The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type

In this paper, we are concerned with the nonsteady Boussinesq system under mixed boundary conditions. *e boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions, velocity, static (or total) pressure, rotation, and stress (or total stress) together, and the boundary conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. Relying on the relations among strain, rotation, normal derivative of velocity, and shape of the boundary surface, we get variational formulation. *e formulations consist of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. For the case of boundary conditions including the static pressure and stress, we prove that if the data of the problem are small enough and compatibility conditions at the initial instance are satisfied, then there exists a unique solution on the given interval. For the case of boundary conditions including the total pressure and total stress, we prove the existence of a solution without restriction on the data and parameters of the problem.


Introduction
In this paper, we are concerned with the Boussinesq equation for heat convection under mixed boundary conditions. Here, v, p, and θ are, respectively, velocity, pressure, and temperature, and α 0 is the parameter for buoyancy effect, f is the body force, g is the heat source, μ is the viscosity, and κ is the thermal conductivity. e strain tensor E(v) is the one with the components which is a mathematical model for nonsteady motion of heat-conducting incompressible Newtonian fluid. Here, α 1 is the parameter for dissipation of energy due to expansion, α 2 is a positive real number, and for two matrices A � a ij } and B � b ij }, A: B � ij a ij b ij and |A| � ( ij a 2 ij ) 1/2 . e term μ(θ)E(v): E(v) represents the dissipation of energy due to viscosity (the Joule effect). Owing to the dissipation of energy due to viscosity μ(θ)E(v): E(v), study of (2) is usually more difficult than the Boussinesq system.
For the papers concerned with (2), we refer to Introduction of [1]. Here, we more mention [2][3][4][5] concerned with (2), where α 1 � 0. In [2], the problem under nonhomogeneous Dirichlet boundary conditions for velocity and temperature in the timedependent domain was studied, and existence of a local-in-time solution or existence of the solution on the given interval for small data was proved. In [3], existence of a strong solution and periodic solution for the 2D problem was studied under the boundary conditions and domain as above. In [4], under homogeneous Dirichlet boundary conditions for velocity and temperature, existence of a strong solution and periodic solution were studied when data of the problem are small enough. Łukaszewicz and Krzyżanowski [5] dealt with the initial boundary value problem on a time-dependent domain with the homogeneous Dirichlet boundary condition for velocity and temperature, and they proved the existence and uniqueness of local weak solutions and the existence of a global weak solution for small initial data.
Several papers are concerned with (1). In [6,7], the existence and uniqueness (for 2D) of a solution to the problem were studied under the homogeneous Dirichlet boundary condition for velocity and mixture of nonhomogeneous Dirichlet and Neumann boundary conditions for temperature. In [8], for the problem with nonhomogeneous Dirichlet boundary conditions for velocity and temperature, the existence of the time periodic solution was proved (see [9]). In [10][11][12][13], problem (1) on the time-dependent domain was studied under the nonhomogeneous Dirichlet boundary condition for velocity and temperature. In [14,15], the problem on exterior domains with the homogeneous Dirichlet boundary condition for velocity and nonhomogeneous Dirichlet boundary condition for temperature was studied. In [16], problem (1) was studied under the mixture of the nonhomogeneous Dirichlet boundary condition and the stress boundary condition for fluid and the mixture of nonhomogeneous Dirichlet, Neumann, and Robin boundary conditions for temperature. ey proved the existence of a unique local-in-time solution under a compatibility condition at the initial instance (see (27) and (31) of [16]). In [17], problem (1) in the cylindrical pipe with inflow and outflow was studied under slip boundary conditions for velocity and the Neumann conditions for temperature. In that, it was proved that there exists a solution on the given interval when norms of derivatives in the direction along the cylinder of the initial velocity, initial temperature, and the external force are small enough. In [18], the existence of a solution to problem (1) on the time-dependent domain was studied under the mixture of the Dirichlet condition of velocity, total pressure, and rotation boundary conditions for fluid and the mixture of Dirichlet, Neumann, and Robin boundary conditions for temperature.
On the contrary, for movement of fluid (v, p), different kinds of boundary conditions are used, and in practice, we deal with the mixture of some kinds of boundary conditions. On some portions of the boundary, we can use boundary conditions with stress or rotation, whereas when there is flux through a portion of the boundary, we can deal with the static pressure p or the total pressure (Bernoulli's pressure) (1/2)|v| 2 + p boundary conditions. ere are many literature studies for the Navier-Stokes problem with mixed boundary conditions (see Introduction of [19,20] and references therein). Recently, Navier-Stokes system with mixed boundary conditions including friction-type conditions was studied (cf. [20,21]).
In [1], problem (2) is studied under mixed boundary conditions, and the boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions, velocity, total pressure, rotation, and total stress together, and the conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. From the result of [1], we can get results for (1) with the boundary conditions as in [1]; however, the result demands that the parameter for buoyancy effect α 0 is small enough in accordance with the data of the problem, and the solution includes "defect measure" as in [22]. Also, for (2) and (1), the problem with a mixed boundary condition including the static pressure (not total pressure) and stress (not total stress) together is not yet considered.
When one of static pressure, stress, or the outflow boundary condition is given on a portion of the boundary, for the initial boundary value problems of the Navier-Stokes equations, the existence of a unique local-in-time solution and a unique solution on a given interval for small data of the problem are proved. From the mathematical point of view, the main difficulty of such problems results from the fact that in a priori estimation of solution, the term arising from the nonlinear term (v · ∇)v is not canceled (cf. Preface in [23]).
In the present paper, we are first concerned with heat convection equation (1) under mixed boundary conditions including the static pressure and stress. e boundary conditions for fluid may include conditions of friction type (Tresca slip, threshold leak, and one-sided leak conditions), velocity, static pressure, rotation, and stress together, and the conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. Due to the boundary conditions of friction type, it is difficult to follow the methods in [16,20]. e main difficulty of this problem is from the estimate of approximate solutions, and due to simultaneous velocity and temperature, the estimate is more difficult than the case of the Navier-Stokes equations. Also, in this paper, we prove the existence of a solution to (1) with the boundary conditions as in [1] without restriction on the parameter for buoyancy effect α 0 . is paper consists of 5 sections. In the last part of Section 1, we give notations.
In Section 2, the problems to study and assumptions for future are stated. According to the boundary conditions for fluid, Problems I and II are distinguished. Problem I includes the static pressure and the stress conditions, whereas Problem II includes the total pressure and the total stress boundary conditions. Assumption for Problem I is stronger than the one for Problem II.
In Section 3, we first get a variational formulation for Problem I which consists of six formulae with six unknown functions, that is, using velocity, tangent stress on slip surface, normal stress on the leak surface, normal stresses on one-sided leak surfaces, and temperature together as unknown functions (Problem I-VE). en, we get a new variational formulation for Problem I consisting of one variational inequality for velocity and a variational equation for temperature (Problem I-VI). e variational formulation for Problem II is obtained in the same way as in [1], and smoothness of the solution with respect to t is weaker than the one in Problem I. In the end of Section 3, the main results of this paper are stated ( eorems 1 and 2). e main result for Problem I asserts that if the data of the problem are small enough and compatibility conditions at 2 International Journal of Differential Equations the initial time (conditions 4 and 6 of eorem 1) hold, then there exists a unique smooth solution. e main result for Problem II asserts the existence of a solution without restriction on the parameter for buoyancy unlike [1]. Section 4 is devoted to the proof of eorem 1. To this end, first in Section 4.1, we consider an approximate problem, where the variational inequality for velocity is replaced by an equation with the gradient of the Moreau regularization of the functional due to the boundary conditions of friction type. Developing the method for the proof of eorem 4.4 of [21], we get existence and estimations of approximate solutions for small data under the compatibility conditions at initial time. In Section 4.2, we complete the proof of the existence and uniqueness of a solution.
Section 5 is devoted to the proof of eorem 2. e existence of solutions to an approximate problem and relative compactness of the set of solutions are studied. en, passing to limit, we get the conclusion.
Let n(x) and τ(x) be, respectively, outward normal and tangent unit vectors at x in zΩ.
with w ≥ 0, then we denote by f ≥ 0( ≤ 0) on Γ i . For convergence in spaces, ⟶ , ⇀, and ⇀ ⇀ mean, respectively, strong, weak, and weak * convergence. Derivative of f(t, x) with respect to t is denoted by f ′ . We also assume that 0 < T < ∞.

Problems and Assumptions
For temperature, we are concerned with the boundary conditions Stress tensor S is the one with components s ij � − pδ ij + 2με ij (v), and stress vector on the boundary surface is σ(v, p) � S · n. e value of the normal stress vector on the boundary surface is , and the total stress vector on the boundary surface is σ t (θ, v, p) � S t · n. e value of the total normal stress vector on the boundary surface is σ t n (θ, v, p) � σ t · n. And σ t τ (θ, v, p) � σ t (θ, v, p) − σ t n (θ, v, p)n. For Problem I, we assume that μ and κ are independent of θ.
us, Problem I is the one with the boundary conditions International Journal of Differential Equations 3 and Problem II is the one with the boundary conditions follows, the problems with boundary conditions (5) and (6) are called, respectively, the case of static pressure and the case of total pressure. Let We assume that g τ ∈ L 2 (Γ 8 ), g n ∈ L 2 (Γ 9 ), g +n ∈ L 2 (Γ 10 ), and that g − n ∈ L 2 (Γ 11 ), and g τ > 0, g n > 0, g +n > 0, and g − n > 0 for a.e. x of the portions of boundary. Also, we use the following assumption.
Assumption 1 (for the case of static pressure). We assume the following: (2) If Γ i , where i is 10 or 11, is nonempty, then at least one of Γ j : j ∈ 2, 4, 7, 9 − 11 International Journal of Differential Equations and there exists diffeomorphism in C 1 between Γ i and Γ j . (3) For the functions of (1), (4) For the functions of (4) and (5), Assumption 2 (for the case of total pressure). We assume (1) and (2) of Assumption 1 and the following: (3′) For the functions of (1), (4′) For the functions of (4) and (6), Remark 1. On Γ 10 (Γ 11 ), only outflow (inflow) is possible, and so (2) of Assumption 1 is used to guarantee div v � 0. In eorems 3.3 and 3.5 of [24], for the proof of equivalence of variational formulations to variational inequalities, this assumption was used via Lemma 3.2 of [24]. In this paper, this assumption is also necessary to guarantee equivalence between Problems I-VE and I-VI in Remark 4.

Variational Formulations and Main Results
Since Γ 1 ≠ ∅ and Γ D ≠ ∅, by the Korn inequality and Poincaré inequality, we use
International Journal of Differential Equations , then Problem I-VE is equivalent to problems (1), (4), and (5) in the following sense.
We will find another variational formulation consisting of a variational inequality and a variational equation, which is equivalent to Problem I-VE if the solution is smooth enough (cf. Remark 4).

(42)
Let functional Ψ be defined by (25)- (28). en, in the same way as Problem I-VI of [21], we find a variational inequality for velocity. en, we get another variational formulation consisting of a variational inequality for velocity and a variational equation for temperature, which is equivalent to Problem II-VE if the solution is smooth enough.

Main Results.
e main results of this paper are as follows.

Theorem 1 (the case of static pressure). Let Assumption 1 be satisfied. Suppose that
(1) e norms of f, ϕ i , i � 2 − 6, g, g R in the spaces they belong to are small enough (2) (compatibility condition at initial time for temperature) e solution satisfying ‖v‖ V ≤ c and ‖θ‖ W 1,2 Γ D ≤ c for a constant c > 0 small enough is unique.
en, there exists a
By the fact that Γ 2j , Γ 3j , and Γ 7j are in C 2.1 (Γ ij ) and 4 of Assumption 1, there exists a constant M such that us, there exists c * such that 10 International Journal of Differential Equations (cf. .5.1.10 of [28]). us, where the operators A 1 , B 1 are the ones in (32). Let u j , j � 1, 2, . . . and φ j , j � 1, 2, . . . be, respectively, bases of the space V and W 1,2 Γ D (Ω). Without loss of generality, we assume that u 1 � v 0 and φ 1 � θ 0 as in [26]. We find a solution v m � m j�1 g jm (t)u j and θ m � m j�1 r jm (t)φ j to the problem which gives us a system for g jm (t) and r jm (t), j � 1 − m. e solutions to (52) depend on ε, but for convenience of notation, here and in what follows, we use subindex m instead of subindex mε. For t m , there exist absolute continuous functions g jm (t) and r jm (t) on [0, t m ). Since f ∈ W 1,∞ (0, T; L 3 (Ω)), f 1 ∈ W 1,∞ (0, T; V * ), g 1 ∈ W 1,2 (0, T; W 1,2 Γ D (Ω) * ), and ∇Φ ε is Lipschitz continuous, g jm ′ (t) and r jm ′ (t) are in fact absolute continuous. If ‖v m (t)‖ and ‖θ m (t)‖ are bounded and v m (t), θ m (t) are integrable, then g jm (t) and r jm (t) are prolonged over t m . Under smallness of the data of the problem and the compatibility condition of the data at the initial instant, we will find estimates for ‖v m (t)‖ and ‖θ m (t)‖ in the following, by which we obtain (111) and see that t m � T.
Multiplying the first and the second equation of (52), respectively, by g jm (t) and φ jm (t) and adding for i � 1, . . . , m, we get International Journal of Differential Equations We will find a priori estimates for Since Φ ε is convex, continuous, and Fréchet differentiable, we have (56) Also, By virtue of (50), (51), (56), and (57), we have from the first equation of (53) where c * and c 2 are, respectively, the ones in (49) and (51), (59) Here and in what follows are the constants independent of the data of problem which are denoted by c with the exceptions of c * and c 2 .

14
International Journal of Differential Equations By the condition of theorem, let ‖f‖ W 1,∞ (0,T;L 3 ) be so small that If are valid, then there exists t m such that on [0, t m ]. erefore, taking into account (86), by the Gronwall inequality, we have on all intervals of t satisfying (90).
Using the estimate, we will obtain a quadratic inequality satisfied by ‖v m (t)‖ V . Put Note that β depends only on the data of the problem. en, when f satisfies (87), we can see from (91) that on [0, t m ], where (90) holds. Let the data of the problem be so small that By (70) and (93), for the small data of the problem, we have erefore, for such small data of the problem that (94) is valid, if then owing to (96), step by step, we have From the above, we see that, for the small data of the problem satisfying (87)-(89) and (94), is valid on the interval where the first inequality of (90) is valid. Put By (66), (93), and (63), for the small data satisfying (87)-(89) and (94), we have a quadratic inequality for ‖v m (t)‖ V , which is the one we want, on the intervals where the first inequality of (90) is satisfied. By the conditions of the theorem, we can assume that the data of the problem are so small that (87)-(89) and (94) are valid, and c satisfies the following inequality: Now, let us prove that if then for any m, Since 2μ − 4c 2 ‖v 0 ‖ V > μ, on an interval [0, t m ], Let us first prove that if the first inequality of (90) is valid on an interval [0, t m ], then more stronger is valid. Putting y � ‖v m (t)‖ V in (101) (which is valid on the interval where the first inequality of (90) holds when (87)-(89) and (94) are valid), we get 0 ≤ c − 3μy + 2c 2 y 2 on 0, t m .
and M is the positive integer, multiply the first equation of (52) by k j (t) and add for j � 1, . . . , M.

Existence and Estimation of a Solution to an Approximate
Problem. We first consider a problem approximating (43). For every 0 < ε < 1, let a functional Φ ε be defined by (45).