜ ̲ , T ˜ ) exist and are unique), from Theorem 10, (4.6) and (4.7) follow. Adding them, it follows that:

(4.15) 1 2 ∣ w ̲ ( t 2 ) ∣ L 2 ( Ω ) 2 − 1 2 ∣ w ̲ ( t 1 ) ∣ L 2 ( Ω ) 2 + 1 2 ∣ δ θ ( t 2 ) ∣ L 2 ( Ω ) 2 + − 1 2 ∣ δ θ ( t 1 ) ∣ L 2 ( Ω ) 2 + ∫ t 1 t 2 μ ϱ 0 ∣ w ̲ ( η ) ∣ H 0 1 ( Ω ) 2 + γ T ∣ δ θ ( η ) ∣ H 0 1 ( Ω ) 2 d η = ∫ t 1 t 2 { − ( ( w ̲ ( η ) ⋅ ∇ ) u ˜ ̲ , w ̲ ( η ) ) L 2 ( Ω ) − α T ( g ̲ ( δ θ ( η ) + d ( η ) − d ˜ ) , w ̲ ( η ) ) L 2 ( Ω ) − ( w ̲ ( η ) ⋅ ∇ θ ˜ , δ θ ( η ) ) L 2 ( Ω ) + T 2 ( η ) − T 2 ˜ 2 − T 1 ( η ) − T 1 ˜ 2 ( w 1 ( η ) , δ θ ( η ) ) L 2 ( Ω ) − ∂ d ∂ η , δ θ ( η ) L 2 ( Ω ) d η .

By (3.28) and (3.22), it can be proved that

(4.16) ∣ ( ( w ̲ ( η ) ⋅ ∇ ) u ˜ ̲ , w ̲ ( η ) ) L 2 ( Ω ) ∣ ⩽ c ˆ 2 c 2 α g ρ 0 μ ∣ T 2 − T 1 ∣ ∣ w ̲ ( η ) ∣ H 0 1 ( Ω ) 2 ,

and, for a suitable h > 0

(4.17) ∣ ( w ̲ ( η ) ⋅ ∇ θ ˜ , δ θ ( η ) ) L 2 ( Ω ) ∣ ⩽ h μ ( T ˜ 2 − T ˜ 1 ) 2 ∣ w ̲ ( η ) ∣ H 0 1 ( Ω ) 2 + h μ ( T ˜ 2 − T ˜ 1 ) 2 ∣ δ θ ( η ) ∣ H 0 1 ( Ω ) 2 ,

(4.18) ∣ α T ( g ̲ δ θ ( η ) , w ̲ ( η ) ) L 2 ( Ω ) ∣ ⩽ α T g ε 2 ∣ w ̲ ( η ) ∣ L 2 ( Ω ) 2 + α T g 4 ε 2 ∣ δ θ ( η ) ∣ L 2 ( Ω ) 2 ,

where ε 1 and ε 2 are the arbitrary positive constants.

From (4.15)–(4.18) it follows that, for a suitable K > 0 ,

(4.19) 1 2 ∣ w ̲ ( t 2 ) ∣ L 2 ( Ω ) 2 − 1 2 ∣ w ̲ ( t 1 ) ∣ L 2 ( Ω ) 2 + 1 2 ∣ δ θ ( t 2 ) ∣ L 2 ( Ω ) 2 − 1 2 ∣ δ θ ( t 1 ) ∣ L 2 ( Ω ) 2 + ∫ t 1 t 2 1 c 2 μ ϱ 0 − K μ ∣ T ˜ 2 − T ˜ 1 ∣ − h μ ( T ˜ 2 − T ˜ 1 ) 2 − α T g ε 2 − 1 4 ( T 2 ( η ) − T 1 ( η ) ) ∣ w ̲ ( η ) ∣ L 2 ( Ω ) 2 + 1 c 2 γ T − h μ ( T ˜ 2 − T ˜ 1 ) 2 − α T g 4 ε 2 − 1 4 ( T 2 ( η ) − T 1 ( η ) ) ∣ δ θ ( η ) ∣ L 2 ( Ω ) 2 d η ⩽ ∫ t 1 t 2 − α T ( g ̲ ( d ( η ) − d ˜ ) , w ̲ ( η ) ) L 2 ( Ω ) + T 2 ( η ) − T 2 ˜ 2 − T 1 ( η ) − T 1 ˜ 2 ( w 1 ( η ) , δ θ ( η ) ) L 2 ( Ω ) − ∂ d ∂ η ( η ) , δ θ ( η ) L 2 ( Ω ) d η .

If in (4.19) we take t 1 = t and t 2 = t + ρ , with t , ρ generic in ( 0 , + ∞ ) , it is clearly possible to show that, if μ is large enough and ∣ T 2 ˜ − T 1 ˜ ∣ is small enough to make the appropriate coefficients positive, for a suitable choice of ε 1 and ε 2 , then

{ ∣ w ̲ ( t ) ∣ L 2 ( Ω ) 2 + ∣ δ θ ( t ) ∣ L 2 ( Ω ) 2 } ⟶ 0 , as t ⟶ + ∞ ,

whence the statement.□

Acknowledgements

I wish to thank Gabriele Grillo, Irene Sabadini and Fabrizio Colombo for support and valuable advice.

  1. Funding information: The author states no funding involved.

  2. Author contributions: The author confirms the sole responsibility for the conception of the study and presented results and manuscript.

  3. Conflict of interest: The author states no conflict of interest.

  4. Data availability statement: There are no data associated with this research.

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Received: 2024-02-26
Revised: 2024-06-24
Accepted: 2024-06-26
Published Online: 2024-08-26

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  131. Corrigendum
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  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0032
Keywords for this article
Oberbeck-Boussinesq model; viscous fluid; heating
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