# Boundary Layer Effects for the Nonlinear Evolution Equations with the Vanishing Diffusion Limit

Authors: Linrui Li; Shu Wang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We study an initial-boundary value problem for a nonlinear evolution system with damping and diffusion; our main purpose is to investigate the boundary layer effects when the vanishing diffusion limitα→0+, especially for the mixed boundary conditions; we prove that the thickness of layer is of the order O(α). Furthermore, the corresponding convergence rates are also obtained.


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## Body

## 1. Introduction and Main Results

The main objective of this article is to study the nonlinear evolution equations with damping and diffusion on the domain[0,1]×[0,∞):(1)ψtα=-k-1αψα-kαθxα+αψxxα,θtα=-1-βθα+μαψxα+2ψαθxα+βθxxα,0<x<1,t>0with initial data(2)ψα,θαx,0=ψ0,θ0x,0≤x≤1,and the mixed boundary conditions, that is, Neumann-Dirichlet boundary conditions(3)ψxα,θα0,t=ψxα,θα1,t=0,0,t≥0,which implies that(4)ψxtα,θtα0,t=ψxtα,θtα1,t=0,0,t≥0,where k, α, β, and μ are positive constants with k>1 and 0<β<1. The limit problem of the vanishing parameter α→0+; we have(5)ψt0=0,θt0=-1-βθ0+2ψ0θx0+βθxx0,0<x<1,t>0with the initial conditions(6)ψ0,θ0x,0=ϕ1x,ϕ2x,0≤x≤1,and the Dirichlet boundary conditions(7)θ00,t=θ01,t=0,0,t≥0,which implies that(8)θt00,t=θt01,t=0,2ψ0θx0+βθxx00,t=2ψ0θx0+βθxx01,t=0,t≥0.We expect to prove that the solution (ψα,θα) of (1)–(3) converges to the solution (ψ0,θ0) of the limit problem (5)–(7), as the diffusion parameter α→0+, and get the boundary layer thickness.The nonlinear interaction between ellipticity and dissipation is involved quite broadly in physical and mechanical systems, such as Rayleigh-Benard problem, superposed fluids, Taylor-Couette instability, dynamic phase transitions, and fluid flow down an inclined plane. [1–3]. As these systems are usually quite complicated, they are far from being well understood. System (1) was originally proposed by Hsieh in [2] to investigate the nonlinear interaction between ellipticity and dissipation. We also refer to [1–3] for the physical background of (1). This study is expected to yield insights into physical systems with similar mechanism, such as the Ginzburg-Landau equation and the Kuramoto-Sivashinsky equation [1, 3, 4]; it may also help us, by comparison, to understand better the nonlinear interaction between other instabilities and dissipation.System (1) has been extensively studied by several authors in different contexts [2, 3, 5–9], such as the well-posedness problems, the nonlinear stability problem, and optimal decay rate of the solutions. However, all the results above need to assume that all parameters are fixed constants. Another interesting problem is the zero diffusion limit: that is, consider the limit problem of solution sequences when one or more of parameters vanishes for the corresponding Cauchy problem or initial-boundary value problem [5–7, 9]. It should be emphasized that flows often move in bounded domains with constraints from boundaries in real world; in the bounded domains case, the boundary effect requires a careful mathematical analysis. Ruan and Zhu [5] consider the initial-boundary value problem (1) with the zero Dirichlet boundary conditions when diffusion parameter β→0+ and they show that the boundary layer thickness δ(β) is of the order O(βγ) with 0<γ<1/2. Subsequently, Peng et al. [6] consider the initial-boundary value problem (1) with the Dirichlet-Neumann boundary conditions when diffusion parameter β→0+ and they show that the boundary layer thickness δ(β) is of the order O(βγ) with 0<γ<3/4. For the parameter α→0+, Chen and Zhu in [10] consider the Cauchy problem for (1); Peng in [7] study the initial-boundary value problem for (1) with the zero Dirichlet boundary conditions and obtain the thickness of layer of the order O(α1/2). However, for the asymptotic limit α→0+, all these results are restricted to the boundary conditions to the Dirichlet boundary conditions. By [6], which motivates our investigation in this paper, we concentrate our efforts to investigate the boundary layer effect and the convergence rates on the initial-boundary value (1)–(3) with the Dirichlet-Neumann boundary conditions when diffusion parameter α→0+, and the solutions of the mixed boundary conditions usually exhibit different behaviors and much rich phenomena comparing with the general boundary conditions.For later presentation, next we state the notation as follows.Notations. Throughout this paper, we denote positive constant independent of α by C. And the constant C may vary from line to line. L2=L2([0,1]) and L∞=L∞([0,1]) denote the usual Lebesgue space in [0,1] with its norms fL2([0,1])=f=(∫01fx2dx)1/2 and fL∞=supx∈[0,1]fx. Hl([0,1]) denotes the usual l-th order Sobolev space with its norm fHl([0,1])=fl=(∑i=0l∂xif2)1/2. For simplicity, f·,tL2, f·,tL∞ and fl, respectively.Before we state the main results, let us recall the definition of boundary layer thickness (BL-thickness) in [11] as follows.Definition 1.
A positive functionδ(α) is called a boundary layer thickness for problem (1)–(3) with vanishing diffusion α, if δ(α)→0 as α→0, and(9)limα→0θα-θ0L∞0,T;L∞0,1=0,(10)limα→0ψα-ψ0L∞0,T;L∞δ,1-δ=0,(11)liminfα→0ψα-ψ0L∞0,T;L∞0,1>0,where 0<δ=δ(β)<1, (ψα,θα), and (ψ0,θ0) is the solution to problem (1)–(3) and the limit problem (5)–(7), respectively.Our first purpose is to show that the initial-boundary value problem (1)–(3) admits a unique global smooth solution (ψα,θα).Theorem 2.
Suppose that the initial data satisfy the conditions:(ψ0,θ0)∈H2, (ψ0,θ0)(0)=(ψ0,θ0)(1)=(0,0) and ψ01+θ01 is sufficiently small, then there exists a unique solution (ψα,θα) to the initial-boundary value problem (1)–(3) satisfying(12)θα∈L∞0,T;H2∩L20,T;H2,θtα∈L∞0,T;H2∩L20,T;H1,where the norms are all uniform in α.Then our second result shows that the initial-boundary value problem (5)–(7) admits a unique global smooth solution (ψ0,θ0).Theorem 3.
Suppose that the initial data satisfy the conditions:(ϕ1,ϕ2)∈H2, ϕ2(0)=ϕ2(1)=0≤ε, and ε is sufficiently small, then there exists a unique solution (ψ0,θ0) to the initial-boundary value problem (5)–(7) satisfying (13)ψ0∈L∞0,T;H2,θ0∈L∞0,T;H2∩L20,T;H2,θt0∈L∞0,T;L2∩L20,T;H1.Furthermore, we give the convergence rates and boundary layer thickness.Theorem 4.
Under the assumptions of Theorems2 and 3, suppose that (14)ψ0x-ϕ1x12≤Cα,θ0x-ϕ2x12≤Cα;then any function δ(α) satisfying the conditions δ(α)→0 and α/δ(α)→0 as α→0+ is a BL-thickness such that(15)θα-θ0L∞0,T;L∞0,1≤Cα,ψα-ψ0L∞0,T;L∞δ,1-δ≤Cαδα,liminfα→0ψα-ψ0L∞0,T;L∞0,1>0.Consequently,(16)limα→0θα-θ0L∞0,T;L∞0,1=0,limα→0ψα-ψ0L∞0,T;L∞δ,1-δ=0.The plan of this paper is as follows. Section2 is devoted to the global existence results on the initial-boundary value problem (1)–(3) and the local existence for the limit problem (5)–(7). Section 3 details convergence rates and the BL-thickness as the diffusion parameter α→0+ for the mixed boundary conditions.


## 2. Proof of Theorems2 and 3

### 2.1. Proof of Theorem2

In this subsection, we shall prove Theorem2 by adapting the elaborate nonlinear energy method. In the following, we devote ourselves to the a priori estimate of solution (ψα(x,t),θα(x,t)) of (1)–(3) under the a priori assumption(17)N1t=sup0<t<Tψα,θαt12≤ε12,where ε1 is a positive constant satisfying 0<ε1≤1, independent of α.By Sobolev inequality, we have(18)ψα,θαtL∞≤ε1.


#### 2.1.1.L2-Estimates

Lemma 5.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(19)12∫01ψα2+θα2dx+∫0t∫01k-1αψα2+1-βθα2+ψxα2+θxα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (1) by ψα and θα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(20)12ddt∫01ψα2+θα2dx+∫01k-1αψα2+1-βθα2dx+α∫01ψxα2dx+β∫01θxα2dx=k+μα∫01ψxαθαdx-∫01ψxαθα2dx.Integrating (20) over (0,t), using Cauchy-Schwarz inequality and (18), we obtain for any p1>0(21)12∫01ψα2+θα2dx+∫0t∫01k-1αψα2+1-βθα2dxdτ+α∫0t∫01ψxα2dxdτ+β∫01θxα2dx=12∫01ψα2+θα2t=0dx+k+μα∫0t∫01ψxαθαdxdτ-∫0t∫01ψxαθα2dxdτ=12∫01ψα2+θα2t=0dx+∫0tk+μα+θαtL∞∫01ψxαθαdxdτ≤12∫01ψ02+θ02dx+p1∫0t∫01ψxα2dxdτ+k+μα+Cε124p1∫0t∫01θα2dxdτ.If we choose p1>0 such that(22)1-β-k+μα+Cε124p1>0,α-p1>0,then we have(23)12∫01ψα2+θα2dx+∫0t∫01ψα2+θα2+ψxα2+θxα2dxdτ≤Cψ0,θ02.


#### 2.1.2. High-Order Energy Estimates

Lemma 6.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(24)12∫01ψxα2+θxα2dx+∫0t∫01ψxα2+θxα2+ψxxα2+βθxxα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (1) by -ψxxα and -θxxα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(25)12ddt∫01ψxα2+θxα2dx+∫01k-1αψxα2+1-βθxα2dxdτ+α∫01ψxxα2dx+β∫01θxxα2dx=k+μα∫01ψxxαθxαdx+∫01ψxαθxα2dx.Integrating (25) over (0,t), using Cauchy-Schwarz inequality and (17)-(18), we obtain for any p1>0(26)12∫01ψxα2+θxα2dx+∫0t∫01k-1αψxα2+1-βθxα2dx+α∫0t∫01ψxxα2dxdτ+β∫0t∫01θxxα2dxdτ=12∫01ψxα2+θxα2t=0dx+k+μα∫0t∫01ψxxαθxαdxdτ+∫0t∫01ψxαθxα2dxdτ≤12∫01ψ0x2+θ0x2dx+p1∫0t∫01ψxxα2dxdτ+k+μ2α2+Cε124p1∫0t∫01θxβ2dxdτ.If we choose p1>0 such that(27)1-β-k+μ2α2+Cε124p1>0,α-p1>0,then we have(28)12∫01ψxα2+θxα2dx+∫0t∫01ψxα2+θxα2+αψxxα2+βθxxα2dxdτ≤Cψ0,θ012.Lemma 7.
Let the assumptions of Theorem2 hold. Then a positive constant C exists independent of α, such that(29)∫01ψtα2+θtα2dx+∫0t∫01ψxtα2dxdτ+θxtα2dxdτ≤C.Proof.
Differentiating (1) with respect to t, we get(30)ψttα=-k-1αψtα-kαθxtα+αψxxtα,θttα=-1-βθtα+μαψxtα+2ψαθxαt+βθxxtα.Multiplying the first and the second equations of (1) by -ψtα and -θtα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(31)12∫01ψtα2+θtα2dx+∫0t∫01k-1αψtα2dxdτ+1-β∫0t∫01θtα2dxdτ+α∫0t∫01ψxtα2dxdτ+β∫01θxtα2dxdτ=k+μα∫0t∫01ψxtαθtαdxdτ+2∫0t∫01ψtαθxαθtαdxdτ-∫0t∫01ψxαθtα2dxdτ=12∫01ψtα2+θtα2t=0dx+∑i=13Ji.We estimate every term as follows:(32)J1≤p1∫0t∫01ψxt2dxdτ+k+μ2α24p1∫0t∫01θtα2dxdτ.Here J2, J3, and J4 are estimated as follows:(33)J2≤Cε1∫0t∫01ψtα2+θtα2dxdτ,J3≤Cε1∫0t∫01θtα2dxdτ.Here we have used Cauchy-Schwarz inequality and Sobolev inequality.
Substituting (32) and (33) into (34), we have from (18) and Lemma 6(34)∫01ψtα2+θtα2dx+∫0t∫01ψtα2+θtα2+ψxtα2+βθxtα2dxdτ≤C.By the similar method, it is easy to obtain(35)∫01ψxxα2+βθxxα2dxdτ≤C.Lemma 8.
Assume that the conditions in Theorem2 hold. Then there exists a positive constant C independent of α, such that(36)∫01θxxα2dx+∫0t∫01θxxα2dxdτ+β∫0t∫01θxxxαdxdτ≤C.Proof.
Differentiating the second equation of (1) with respect to x, we get(37)θxtα=-1-βθxα+μαψxxα+2ψαθxαx+βθxxxα.Multiplying the equation (37) by -θxxxα,(38)∫01θxxα2dx+∫0t∫011-βθxxα2dxdτ+β∫0t∫01θxxxαdxdτ=∫01θ0xx2dx-μ∫0t∫01ψtα+k-1αψα+kαθxαθxxxαdxdτ-2∫0t∫01ψxαθxαθxxxαdxdτ-2∫0t∫01ψαθxxαθxxxαdxdτ=∫01θ0xx2dx+μkα∫0t∫01θxxα2dxdτ+∫0t∫01μψxtα+k-1αψxαθxxαdxdτ+2∫0t∫01ψxxαθxαθxxαdxdτ+3∫0t∫01ψαθxxα2dxdτ=∫01θ0xx2dx+μkα∫0t∫01θxxα2dxdτ+∑i=46Ji.We estimate every term as follows:(39)J4≤C∫0t∫01ψxtα2+ψxα2dxdτ+p1∫0t∫01θxxα2dxdτ,J5≤Cε1∫0t∫01ψxxα2dxdτ+Cε1∫0t∫01θxxα2dxdτ,J6≤Cε1∫0t∫01θxxα2dxdτ.
Substituting (39) into (45), we have(40)∫01θxxα2dx+∫0t∫01θxxα2dxdτ+β∫0t∫01θxxxαdxdτ≤C.
Now we notice that the priori assumption (18) can be closed. Since, under this priori assumption (18), we deduced that (19), (24), (29), and (36) hold provided that ε1 is sufficiently small. Therefore, assumption (18) is always true provided that ψ0,θ0L22 is sufficiently small.Lemma 9.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(41)12∫01ψxtα2dx+θxtα2dx+∫0t∫01ψxtα2+θxtα2dxdτ+α∫0t∫01ψxxtα2dxdτ+β∫0t∫01θxxtα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (30) by -ψxxtα and -θxxtα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(42)12∫01ψxtα2dx+θxtα2dx+∫0t∫01k-1αψxtα2+1-βθxtα2dxdτ+α∫0t∫01ψxxtαdxdτ+β∫0t∫01θxxtαdxdτ=12∫01ψxtα2dx+θxtα2t=0dx+kα∫0t∫01ψxxtαθxtαdxdτ+μα∫0t∫01ψxxtαθxtαdxdτ+2∫0t∫01ψαθxαt-θxxtαdxdτ=12∫01ψxtα2dx+θxtα2t=0dx+k+μα∫0t∫01ψxxtαθxtαdxdτ-2∫0t∫01ψtαθxαθxxtαdxdτ-2∫0t∫01ψαθxtαθxxtαdxdτ=M+2∫0t∫01ψxtαθxαθxtαdxdτ+2∫0t∫01ψtαθxxαθxtαdx+∫0t∫01ψxαθxtα2dxdτ=M+∑i=710Ji.We estimate every term as follows:(43)J7≤p1∫0t∫01ψxxtα2dxdτ+k+μα24p1∫0t∫01θxt2dxdτ,J8≤Cε1∫0t∫01ψxtα2dxdτ+Cε1∫0t∫01θxtα2dxdτ,J9≤2∫0t∫01ψtαθxtα2dxdτ+12∫0t∫01θxxα2dxdτ≤C∫0t∫01ψtα2+ψxtα2dx∫01θxtα2dxdτ+12∫0t∫01θxxα2dxdτ,J10≤Cε1∫0t∫01θxtα2dxdτ.Substituting (43) into (45), we obtain(44)12∫01ψxtα2dx+θxtα2dx+∫0t∫01ψxtα2+θxtα2dxdτ+α∫0t∫01ψxxtα2dxdτ+β∫0t∫01θxxtα2dxdτ≤M.By the similar method, we have(45)∫01ψxxxα2dx+θxxxα2dx+∫0t∫01ψxxxα2+βθxxxα2dxdτ≤M.This completes the proof of Lemma 9.Combination of Lemmas5–9 and a well-known result on the local existence of the solutions, we can get the global existence of the solutions; this proves Theorem 2.


### 2.2. Proof of Theorem3

Suppose a priori assumption:(46)A2T=sup0<t<Tψ0,θ0t32≤ε2,where 0<ε2≪1. We obtain the following energy estimates.Lemma 10.
Suppose that the initial data satisfy the conditions:ϕ1∈H2, ϕ2∈H2, ϕ2(0)=ϕ2(1)=0, and ϕ1L2+ϕ1L2≤ε, and ε is sufficiently small. Then there exists a unique solution (ψ0,θ0) to the initial-boundary value problem (5)–(7) satisfying(47)12∫01θ02dx+∫0t∫01θ02+θx02dxdτ≤Cϕ22,(48)∫01θx02dx+∫0t∫01θx02dxdτ+β∫0t∫01θxx02dxdτ≤Cϕ212,(49)∫01θt02+θxx02dx+∫0t∫01θt02+θxt02dxdτ≤Cϕ12+ϕ222.Proof.
Multiplying the second equation of (5) by θ0, integrating the resulting equation over (0,t)×(0,1), using integration by part and the boundary conditions (7), and then adding them, we arrive at(50)12∫01θ02dx+1-β∫0t∫01θ02dxdτ+β∫0t∫01θx02dxdτ=12∫01ϕ22dx+2∫0t∫01ψ0θx0θ2dxdτ=Cϕ22+2ψ0L∞∫0t∫01θ0θx0dxdτ=Cϕ22+2ε2∫0t∫01θ02dxdτ+Cε∫0t∫01θx02dxdτ.Then we have(51)12∫01θ02dx+∫0t∫01θ02+θx02dxdτ≤Cϕ22.Differentiating the second equation of (5) with respect to x, we get(52)θxt0=-1-βθx0+2ψ0θx0x+βθxxx0.Multiplying (52) by θx0, using integration by part and the boundary conditions, we obtain(53)12∫01θx02dx+1-β∫0t∫01θx02dxdτ+β∫0t∫01θxxα2dxdτ=12∫01θx02t=0dx+2∫0t∫01ψx0θx02dxdτ+2∫0t∫01ψ0θxx0θx0dxdτ≤12∫01ϕ2xxdx+2ψx0L∞∫0t∫01θx02dxdτ+2ψ0L∞∫0t∫01θxx02dxdτ+∫0t∫01θx02dxdτ≤Cϕ212.Therefore, we have(54)∫01θx02dx+∫0t∫01θx02dxdτ+β∫0t∫01θxx02dxdτ≤Cϕ212.Moreover, if we differentiate the second equation of (5) with respect to t, we have(55)θtt0=-1-βθt0+2ψ0θx0t+βθxx0t,and then multiplying (55) by θt0 and integrating the resulting equation over (0,t)×(0,1), using integration by part and the boundary conditions (7), we obtain(56)12∫01θt02dx+1-β∫0t∫01θt02dxdτ+β∫0t∫01θxt02dxdτ=12∫01θ0tdx+2∫0t∫01ψ0θx0tθt0≤12ϕ2t2+2∫0t∫01ψt0θx0+ψ0θxt0dxdτ=12C+2ψt0L∞∫0t∫01θx02dxdτ+2ψ0L∞∫0t∫01θxt02dxdτ;then we have(57)∫01θt02dx≤C.Utilizing the above estimates and the similar produces, by the assumption of ε, we have(58)12∫01θt02dx+1-β∫0t∫01θt02dxdτ+β∫0t∫01θxt02dxdτ≤Cϕ122+ϕ222.Therefore,(59)∫01θxx02dx≤C∫01θt02+θ02+ψ0θx02dx≤C∫01θt02+θ02+ψ02+θx02dx≤Cϕ122+ϕ222.Here we have used (47), (48), (58), and (59). The proof of Lemma 10 is finished.Finally, based on Lemma10, we can prove Theorem 3.


## 2.1. Proof of Theorem2

In this subsection, we shall prove Theorem2 by adapting the elaborate nonlinear energy method. In the following, we devote ourselves to the a priori estimate of solution (ψα(x,t),θα(x,t)) of (1)–(3) under the a priori assumption(17)N1t=sup0<t<Tψα,θαt12≤ε12,where ε1 is a positive constant satisfying 0<ε1≤1, independent of α.By Sobolev inequality, we have(18)ψα,θαtL∞≤ε1.


### 2.1.1.L2-Estimates

Lemma 5.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(19)12∫01ψα2+θα2dx+∫0t∫01k-1αψα2+1-βθα2+ψxα2+θxα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (1) by ψα and θα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(20)12ddt∫01ψα2+θα2dx+∫01k-1αψα2+1-βθα2dx+α∫01ψxα2dx+β∫01θxα2dx=k+μα∫01ψxαθαdx-∫01ψxαθα2dx.Integrating (20) over (0,t), using Cauchy-Schwarz inequality and (18), we obtain for any p1>0(21)12∫01ψα2+θα2dx+∫0t∫01k-1αψα2+1-βθα2dxdτ+α∫0t∫01ψxα2dxdτ+β∫01θxα2dx=12∫01ψα2+θα2t=0dx+k+μα∫0t∫01ψxαθαdxdτ-∫0t∫01ψxαθα2dxdτ=12∫01ψα2+θα2t=0dx+∫0tk+μα+θαtL∞∫01ψxαθαdxdτ≤12∫01ψ02+θ02dx+p1∫0t∫01ψxα2dxdτ+k+μα+Cε124p1∫0t∫01θα2dxdτ.If we choose p1>0 such that(22)1-β-k+μα+Cε124p1>0,α-p1>0,then we have(23)12∫01ψα2+θα2dx+∫0t∫01ψα2+θα2+ψxα2+θxα2dxdτ≤Cψ0,θ02.


### 2.1.2. High-Order Energy Estimates

Lemma 6.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(24)12∫01ψxα2+θxα2dx+∫0t∫01ψxα2+θxα2+ψxxα2+βθxxα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (1) by -ψxxα and -θxxα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(25)12ddt∫01ψxα2+θxα2dx+∫01k-1αψxα2+1-βθxα2dxdτ+α∫01ψxxα2dx+β∫01θxxα2dx=k+μα∫01ψxxαθxαdx+∫01ψxαθxα2dx.Integrating (25) over (0,t), using Cauchy-Schwarz inequality and (17)-(18), we obtain for any p1>0(26)12∫01ψxα2+θxα2dx+∫0t∫01k-1αψxα2+1-βθxα2dx+α∫0t∫01ψxxα2dxdτ+β∫0t∫01θxxα2dxdτ=12∫01ψxα2+θxα2t=0dx+k+μα∫0t∫01ψxxαθxαdxdτ+∫0t∫01ψxαθxα2dxdτ≤12∫01ψ0x2+θ0x2dx+p1∫0t∫01ψxxα2dxdτ+k+μ2α2+Cε124p1∫0t∫01θxβ2dxdτ.If we choose p1>0 such that(27)1-β-k+μ2α2+Cε124p1>0,α-p1>0,then we have(28)12∫01ψxα2+θxα2dx+∫0t∫01ψxα2+θxα2+αψxxα2+βθxxα2dxdτ≤Cψ0,θ012.Lemma 7.
Let the assumptions of Theorem2 hold. Then a positive constant C exists independent of α, such that(29)∫01ψtα2+θtα2dx+∫0t∫01ψxtα2dxdτ+θxtα2dxdτ≤C.Proof.
Differentiating (1) with respect to t, we get(30)ψttα=-k-1αψtα-kαθxtα+αψxxtα,θttα=-1-βθtα+μαψxtα+2ψαθxαt+βθxxtα.Multiplying the first and the second equations of (1) by -ψtα and -θtα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(31)12∫01ψtα2+θtα2dx+∫0t∫01k-1αψtα2dxdτ+1-β∫0t∫01θtα2dxdτ+α∫0t∫01ψxtα2dxdτ+β∫01θxtα2dxdτ=k+μα∫0t∫01ψxtαθtαdxdτ+2∫0t∫01ψtαθxαθtαdxdτ-∫0t∫01ψxαθtα2dxdτ=12∫01ψtα2+θtα2t=0dx+∑i=13Ji.We estimate every term as follows:(32)J1≤p1∫0t∫01ψxt2dxdτ+k+μ2α24p1∫0t∫01θtα2dxdτ.Here J2, J3, and J4 are estimated as follows:(33)J2≤Cε1∫0t∫01ψtα2+θtα2dxdτ,J3≤Cε1∫0t∫01θtα2dxdτ.Here we have used Cauchy-Schwarz inequality and Sobolev inequality.
Substituting (32) and (33) into (34), we have from (18) and Lemma 6(34)∫01ψtα2+θtα2dx+∫0t∫01ψtα2+θtα2+ψxtα2+βθxtα2dxdτ≤C.By the similar method, it is easy to obtain(35)∫01ψxxα2+βθxxα2dxdτ≤C.Lemma 8.
Assume that the conditions in Theorem2 hold. Then there exists a positive constant C independent of α, such that(36)∫01θxxα2dx+∫0t∫01θxxα2dxdτ+β∫0t∫01θxxxαdxdτ≤C.Proof.
Differentiating the second equation of (1) with respect to x, we get(37)θxtα=-1-βθxα+μαψxxα+2ψαθxαx+βθxxxα.Multiplying the equation (37) by -θxxxα,(38)∫01θxxα2dx+∫0t∫011-βθxxα2dxdτ+β∫0t∫01θxxxαdxdτ=∫01θ0xx2dx-μ∫0t∫01ψtα+k-1αψα+kαθxαθxxxαdxdτ-2∫0t∫01ψxαθxαθxxxαdxdτ-2∫0t∫01ψαθxxαθxxxαdxdτ=∫01θ0xx2dx+μkα∫0t∫01θxxα2dxdτ+∫0t∫01μψxtα+k-1αψxαθxxαdxdτ+2∫0t∫01ψxxαθxαθxxαdxdτ+3∫0t∫01ψαθxxα2dxdτ=∫01θ0xx2dx+μkα∫0t∫01θxxα2dxdτ+∑i=46Ji.We estimate every term as follows:(39)J4≤C∫0t∫01ψxtα2+ψxα2dxdτ+p1∫0t∫01θxxα2dxdτ,J5≤Cε1∫0t∫01ψxxα2dxdτ+Cε1∫0t∫01θxxα2dxdτ,J6≤Cε1∫0t∫01θxxα2dxdτ.
Substituting (39) into (45), we have(40)∫01θxxα2dx+∫0t∫01θxxα2dxdτ+β∫0t∫01θxxxαdxdτ≤C.
Now we notice that the priori assumption (18) can be closed. Since, under this priori assumption (18), we deduced that (19), (24), (29), and (36) hold provided that ε1 is sufficiently small. Therefore, assumption (18) is always true provided that ψ0,θ0L22 is sufficiently small.Lemma 9.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(41)12∫01ψxtα2dx+θxtα2dx+∫0t∫01ψxtα2+θxtα2dxdτ+α∫0t∫01ψxxtα2dxdτ+β∫0t∫01θxxtα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (30) by -ψxxtα and -θxxtα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(42)12∫01ψxtα2dx+θxtα2dx+∫0t∫01k-1αψxtα2+1-βθxtα2dxdτ+α∫0t∫01ψxxtαdxdτ+β∫0t∫01θxxtαdxdτ=12∫01ψxtα2dx+θxtα2t=0dx+kα∫0t∫01ψxxtαθxtαdxdτ+μα∫0t∫01ψxxtαθxtαdxdτ+2∫0t∫01ψαθxαt-θxxtαdxdτ=12∫01ψxtα2dx+θxtα2t=0dx+k+μα∫0t∫01ψxxtαθxtαdxdτ-2∫0t∫01ψtαθxαθxxtαdxdτ-2∫0t∫01ψαθxtαθxxtαdxdτ=M+2∫0t∫01ψxtαθxαθxtαdxdτ+2∫0t∫01ψtαθxxαθxtαdx+∫0t∫01ψxαθxtα2dxdτ=M+∑i=710Ji.We estimate every term as follows:(43)J7≤p1∫0t∫01ψxxtα2dxdτ+k+μα24p1∫0t∫01θxt2dxdτ,J8≤Cε1∫0t∫01ψxtα2dxdτ+Cε1∫0t∫01θxtα2dxdτ,J9≤2∫0t∫01ψtαθxtα2dxdτ+12∫0t∫01θxxα2dxdτ≤C∫0t∫01ψtα2+ψxtα2dx∫01θxtα2dxdτ+12∫0t∫01θxxα2dxdτ,J10≤Cε1∫0t∫01θxtα2dxdτ.Substituting (43) into (45), we obtain(44)12∫01ψxtα2dx+θxtα2dx+∫0t∫01ψxtα2+θxtα2dxdτ+α∫0t∫01ψxxtα2dxdτ+β∫0t∫01θxxtα2dxdτ≤M.By the similar method, we have(45)∫01ψxxxα2dx+θxxxα2dx+∫0t∫01ψxxxα2+βθxxxα2dxdτ≤M.This completes the proof of Lemma 9.Combination of Lemmas5–9 and a well-known result on the local existence of the solutions, we can get the global existence of the solutions; this proves Theorem 2.


## 2.1.1.L2-Estimates

Lemma 5.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(19)12∫01ψα2+θα2dx+∫0t∫01k-1αψα2+1-βθα2+ψxα2+θxα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (1) by ψα and θα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(20)12ddt∫01ψα2+θα2dx+∫01k-1αψα2+1-βθα2dx+α∫01ψxα2dx+β∫01θxα2dx=k+μα∫01ψxαθαdx-∫01ψxαθα2dx.Integrating (20) over (0,t), using Cauchy-Schwarz inequality and (18), we obtain for any p1>0(21)12∫01ψα2+θα2dx+∫0t∫01k-1αψα2+1-βθα2dxdτ+α∫0t∫01ψxα2dxdτ+β∫01θxα2dx=12∫01ψα2+θα2t=0dx+k+μα∫0t∫01ψxαθαdxdτ-∫0t∫01ψxαθα2dxdτ=12∫01ψα2+θα2t=0dx+∫0tk+μα+θαtL∞∫01ψxαθαdxdτ≤12∫01ψ02+θ02dx+p1∫0t∫01ψxα2dxdτ+k+μα+Cε124p1∫0t∫01θα2dxdτ.If we choose p1>0 such that(22)1-β-k+μα+Cε124p1>0,α-p1>0,then we have(23)12∫01ψα2+θα2dx+∫0t∫01ψα2+θα2+ψxα2+θxα2dxdτ≤Cψ0,θ02.


## 2.1.2. High-Order Energy Estimates

Lemma 6.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(24)12∫01ψxα2+θxα2dx+∫0t∫01ψxα2+θxα2+ψxxα2+βθxxα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (1) by -ψxxα and -θxxα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(25)12ddt∫01ψxα2+θxα2dx+∫01k-1αψxα2+1-βθxα2dxdτ+α∫01ψxxα2dx+β∫01θxxα2dx=k+μα∫01ψxxαθxαdx+∫01ψxαθxα2dx.Integrating (25) over (0,t), using Cauchy-Schwarz inequality and (17)-(18), we obtain for any p1>0(26)12∫01ψxα2+θxα2dx+∫0t∫01k-1αψxα2+1-βθxα2dx+α∫0t∫01ψxxα2dxdτ+β∫0t∫01θxxα2dxdτ=12∫01ψxα2+θxα2t=0dx+k+μα∫0t∫01ψxxαθxαdxdτ+∫0t∫01ψxαθxα2dxdτ≤12∫01ψ0x2+θ0x2dx+p1∫0t∫01ψxxα2dxdτ+k+μ2α2+Cε124p1∫0t∫01θxβ2dxdτ.If we choose p1>0 such that(27)1-β-k+μ2α2+Cε124p1>0,α-p1>0,then we have(28)12∫01ψxα2+θxα2dx+∫0t∫01ψxα2+θxα2+αψxxα2+βθxxα2dxdτ≤Cψ0,θ012.Lemma 7.
Let the assumptions of Theorem2 hold. Then a positive constant C exists independent of α, such that(29)∫01ψtα2+θtα2dx+∫0t∫01ψxtα2dxdτ+θxtα2dxdτ≤C.Proof.
Differentiating (1) with respect to t, we get(30)ψttα=-k-1αψtα-kαθxtα+αψxxtα,θttα=-1-βθtα+μαψxtα+2ψαθxαt+βθxxtα.Multiplying the first and the second equations of (1) by -ψtα and -θtα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(31)12∫01ψtα2+θtα2dx+∫0t∫01k-1αψtα2dxdτ+1-β∫0t∫01θtα2dxdτ+α∫0t∫01ψxtα2dxdτ+β∫01θxtα2dxdτ=k+μα∫0t∫01ψxtαθtαdxdτ+2∫0t∫01ψtαθxαθtαdxdτ-∫0t∫01ψxαθtα2dxdτ=12∫01ψtα2+θtα2t=0dx+∑i=13Ji.We estimate every term as follows:(32)J1≤p1∫0t∫01ψxt2dxdτ+k+μ2α24p1∫0t∫01θtα2dxdτ.Here J2, J3, and J4 are estimated as follows:(33)J2≤Cε1∫0t∫01ψtα2+θtα2dxdτ,J3≤Cε1∫0t∫01θtα2dxdτ.Here we have used Cauchy-Schwarz inequality and Sobolev inequality.
Substituting (32) and (33) into (34), we have from (18) and Lemma 6(34)∫01ψtα2+θtα2dx+∫0t∫01ψtα2+θtα2+ψxtα2+βθxtα2dxdτ≤C.By the similar method, it is easy to obtain(35)∫01ψxxα2+βθxxα2dxdτ≤C.Lemma 8.
Assume that the conditions in Theorem2 hold. Then there exists a positive constant C independent of α, such that(36)∫01θxxα2dx+∫0t∫01θxxα2dxdτ+β∫0t∫01θxxxαdxdτ≤C.Proof.
Differentiating the second equation of (1) with respect to x, we get(37)θxtα=-1-βθxα+μαψxxα+2ψαθxαx+βθxxxα.Multiplying the equation (37) by -θxxxα,(38)∫01θxxα2dx+∫0t∫011-βθxxα2dxdτ+β∫0t∫01θxxxαdxdτ=∫01θ0xx2dx-μ∫0t∫01ψtα+k-1αψα+kαθxαθxxxαdxdτ-2∫0t∫01ψxαθxαθxxxαdxdτ-2∫0t∫01ψαθxxαθxxxαdxdτ=∫01θ0xx2dx+μkα∫0t∫01θxxα2dxdτ+∫0t∫01μψxtα+k-1αψxαθxxαdxdτ+2∫0t∫01ψxxαθxαθxxαdxdτ+3∫0t∫01ψαθxxα2dxdτ=∫01θ0xx2dx+μkα∫0t∫01θxxα2dxdτ+∑i=46Ji.We estimate every term as follows:(39)J4≤C∫0t∫01ψxtα2+ψxα2dxdτ+p1∫0t∫01θxxα2dxdτ,J5≤Cε1∫0t∫01ψxxα2dxdτ+Cε1∫0t∫01θxxα2dxdτ,J6≤Cε1∫0t∫01θxxα2dxdτ.
Substituting (39) into (45), we have(40)∫01θxxα2dx+∫0t∫01θxxα2dxdτ+β∫0t∫01θxxxαdxdτ≤C.
Now we notice that the priori assumption (18) can be closed. Since, under this priori assumption (18), we deduced that (19), (24), (29), and (36) hold provided that ε1 is sufficiently small. Therefore, assumption (18) is always true provided that ψ0,θ0L22 is sufficiently small.Lemma 9.
Assume that the conditions in Theorem2 hold. Then a positive constant C exists independent of α, such that(41)12∫01ψxtα2dx+θxtα2dx+∫0t∫01ψxtα2+θxtα2dxdτ+α∫0t∫01ψxxtα2dxdτ+β∫0t∫01θxxtα2dxdτ≤C.Proof.
Multiplying the first and the second equations of (30) by -ψxxtα and -θxxtα, respectively, then integrating the resulting equations over (0,1), using integration by parts and the boundary conditions (3), and then adding them, we have(42)12∫01ψxtα2dx+θxtα2dx+∫0t∫01k-1αψxtα2+1-βθxtα2dxdτ+α∫0t∫01ψxxtαdxdτ+β∫0t∫01θxxtαdxdτ=12∫01ψxtα2dx+θxtα2t=0dx+kα∫0t∫01ψxxtαθxtαdxdτ+μα∫0t∫01ψxxtαθxtαdxdτ+2∫0t∫01ψαθxαt-θxxtαdxdτ=12∫01ψxtα2dx+θxtα2t=0dx+k+μα∫0t∫01ψxxtαθxtαdxdτ-2∫0t∫01ψtαθxαθxxtαdxdτ-2∫0t∫01ψαθxtαθxxtαdxdτ=M+2∫0t∫01ψxtαθxαθxtαdxdτ+2∫0t∫01ψtαθxxαθxtαdx+∫0t∫01ψxαθxtα2dxdτ=M+∑i=710Ji.We estimate every term as follows:(43)J7≤p1∫0t∫01ψxxtα2dxdτ+k+μα24p1∫0t∫01θxt2dxdτ,J8≤Cε1∫0t∫01ψxtα2dxdτ+Cε1∫0t∫01θxtα2dxdτ,J9≤2∫0t∫01ψtαθxtα2dxdτ+12∫0t∫01θxxα2dxdτ≤C∫0t∫01ψtα2+ψxtα2dx∫01θxtα2dxdτ+12∫0t∫01θxxα2dxdτ,J10≤Cε1∫0t∫01θxtα2dxdτ.Substituting (43) into (45), we obtain(44)12∫01ψxtα2dx+θxtα2dx+∫0t∫01ψxtα2+θxtα2dxdτ+α∫0t∫01ψxxtα2dxdτ+β∫0t∫01θxxtα2dxdτ≤M.By the similar method, we have(45)∫01ψxxxα2dx+θxxxα2dx+∫0t∫01ψxxxα2+βθxxxα2dxdτ≤M.This completes the proof of Lemma 9.Combination of Lemmas5–9 and a well-known result on the local existence of the solutions, we can get the global existence of the solutions; this proves Theorem 2.


## 2.2. Proof of Theorem3

Suppose a priori assumption:(46)A2T=sup0<t<Tψ0,θ0t32≤ε2,where 0<ε2≪1. We obtain the following energy estimates.Lemma 10.
Suppose that the initial data satisfy the conditions:ϕ1∈H2, ϕ2∈H2, ϕ2(0)=ϕ2(1)=0, and ϕ1L2+ϕ1L2≤ε, and ε is sufficiently small. Then there exists a unique solution (ψ0,θ0) to the initial-boundary value problem (5)–(7) satisfying(47)12∫01θ02dx+∫0t∫01θ02+θx02dxdτ≤Cϕ22,(48)∫01θx02dx+∫0t∫01θx02dxdτ+β∫0t∫01θxx02dxdτ≤Cϕ212,(49)∫01θt02+θxx02dx+∫0t∫01θt02+θxt02dxdτ≤Cϕ12+ϕ222.Proof.
Multiplying the second equation of (5) by θ0, integrating the resulting equation over (0,t)×(0,1), using integration by part and the boundary conditions (7), and then adding them, we arrive at(50)12∫01θ02dx+1-β∫0t∫01θ02dxdτ+β∫0t∫01θx02dxdτ=12∫01ϕ22dx+2∫0t∫01ψ0θx0θ2dxdτ=Cϕ22+2ψ0L∞∫0t∫01θ0θx0dxdτ=Cϕ22+2ε2∫0t∫01θ02dxdτ+Cε∫0t∫01θx02dxdτ.Then we have(51)12∫01θ02dx+∫0t∫01θ02+θx02dxdτ≤Cϕ22.Differentiating the second equation of (5) with respect to x, we get(52)θxt0=-1-βθx0+2ψ0θx0x+βθxxx0.Multiplying (52) by θx0, using integration by part and the boundary conditions, we obtain(53)12∫01θx02dx+1-β∫0t∫01θx02dxdτ+β∫0t∫01θxxα2dxdτ=12∫01θx02t=0dx+2∫0t∫01ψx0θx02dxdτ+2∫0t∫01ψ0θxx0θx0dxdτ≤12∫01ϕ2xxdx+2ψx0L∞∫0t∫01θx02dxdτ+2ψ0L∞∫0t∫01θxx02dxdτ+∫0t∫01θx02dxdτ≤Cϕ212.Therefore, we have(54)∫01θx02dx+∫0t∫01θx02dxdτ+β∫0t∫01θxx02dxdτ≤Cϕ212.Moreover, if we differentiate the second equation of (5) with respect to t, we have(55)θtt0=-1-βθt0+2ψ0θx0t+βθxx0t,and then multiplying (55) by θt0 and integrating the resulting equation over (0,t)×(0,1), using integration by part and the boundary conditions (7), we obtain(56)12∫01θt02dx+1-β∫0t∫01θt02dxdτ+β∫0t∫01θxt02dxdτ=12∫01θ0tdx+2∫0t∫01ψ0θx0tθt0≤12ϕ2t2+2∫0t∫01ψt0θx0+ψ0θxt0dxdτ=12C+2ψt0L∞∫0t∫01θx02dxdτ+2ψ0L∞∫0t∫01θxt02dxdτ;then we have(57)∫01θt02dx≤C.Utilizing the above estimates and the similar produces, by the assumption of ε, we have(58)12∫01θt02dx+1-β∫0t∫01θt02dxdτ+β∫0t∫01θxt02dxdτ≤Cϕ122+ϕ222.Therefore,(59)∫01θxx02dx≤C∫01θt02+θ02+ψ0θx02dx≤C∫01θt02+θ02+ψ02+θx02dx≤Cϕ122+ϕ222.Here we have used (47), (48), (58), and (59). The proof of Lemma 10 is finished.Finally, based on Lemma10, we can prove Theorem 3.


## 3. Convergence Rates and BL-Thickness

In this section, we turn another interesting problem, which is concerned withL2 convergence rates of the vanishing diffusion viscosity parameter α→0+ and the BL-thickness. That is, we will give the proof of Theorem 4, and it suffices to show the following two lemmas.Lemma 11.
Assume that the conditions in Theorem4 hold. Then we have(60)∫01ψα-ψ02+θα-θ02dx+∫0t∫01αψα-ψ02+θα-θ02+θα-θ0x2dxdτ≤Cα,∫01θα-θ0x2dx+∫0t∫01θα-θ0t2dxdτ≤Cα,where C is a positive constant, independent of α.Proof.
Setuα=ψα-ψ0, vα=θα-θ0; then we have(61)utα=-k-1αuα-k-1αψ0-kαθxα+αψxxα,vtα=-1-βvα+μαψxα+2ψαvxα+2uαθx0+βvxxαwith initial data(62)uα,vαx,0=ψ0x-ϕ1x,θ0x-ϕ2x,0≤x≤1,and boundary condition(63)vα0,t=vα1,t=0,which implies that(64)vtα0,t=vtα1,t=0,t≥0.Multiplying the first equation of (61) by uα and the second equation of (61) by vα, respectively, and then adding them, we have(65)12∫01uα2+vα2dx+∫0t∫01k-1αuα2+1-βvα2+βvxα2dxdτ=12∫01uα2+vα2t=0dx-k-1α∫0t∫01uαψ0dxdτ-kα∫0t∫01θxαuαdxdτ+μα∫0t∫01ψxαvαdxdτ+2∫0t∫01ψαvxαvαdxdτ+2∫0t∫01uαθx0vαdxdτ+α∫0t∫01ψxxαuαdxdτ=12∫01ψ0x-ϕ1x2+θ0x-ϕ2x2dx-k-1α∫0t∫01ψ0uαdxdτ-kα∫0t∫01θxαuαdxdτ+μα∫0t∫01ψxαvαdxdτ-∫0t∫01ψxαvα2dxdτ=12∫01ψ0x-ϕ1x2+θ0x-ϕ2x2dx+∑i=16Mi.Then we estimate every term M1–M6 as follows:(66)M1≤k-1α2∫0t∫01ψ02dxdτ+k-1α2∫0t∫01uα2dxdτ,M2≤k∫0t∫01θxα2dxdτ+k-1α2k∫0t∫01uα2dxdτ,M3≤μ2α24λ∫0t∫01ψxα2dxdτ+Cλ∫0t∫01vα2dxdτ,M4+M5=2∫0t∫01ψvxαvαdxdτ+2∫0t∫01uαθx0vαdxdτ≤Cε∫0t∫01vxα2dxdτ+Cε∫0t∫01vα2dxdτ+Cδ∫0t∫01uα2dxdτ+Cδ∫0t∫01vα2dxdτ,M6≤α24λ∫ψxxα2dxdτ+λ∫0t∫01uα2dxdτ.Putting (66) into (65), we obtain(67)∫01uα2+vα2dx+∫0t∫01αuα2+vα2+vxα2dxdτ≤Cα+λ+Cδ∫0t∫01uα2dxdτ.By Gronwall’s inequality, we have(68)∫01uα2+uα2dx+∫0t∫01αuα2+vα2+vxα2dxdτ≤Cα.Multiplying the second equation of (61) by vtα, we have(69)121-β∫01vα2dx+12β∫01vxα2dx+∫0t∫01vtα2dxdτ=12∫011-βvα2+βvxα2t=0dx+∫0t∫01μαψxαvtαdxdτ+2∫0t∫01ψαvxαvtαdxdτ+2∫0t∫01uαθx0vtαdxdτ=∑i=710Mi.We estimate every term M7–M10 as follows:(70)M7=12∫011-βθ0x-ϕ2x2+βθ0x-ϕ2xx2dx≤Cθ0-ϕ2x12≤Cα,(71)M8≤μ2α22∫0t∫01ϕxα2+12vtα2dxdτ≤Cα+12∫0t∫01vtα2dxdτ,(72)M9≤Cε∫0t∫01vxα2dxdτ+Cε∫0t∫01vtα2dxdτ,(73)M10≤Cδ∫0t∫01uα2dxdτ+Cδvtα2dxdτ≤Cδα+Cδ∫0t∫01vtα2dxdτ.Collecting (70)–(73) and putting into (69), we have(74)12∫01vα2+vxα2dx+∫0t∫01vtα2dxdτ≤Cα.
Utilizing the similar step, multiplying the second equation of (61) by vtα, then integrating the resulting equation over (0,t)×(0,1), and then estimating every term, finally we obtain(75)∫01vα2+vxα2dx+∫0t∫01vtα2dxdτ≤Cα.The following lemma will be devoted to the boundary layer thickness.Lemma 12.
Assume that the conditions in Theorem2 hold. Then we have(76)δ∫δ1-δψxα-ψx0dx≤Cα.Proof.
Differentiating (62) with respect to x, we get(77)uxtα=-k-1αuxα-k-1αψx0-kαθxxα+αψxxxα.Let uxα=ω; then we have(78)ωt=-k-1αω-k-1αψx0-kαθxxα+αψxxxα.To construct a cut-off function as in [12], we define φη, for η∈(0,1), and ξδ for δ∈(0,1/2), by(79)φηs=s2+η2,ξδ=x,0≤x≤δ,δ,δ≤x≤1-δ,1-x,1-δ≤x≤1.It is easy to check that φη satisfies(80)s≤φηs≤s+1,φη′s≤1,0≤sφη′s≤φηs,φη′′s≥0,s2φη′′s≥0.ξδ satisfies 0≤ξδ≤δ, ξδ(1)=ξδ(0)=0, multiplying (78) by φη′(ω)ξδ and then integrating over(81)∫0t∫01φη′ωξδdx-ε∫01ξδdx=-k-1α∫0t∫01ωφη′ωξδdxdτ-k-1α∫0t∫01ψx0φη′ωξδdxdτ-kα∫0t∫01θxxαφη′ωξδdxdτ+α∫0t∫01ψxxxαφε′ωξδdxdτ=∑i=14Mi.By the fact that 0≤sφη′(s)≤φη(s), we have(82)M1≤∫0t∫01φεωξδdxdτ.By φη′(ω)≤1, 0≤ξδ≤δ, we obtain(83)M2≤Cδ∫0t∫01φx02dxdτ1/2≤Cδα,M3≤Cαδ∫0t∫01θxxα2dxdτ1/2≤Cδα,M4≤Cαδ∫0t∫01θxxxα2dxdτ≤Cα.Combining (82) and (83), we notice that (84)ε∫01ξδdx≤εδ,∫0t∫01φη′ωξδdxdτ≤C∫0t∫01φηωξδdxdτ+Cαδ+ϵδ.By Gronwall’s inequality, we have (85)∫01φηωξδdx≤Cεδ+αδ.By the definition of ξδ,|z|≤φη(ω), we have(86)∫δ1-δωdx≤Cεδ+αδ<Cε+α.Letting ε→0,(87)∫δ1-δωdx≤Cα.Because W1,1([δ,1-δ])↪L∞([δ,1-δ]), it follows that(88)ψα-ψ0L∞δ,1-δ≤∫01ψα-ψ0dx+∫δ1-δψxα-ψx0dx≤Cα.As in [5, 6, 13], we observe that inequality (11) holds. This completes the proof of Theorem 4.



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*Source: 2581953-2018-05-06.xml*