Journal of Function Spaces
Volume 2018 (2018), Article ID 5138414, 5 pages
https://doi.org/10.1155/2018/5138414

Research Article

Approximation Property of the Stationary Stokes Equations with the Periodic Boundary Condition

Soon-Mo Jung¹ and Jaiok Roh²

¹Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea
²Department of Finance and Information Statistics, Institute of Statistics, Hallym University, Chuncheon, Kangwon-Do 24252, Republic of Korea

Correspondence should be addressed to Jaiok Roh; joroh@hallym.ac.kr

Received 29 June 2018; Accepted 12 September 2018; Published 9 October 2018

Academic Editor: Mark A. McKibben

Copyright © 2018 Soon-Mo Jung and Jaiok Roh. 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.

In this paper, we will consider the stationary Stokes equations with the periodic boundary condition and we will study approximation property of the solutions by using the properties of the Fourier series. Finally, we will discuss that our estimation for approximate solutions is optimal.

1. Introduction

The study of stability problems for various functional equations originated from a famous talk presented by Ulam in 1940. In this talk, he discussed a problem concerning the stability of homomorphisms. And Obłoza [1, 2] first investigated the Hyers-Ulam stability of the linear differential equations which have the form . Thereafter, a number of mathematicians have dealt with this subject for different types of differential equations (see [3–8]).

For an open interval of with , we consider the linear differential equation of th order

defined on , where is an times continuously differentiable function.

We say that the differential equation (1) satisfies the Hyers-Ulam stability provided the following statement is true for any : if an times continuously differentiable function satisfies the differential inequality

for all , then there exists a solution to (1) such that

for all , where depends on only and satisfies .

Recently, several mathematicians investigated the Hyers-Ulam stability for the partial differential equations. One can refer to [9–14].

In this paper, we will investigate approximate properties of the solutions for the stationary Stokes equations with the periodic boundary condition. The stationary Stokes problem associated with the space periodicity condition is the following one: For a given , find and such that

where is the canonical basis of , is the period in the -th direction, and is the cube of the period.

The advantage of the boundary condition (6) is that it leads to a simple functional setting, while many of the mathematical difficulties remain unchanged. In fact, in the next section we will introduce in detail the corresponding functional setting of the problem.

Finally, we will discuss that our estimation for approximate solutions is optimal.

2. Preliminary Results

In this section, we will introduce the useful functional settings and preliminary results for the solutions of the stationary Stokes equations with the periodic boundary condition. For the materials of this section, we totally refer to the book by Roger Temam [15]. So if the reader wants to understand more deeply, one can refer to this book.

For the functional spaces of the solutions, we will consider the Lebesque space with the periodic boundary condition. We set by the Sobolev space of functions which are in , with all their derivatives of order . Then, is a Hilbert space with the inner product and the norm

where , , , and

We also set by , , the space of functions which are periodic with period :

For , means simply . Then, for an arbitrary , is a Hilbert space with the inner product

And the functions in are characterized by their Fourier series expansion

We also denote

Then, for , is a Hilbert space for the norm , and and are in duality for all .

Now, we introduce two important function spaces,

where . We also introduce the inner product and the norm

One notes that is a Hilbert space with this norm. Also, the dual of is

will denote the dual norm of on . For the boundary value, due to trace theorem we have that if and only if its restriction to belongs to

where we have numbered the faces of as follows:

and is an improper notation for the trace of on . And if and only if belongs to

Now, let us look at the stationary Stokes problem (4) with the periodic boundary condition (6); given or , find and such that

Here, to solve the above problem we use the Fourier series. Let us introduce the Fourier expansions of , , and ;

Equation (20) reduces for every to

and

Taking the scalar product of (22) with and using (23) we find the ’s:

then (22) provided the ’s;

By definition (11) of , if then and ; if then and . Now if belongs to , then for every so that and .

3. Approximate Properties for the Solutions

In this section, we will discuss approximate properties for the solutions of the stationary Stokes equations with the periodic boundary condition. In this paper, we will prove theorems for while one can extend our result to .

Theorem 1. Let the functions and satisfy the equations

where and . Then there exist and satisfying

such that

for some constants and .

Proof. For existence of the solutions and , One can prove by (24) and (25). Next, to obtain (28) and (29) we denote the Fourier expansions of , , and as the following;

Then, by (24), (25), and (26) we obtain

and

Also, by (24), (25), and (27) we obtain

and

So, by (31)–(34), for the Fourier expansions of and we have

Then, for , from (35) we have

and for and , we have

which implies

Hence, we have

and

Similarly, for -norm of and , we obtain

and

Also, for -norm of , we have

Therefore, by (39)–(43), we complete the proof.

Remark 2. We consider the function as for and for . Then, we have

And we consider the function as for and for . Then we have

Hence, our estimation for and is optimal.

Corollary 3. Let the functions and satisfy the equations

where and with . Then there exist and satisfying

such that

Corollary 4. Let the functions and satisfy the equations

where with . Then there exist and satisfying

such that

Corollary 5. Let the functions and satisfy the equations

where and with . Then there exist and satisfying

such that

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Hallym University Research Fund (HRF-201805-008).

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