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

with initial data (ψα, θα) (x, 0) = (ψ0, θ0) (x) , 0 ≤ x ≤ 1, (2) and the mixed boundary conditions, that is, Neumann-Dirichlet boundary conditions (ψα x , θα) (0, t) = (ψα x , θα) (1, t) = (0, 0) , t ≥ 0, (3) which implies that (ψα xt, θα t ) (0, t) = (ψα xt, θα t ) (1, t) = (0, 0) , t ≥ 0, (4) 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 ψ0 t = 0, θ0 t = − (1 − β) θ0 + 2ψ0θ0 x + βθ0 xx, 0 < x < 1, t > 0 (5) with the initial conditions

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][2][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][2][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][6][7][8][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][6][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 initialboundary value problem (1) with the zero Dirichlet boundary conditions when diffusion parameter  → 0 + and they show that the boundary layer thickness () is of the order (  ) with 0 <  < 1/2.Subsequently, Peng et al. [6] consider the initial-boundary value problem (1) with the Dirichlet-Neumann boundary conditions when diffusion parameter  → 0 + and they show that the boundary layer thickness () is of the order (  ) with 0 <  < 3/4.For the parameter  → 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 ( 1/2 ).However, for the asymptotic limit  → 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  → 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  by .And the constant  may vary from line to line.
Before we state the main results, let us recall the definition of boundary layer thickness (BL-thickness) in [11] as follows.
Our first purpose is to show that the initial-boundary value problem (1)-(3) admits a unique global smooth solution (  ,   ).
Then our second result shows that the initial-boundary value problem ( 5)-( 7) admits a unique global smooth solution ( 0 ,  0 ).Theorem 3. Suppose that the initial data satisfy the conditions: ( 1 ,  2 ) ∈  2 ,  2 (0) =  2 (1) = 0 ≤ , and  is sufficiently small, then there exists a unique solution ( 0 ,  0 ) to the initialboundary value problem ( 5)- (7) satisfying Furthermore, we give the convergence rates and boundary layer thickness.(15) Consequently, The plan of this paper is as follows.Section 2 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 BLthickness as the diffusion parameter  → 0 + for the mixed boundary conditions.

Proof of Theorems 2 and 3
2.1.Proof of Theorem 2. In this subsection, we shall prove Theorem 2 by adapting the elaborate nonlinear energy method.In the following, we devote ourselves to the a priori estimate of solution (  (, ),   (, )) of ( 1)-( 3) under the a priori assumption where  1 is a positive constant satisfying 0 <  1 ≤ 1, independent of .By Sobolev inequality, we have Assume that the conditions in Theorem 2 hold.Then a positive constant  exists independent of , such that Proof.Multiplying the first and the second equations of (1) by   and   , respectively, then integrating the resulting equations over (0, 1), using integration by parts and the boundary conditions (3), and then adding them, we have Integrating ( 20) over (0, ), using Cauchy-Schwarz inequality and (18), we obtain for any  1 > 0 If we choose  1 > 0 such that then we have (23)

High-Order Energy Estimates
Lemma 6. Assume that the conditions in Theorem 2 hold.Then a positive constant  exists independent of , such that Proof.Multiplying the first and the second equations of (1) by −   and −   , respectively, then integrating the resulting equations over (0, 1), using integration by parts and the boundary conditions (3), and then adding them, we have Integrating (25) over (0, ), using Cauchy-Schwarz inequality and ( 17)-(18), we obtain for any  1 > 0 If we choose  1 > 0 such that then we have (28) Lemma 7. Let the assumptions of Theorem 2 hold.Then a positive constant  exists independent of , such that Proof.Differentiating (1) with respect to , we get Multiplying the first and the second equations of (1) by −   and −   , respectively, then integrating the resulting equations over (0, 1), using integration by parts and the boundary conditions (3), and then adding them, we have We estimate every term as follows: Here  2 ,  3 , and  4 are estimated as follows: Here we have used Cauchy-Schwarz inequality and Sobolev inequality.Substituting (32) and ( 33) into (34), we have from (18) and Lemma 6 By the similar method, it is easy to obtain Lemma 8. Assume that the conditions in Theorem 2 hold.Then there exists a positive constant  independent of , such that Proof.Differentiating the second equation of (1) with respect to , we get Multiplying the equation (37) by −   , We estimate every term as follows: Substituting (39) into (45), we have 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  1 is sufficiently small.Therefore, assumption (18) is always true provided that ‖ 0 ,  0 ‖ 2  2 is sufficiently small.

Lemma 9.
Assume that the conditions in Theorem 2 hold.Then a positive constant  exists independent of , such that Proof.Multiplying the first and the second equations of (30) by −   and −   , respectively, then integrating the resulting equations over (0, 1), using integration by parts and the boundary conditions (3), and then adding them, we have We estimate every term as follows: Substituting ( 43) into (45), we obtain By the similar method, we have This completes the proof of Lemma 9.
Combination of Lemmas 5-9 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.

Proof of Theorem 3. Suppose a priori assumption:
where 0 <  2 ≪ 1.We obtain the following energy estimates.
Lemma 10.Suppose that the initial data satisfy the conditions: , and  is sufficiently small.Then there exists a unique solution ( 0 ,  0 ) to the initial-boundary value problem ( 5)-( 7) satisfying 2 ) . (49) Proof.Multiplying the second equation of ( 5) by  0 , integrating the resulting equation over (0, )×(0, 1), using integration by part and the boundary conditions (7), and then adding them, we arrive at Then we have Differentiating the second equation of ( 5) with respect to , we get Multiplying (52) by  0  , using integration by part and the boundary conditions, we obtain Therefore, we have Moreover, if we differentiate the second equation of ( 5) with respect to , we have and then multiplying (55) by  0  and integrating the resulting equation over (0, ) × (0, 1), using integration by part and the boundary conditions (7), we obtain then we have Utilizing the above estimates and the similar produces, by the assumption of , we have Therefore, 2 ) . ( Here we have used (47), (48), (58), and (59).The proof of Lemma 10 is finished.
Finally, based on Lemma 10, we can prove Theorem 3.

Convergence Rates and BL-Thickness
In this section, we turn another interesting problem, which is concerned with  2 convergence rates of the vanishing diffusion viscosity parameter  → 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 Theorem 4 hold.Then we have where  is a positive constant, independent of .
Proof.Set   =   −  0 , V  =   −  0 ; then we have with initial data and boundary condition which implies that Multiplying the first equation of (61) by   and the second equation of (61) by V  , respectively, and then adding them, we have Then we estimate every term  1 - 6 as follows: Putting (66) into (65), we obtain By Gronwall's we have Multiplying the second equation of (61) by V   , we have We estimate every term  7 - 10 as follows:  Utilizing the similar step, multiplying the second equation of (61) by V   , then integrating the resulting equation over (0, )×(0, 1), and then estimating every term, finally we obtain The following lemma will be devoted to the boundary layer thickness.
Lemma 12. Assume that the conditions in Theorem 2 hold.Then we have  ∫ (81)