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 [3–8]).

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:I→C 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:I→C satisfies the differential inequality(2)Fyn,yn-1,…,y′,y,x≤εfor all x∈I, then there exists a solution y0:I→C to (1) such that(3)yx-y0x≤Kεfor all x∈I, where K(ε) depends on ε only and satisfies limε→0K(ε)=0.

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 f, find u and p such that(4)-Δu+∇p=f in Q(5)∇·u=0 in Q(6)ux+Lei=ux ∀x∈Rn,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 [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 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=∑α≤mDαu,Dαv,um=u,um1/2,where α=(α1,…,αn), αi∈N0, [α]=α1+⋯+αn, and (8)Dα=D1α1⋯Dnαn=∂α∂x1α1⋯∂xnαn.

We also set by Hpm(Q), m∈N0, the space of functions which are periodic with period Q:(9)ux+Lei=ux ∀i=1,…,n.For m=0, Hp0(Q) means simply L2(Q). Then, for an arbitrary m∈N, Hpm(Q) is a Hilbert space with the inner product (10)u,vm=∑α≤m∫QDαuxDαvxdx.And the functions in Hpm(Q) are characterized by their Fourier series expansion(11)HpmQ=u:u=∑k∈Zncke2iπk·x/L, c¯k=c-k, ∑k∈Znk2mck2<∞.We also denote(12)H˙pmQ=u∈HpmQ of type 11:c0=0.Then, for m∈N, H˙pm(Q) is a Hilbert space for the norm ∑k∈Zn|k|2m|ck|21/2, and H˙pm(Q) and H˙p-m(Q) are in duality for all m∈N.

Now, we introduce two important function spaces,(13)V=u∈Hp1Q:∇·u=0 in Rn,(14)H=u∈Hp0Q:∇·u=0 in Rn,where Hpm(Q)={Hpm(Q)}n. We also introduce the inner product and the norm (15)u,v=∑i=1n∂u∂xi,∂v∂xi,u=u,u1/2.One notes that V is a Hilbert space with this norm. Also, the dual V′ of V is (16)V′=u∈Hp-1Q=Hp1Q′:∇·u=0 in Rn;·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|Q to Q belongs to (17)v∈H1Q:∇·v=0, vΓj+n=vΓj j=1,…,nwhere we have numbered the faces Γ1,…,Γ2n of Q as follows: (18)Γj=∂Q∩xj=0,Γj+n=∂Q∩xj=L,and v|Γi is an improper notation for the trace of v on Γi. And u∈H if and only if u belongs to (19)v∈L2Q:∇·v=0, v·νΓj+n=-v·νΓj j=1,…,n.

Now, let us look at the stationary Stokes problem (4) with the periodic boundary condition (6); given f∈H˙p0(Q) or H˙p-1(Q), find u∈H˙p1(Q) and p∈L2(Q) such that(20)-Δu+∇p=f in Q,∇·u=0 in Q.

Here, to solve the above problem we use the Fourier series. Let us introduce the Fourier expansions of u, p, and f; (21)u=∑k∈Znuke2πik·x/L,p=∑k∈Znpke2πik·x/L,f=∑k∈Znfke2πik·x/L.Equation (20) reduces for every k≠0 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·fk2πik2 for k∈Zn, k≠0then (22) provided the uk’s;(25)uk=L24π2k2fk-k·fkkk2 for k∈Zn, k≠0.By definition (11) of Hpm(Q), if f∈H˙p0(Q) then u∈H˙p2(Q) and p∈H˙p1(Q); if f∈H˙p-1(Q) then u∈H˙p1(Q) and p∈H˙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 v∈H˙p2(Q) and q∈H˙p1(Q) satisfy the equations(26)-Δv+∇q-f=g in Q,∇·v=0 in Q,where gL2≤ε and f,g∈H˙p0(Q). Then there exist u∈H˙p2(Q) and p∈H˙p1(Q) satisfying(27)-Δu+∇p-f=0 in Q,∇·u=0 in Qsuch that (28)u-vHi≤KigL2≤Kiε for i=0,1,2,(29)p-qHi≤MigL2≤Miε for i=0,1for some constants Ki and Mi.

Proof. For existence of the solutions u∈H˙p2(Q) and p∈H˙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=∑k∈Z2vke2πik·x/L,q=∑k∈Z2qke2πik·x/L,g=∑k∈Z2gke2πik·x/L.Then, by (24), (25), and (26) we obtain(31)qk=Lk·fk+gk2πik2 for k∈Z2, k≠0and(32)vk=L24π2k2fk+gk-k·fk+gkkk2 for k∈Z2, k≠0.Also, by (24), (25), and (27) we obtain(33)pk=Lk·fk2πik2 for k∈Z2, k≠0and(34)uk=L24π2k2fk-k·fkkk2 for k∈Z2, k≠0.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-vk≤L28π2gkand 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-vk≤L24π2gk.

Hence, we have(39)u-vL2=∑k∈Znuk-vk21/2≤L24π2gL2≤K1εand(40)p-qL2=∑k∈Znpk-qk21/2≤L2πgL2≤M1ε.

Similarly, for H1-norm of u-v and p-q, we obtain(41)u-vH1=∑k∈Znk2uk-vk21/2≤L24π2gL2≤K2εand(42)p-qH1=∑k∈Znk2pk-qk21/2≤L2πgL2≤M2ε.Also, for H2-norm of u-v, we have(43)u-vH2=∑k∈Znk4uk-vk21/2≤L24π2gL2≤K3ε.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=∑k∈Z2uk-vk21/2=L24π2gL2,u-vH1=∑k∈Z2k2uk-vk21/2=L24π2gL2,u-vH2=∑k∈Z2k4uk-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=∑k∈Z2pk-qk21/2=L2πgL2,p-qH1=∑k∈Z2k2pk-qk21/2=L2πgL2.Hence, our estimation for Ki and Mi is optimal.

Corollary 3. Let the functions v∈H˙p2(Q) and q∈H˙p1(Q) satisfy the equations (46)-Δv+∇q-f=g in Q,∇·v=0 in Q,where f∈H and g∈H˙p0(Q) with gL2≤ε. Then there exist u∈H˙p2(Q) and p∈H˙p1(Q) satisfying (47)-Δu+∇p-f=0 in Q,∇·u=0 in Q.such that (48)u-vHi≤KigL2≤Kiε for i=0,1,2,p-qHi≤MigL2≤Miε for i=0,1

Corollary 4. Let the functions v∈H˙p2(Q) and q∈H˙p1(Q) satisfy the equations (49)-Δv+∇q-f=g in Q,∇·v=0 in Q,where f,g∈H with gL2≤ε. Then there exist u∈H˙p2(Q) and p∈H˙p1(Q) satisfying (50)-Δu+∇p-f=0 in Q,∇·u=0 in Q.such that (51)u-vHi≤KigL2≤Kiε for i=0,1,2,p-qHi=0 for i=0,1.

Corollary 5. Let the functions v∈H˙p2(Q) and q∈H˙p1(Q) satisfy the equations(52)-Δv+∇q-f=g in Q,∇·v=0 in Q,where f∈H˙p0(Q) and g∈H with gL2≤ε. Then there exist u∈H˙p2(Q) and p∈H˙p1(Q) satisfying(53)-Δu+∇p-f=0 in Q,∇·u=0 in Q.such that(54)u-vHi≤KigL2≤Kiε for i=0,1,2,(55)p-qHi=0 for i=0,1.