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

Authors: Yuanfei Li
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

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 [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, 12–14] 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 [24]) are considered(1)∂u∂t+u∂u∂x1+w∂u∂x2−1ε′v+1ε′∂p∂x1=γ1Δu,∂v∂t+u∂v∂x1+w∂v∂x2+1ε′u=γ2Δv,∂p∂x2+ρ=0,∂u∂x1+∂w∂x2=0,∂T∂t+u∂T∂x1+w∂T∂x2=γ3ΔT,ρ=ρref1−βTT−Tref,in a cylindrical domain Ω=0,1×−h,0,h>0. In (1), the unknown functions u,v, w, ρ,p,andT are the horizontal velocity field, the vertical velocity, the density, the pressure, and the temperature, respectively; ε′ is the Rossby number; γi>0i=1,2,3 are the viscosity coefficients; ρrefandTref are the reference values of the density and the temperature; βT is the expansion coefficient (constants); Δ=∂x12+∂x22.The boundary ofΩ is defined by ∂Ω which can be partitioned into(2)Γ0=x1,x2∈Ω¯:0<x1<1,x2=0,Γ−h=x1,x2∈Ω¯:0<x1<1,x2=−h,Γs=x1,x2∈Ω¯:x1=0orx1=1,−h≤x2≤0.System (1) also has the following boundary conditions:(3)∂u∂x2=α1g1x1,t,∂v∂x2=α2g2x1,t,w=0,∂T∂x2=βT∗x1,t,onΓ0,∂u∂x2=∂v∂x2=0,w=0,∂T∂x2=0,onΓ−h,u=v=w=0,∂T∂x1=0,onΓs,where g1x1,tandg2x1,t are the wind stress on the ocean surface, α1,α2,andβ are the positive constants, and T∗x1,t is the typical temperature distribution of the top surface of the ocean. g1x1,t,g2x1,t, and T∗x1,t also satisfy the compatibility boundary conditions:(4)g10,t=g20,t=T∗0,t=g11,t=g21,t=T∗1,t=0.In addition, the initial conditions can be written as(5)ux1,x2,0=u0x1,x2,vx1,x2,0=v0x1,x2,Tx1,x2,0=T0x1,x2,inΩ.The present paper is organized as follows. In Section2, 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

Equations (1)–(5) are formulated as in [7–9]. Realizing the boundary conditions (4), equation (1)4 is integrated from −h to x2 to obtain(6)wx1,x2,t=wx1,−h,t−∫−hx2∂∂x1ux1,ζ,tdζ=−∂∂x1∫−hx2ux1,ζ,tdζ,(7)∫−h0∂∂x1ux1,ζ,tdζ=−∂∂x1∫−h0ux1,ζ,tdζ=0.By integrating (1)3 and (1)4, the following is obtained:(8)px1,x2,t=ps−∫x20ρx1,ζ,tdζ=ps−ρref∫x201−βTTx1,ζ,t−Trefdζ,where ps=px1,0,t is the pressure on the surface of the ocean. For convenience, suppose ρrefβT=ε′=γ1=γ2=γ3=1. Inserting (6) and (7) into (1)–(5), the problem can be rewritten as(9)∂u∂t+u∂u∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂u∂x2−v+∂ps∂x1−∫x20∂∂x1Tx1,ζ,tdζ=Δu,∂v∂t+u∂v∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂v∂x2+u=Δv,∂T∂t+u∂T∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂T∂x2=ΔT,with the following boundary conditions:(10)∂u∂x2x2=0=α1g1x1,t,∂v∂x2x2=0=α2g2x1,t,∂u∂x2x2=−h=∂v∂x2x2=−h=0,u,vΓs=0,∂T∂x2x2=0=−βT∗x1,t,∂T∂x2x2=−h=0,∂T∂x1Γs=0,and the initial conditions:(11)u,v,Tt=0=u0,v0,T0.In this paper, some well-known inequalities are used throughout this paper.Lemma 1.
Ifωx1∈C10,1 and ω0=ω1=0, then(12)∫01ω2dx1≤1π2∫01∂ω∂x12dx1.Lemma 2.
Ifωx2∈C1−h,0 and ω−h=ω0=0, then(13)∫−h0ω2dx2≤h2π2∫−h0∂ω∂x22dx2.
For proofs of these inequalities, see references [28, 29].Lemma 3.
Ifωx1,x2,t is a sufficiently smooth function in Ω=0,1×−h,0 and ω0,x2,t=ω1,x2,t=0, then(14)∫Ωω4dA1/2≤C∫Ωω2dA1/2∫Ω∇ω2dA1/2+∫Ωω2dA1/4∫Ω∇ω2dA3/4,or(15)∫Ωω4dA1/2≤C∫Ωω2dA+δ∫Ω∇ω2dA,where ∇=∂x1,∂x2, C is a positive computable constant, and δ is a positive arbitrary constant.Proof.
By the Hölder inequality, one can write(16)∫Ωω4dA≤∫−h0∫01ω6dx11/2∫01ω2dx11/2dx2.
Sinceω0,x2,t=ω1,x2,t=0, the following is obtained:(17)ω3=3∫0x1ω2ξ,x2,t∂ωξ,x2,t∂ξdξ=−3∫x11ω2ξ,x2,t∂ωξ,x2,t∂ξdξ.
Therefore,(18)ω3≤32∫01ω2x1,x2,t∂ωx1,x2,t∂x1dx1.
Then,(19)∫01ω6dx11/2≤32∫01ω2∂ω∂x1dx1.
Inserting (19) into (16), the following is obtained:(20)∫Ωω4dA≤32∫−h0∫01ω2∂ω∂x1dx1∫01ω2dx11/2dx2≤32max−h≤x2≤0∫01ω2dx11/2∫Ωω2∂ω∂x1dA.
Obviously,(21)ω2=2∫−hx2ωx1,ζ,t∂ωx1,ζ,t∂ζdζ+ω2x1,−h,t=−2∫x20ωx1,ζ,t∂ωx1,ζ,t∂ζdζ+ω2x1,0,t,so,(22)ω2≤∫−h0ω∂ω∂x2dx2+12ω2x1,0,t+ω2x1,−h,t.
To bound the last term of (22), a new known function fx2 is defined, satisfying(23)f0>0,f−h<0,f′x2≤m1,fx2≤m2,for−h≤x2≤0,where m1andm2 are the positive constants. For example, fx2=m1/2x2+h/2,m1h<4m2, satisfies all the conditions in (23). Using the above estimates and employing the divergence theorem allow one to write(24)minf0,−f−hω2x1,0,t+ω2x1,−h,t≤f0ω2x1,0,t−f−hω2x1,−h,t=∫−h0∂∂x2fω2dx2=∫−h0f′x2ω2dx2+2∫−h0fω∂ω∂x2dx2≤m1∫−h0ω2dx2+2m2∫−h0ω∂ω∂x2dx2.
Inserting (24) into (22),(25)ω2≤m3∫−h0ω2dx2+m4∫−h0ω∂ω∂x2dx2,where(26)m3=m12minf0,−f−h,m4=1+m2minf0,−f−h.
Therefore,(27)max−h≤x2≤0∫01ω2dx11/2≤m3∫Ωω2dA+m4∫Ωω∂ω∂x2dA1/2.
Thus, from (20) and (27) and by the Hölder inequality, one has(28)∫Ωω4dA≤32m3∫Ωω2dA+m4∫Ωω2dA1/2∫Ω∂ω∂x22dA1/21/2∫Ωω4dA1/2∫Ω∂ω∂x12dA1/2.
After simplification,(29)∫Ωω4dA1/2≤C∫Ωω2dA1/2∫Ω∇ω2dA1/2+∫Ωω2dA1/4∫Ω∇ω2dA3/4.


## 3. A Priori Estimates

Now, some a priori estimates for the solutions of (9)–(11) are derived.Lemma 4.
AssumeT be the solutions to (9)–(11) with T0∈L2Ω,T∗∈H1Ω. Then,(30)∫0t∫Ω∇T2dAdη≤F1t,where F1t will be defined later.Proof.
Taking the inner product of equation (9)3 with T, in L2Ω, the following is obtained:(31)12ddt∫ΩT2dA+∫Ω∇T2dA=∫01∂T∂x2x2=0Tx1,0,tdx1−∫Ωu∂T∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂T∂x2TdA.
A functionHx1,x2,t is defined as(32)∂H∂x2=βx2+hhT∗x1,t.
Therefore,(33)∫01∂T∂x2x2=0Tx1,0,tdx1=∫01∂H∂x2x2=0Tx1,0,tdx1=∫Ω∂∂x2∂H∂x2TdA=∫Ω∂2H∂x22TdA+∫Ω∂H∂x2∂T∂x2dA=∫Ω∂2H∂x22TdA+∫Ω∂H∂x2∂T∂x2dA+12∫Ω∂H∂x22dA+12∫Ω∂T∂x22dA.
Integrating by parts,(34)−∫Ωu∂T∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂T∂x2TdA=0.
By the above results, the following is obtained:(35)ddt∫ΩT2dA+∫Ω∇T2dA=∫ΩT2dA+a1t,where a1t=∫Ω∂2H/∂x222dA+∫Ω∂H/∂x22dA. Using inequality (35) and the Gronwall inequality, the following is obtained:(36)∫ΩT2dA≤∫ΩT02dA⋅et+∫0ta1ηet−ηdη≐a2t.
Moreover,(37)∫0t∫Ω∇T2dAdη≤F1t,where F1t=∫0ta1η+a2ηdη+∫ΩT02dA.Lemma 5.
Letu,v be the solution of (9)–(11) with u0,v0,T0∈L2Ω. Then,(38)∫Ωu2dA+∫Ωv2dA≤F2t,∫0t∫Ω∇u2dAdη+∫0t∫Ω∇v2dAdη≤F3t,where F2t and F3t will be defined later.Proof.
Taking the inner product of equation (9)1 with u, in L2Ω, the following is obtained:(39)12ddt∫Ωu2dA+∫Ω∇u2dA=∫01∂u∂x2x2=0ux1,0,tdx1−∫ΩuvdA+∫Ω∫x20∂∂x1Tx1,ζ,tdζudA.
A functionS1x1,x2,t is defined as(40)∂S1∂x2=α1x2+hhg1x1,t.
Obviously,S1 has the same boundary conditions of u. Therefore,(41)∫01∂u∂x2x2=0ux1,0,tdx1=∫01∂S1∂x2x2=0ux1,0,tdx1=∫Ω∂∂x2∂S1∂x2udA=∫Ω∂2S1∂x22udA+∫Ω∂S1∂x2∂u∂x2dA≤12∫Ω∂2S1∂x222dA+12∫Ωu2dA+12∫Ω∂S1∂x22dA+12∫Ω∂u∂x22dA.
By the Cauchy–Schwarz inequality, the following is obtained:(42)∫Ω∫x20∂∂x1Tx1,ζ,tdζudA≤h∫Ω∂T∂x12dA1/2∫Ωu2dA1/2≤h2∫Ω∂T∂x12dA+h2∫Ωu2dA.
By the above results, the following is obtained:(43)ddt∫Ωu2dA+∫Ω∇u2dA≤−2∫ΩuvdA+1+h∫Ωu2dA+h∫Ω∂T∂x12dA+∫Ω∂2S1∂x222dA+∫Ω∂S1∂x22dA.
Similarly, from (9)2,(44)ddt∫Ωv2dA+∫Ω∇v2dA≤2∫ΩuvdA+∫Ωv2dA+∫Ω∂2S2∂x222dA+∫Ω∂S2∂x22dA,where(45)∂S2∂x2=α1x2+hhg2x1,t.
Combining (43) and (44), the following is obtained:(46)ddt∫Ωu2dA+∫Ωv2dA+∫Ω∇u2dA+∫Ω∇v2dA≤1+h∫Ωu2dA+∫Ωv2dA+h∫Ω∂T∂x12dA+a3t,where a3t=∫Ω∂2S1/∂x222dA+∫Ω∂S1/∂x22dA+∫Ω∂2S2/∂x222dA+∫Ω∂S2/∂x22dA. By the Gronwall inequality and Lemma 2, one gets(47)∫Ωu2dA+∫Ωv2dA≤F2t,where F2t=he1+ht∫Ωu02dA+∫Ωv02dA+F1t+∫0ta3ηe1+ht−ηdη. Moreover,(48)∫0t∫Ω∇u2dAdη+∫0t∫Ω∇v2dAdη≤F3t,where F3t=1+h∫0tF2ηdη+hF1t+∫0ta3ηdη+∫Ωu02dA+∫Ωv02dA.Lemma 6.
IfT0,T∗∈L∞Ω, then the solution T of (2.4)3 satisfies(49)supΩT≤Tm,where Tm=supΩT∗∞,T0∞.Proof.
Multiplying (2.4)3 by Tp−1 and integrating by parts, one gets(50)1pddt∫ΩTpdA+p−1p2∫Ω∇Tp/22dA=∫01∂T∂x2x2=0Tp−1x1,0,tdx1.
By the Hölder inequality and the Cauchy–Schwarz inequality, the following is obtained:(51)∫01∂T∂x2x2=0Tp−1x1,0,tdx=∫01∂H∂x2Tp−1dx1=∫Ω∂∂x2∂H∂x2Tp−1dA=∫Ω∂2H∂x22Tp−1dA+p−1∫Ω∂H∂x2Tp−2∂T∂x2dA≤1p∫Ω∂2H∂x22pdA+p−1p∫ΩTpdA+2p−1p∫Ω∂H∂x2pdA+p−1p−2p∫ΩTpdA+p−1p2∫Ω∇Tp/22dA.
Therefore, from (50) and (51), one has(52)ddt∫ΩTpdA≤p−12∫ΩTpdA+2p−1∫Ω∂H∂x2pdA+p−1p−2∫Ω∂2H∂x22pdA.
By the Gronwall inequality, the following is obtained:(53)∫ΩTpdA≤∫ΩT0pdA⋅ep−12t+∫0tep−12t−η⋅2p−1∫Ω∂H∂x2pdA+p−1p−2∫Ω∂2H∂x22pdAdη.
Therefore,(54)∫ΩTpdA1/p≤∫ΩT0pdA⋅ep−12t+∫0tep−12t−η⋅2p−1∫Ω∂H∂x2pdA+p−1p−2∫Ω∂2H∂x22pdAdη1/p.
Letting nowp⟶∞ in (54), one obtains(55)supΩT≤Tm.Lemma 7.
Letu,v be the solution of (9)–(11) with u0,v0,T0∈H1Ω. Then,(56)∫Ω∂u∂x22dA+∫Ω∂v∂x22dA+∫0t∫Ω∇∂u∂x22dAdη+∫0t∫Ω∇∂v∂x22dAdη≤F5t.
Moreover,(57)∫0t∫Ω∂u∂x24dAdη+∫0t∫Ω∂v∂x24dAdη≤F6t,where F5t and F6t will be defined later.Proof.
Using (2.4)1, one starts with(58)∫0t∫Ω∂∂x2∂u∂t+u∂u∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂u∂x2−v+∂ps∂x1−∫x20∂∂x1Tx1,ζ,tdζ−Δu∂u∂x2dAdη=0.
Integrating by parts, the following is obtained:(59)12∫Ω∂u∂x22dA+∫0t∫Ω∇∂u∂x22dAdη=∫0t∫01∂u∂x2∂2u∂x22dx1dη+12∫Ω∂u0∂x22dA+∫0t∫Ω∂v∂x2∂u∂x2dAdη−∫0t∫Ω∂T∂x1∂u∂x2dAdη.
By Lemmas4 and 5 and the Hölder inequality, one has(60)−∫0t∫Ω∂T∂x1∂u∂x2dAdη≤∫0t∫Ω∂T∂x12dAdη1/2∫0t∫Ω∂u∂x22dAdη1/2≤F1tF3t.
By (40) and using the divergence theorem,(61)∫0t∫01∂u∂x2∂2u∂x22dx1dη=∫0t∫Ω∂∂x2∂S1∂x2∂2u∂x22dAdη=∫0t∫Ω∂2S1∂x22∂2u∂x22dAdη+∫0t∫Ω∂S1∂x2∂3u∂x23dAdη≤12∫0t∫Ω∂2S1∂x222dAdη+12∫0t∫Ω∂2u∂x222dAdη+∫0t∫Ω∂S1∂x2∂∂x2∂u∂t+u∂u∂x1−∫−hx2∂∂x1ux1,ζ,tdζ∂u∂x2−v+∂ps∂x1−∫x20∂∂x1Tx1,ζ,tdζ−∂2∂x12udAdη=12∫0t∫Ω∂2S1∂x222dAdη+12∫0t∫Ω∇∂u∂x22dAdη+∫Ω∂S1∂x2∂u∂x2dA+∫0t∫Ω∂S1t∂x2∂u∂x2dAdη+∫0t∫Ω∂S1∂x2∂v∂x2dAdη+∫0t∫Ω∂S1∂x2u∂2u∂x1∂x2−∫−hx2∂∂x1ux1,ζ,ηdζ∂2u∂x22dAdη+∫0t∫Ω∂S1∂x2∂T∂x1dAdη+∫0t∫Ω∂2S1∂x1∂x2∂2u∂x1∂x2dAdη.
By the Cauchy–Schwarz inequality and Lemmas4 and 5, the following is obtained:(62)∫Ω∂S1∂x2∂u∂x2dA≤∫Ω∂S1∂x22dAdη+14∫Ω∂u∂x22dA,∫0t∫Ω∂S1t∂x2∂u∂x2dAdη≤∫0t∫Ω∂S1t∂x22dAdη∫0t∫Ω∂u∂x22dAdη≤F3t∫0t∫Ω∂S1t∂x22dAdη,∫0t∫Ω∂S1∂x2∂v∂x2dAdη≤∫0t∫Ω∂S1∂x22dAdη∫0t∫Ω∂v∂x22dAdη≤F3t∫0t∫Ω∂S1∂x22dAdη,∫0t∫Ω∂S1∂x2∂T∂x1dAdη≤∫0t∫Ω∂S1∂x22dAdη∫0t∫Ω∂T∂x12dAdη≤F1t∫0t∫Ω∂S1∂x22dAdη,∫0t∫Ω∂2S1∂x1∂x2∂2u∂x1∂x2dAdη≤2∫0t∫Ω∂2S1∂x1∂x22dAdη+18∫0t∫Ω∇∂u∂x22dAdη,and by (13) with δ=1 and Lemma 5,(63)∫0t∫Ω∂S1∂x2u∂2u∂x1x2−∫−hx2∂∂x1ux1,ζ,ηdζ∂2u∂x22dAdη=−∫0t∫Ω∂2S1∂x1∂x2u∂u∂x2dAdη+∫0t∫Ω∂2S1∂x22∫−hx2∂∂x1ux1,ζ,ηdζ∂u∂x2dAdη≤∫0t∫Ω∂u∂x22dA1/2∫0t∫Ωu4dA1/2∫Ω∂2S1∂x1∂x24dA1/2dη1/2+∫0t∫Ω∫−hx2∂∂x1ux1,ζ,ηdζ2dA1/2∫0t∫Ω∂2S1∂x224dA1/2∫Ω∂u∂x24dA1/2dη1/2≤maxt∫Ω∂2S1∂x1∂x24dA4CF3t∫0t∫Ωu2dA+∫Ω∇u2dAdη1/2+hπmaxt∫Ω∂2S1∂x224dA4CF3t∫0t∫Ω∂u∂x22dA+∫Ω∇∂u∂x22dAdη1/2≤maxt∫Ω∂2S1∂x1∂x24dA4CF3t∫0tF2ηdη+F3t1/2+2h2π2maxt∫Ω∂2S1∂x224dACF3t+18F3t+∫0t∫Ω∇∂u∂x22dAdη.
Inserting the above two inequalities into (61), one obtains(64)∫0t∫01∂u∂x2∂2u∂x22dx1dη≤14∫Ω∂u∂x22dA+34∫0t∫Ω∇∂u∂x22dAdη+a4t,where(65)a4t=12∫0t∫Ω∂2S1∂x222dAdη+∫Ω∂S1∂x22dAdη+F3t∫0t∫Ω∂S1t∂x22dAdη+F3t∫0t∫Ω∂S1∂x22dAdη+F1t∫0t∫Ω∂S1∂x22dAdη+2∫0t∫Ω∂2S1∂x1∂x22dAdη+maxt∫Ω∂2S1∂x1∂x24dA4CF3t∫0tF2ηdη+F3t1/2+2h2π2maxt∫Ω∂2S1∂x224dACF3t+18F3t+12∫Ω∂u0∂x22dA.
Combining the inequalities (59), (60), and (64), the following is obtained:(66)∫Ω∂u∂x22dA+∫0t∫Ω∇∂u∂x22dAdη≤4∫0t∫Ω∂v∂x2∂u∂x2dAdη+a5t,where a5t=4a4t+4F1tF3t. Similar to (58) and (59), by (2.4)2, one has(67)12∫Ω∂v∂x22dA+∫Ω∇∂v∂x22dAdη=∫0t∫01∂v∂x2∂2v∂x22dx1dη−∫Ω∂v∂x2∂u∂x2dAdη+12∫Ω∂v0∂x22dA+∫Ω∂u∂x2∂v∂x1−∂u∂x1∂v∂x2∂v∂x2dA−∫Ωu∂2v∂x1x2−∫−hx2∂∂x1ux1,ζ,tdζ∂2v∂x22∂v∂x2dAdη.
In a similar way,(68)∫Ω∂v∂x22dA+∫0t∫Ω∇∂v∂x22dAdη≤−4∫0t∫Ω∂v∂x2∂u∂x2dAdη+a6t,for some computable positive function a6t. Combining (66) and (67), one obtains(69)∫Ω∂u∂x22dA+∫Ω∂v∂x22dA+∫0t∫Ω∇∂u∂x22dAdη+∫0t∫Ω∇∂v∂x22dAdη≤F5t,where F5t=a5t+a6t. Using Lemma 3 with δ=1, Lemma 5, and (69),(70)∫0t∫Ω∂u∂x24dAdη+∫0t∫Ω∂v∂x24dAdη≤C∫0t∫Ω∂u∂x22dAdη+∫0t∫Ω∂v∂x22dAdη+∫0t∫Ω∇∂u∂x22dAdη+∫0t∫Ω∇∂v∂x22dAdη2≤CF3t+F5t2≐F6t.


## 4. Convergence Results on Boundary Conditions

Supposingu∗,v∗,T∗ also be the solutions of (9)–(11) with the boundary conditions g1x1,t=g2x1,t=T∗x1,t=0. Let(71)u˜=u−u∗,v˜=v−v∗,T˜=T−T∗,πs=ps−ps∗,then u˜,v˜,T˜ satisfies(72)∂u˜∂t+u˜∂u∂x1−∫−hx2∂∂x1u˜x1,ζ,tdζ∂u∂x2+u∗∂u˜∂x1−∫−hx2∂∂x1u∗x1,ζ,tdζ∂u˜∂x2−v˜+∂πs∂x1−∫x20∂∂x1T˜x1,ζ,tdζ=Δu˜,∂v˜∂t+u˜∂v∂x1−∫−hx2∂∂x1u˜x1,ζ,tdζ∂v∂x2+u∗∂v˜∂x1−∫−hx2∂∂x1u∗x1,ζ,tdζ∂v˜∂x2−u˜=Δv˜,∂T˜∂t+u˜∂T∂x1−∫−hx2∂∂x1u˜x1,ζ,tdζ∂T∂x2+u∗∂T˜∂x1−∫−hx2∂∂x1u∗x1,ζ,tdζ∂T˜∂x2=ΔT˜,with the following boundary conditions:(73)∂u˜∂x2x2=0=α1g1x1,t,∂v˜∂x2x2=0=α2g2x1,t,∂u˜∂x2x2=−h=∂v˜∂x2x2=−h=0,u˜,v˜Γs=0,∂T˜∂x2x2=0=−βT∗x1,t,∂T˜∂x2x2=−h=0,∂T˜∂x1Γs=0,and the initial conditions:(74)u˜,v˜,T˜t=0=0,0,0.The following theorem is obtained.Theorem 1.
Letu˜,v˜,T˜ be the solutions of (71)–(73) with g1,g2,T∗∈H1Ω, T∗,T0∈L∞Ω, and u0,v0∈L2Ω. Then, u˜,v˜,T˜ satisfies the inequality(75)3h2π2Tm2∫Ωu˜2dA+3h2π2Tm2∫Ωv˜2dA+h2π2Tm2∫0t∫Ω∇u˜2dAdη+3h2π2Tm2∫0t∫Ω∇v˜2dAdη+∫ΩT˜2dA+12∫0t∫Ω∇T˜2dAdη≤∫0t∫013h2α1π2Tm2g12x1,η+3h2α2π2Tm2g22x1,η+βT∗2x1,ηdx1dη+bt∫0t∫0s∫01e∫stbηdη3h2α1π2Tm2g12x1,η+3h2α2π2Tm2g22x1,η+βT∗2x1,ηdx1dηds,which are the convergence results on boundary conditions g1,g2,T∗.Proof.
Now taking the inner product of equation (72)1 with u˜, in L2Ω, the following is obtained:(76)12∫Ωu˜2dA+∫0t∫Ω∇u˜2dAdη=∫0t∫Ωu˜v˜dAdη−∫0t∫Ω∂πs∂x1u˜dAdη+∫0t∫01∂u˜∂x2u˜dx1dη+∫0t∫Ω∫x20∂∂x1T˜x1,ζ,ηdζu˜dAdη−∫0t∫Ωu˜∂u∂x1−∫−hx2∂∂x1u˜x1,ζ,ηdζ∂u∂x2u˜dAdη.
By the Hölder inequality, Lemmas1, 5, and 7, and the A-G mean inequality, the following is obtained:(77)−∫0t∫Ωu˜∂u∂x1−∫−hx2∂∂x1u˜x1,ζ,ηdζ∂u∂x2u˜dAdη≤∫0t∫Ω∂u∂x12dA1/2∫0t∫Ωu˜4dAdη1/2+∫0t∫Ω∫−hx2∂∂x1u˜x1,ζ,ηdζ2dAdη1/2∫0t∫Ω∂u∂x24dAdη1/2∫0t∫Ωu˜4dAdη1/4≤F3tC∫0t∫Ωu˜2dAdη+δ1∫0t∫Ω∇u˜2dAdη+ChπF6t4∫0t∫Ω∂u˜∂x12dAdη1/2∫0t∫Ωu˜2dAdη+δ1∫0t∫Ω∇u˜2dAdη1/2≤F3tC∫0t∫Ωu˜2dAdη+δ1∫0t∫Ω∇u˜2dAdη+ChπF6t4∫0t∫Ω∂u˜∂x12dAdη1/2∫0t∫Ωu˜2dAdη1/2+Cδ1hπF6t4∫0t∫Ω∂u˜∂x12dAdη1/2∫0t∫Ω∇u˜2dAdη1/2≤F3tC+Ch24δ2π2F6t∫0t∫Ωu˜2dAdη+F3tCδ1+δ2+Cδ1hπF6t4∫0t∫Ω∇u˜2dAdη,for arbitrary positive constants δ1 and δ2. By the Cauchy–Schwarz inequality again(78)∫0t∫Ω∫x20∂∂x1T˜x1,ζ,ηdζu˜dAdη≤π212h2Tm2∫0t∫Ω∂∂x1T˜2dAdη+3h4π2Tm2∫0t∫Ωu˜2dAdη,(79)∫0t∫01∂u˜∂x2u˜dx1dη=α1∫0t∫01g1x1,tu˜x1,0,tdx1dη≤α12∫0t∫01g12x1,tdx1dη+α12∫0t∫01u˜2x1,0,tdx1dη.
In view of (25), the following is obtained:(80)∫0t∫01∂u˜∂x2u˜dx1dη≤α12∫0t∫01g12x1,tdx1dη+m3α12∫0t∫Ωu˜2dAdη+m4α12∫0t∫Ωu˜∂u˜∂x1dAdη≤α12∫0t∫01g12dx1dη+2m3+m4δ3−1α14∫0t∫Ωu˜2dAdη+m4α14δ3∫0t∫Ω∇u˜2dAdη.
Inserting (77)–(80) into (76), one has(81)12∫Ωu˜2dA+∫0t∫Ω∇u˜2dAdη≤∫0t∫Ωu˜v˜dAdη+α12∫0t∫01g12dx1dη+π212h2Tm2∫0t∫Ω∇T˜2dAdη+F3tC+Ch24δ2π2F6t+2m3+m4δ3−1α14+3h4π2Tm2∫0t∫Ωu˜2dAdη+F3tCδ1+δ2+Cδ1hπF6t4+m4α14δ3∫0t∫Ω∇u˜2dAdη.
Now taking the inner product of the second equation of (72) with v˜, the following is obtained:(82)12∫Ωv˜2dA+∫0t∫Ω∇v˜2dAdη=−∫0t∫Ωu˜v˜dAdη+∫0t∫01∂v˜∂x2v˜dx1dη−∫0t∫Ωu˜∂v∂x1−∫−hx2∂∂x1u˜x1,ζ,ηdζ∂v∂x2v˜dAdη.
Computing as (80),(83)∫0t∫01∂v˜∂x2v˜dx1dη≤α22∫0t∫01g22dx1dη+2m3+m4δ4−1α24∫0t∫Ωv˜2dAdη+m4α24δ4∫0t∫Ω∇v˜2dAdη.
Computing as (77),(84)−∫0t∫Ωu˜∂v∂x1−∫−hx2∂∂x1u˜x1,ζ,ηdζ∂v∂x2v˜dAdη≤CF3t2∫0t∫Ωu˜2dAdη+δ5∫0t∫Ω∇u˜2dAdη+CF3t2∫0t∫Ωv˜2dAdη+δ5∫0t∫Ω∇v˜2dAdη+CF6thδ6π∫0t∫Ωv˜2dAdη+δ6∫0t∫Ω∇u˜2dAdη+δ6∫0t∫Ω∇v˜2dAdη,where δ5andδ6 are the arbitrary constants. Inserting (83) and (84) into (82), the following is obtained:(85)12∫Ωv˜2dA+∫0t∫Ω∇v˜2dAdη≤α22∫0t∫01g22dx1dη+CF3t2∫0t∫Ωu˜2dAdη+2m3+m4δ4−1α24+CF3t2+CF6thδ6π∫0t∫Ωv˜2dAdη+m4α24δ4+CF3t2δ5+Cδ6F6thπ∫0t∫Ω∇v˜2dAdη+Cδ6F6thπ∫0t∫Ω∇u˜2dAdη.
(81) and (85) are combined and suitable δi,i=1,2,…,6 are chosen such that(86)F3tCδ1+δ2+Cδ1hπF6t4+m4α14δ3+Cδ6F6thπ=12,m4α24δ4+CF3t2δ5+Cδ6F6thπ=12,to have(87)∫Ωu˜2dA+∫Ωv˜2dA+∫0t∫Ω∇u˜2dAdη+∫0t∫Ω∇v˜2dAdη≤α1∫0t∫01g12dx1dη+α2∫0t∫01g22dx1dη+π26h2Tm2∫0t∫Ω∇T˜2dAdη+2F3tC+Ch22δ2π2F6t+2m3+m4δ3−1α12+CF3t+6h4π2Tm2∫0t∫Ωu˜2dAdη+2m3+m4δ4−1α22+CF3t+2CF6thδ6π∫0t∫Ωv˜2dAdη.
The inner product of equation (72)3 is taken with T˜ to have(88)12∫ΩT˜2dA+∫0t∫ΩT2dAdη=∫0t∫01∂T˜∂x2T˜dx1dη−∫0t∫Ωu˜∂T∂x1−∫−hx2∂∂x1u˜x1,ζ,ηdζ∂T∂x2T˜dAdη.
Similar to the computation in (79) and (80), the following is obtained:(89)∫0t∫01∂T˜∂x2T˜dx1dη≤β2∫0t∫01T∗2dx1dη+2m3+m4δ7−1β4∫0t∫ΩT˜2dAdη+m4β4δ7∫0t∫Ω∇T˜2dAdη,for an arbitrary positive constant δ7. On integrating by parts, one has(90)−∫0t∫Ωu∗∂T˜∂x1−∫−hx2∂∂x1u∗x1,ζ,ηdζ∂T˜∂x2T˜dAdη=0.
By the Hölder inequality and Lemmas2 and 6, the following is obtained:(91)−∫0t∫Ωu˜∂T∂x1−∫−hx2∂∂x1u˜x1,ζ,ηdζ∂T∂x2T˜dAdη=∫0t∫Ωu˜T∂T˜∂x1dAdη−∫0t∫Ω∫−hx2∂∂x1u˜x1,ζ,ηdζT∂T˜∂x2dAdη≤Tm∫0t∫Ω∂T˜∂x12dAdη1/2∫0t∫Ωu˜2dAdη1/2+Tm∫0t∫Ω∫−hx2∂∂x1u˜x1,ζ,ηdζ2dAdη1/2∫0t∫Ω∂T˜∂x22dAdη1/2≤Tm2∫0t∫Ωu˜2dAdη+h2π2Tm2∫0t∫Ω∇u˜2dAdη+14∫0t∫Ω∇T˜2dAdη.
Inserting (89)–(91) into (88) and choosing δ7=1/m4β, one gets(92)∫ΩT˜2dA+∫0t∫Ω∇T˜2dAdη≤β∫0t∫01T∗2dx1dη+2m3+m4δ7−1β2∫0t∫ΩT˜2dAdη+2Tm2∫0t∫Ωu˜2dAdη+2h2π2Tm2∫0t∫Ω∇u˜2dAdη.
Combining (87) and (92), the following is obtained:(93)3h2π2Tm2∫Ωu˜2dA+3h2π2Tm2∫Ωv˜2dA+h2π2Tm2∫0t∫Ω∇u˜2dAdη+3h2π2Tm2∫0t∫Ω∇v˜2dAdη+∫ΩT˜2dA+12∫0t∫Ω∇T˜2dAdη≤∫0t∫013h2α1π2Tm2g12+3h2α2π2Tm2g22+βT∗2dx1dη+bt3h2π2Tm2∫0t∫Ωu˜2dAdη+3h2π2Tm2∫0t∫Ωv˜2dAdη+∫0t∫ΩT˜2dAdη,where(94)bt=max2F3tC+Ch22δ2π2F6t+2m3+m4δ3−1α12+CF3t+6h4π2Tm2+2π23h2,2m3+m4δ4−1α22+CF3t+2CF6thδ6π,2m3+m4δ7−1β2.
Therefore, by the Gronwall inequality,(95)3h2π2Tm2∫0t∫Ωu˜2dAdη+3h2π2Tm2∫0t∫Ωv˜2dAdη+∫0t∫ΩT˜2dAdη≤∫0t∫0s∫01e∫stbηdη3h2α1π2Tm2g12x1,η+3h2α2π2Tm2g22x1,η+βT∗2x1,ηdx1dηds.
Then, returning to (93) it can be seen that(96)3h2π2Tm2∫Ωu˜2dA+3h2π2Tm2∫Ωv˜2dA+h2π2Tm2∫0t∫Ω∇u˜2dAdη+3h2π2Tm2∫0t∫Ω∇v˜2dAdη+∫ΩT˜2dA+12∫0t∫Ω∇T˜2dAdη≤∫0t∫013h2α1π2Tm2g12x1,η+3h2α2π2Tm2g22x1,η+βT∗2x1,ηdx1dη+bt∫0t∫0s∫01e∫stbηdη3h2α1π2Tm2g12x1,η+3h2α2π2Tm2g22x1,η+βT∗2x1,ηdx1dηds.
It is obvious thatu⟶u∗,v⟶v∗,andT⟶T∗ in L2Ω, as well as H1Ω, when g1,g2,T∗⟶0 from inequality (96).



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*Source: 6919725-2020-07-18.xml*