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 orderdefined 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 inequalityfor all , then there exists a solution to (1) such thatfor 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 thatwhere 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 expansionWe also denoteThen, 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 toandTaking 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 equationswhere and . Then there exist and satisfyingsuch 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 obtainandAlso, by (24), (25), and (27) we obtainandSo, by (31)–(34), for the Fourier expansions of and we haveThen, for , from (35) we have and for and , we have which implies Hence, we haveandSimilarly, for -norm of and , we obtainandAlso, for -norm of , we haveTherefore, 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 equationswhere and with . Then there exist and satisfyingsuch 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).