Continuous Dependence on the Heat Source of 2D Large-Scale Primitive Equations in Oceanic Dynamics

In this paper, we consider the initial-boundary value problem for the two-dimensional primitive equations of the large-scale oceanic dynamics. These models are often used to predict weather and climate change. Using the differential inequality technique, rigorous a priori bounds of solutions and the continuous dependence on the heat source are established. We show the application of symmetry in mathematical inequalities in practice.


Introduction
Primitive equations are very useful models which are often used to study the climate and weather prediction. It was Lions, Teman and Wang (see [1][2][3][4]) who first started the mathematical study of the primitive equations of the atmosphere, the ocean and the coupled atmosphere-ocean. Assuming that all unknown functions are independent of the latitude y, Petcu et al. [5] obtained the two-dimensional primitive equations of the ocean from the three-dimensional primitive equations. The existence and uniqueness of strong solutions of the primitive equations were derived. In a following paper, Huang and Guo [6] considered the two-dimensional primitive equations of large-scale oceanic motion. They obtained the the existence and uniqueness of global strong solutions. Huang et al. [7] studied the two-dimensional primitive equations of large-scale ocean in geophysics driven by degenerate noise. They proved the asymptotically strong Feller property of the probability transition semigroups. Due to the importance of primitive equations, there are many papers to study the problems (see, e.g., [8][9][10][11][12][13][14]).
Recently, many authors began to study the structural stability of large-scale primitive equations. Li [15] obtained the continuous dependence on the viscosity coefficient of primitive equations of the atmosphere with vapor saturation. By using the energy analysis methods, Li [16] proved that the primitive equations of the coupled atmosphere-ocean depended continuously on the boundary parameters. The inspiration of the study came from the fluid equations. There have been a lot of articles in the literature to study the stability of fluid equations (for interest, see [17][18][19][20][21][22][23][24][25][26][27][28][29]).
In this paper, we also assume that all the unknown functions are independent of the latitude y as in [5,6]. We consider the following two-dimensional large-scale primitive equations with heat source: (1) The domain is defined as where h is the depth of the oceanic which is always assumed to be a positive constant in this paper. In (1) the unknown functions (u, v), w, ρ, p, T are the horizontal velocity field, the vertical velocity, the density, the pressure, the temperature, respectively. Q is the heat source function which is given. f is a function of the Earth's rotation which is taken to be constant here, and µ i > 0(i = 1, 2, 3) are the viscosity coefficients. ρ 0 , T re f are the reference values of the density and the temperature. β T is the expansion coefficient (constants), ∆ = ∂ 2 x + ∂ 2 z . We observe that, in the case of ocean dynamics, one has to add the diffusion-transport equation of the salinity to the system (1). The salinity equation is not present in (1), but this would raise little additional difficulty to take into account the salinity.
The boundary of Ω is denoted by ∂Ω which can be partitioned into The system (1) also has the following boundary conditions: where β is a positive constant. In addition, the initial conditions can be written as The aim of this paper is to prove the continuous dependence on the heat source of problem (1)-(3) by using the energy methods. This type of study is devoted to know whether a small change in the equation can cause a large change in the solutions. While we take advantage of the mathematical analysis and the symmetry in mathematical inequalities to study these equations, it is helpful for us to know their applicability in physics. Since there will appear some inevitable errors in reality, the study of continuous dependence or convergence results becomes more and more significant. At present, most articles in the literature mainly focused on the existence and long-time behavior of the solutions of the primitive equations. Obviously, the structural stability of the primitive equations has not been paid enough attentions. The research of this paper will bring reference to the study of structural stability of other types of primitive equations.
The present paper is organized as follows. In next section we give some preliminaries of the problem and some well-known inequalities which will be used in the whole paper. We establish rigorous a priori bounds of the solutions in Section 2. In Section 3 we want to prove that the energy is exponential decay with time. Finally, we show how to derive a continuous dependence on the the heat source of our problem in Section 4.

Preliminaries of the Problem
We formulate the Equations (1)-(3). Since w| z=−h = 0, we integrate the Equation (1) 4 from −h to z to obtain In view of w| z=0 = 0 This means that 0 −h u(x, ζ, t)dζ is a constant for arbitrary 0 ≤ x ≤ 1. Realizing the boundary conditions (2) 3 we deduce that By integrating (1) 3 and using (1) 6 we have where p s = p(x, 0, t) is the pressure on the surface of the ocean which is unknown and a function of the horizontal variable only, and µ = ρ 0 β T . Inserting (4)-(6) into (1)-(3), our problem can be rewritten as with the following boundary conditions and the initial conditions In this paper, we also use some well-known inequalities. We list them here.
where ∇ = (∂ x , ∂ z ), C is a positive computable constant and δ is a positive arbitrary constant.
Proof. By the Hölder inequality, we then write Since Therefore Then we have Inserting (15) into (12) we get Obviously, we have so, To bound the last term of (18), we define a new known function, f (z), satisfying where m 1 , m 2 are positive constants. For example, f (z) = m 1 2 (z + h 2 ), m 1 h < 4m 2 satisfies all the conditions in (19). Using the above estimates and employing the divergence theorem allow us to write Inserting (20) into (18), we have where Therefore max −h≤z≤0 Thus, from (16) and (23), by the Hölder inequality we have We have after simplification

A priori Estimates
Now we derive some a priori estimates for the solutions of (7)-(9). Multiplying Equation (7) 3 with T and integrating over Ω and using (2.5) 2 we find Integrating by parts we have By the Cauchy-Schwarz inequality and the Hölder inequality we deduce By (26)- (28) we have By the Gronwall inequality, we have Taking the inner product of Equation (7) 1 with u, in L 2 (Ω), we have An integration leads to Integrating by parts and using (7) By the Cauchy-Schwarz inequality we have Similarly, we can have from (7) 2 Combining (35) and (36) and using (30) we get We integrate (37) from 0 to t to find

Estimate for |T|
We multiply (7) 3 by T p−1 , and integrate by parts to find After integrating by parts on the third term of (39) and realizing the boundary condi- By the Hölder inequality and the Cauchy-Schwarz inequality we have Therefore, By the Gronwall inequality we have Therefore Letting now p → ∞ in (43) we can obtain where T m = sup Ω {||Q|| ∞ , ||T 0 || ∞ }.
3.3. Estimate for || ∂u ∂z || L 4 (Ω) Using (7) 1 we start with Integrating by parts we have Upon integrating by parts we get By (30), (38) and the Hölder inequality we have Inserting the above bounds into (46) we write We now carry out a similar procedure starting from (7) 2 to obtain Upon integrating by parts we get where δ 1 , δ 2 , δ 3 are positive constants which will be given later.

Exponential Decay Estimates with Time When Q = 0
In this section we want to prove the following theorem basis on Section 3.

Continuous Dependence on the Heat Source
Supposing (u * , v * , T * , p * s ) also be the solutions of (7)-(9) with the same initial-boundary conditions as (u, v, T, p s ), but with different heat source Q * . Let then ( u, v, π s ) satisfies the following initial-boundary problem We have the following theorem: Theorem 2. Let ( u, v, T) be the solutions of (63)-(65) with Q, T 0 ∈ L ∞ (Ω) and T 0 , u 0 , v 0 ∈ L 2 (Ω). Then ( u, v, T) satisfy the inequality for θ > 0, γ 1 (t) > 0 which is the continuous dependence result on the heat source Q.
Proof. Now taking the inner product of the first equation of (63) with u, in L 2 (Ω), we have An integration by parts leads to for computable b 1 (t), b 2 (t) and positive arbitrary constant δ 1 .

Conclusions
In this paper, we obtain the continuous dependence of the two-dimensional large-scale primitive equations in oceanic dynamics, where the depth of the ocean is assumed to be a positive constant. When the depth of the ocean is positive but not always a constant, Huang and Guo [32] have obtained the existence and uniqueness of a global strong solution for the problem. The study of the continuous dependence of the primitive equations in this case may be more interesting.