JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2018/5138414 5138414 Research Article Approximation Property of the Stationary Stokes Equations with the Periodic Boundary Condition http://orcid.org/0000-0002-6812-016X Jung Soon-Mo 1 http://orcid.org/0000-0003-2937-3698 Roh Jaiok 2 McKibben Mark A. 1 Mathematics Section College of Science and Technology Hongik University 30016 Sejong Republic of Korea hongik.ac.kr 2 Department of Finance and Information Statistics Institute of Statistics Hallym University Chuncheon Kangwon-Do 24252 Republic of Korea hallym.ac.kr 2018 9102018 2018 29 06 2018 12 09 2018 9102018 2018 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.

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.

Hallym University HRF-201805-008
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 y(x)+g(x)y(x)=r(x). Thereafter, a number of mathematicians have dealt with this subject for different types of differential equations (see ).

For an open interval I=(a,b) of R with -a<b+, we consider the linear differential equation of nth order(1)Fyn,yn-1,,y,y,x=0defined on I, where y:IC is an n times continuously differentiable function.

We say that the differential equation (1) satisfies the Hyers-Ulam stability provided the following statement is true for any ε>0: if an n times continuously differentiable function y:IC satisfies the differential inequality(2)Fyn,yn-1,,y,y,xεfor all xI, then there exists a solution y0:IC to (1) such that(3)yx-y0xKεfor all xI, where K(ε) depends on ε only and satisfies limε0K(ε)=0.

Recently, several mathematicians investigated the Hyers-Ulam stability for the partial differential equations. One can refer to .

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 p such that(4)-Δu+p=finQ(5)·u=0inQ(6)ux+Lei=uxxRn,where {e1,,en} is the canonical basis of Rn, L is the period in the i-th direction, and Q=(0,L)n 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 . 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 L2(Rn) with the periodic boundary condition. We set by Hm(Ω) the Sobolev space of functions which are in L2(Ω), with all their derivatives of order m. Then, Hm(Ω) is a Hilbert space with the inner product and the norm (7)u,vm=αmDαu,Dαv,um=u,um1/2,where α=(α1,,αn), αiN0, [α]=α1++αn, and (8)Dα=D1α1Dnαn=αx1α1xnαn.

We also set by Hpm(Q), mN0, the space of functions which are periodic with period Q:(9)ux+Lei=uxi=1,,n.For m=0, Hp0(Q) means simply L2(Q). Then, for an arbitrary mN, Hpm(Q) is a Hilbert space with the inner product (10)u,vm=αmQDαuxDαvxdx.And the functions in Hpm(Q) are characterized by their Fourier series expansion(11)HpmQ=u:u=kZncke2iπk·x/L,c¯k=c-k,kZnk2mck2<.We also denote(12)H˙pmQ=uHpmQ of  type 11:c0=0.Then, for mN, H˙pm(Q) is a Hilbert space for the norm kZn|k|2m|ck|21/2, and H˙pm(Q) and H˙p-m(Q) are in duality for all mN.

Now, we introduce two important function spaces,(13)V=uHp1Q:·u=0inRn,(14)H=uHp0Q:·u=0inRn,where Hpm(Q)={Hpm(Q)}n. We also introduce the inner product and the norm (15)u,v=i=1nuxi,vxi,u=u,u1/2.One notes that V is a Hilbert space with this norm. Also, the dual V of V is (16)V=uHp-1Q=Hp1Q:·u=0inRn;·V will denote the dual norm of · on V. For the boundary value, due to trace theorem we have that uV if and only if its restriction u|Q to Q belongs to (17)vH1Q:·v=0,vΓj+n=vΓjj=1,,nwhere we have numbered the faces Γ1,,Γ2n of Q as follows: (18)Γj=Qxj=0,Γj+n=Qxj=L,and v|Γi is an improper notation for the trace of v on Γi. And uH if and only if u belongs to (19)vL2Q:·v=0,v·νΓj+n=-v·νΓjj=1,,n.

Now, let us look at the stationary Stokes problem (4) with the periodic boundary condition (6); given fH˙p0(Q) or H˙p-1(Q), find uH˙p1(Q) and pL2(Q) such that(20)-Δu+p=finQ,·u=0inQ.

Here, to solve the above problem we use the Fourier series. Let us introduce the Fourier expansions of u, p, and f; (21)u=kZnuke2πik·x/L,p=kZnpke2πik·x/L,f=kZnfke2πik·x/L.Equation (20) reduces for every k0 to(22)4π2k2L2uk+2πikLpk=fkand(23)k·uk=0.Taking the scalar product of (22) with k and using (23) we find the pk’s:(24)pk=Lk·fk2πik2for  kZn,k0then (22) provided the uk’s;(25)uk=L24π2k2fk-k·fkkk2for  kZn,k0.By definition (11) of Hpm(Q), if fH˙p0(Q) then uH˙p2(Q) and pH˙p1(Q); if fH˙p-1(Q) then uH˙p1(Q) and pH˙p0(Q). Now if f belongs to H, then k·fk=0 for every k so that p=0 and uk=fkL2/4π2|k|2.

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 n=2 while one can extend our result to Rn.

Theorem 1.

Let the functions vH˙p2(Q) and qH˙p1(Q) satisfy the equations(26)-Δv+q-f=ginQ,·v=0inQ,where gL2ε and f,gH˙p0(Q). Then there exist uH˙p2(Q) and pH˙p1(Q) satisfying(27)-Δu+p-f=0inQ,·u=0inQsuch that (28)u-vHiKigL2Kiεfor  i=0,1,2,(29)p-qHiMigL2Miεfor  i=0,1for some constants Ki and Mi.

Proof.

For existence of the solutions uH˙p2(Q) and pH˙p1(Q), One can prove by (24) and (25). Next, to obtain (28) and (29) we denote the Fourier expansions of v, q, and g as the following; (30)v=kZ2vke2πik·x/L,q=kZ2qke2πik·x/L,g=kZ2gke2πik·x/L.Then, by (24), (25), and (26) we obtain(31)qk=Lk·fk+gk2πik2for  kZ2,k0and(32)vk=L24π2k2fk+gk-k·fk+gkkk2for  kZ2,k0.Also, by (24), (25), and (27) we obtain(33)pk=Lk·fk2πik2for  kZ2,k0and(34)uk=L24π2k2fk-k·fkkk2for  kZ2,k0.So, by (31)–(34), for the Fourier expansions of u-v and p-q we have(35)uk-vk=-L24π2k2gk-k·gkkk2,pk-qk=-Lk·gk2πik2.Then, for |k|2, from (35) we have (36)uk-vkL28π2gkand for |k|=1 and gk=(gk1,gk2), we have (37)gk-k·gkkk2=0,gk2,for  k=1,0,gk-k·gkkk2=0,gk2,for  k=-1,0,gk-k·gkkk2=gk1,0,for  k=0,1,gk-k·gkkk2=gk1,0,for  k=0,-1which implies (38)uk-vkL24π2gk.

Hence, we have(39)u-vL2=kZnuk-vk21/2L24π2gL2K1εand(40)p-qL2=kZnpk-qk21/2L2πgL2M1ε.

Similarly, for H1-norm of u-v and p-q, we obtain(41)u-vH1=kZnk2uk-vk21/2L24π2gL2K2εand(42)p-qH1=kZnk2pk-qk21/2L2πgL2M2ε.Also, for H2-norm of u-v, we have(43)u-vH2=kZnk4uk-vk21/2L24π2gL2K3ε.Therefore, by (39)–(43), we complete the proof.

Remark 2.

We consider the function g as gk=(0,0) for k(1,0) and gk=(0,1) for k=(1,0). Then, we have (44)u-vL2=kZ2uk-vk21/2=L24π2gL2,u-vH1=kZ2k2uk-vk21/2=L24π2gL2,u-vH2=kZ2k4uk-vk21/2=L24π2gL2.And we consider the function g as gk=(0,0) for k(1,0) and gk=(1,0) for k=(1,0). Then we have (45)p-qL2=kZ2pk-qk21/2=L2πgL2,p-qH1=kZ2k2pk-qk21/2=L2πgL2.Hence, our estimation for Ki and Mi is optimal.

Corollary 3.

Let the functions vH˙p2(Q) and qH˙p1(Q) satisfy the equations (46)-Δv+q-f=ginQ,·v=0inQ,where fH and gH˙p0(Q) with gL2ε. Then there exist uH˙p2(Q) and pH˙p1(Q) satisfying (47)-Δu+p-f=0inQ,·u=0inQ.such that (48)u-vHiKigL2Kiεfor  i=0,1,2,p-qHiMigL2Miεfor  i=0,1

Corollary 4.

Let the functions vH˙p2(Q) and qH˙p1(Q) satisfy the equations (49)-Δv+q-f=ginQ,·v=0inQ,where f,gH with gL2ε. Then there exist uH˙p2(Q) and pH˙p1(Q) satisfying (50)-Δu+p-f=0inQ,·u=0inQ.such that (51)u-vHiKigL2Kiεfor  i=0,1,2,p-qHi=0for  i=0,1.

Corollary 5.

Let the functions vH˙p2(Q) and qH˙p1(Q) satisfy the equations(52)-Δv+q-f=gin  Q,·v=0in  Q,where fH˙p0(Q) and gH with gL2ε. Then there exist uH˙p2(Q) and pH˙p1(Q) satisfying(53)-Δu+p-f=0in  Q,·u=0in  Q.such that(54)u-vHiKigL2Kiεfor  i=0,1,2,(55)p-qHi=0for  i=0,1.

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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