Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 2581953, 11 pages
https://doi.org/10.1155/2018/2581953

Research Article

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

Linrui Li¹ and Shu Wang²

¹Basic Courses Department, Institute of Disaster Prevention, 465 Xueyuan Street, Sanhe, Hebei 065201, China
²College of Applied Sciences, Beijing University of Technology, Pingleyuan 100, Chaoyang District, Beijing 100124, China

Correspondence should be addressed to Linrui Li; linrui020213@163.com

Received 16 January 2018; Accepted 25 March 2018; Published 6 May 2018

Academic Editor: Gabriella Bretti

Copyright © 2018 Linrui Li and Shu Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 limit , especially for the mixed boundary conditions; we prove that the thickness of layer is of the order . Furthermore, the corresponding convergence rates are also obtained.

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 :

with initial data

and the mixed boundary conditions, that is, Neumann-Dirichlet boundary conditions

which implies that

where , , , and are positive constants with and . The limit problem of the vanishing parameter ; we have

with the initial conditions

and the Dirichlet boundary conditions

which implies that

We expect to prove that the solution of (1)–(3) converges to the solution of the limit problem (5)–(7), as the diffusion parameter , 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 and they show that the boundary layer thickness is of the order with . Subsequently, Peng et al. [6] consider the initial-boundary value problem (1) with the Dirichlet-Neumann boundary conditions when diffusion parameter and they show that the boundary layer thickness is of the order with . For the parameter , 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 . However, for the asymptotic limit , 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 , 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. and denote the usual Lebesgue space in with its norms and . denotes the usual -th order Sobolev space with its norm . For simplicity, , and , 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 function is called a boundary layer thickness for problem (1)–(3) with vanishing diffusion , if as , and

where , , and 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 .

Theorem 2. Suppose that the initial data satisfy the conditions: , and is sufficiently small, then there exists a unique solution to the initial-boundary value problem (1)–(3) satisfying

where the norms are all uniform in .

Then our second result shows that the initial-boundary value problem (5)–(7) admits a unique global smooth solution .

Theorem 3. Suppose that the initial data satisfy the conditions: , , and is sufficiently small, then there exists a unique solution to the initial-boundary value problem (5)–(7) satisfying

Furthermore, we give the convergence rates and boundary layer thickness.

Theorem 4. Under the assumptions of Theorems 2 and 3, suppose that

then any function satisfying the conditions and as is a BL-thickness such that

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 BL-thickness as the diffusion parameter for the mixed boundary conditions.

2. 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 is a positive constant satisfying , independent of

By Sobolev inequality, we have

2.1.1. -Estimates

Lemma 5. 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 , using integration by parts and the boundary conditions (3), and then adding them, we have

Integrating (20) over , using Cauchy-Schwarz inequality and (18), we obtain for any

If we choose such that

then we have

2.1.2. 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 , using integration by parts and the boundary conditions (3), and then adding them, we have

Integrating (25) over , using Cauchy-Schwarz inequality and (17)-(18), we obtain for any

If we choose such that

then we have

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 , using integration by parts and the boundary conditions (3), and then adding them, we have

We estimate every term as follows:

Here , , and 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 is sufficiently small. Therefore, assumption (18) is always true provided that 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 , 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.

2.2. Proof of Theorem 3

Suppose a priori assumption:

where . We obtain the following energy estimates.

Lemma 10. Suppose that the initial data satisfy the conditions: , , , and , and is sufficiently small. Then there exists a unique solution to the initial-boundary value problem (5)–(7) satisfying

Proof. Multiplying the second equation of (5) by , integrating the resulting equation over , 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 , 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 and integrating the resulting equation over , 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,

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.

3. Convergence Rates and BL-Thickness

In this section, we turn another interesting problem, which is concerned with convergence rates of the vanishing diffusion viscosity parameter 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 Theorem 4 hold. Then we have

where is a positive constant, independent of

Proof. Set , ; 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 , respectively, and then adding them, we have

Then we estimate every term – as follows:

Putting (66) into (65), we obtain

By Gronwall’s inequality, we have

Multiplying the second equation of (61) by , we have

We estimate every term – as follows:

Collecting (70)–(73) and putting into (69), we have

Utilizing the similar step, multiplying the second equation of (61) by , then integrating the resulting equation over , 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

Proof. Differentiating (62) with respect to , we get

Let ; then we have

To construct a cut-off function as in [12], we define , for , and for , by

It is easy to check that satisfies

satisfies , , multiplying (78) by and then integrating over

By the fact that , we have

By , , we obtain

Combining (82) and (83), we notice that

By Gronwall’s inequality, we have

By the definition of , we have

Letting ,

Because , it follows that

As in [5, 6, 13], we observe that inequality (11) holds. This completes the proof of Theorem 4.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The paper was finished in collaboration among all authors. Linrui Li found the motivation of this paper and suggested the outline of the proofs. Shu Wang provided good ideas and gave some valuable suggestions for completing the paper. All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript.

Acknowledgments

This research was supported by the Science and Technology Planning Project of Hebei Province (Grant no. 16275432) and the National Science Foundation of China (Grant no. 11371042). This research was done during Dr. Linrui Li’s visit to College of Applied Sciences in Beijing University of Technology. She would like to thank Professor Shu Wang for his many useful suggestions and stimulating discussions on this work.

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