Convergence Results on the Boundary Conditions for 2D Large-Scale Primitive Equations in Oceanic Dynamics

In this paper, the initial boundary value problem for the two-dimensional large-scale primitive equations of large-scale oceanic motion in geophysics is considered, which are fundamental models for weather prediction. By establishing rigorous a priori bounds with coefficients and deriving some useful inequalities, the convergence result for the boundary conditions is obtained.


Introduction
e primitive equations are fundamental models for weather prediction, which are derived from the Boussinesq system of incompressible flow (see, e.g., [1][2][3][4][5]). Due to their importance, many authors have considered the primitive equations analytically by using many new methods (see Zeng [6], Lions and Temam [7,8], Sun and Cui [9], Hieber et al. [10], and You and Li [11]). For more papers, one can see [9,[12][13][14] and the references therein. It is very obvious that the papers in the literature mainly concern the well posedness of the 2D or 3D primitive equations and the properties of solutions.
Different from the results above, the aim of this paper is to establish the convergence result of the solution when the boundary data tend to zero. It is very important to know whether a small change in the equation can cause a large change in the solution. By taking advantage of the mathematical analysis to study these equations, it is helpful to know their applicability in physics. Since some inevitable errors will appear in reality, the study of continuous dependence or convergence results becomes more and more significant. ere have been many papers in the literature to study the continuous dependence or convergence for varieties of equations (e.g., Brinkman, Darcy, and Forchheimer equations) (see [15][16][17][18][19][20][21]). For some type of primitive equations, one can see [22,23].
In this paper, the two-dimensional large-scale primitive equations (see [24]) are considered zu zt (1) in a cylindrical domain Ω � (0, 1) × (−h, 0), h > 0. In (1), the unknown functions (u, v), w, ρ, p, and T are the horizontal velocity field, the vertical velocity, the density, the pressure, and the temperature, respectively; ε ′ is the Rossby number; c i > 0(i � 1, 2, 3) are the viscosity coefficients; ρ ref and T ref are the reference values of the density and the temperature; β T is the expansion coefficient (constants); Δ � z 2 e boundary of Ω is defined by zΩ which can be partitioned into System (1) also has the following boundary conditions: where g 1 (x 1 , t) and g 2 (x 1 , t) are the wind stress on the ocean surface, α 1 , α 2 , and β are the positive constants, and T * (x 1 , t) is the typical temperature distribution of the top surface of the ocean. g 1 (x 1 , t), g 2 (x 1 , t), and T * (x 1 , t) also satisfy the compatibility boundary conditions: In addition, the initial conditions can be written as in Ω.

(5)
e present paper is organized as follows. In Section 2, some preliminaries of the problem and some well-known inequalities which will be used in the whole paper are given. Inspired by [25][26][27], rigorous a priori bounds with coefficients are established. Finally, the convergence result on the boundary data of our problem is derived in Section 4.

Preliminaries of the Problem
Equations (1)-(5) are formulated as in [7][8][9]. Realizing the boundary conditions (4), equation (1) 4 is integrated from −h to x 2 to obtain By integrating (1) 3 and (1) 4 , the following is obtained: where p s � p(x 1 , 0, t) is the pressure on the surface of the ocean. For convenience, suppose (6) and (7) into (1)-(5), the problem can be rewritten as with the following boundary conditions: and the initial conditions: In this paper, some well-known inequalities are used throughout this paper.
us, from (20) and (27) and by the Hölder inequality, one has Discrete Dynamics in Nature and Society 3 After simplification, □

A Priori Estimates
Now, some a priori estimates for the solutions of (9)-(11) are derived.

Lemma 4.
Assume T be the solutions to (9)- (11) with where F 1 (t) will be defined later.
Proof. Taking the inner product of equation (9) 3 with T, in L 2 (Ω), the following is obtained: Integrating by parts, (34) By the above results, the following is obtained: where a 1 (t) � Ω (z 2 H/zx 2 2 ) 2 dA + Ω (zH/zx 2 ) 2 dA. Using inequality (35) and the Gronwall inequality, the following is obtained: where where F 2 (t) and F 3 (t) will be defined later.
Proof. Taking the inner product of equation (9) 1 with u, in L 2 (Ω), the following is obtained: A function S 1 (x 1 , x 2 , t) is defined as Obviously, S 1 has the same boundary conditions of u. erefore, By the Cauchy-Schwarz inequality, the following is obtained: By the above results, the following is obtained: Similarly, from (9) where Combining (43) and (44), the following is obtained: where By the Gronwall inequality and Lemma 2, one gets where where where

Proof. Multiplying (2.4) 3 by T p−1 and integrating by parts, one gets
By the Hölder inequality and the Cauchy-Schwarz inequality, the following is obtained: Discrete Dynamics in Nature and Society 5 erefore, from (50) and (51), one has By the Gronwall inequality, the following is obtained: erefore, Letting now p ⟶ ∞ in (54), one obtains Moreover, where F 5 (t) and F 6 (t) will be defined later.
Proof. Using (2.4) 1 , one starts with Integrating by parts, the following is obtained: 6 Discrete Dynamics in Nature and Society (59) By Lemmas 4 and 5 and the Hölder inequality, one has (60) By (40) and using the divergence theorem, Discrete Dynamics in Nature and Society 7 By the Cauchy-Schwarz inequality and Lemmas 4 and 5, the following is obtained: 8 Discrete Dynamics in Nature and Society and by (13) with δ � 1 and Lemma 5, Inserting the above two inequalities into (61), one obtains Combining the inequalities (59), (60), and (64), the following is obtained: where a 5 (t) � 4a 4 (t) + 4 ��������� F 1 (t)F 3 (t). Similar to (58) In a similar way, for some computable positive function a 6 (t). Combining (66) and (67), one obtains where F 5 (t) � a 5 (t) + a 6 (t). Using Lemma 3 with δ � 1, Lemma 5, and (69), 12 Discrete Dynamics in Nature and Society for arbitrary positive constants δ 1 and δ 2 . By the Cauchy-Schwarz inequality again In view of (25), the following is obtained: Inserting (77)-(80) into (76), one has Now taking the inner product of the second equation of (72) with v, the following is obtained: Discrete Dynamics in Nature and Society 13 Computing as (80), Computing as (77), where δ 5 and δ 6 are the arbitrary constants. Inserting (83) and (84) into (82), the following is obtained: e inner product of equation (72) 3 is taken with T to have