On Stability of Solutions to Equations Describing Incompressible Heat-Conducting Motions Under Navier’s Boundary Conditions

In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove existence of 3d global strong solutions via stability.


Introduction
We consider a complete thermodynamical model of unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional cylinder. The model comprises evolutionary equations for the velocity, pressure and temperature fields. We restrict our considerations to temperature independent material coefficients, that is, the viscosity and heat conductivity coefficients. We assume that the external force depends on temperature in a very special way (see Theorem 1). The heat equation contains the dissipative heating term which follows from the general theory of incompressible heat-conducting Navier-Stokes motions. We assume that the fluid is Newtonian, the internal energy is proportional to the temperature and the heat flux is proportional to the temperature gradient.
Let D = Ω × (−a, a) be a bounded cylinder, where Ω ⊂ R 2 be an open domain, L = 2a is the length of the cylinder which is parallel to the x 3 -axis. The three-dimensional problem under the consideration is as follows v t + v · ∇v − ν rot 2 where S = ∂D, v = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) ∈ R 3 is the velocity of the fluid, x = (x 1 , x 2 , x 3 ) are given Cartesian coordinates, p = p(x, t) ∈ R is the pressure, θ = θ(x, t) ∈ R + is the temperature of the fluid, f = (f 1 (x, t), f 2 (x, t), f 3 (x, t)) ∈ R 3 is the external force field. The Cartesian coordinates x are such that x 3 -axis is parallel to the cylinder and it is located inside it. By the dot we denote the scalar product in R 3 . By D(v) = {v i,x j + v j,x j } i,j =1,2,3 = ∇v + ∇v T we denote the dilatation tensor, rot 2 = rot rot = − which holds on any divergence free vector. Moreover, ν > 0 is the constant viscosity coefficient, κ > 0 is also the constant heat conductivity coefficient and θ * > 0 is a constant. Moreover,n is the unit outward normal vector to S. The boundary S is split into two parts, i.e. S = S 1 ∪ S 2 , where S 1 is parallel to the x 3 -axis and S 2 is perpendicular to it. Additionally, S 2 = S 2 (−a) ∪ S 2 (a), where S 2 (b) meets x 3 -axis at x 3 = b.
The function α = α(θ) from the r.h.s. of (1.1) 1 will be determined later. The term α(θ)f generates the motion by heating and mechanically by the external force f . Similarly, the term ν|D(v)| 2 from the r.h.s. of (1.1) 3 heats the fluid by the viscous dissipation.
The aim of this paper is to prove the existence and uniqueness of a two-dimensional solution to problem (1), the global existence of sufficiently small three-dimensional perturbations of the 2d-solution and the existence of a global solution to problem (1) close to the two-dimensional solution assuming that the differences of external forces and initial data are sufficiently small in appropriate norms.
The important assumption is such that the external force f does not decay in time. This implies that the solution does not vanish as time tends to infinity. Then, via stability and periodicity, the existence of stationary 2d and 3d solutions can be proved.
The stability of a two-dimensional solution to the Navier-Stokes system with a nondecaying in time external force has been examined in [21] and [23]. In [21] the system supplemented by the periodic boundary conditions is considered, while the paper [23] is devoted to the system in a cylinder with the Navier boundary conditions. Problem (1) is much more difficult than the problem considered in [19], where the r.h.s. of (1) 3 is assumed to be zero. Then in [19] it is proved that there exists a global solution under the assumption that v ,x 3 (0), θ ,x 3 (0), f ,x 3 are sufficiently small in L 2 -norms. Hence the obtained solution also has the same property. This means that the solution in [19] is close to the two-dimensional solution. However, in [19], contrary to this paper, it is not possible to control a behavior of solutions as time tends to infinity.
In paper [22] we proved the existence of a strong global solution to (1) 1,2,3,6,7 with the slip boundary conditions for velocity which are close for all time to a two-dimensional solution.
Therefore the main operators in (1) 1 and (2) 1 are compatible. Moreover, it is shown in [23] thatn × rot w| S 0 =τ rot (2) w| S 0 andn × rot w| S 2 = 0, wheren = (n 1 , n 2 , 0),τ = (n 2 , −n 1 , 0). To show the stability of two-dimensional solutions of (2) we introduce the quantities To formulate main results of the paper we introduce the following notation: The norms introduced above as well as norms occurring in formulations of the main results are defined in Sect. 2.
Theorem 1 is proved in Sect. 3. The proof of Theorem 1 is based on the Faedo-Galerkin approximations combined with a step by step method. Since we need in the proof H 2estimate for the velocity and H 1 -estimate for the temperature, we assume that |α(ϑ)| ≤ a 1 + a 2 ϑ σ for ϑ ≥ θ * , where 0 < σ < 1 2 , a 1 , a 2 > 0. This restriction is generated by the dissipative heating term appearing in the r.h.s. of (1) 3 . In the case of the lack of the dissipative term, the L p -estimate for the temperature follows immediately, so the above restriction on α is not necessary. To obtain estimate (4) we show that the data in time intervals [kT , (k + 1)T ], k ∈ N 0 do not increase with k.
Our next result is the theorem about the stability of the two-dimensional solution obtained in Theorem 1 in the set of three-dimensional solutions.
To prove Theorem 2 we derive a nonlinear differential inequality (see the proof of Lemma 9) and then apply a contradiction argument.
Theorems 1 and 2 imply the following global existence result.
In view of Theorem 1 we prove Theorem 2 by using estimates (7)-(8) derived for the Faedo-Galerkin approximations.
There is a wide literature concerning equations describing motions of incompressible heat conducting fluids. Papers [6,8,10,16,19] are concerned with solvability results for the Boussinesq equations, that is, for equations (1) 1,2,3 , where ν|D(v)| 2 disappears. Kagei in [10] and Morimoto in [16] proved the existence of weak solutions to the three-dimensional Boussinesq equations with temperature -independent viscosity, while in [8] the weak solvability of the three-dimensional problem with temperature -dependent viscosity has been examined. Diaz and Galiano [6] showed existence of weak solutions to the following initialboundary value problem under some restrictions on the functions ν(θ), ϕ(θ), F (θ).
The question of the uniqueness of a solution to the two-dimensional Boussinesq equations with the Dirichlet boundary conditions and with temperature dependent viscosity has been studied in [9].
The recent papers concerning the Boussinesq system are [2] and [15]. In [2] Brandolese and Schonbek examine a large time behavior of both weak and strong solutions to the threedimensional system. Liu and Li [15] study the stability question for the Cauchy problem to equations of type (1), where the r.h.s. of (1) 3 vanishes and the r.h.s. of (1) 1 equals θē 3 are sufficiently small. The following stability is shown: if initially they are small. Since the Cauchy problem is considered, the technique of used function spaces is in the spirit of the French School. In contrast to our result solutions in [14] converge to zero as time tends to infinity. Papers of Kagei [11,12] are devoted to the two-dimensional system (1) 1,2,3 where the lefthand side of equation (1) 3 contains also the term −e 2 · v, e 2 = (0, 1). Considering both the periodic and Dirichlet boundary conditions the author discusses the existence, uniqueness and large time behavior questions in dependence on assumptions about the right-hand side of (1) 1 and initial data. Kagei [11] touched the very difficult 2d Bérnard convection problem. For a strongly nonhomogeneous boundary condition for the temperature the motion becomes very chaotic, so the concept of the strange attractor appears. The author relaxes the boundary condition on the temperature assuming that it is more homogeneous. Then he is able to derive many qualitative properties for the obtained weak solutions.
The Navier-Stokes system with temperature dependent viscosity coupled with the equation of conservation of internal energy has been studied in [3,5,7,17].
Buliček, Feireisl, Málek [3] and Buliček, Kaplicky, Málek [4] proved the existence of suitable weak solutions to the Navier-Stokes-Fourier system in [3] and to the Stokes-Fourier in [4]. Moreover, Feireisl and Málek [7] studied the existence of weak solutions to the Navier-Stokes-Fourier system with temperature dependent viscosity. Naumann [17] proved the existence of weak solutions to system (1) 1 -(1) 3 with the Dirichlet boundary conditions and with temperature-dependent viscosity and heat conductivity coefficients. Moreover, it is assumed that the r.h.s. of (1) 1 disappears. Consighlieri, Rodrigues, Shilkin [5] consider problem similar to (1) 1 -(1) 3 with temperature-dependent viscosity in 2d-torus, that is, with periodic boundary conditions. The existence of weak solutions is proved. Moreover, the viscosity coefficient ν = ν(θ) satisfies ν 1 , ν 2 , ν 1 ≤ ν(θ) ≤ ν 2 , where ν 1 , ν 2 are two positive constants. Kagei, Ružička, Thäter [13] proved the existence, uniqueness and stability of the motionless state for system of type (1), where v · ∇v and v · ∇θ disappear. But in (1) 3 additionally appears the term θv · f . They proved the existence of an unique solution in the class where P is the Helmholtz projection. Shilkin in [18] studies the classical solvability of the coupled system of two nonlinear parabolic equations with temperature-dependent viscosity which models a Poiseuille-type flow of an incompressible viscous fluid. He shows the existence of solutions to equations simpler than (1) because the terms v · ∇v and v · ∇θ disappear. Moreover, it is supposed that α(θ) = 1. The considered system is complemented by the Dirichlet conditions for the velocity and temperature. The problem is considered in a bounded domain.
The paper consists of four sections. In Sect. 2 notation and some auxiliary results are introduced. In Sect. 3 the two-dimensional problem is considered. Some useful a priori estimates are derived and Theorem 1 is proved. Section 4 contains the proofs of Theorems 2 and 3.

Notation and Auxiliary Results
Let  Let u = u(x 1 , . . . , x n ) be a scalar-valued function defined on Ω ⊂ R n , n = 2, 3. Then by ∇u or u x we denote the gradient of u. For u : Ω → R n , n = 2, 3, by ∇u or u x we denote the matrix [u ix j ] i,j =1,...,n . Moreover, then |u x | = n ij =1 |u ix j | 2 . We need the following lemma.
Consider the problem where Ω ⊂ R 2 is an open domain, S 0 is the boundary of Ω.
Then there exists a solution w ∈ H s+1 (Ω) to problem (10) and Now, consider the elliptic overdetermined problem where D ⊂ R 3 is a cylinder.

Lemma 4 Let w be a sufficiently regular function satisfying
Then Ω w · ∇w ·r ot rot w (2) Proof Since rot (2) By condition (12) 2 we get that I 1 = 0. Next, by (12) 1 Finally, This concludes the proof.

Lemma 7 Let the assumptions of Lemma 5 be satisfied. Let
where c does not depend on k.

Proof of Theorem 1
We prove the existence of a strong-weak solution to problem (13) by applying the Faedo-Galerkin approximations. Let which satisfies the following system of ordinary differential equations Ωθ where w kT m ∈ span{φ 1 , . . . , φ m }, w kT m → w(kT ) strongly in V as m → ∞,θ kT m ∈ span{ψ 1 , . . . , ψ m },θ kT m →θ(kT ) strongly in L 2 (Ω). We choose a special basis {φ j } ∞ j =1 which consists of the eigenfunctions of the Stokes system with Navier's boundary conditions, that is, φ j and some p j satisfy where λ j → ∞ as j → ∞. In what follows we will use the weak formulation of (29), that is, Now, we multiply the both sides of (27) by λ j and use (30) to the first and second term on the left-hand side of (27). We also use the equality Ω rot w m (t) · rot (2) To the other two terms in (27) we apply (30) and the identity Next, multiplying the resulting system of equations by c jm and summing for j = 1, . . . , m we get where we used that by Lemma 4 Similarly, multiplying (28) by d jm (t) and summing for j = 1, . . . , m we obtain Then repeating the considerations from the proofs of Lemmas 5-7 we derive the following estimate where c(A 1 , A 2 ) does not depend on k. By (31) and the Aubin-Lions lemma (see [14], The above convergences enable passing to the limit in (27)-(28) in the standard way (after multiplying (27)-(28) by appropriate functions of t and integrating them in [kT , (k + 1)T ]). In particular passing to the limit in the term on the r.h.s. of (27) is possible thanks to the strong convergence ofθ m in L 2 (kT , (k + 1)T ; L 2 (Ω)), which yields that for a subsequence ofθ m we haveθ m →θ a.e. in Ω × kT , (k + 1)T . Therefore, Hence (see [14], Lemma 1.3) The last convergence suffices to pass to the limit in the r.h.s. of (27). This way we obtain that (w, q,θ) is the strong-weak solution to problem (13). This ends the proof of Theorem 1.

Stability of Two-Dimensional Solution
To prove the stability of the two-dimensional solution which exists in virtue of Theorem 1 we need some a priori estimates of solutions to problem (3). Assume that where θ * > 0 is the constant from assumption (14) and where a 3 > 0 is a constant. First, we formulate the lemma Lemma 8 Let (32)-(33) and the assumptions of Theorem 1 be satisfied. Assume that f ∈ L 2,loc (R + ; L ∞ (D)). Moreover, assume that (w, ϑ, q) is the two-dimensional solutions of problem (1) which exists in view of Theorem 1. Let (u, χ, η) be a sufficiently regular solution to problem (3). Then for t > 0.
Proof Multiplying equation (3) 1 by rot 2 u and integrating over D gives where where the summation convention over repeated indices is applied. By the boundary conditions . Therefore, the Hölder and Young inequalities applied in (35) yield Consider the terms on the right-hand side of (36). First, using the interpolation inequality (see [1]) together with the estimates and where the Young inequality has been also used. Next, where we used that w only depends on x 1 , x 2 . Finally, the third term on the right-hand side of (36) is bounded by Hence inequality (36) yields Now, multiply (3) 3 by χ and integrate the result over D. We get The first term on the right-hand side of (40) is bounded by The second term is estimated by We estimate the third term as follows where the interpolation inequality has been used (see [1]). Finally, Using the above estimates in (36) we obtain Inequalities (37)-(39) and (41) imply (34).
Introduce the notation wherec 0 > 0 does not depend on k.

Lemma 9
Let the assumptions of Lemma 8 hold. Assume that wherec 1 ,c 2 > 0 are the constants from (48). Let γ and δ be so small that where M > 0 is the constant from (49). Moreover, let T be so large and δ be so small that If X(0) ≤ γ and G(t) ≤ δγ for all t ∈ R + , then X(t) ≤cγ for all t ∈ R + , wherec =c(T ) does not depend on k.
Proof By Lemma 5 where c > 0 does not depend on k. Now, by (42) and estimate (47), inequality (34) can be written in the form dX dt +c 1 X ≤c 2 X 3 + AX + G for t ∈ kT , (k + 1)T , k ∈ N 0 .
Assume that X(kT ) ≤ γ for some k ∈ N 0 .