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.
The primitive equations are fundamental models for weather prediction, which are derived from the Boussinesq system of incompressible flow (see, e.g., [1–5]). Due to their importance, many authors have considered the primitive equations analytically by using many new methods (see Zeng , Lions and Temam [7, 8], Sun and Cui , Hieber et al. , and You and Li ). For more papers, one can see [9, 12–14] 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. There 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 [15–21]). For some type of primitive equations, one can see [22, 23].
In this paper, the two-dimensional large-scale primitive equations (see ) are consideredin a cylindrical domain . In (1), the unknown functions , , are the horizontal velocity field, the vertical velocity, the density, the pressure, and the temperature, respectively; is the Rossby number; are the viscosity coefficients; are the reference values of the density and the temperature; is the expansion coefficient (constants); .
The boundary of is defined by which can be partitioned into
System (1) also has the following boundary conditions:where are the wind stress on the ocean surface, are the positive constants, and is the typical temperature distribution of the top surface of the ocean. , and also satisfy the compatibility boundary conditions:
In addition, the initial conditions can be written as
The 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 [25–27], rigorous a priori bounds with coefficients are established. Finally, the convergence result on the boundary data of our problem is derived in Section 4.
2. Preliminaries of the Problem
By integrating (1)3 and (1)4, the following is obtained:where is the pressure on the surface of the ocean. For convenience, suppose . Inserting (6) and (7) into (1)–(5), the problem can be rewritten aswith the following boundary conditions:and the initial conditions:
In this paper, some well-known inequalities are used throughout this paper.
Lemma 1. If and , then
Lemma 3. If is a sufficiently smooth function in and , thenorwhere , is a positive computable constant, and is a positive arbitrary constant.
Proof. By the Hölder inequality, one can writeSince , the following is obtained:Therefore,Then,Inserting (19) into (16), the following is obtained:Obviously,so,To bound the last term of (22), a new known function is defined, satisfyingwhere are the positive constants. For example, , satisfies all the conditions in (23). Using the above estimates and employing the divergence theorem allow one to writeInserting (24) into (22),whereTherefore,Thus, from (20) and (27) and by the Hölder inequality, one hasAfter simplification,
3. A Priori Estimates
Proof. Taking the inner product of equation (9)3 with , in , the following is obtained:A function is defined asTherefore,Integrating by parts,By the above results, the following is obtained:where . Using inequality (35) and the Gronwall inequality, the following is obtained:Moreover,where .
Proof. Taking the inner product of equation (9)1 with , in , the following is obtained:A function is defined asObviously, has the same boundary conditions of . Therefore,By the Cauchy–Schwarz inequality, the following is obtained:By the above results, the following is obtained:Similarly, from (9)2,whereCombining (43) and (44), the following is obtained:where . By the Gronwall inequality and Lemma 2, one getswhere . Moreover,where .
Lemma 6. If , then the solution of (2.4)3 satisfieswhere .
Proof. Multiplying (2.4)3 by and integrating by parts, one getsBy the Hölder inequality and the Cauchy–Schwarz inequality, the following is obtained:Therefore, from (50) and (51), one hasBy the Gronwall inequality, the following is obtained:Therefore,Letting now in (54), one obtains
Proof. Using (2.4)1, one starts withIntegrating by parts, the following is obtained:By Lemmas 4 and 5 and the Hölder inequality, one hasBy (40) and using the divergence theorem,By the Cauchy–Schwarz inequality and Lemmas 4 and 5, the following is obtained:and by (13) with and Lemma 5,Inserting the above two inequalities into (61), one obtainswhereCombining the inequalities (59), (60), and (64), the following is obtained:where . Similar to (58) and (59), by (2.4)2, one hasIn a similar way,for some computable positive function . Combining (66) and (67), one obtainswhere . Using Lemma 3 with , Lemma 5, and (69),