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

In this paper


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][4][5][6][7][8]).
We say that the differential equation (1) satisfies the Hyers-Ulam stability provided the following statement is true for any  > 0: if an  times continuously differentiable function  :  → C satisfies the differential inequality      F ( () , for all  ∈ , where () depends on  only and satisfies lim →0 () = 0.
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 f, find u and  such that where { 1 , . . .,   } is the canonical basis of R  ,  is the period in the -th direction, and  = (0, )  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.

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  2 (R  ) with the periodic boundary condition.We set by   (Ω) the Sobolev space of functions which are in  2 (Ω), with all their derivatives of order ≤ .Then,   (Ω) is a Hilbert space with the inner product and the norm where  = ( 1 , . . .,   ),   ∈ N 0 , [] =  1 + ⋅ ⋅ ⋅ +   , and We also set by    (),  ∈ N 0 , the space of functions which are periodic with period : For  = 0,  0  () means simply  2 ().Then, for an arbitrary  ∈ N,    () is a Hilbert space with the inner product And the functions in    () are characterized by their Fourier series expansion We also denote Then, for  ∈ N, Ḣ  () is a Hilbert space for the norm where H   () = {   ()}  .We also introduce the inner product and the norm One notes that V is a Hilbert space with this norm.Also, the dual V  of V is ‖⋅‖ V  will denote the dual norm of ‖⋅‖ on V  .For the boundary value, due to trace theorem we have that u ∈ V if and only if its restriction u|  to  belongs to where we have numbered the faces Γ 1 , . . ., Γ 2 of  as follows: and k| Γ  is an improper notation for the trace of k on Γ  .And u ∈ H if and only if u belongs to Now, let us look at the stationary Stokes problem (4) with the periodic boundary condition (6); given f ∈ Ḣ0  () or Ḣ−1  (), find u ∈ Ḣ1  () and  ∈  2 () such that Here, to solve the above problem we use the Fourier series.Let us introduce the Fourier expansions of u, , and f; and Taking the scalar product of ( 22) with k and using (23) we find the  k 's: then ( 22) provided the  k 's; By definition (11) of

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  = 2 while one can extend our result to R  .