Stokes operators with variable coefficients and applications

R E S E A R C H

Open Access

Stokes operators with variable coefficients

and applications

Veli B Shakhmurov

*_{Correspondence:}

[email protected] Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul 34959, Turkey Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

Abstract

The stationary and instationary Stokes problems with variable coefficients in abstract Lp_{spaces are considered. The problems contain abstract operators and nonlocal} boundary conditions. The well-posedness of these problems is derived.

MSC: 35Q30; 76D05; 34G10; 35J25

Keywords: Stokes systems; Navier-Stokes equations; abstract differential equations with variable coefficients; semigroups of operators; boundary value problems; maximalLp_regularity

1 Introduction

We consider the initial and boundary value problem (BVP) for the following Stokes type equation with variable coefficients:

∂u

∂t +

n

k= ak(x)

∂u

∂x_k +A(x)u+

n

k= Ak(x)

∂u

∂xk

+∇ϕ=f(x,t), divu= , (.)

mkj

i=

αkji

∂iu

∂xi_k(Gk) +βkji

∂iu

∂xi_k(Gkb)

= , k= , , . . . ,n,j= , , (.)

u(x, ) =a(x), x∈G,t∈(,T), (.)

where

x= (x,x, . . . ,xn)∈G= n

k=

(,bk), mkj∈ {, },

Gk= (x,x, . . . ,xk–, ,xk+, . . . ,xn),

Gkb= (x,x, . . . ,xk–,bk,xk+, . . . ,xn),

A=A(x) andAk=Ak(x) are linear operators in a Banach spaceE,αkji,βkjiare complex

numbers, andakare complex-valued functions. Heref= (f(x,t),f(x,t), . . . ,fn(x,t))

repre-sents a given,adenotes the initial data and

u=u(x,t),u(x,t), . . . ,un(x,t)

, ϕ=ϕ(x,t)

represent the unknown functions. Moreover,uj(x,t),fj(x,t), andϕ(x,t) areE-valued

func-tions. This problem is characterized by the presence of abstract operator functions and complex-valued variable coefficients in the principal part. Moreover, boundary conditions are nonlocal, generally. The existence, uniqueness, and coercive estimates of maximal reg-ular solution of problem (.)-(.) are obtained. Since the Banach spaceEis arbitrary and Ais a possible linear operator, by choosingEand operatorsA,Akwe can obtain maximal

regularity properties for numerous class of Stokes type problems with variable coefficients. ForE=C,ak(x)≡–,A=κ>  problem (.)-(.) is reduced to nonlocal Stokes problem

∂u

∂t –u+κu+∇ϕ=f(x,t), divu= ,

mkj

i=

αkji

∂iu

∂xi_k(Gk,t) +βkji

∂iu

∂xi_k(Gkb,t)

= , (.)

u(x, ) =a(x), x∈G,t∈(,T),k= , , . . . ,n,j= , ,

whereCis the set of complex numbers. Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems were extensively studied,e.g., in [–]. There is an extensive literature on the solvability of the initial value problems (IVPs) for the Stokes equation (see,e.g., [, , ] and further papers cited there). Solonnikov [] proved that for everyf ∈Lp_(×(,T);R_{) =B(p),p∈(,∞) the instationary Stokes} problem

∂u

∂t –u+∇ϕ=f(x,t), divu= , u|∂= , (.)

u(x, ) = , x∈,t∈(,T)

has a unique solution (u,∇ϕ) so that

∂_∂u_t

B(p)

+∇u_B(p)+∇ϕB(p,q)≤CfB(p,q).

Then Giga and Sohr [] improved the result of Sollonikov for spaces with different ex-ponents in space and time,i.e., they proved that for everyf ∈Lp_{(,T; (L}q₍₎₎n_{) there is a}

unique solution (u,∇ϕ) of problem (.) so that

∂_∂u_t

B(p,q)

+∇u_B(p,q)+∇ϕB(p,q)≤CfB(p,q), (.)

where

B(p,q) =Lp,T;Lq()n, p,q∈(,∞).

Moreover, the estimate obtained was global in time,i.e., the constantC=C(,p,q) is inde-pendent ofTandf. To derive the globalLp_-Lq_{estimates (.), Giga and Sohr used abstract}

(see,e.g., []). Consider first at all, the stationary version of problem (.)-(.),i.e., con-sider the abstract Stokes problem

n

k= ak(x)

∂u

∂x

k

+A(x) +λu+

n

k= Ak(x)

∂u

∂xk

+∇ϕ=f(x), divu= , x∈G, (.)

mkj

i=

αkji

∂iu

∂xi_k(Gk) +βkji

∂iu

∂xi_k(Gkb)

= , k= , , . . . ,n,j= , , (.)

λis a complex number. By applying the corresponding projection transformationP, (.)-(.) can be reduced to the following problem:

P

n

k= ak(x)

∂_u

∂x_k +P

A(x) +λu+P

n

k= Ak(x)

∂u

∂xk

=f(x), x∈G, (.)

Lkju= mkj

i=

αkji

∂i_u ∂xi k

(Gk) +βkji ∂i_u ∂xi k

(Gkb)

= , k= , , . . . ,n,j= , . (.)

LetOqdenote the operator generated by problem (.)-(.),i.e., letOqbe a Stokes

oper-ator in theE-valued solenoidal spaceLqσ(G;E) defined by

D(Oq) =

W,q σ

G;E(A),En= u∈W,q_G;E(A),En_,L

kju= ,divu= 

,

Oqu=P n

k= ak(x)

∂_u

∂xu+PAu+P

n

k= Ak(x)

∂u

∂xk

.

We prove thatOqis a positive operator and –Oqis a generator of an analytic semigroup

inLqσ(G;E). In other words, we consider the instationary Stokes problem (.)-(.) and

prove the well-posedness of this problem. We prove that there is a unique solution (u,∇ϕ) of problem (.)-(.) forf ∈(Lp_(,T;X

q))n,a∈Yp,q, and the following estimate holds:

∂u

∂t

Lp,q

+

n

k=

∂u

∂x

k

Lp,q

+AuLp,q+∇ϕLp,q≤CfLp,q+aY_p,q, (.)

whereXqis the class ofE-valuedLq-spaces andYp,qis a corresponding interpolation space.

The estimate (.) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem. Finally, we give some application of this ab-stract Stokes problem to anisotropic Stokes equations and systems of equations. Note that the abstract Stokes problem with constant coefficients was studied in [].

2 Definitions and background

LetEbe a Banach space.Lp_{(;E) denotes the space of strongly measurableE-valued}

func-tions that are defined on the measurable subset⊂Rnwith the norm

fLp=fLp_(;E)=

f(x)p_Edx



p

The Banach spaceEis called an UMD space if the Hilbert operator

(Hf)(x) =lim ε→

|x–y|>ε

f(y) x–ydy

is bounded inLp(R,E),p∈(,∞) (see,e.g., []). UMD spaces includee.g. Lp,lp spaces,

and Lorentz spacesLpq,p,q∈(,∞).

Let

Sψ= λ∈C,|argλ| ≤ψ, ≤ψ<π

,

Sψ,κ= λ∈Sψ,|λ|>κ> 

.

A linear operatorAis said to beψ-positive in a Banach spaceEwith boundM>  if D(A) is dense onEand(A+λI)–B(E)≤M( +|λ|)–for anyλ∈Sψ, ≤ψ<π, where

Iis the identity operator inE, andB(E) is the space of bounded linear operators inE. It is known [, §..] that there exist fractional powersAθ _{of a positive operatorA. Let}

E(Aθ_{) denote the spaceD(A}θ_{) with norm}

uE_(Aθ₎=

up+Aθupp_{, ≤p<∞,  <θ<∞.}

LetEandEbe two Banach spaces. By (E,E)θ,p,  <θ< , ≤p≤ ∞, will be denoted

the interpolation spaces obtained from{E,E}by theK-method [, §..].

Let N denote the set of natural numbers. A set ⊂B(E,E) is called R-bounded (see, e.g., []) if there is a positive constantC such that for allT,T, . . . ,Tm ∈and

u,u, . . . ,um∈E,m∈N,

m

j=

rj(y)Tjuj E

dy≤C

m

j= rj(y)uj

E

dy,

where{rj}is a sequence of independent symmetric{–, }-valued random variables on. The smallestCfor which the above estimate holds is called aR-bound of the collection

and denoted byR().

A seth⊂B(E,E) is called uniformR-bounded inh, if there is a constantC

indepen-dent onh∈σ⊂Rsuch that

m

j=

rj(y)Tj(h)uj E

dy≤C

m

j= rj(y)uj

E

dy

for all T(h),T(h), . . . ,Tm(h) ∈h and u,u, . . . ,um ∈ E, m∈ N. It is implied that sup_h∈QR(h)≤C.

Theψ-positive operatorAis said to beR-positive in a Banach spaceEif the setLA= {ξ(A+ξ)–_:ξ∈S

ψ}, ≤ψ<πisR-bounded.

The operatorA(t) is said to beψ-positive inEuniformly with respect totwith bound M>  ifD(A(t)) is independent ont,D(A(t)) is dense inEand(A(t) +λ)– _≤ M

+|λ|for all

LetEandEbe two Banach spaces andEcontinuously and densely embedded intoE. Letbe a domain inRnandmis a positive integer.Wm,p(;E,E) denotes the space of all functionsu∈Lp_(;E

) that have generalized derivatives∂

m_u

∂xm_k ∈L

p_{(;E) with the norm}

uWm,p_(;E,E)=uLp_(;E)+

n

k=

∂mu

∂xm k

Lp_(;E)

<∞.

Forn= ,= (a,b),a,b∈Rthe spaceWm,p_{(;E,E) will be denoted byW}m,p_{(a,b;E,E).}

ForE=Ethe spaceWm,p_{(;E,E) is denoted byW}m,p_(;E).

Let Ls,p(;E), –∞<s<∞, denote theE-valued Liouville space of order ssuch that L,p_{(;E) =L}p_{(;E). It is known that ifEis a UMD space, thenL}m,p_{(;E) =W}m,p_(;E)

for positive integerm(see,e.g., [, §].

LetLs,p_{(;E,E) denote a Liouville-Lions type space,i.e.,}

Ls,p(;E,E) = u∈Ls,p(;E)∩Lq(;E),

uLs,p_(;E,E)=uLp_(;E)+uLs,p_(;E)<∞.

Lqσ(;E) denote the E-valued solenoidal space, i.e., the closure of (C∞σ(;E))n in

(Lq_(;E))n_{, where}

C_∞σ(;E) = u∈C∞(;E),divu= 

.

Sometimes we use one and the same symbolCwithout distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, sayα, we writeCα.

The embedding theorems in vector-valued spaces play a key role in the theory of DOEs. For estimating lower order derivatives we use the following embedding theorems from [].

Theorem A Suppose the following conditions are satisfied:

() Eis a UMD space andAis anR-positive operator inE; () α= (α,α, . . . ,αn)andmis a positive integer such thatκ=

n

k=|α|m ≤,

≤μ≤ –κ, <p<∞, <h≤h,his a fixed positive number;

() ⊂Rnis a region such that there exists a bounded linear extension operator from Wm,p_(;E(A),E)toWm,p_(Rn_;E(A),E).

Then the embedding Dα_Wm,p_{(;E(A),E)⊂L}p_(;E(A–κ–μ_{))is continuous and for all}

u∈Wm,p_{(;E(A),E)the following uniform estimate holds:}

Dαu_Lp_(;E(A–κ–μ₎₎≤h

μ

uWm,p_(;E(A),E)+h–(–μ)uLp_(;E).

Remark . If⊂Rnis a region satisfying the strongl-horn condition (see [, §]),E=R, A=I, then forp∈(,∞) there exists a bounded linear extension operator fromWm,p_{() =}

Theorem A Suppose all conditions of TheoremAare satisfied and <μ≤ –κ. More-over,letbe a bounded region and A–_∈σ

∞(E).Then the embedding

Dα_Wm,p_;E(A),E⊂Lp_;EA–κ–μ

is compact.

Theorem A Suppose all conditions of TheoremAsatisfied and <μ≤ –κ.Then the embedding

Dα_Wm,p_;E(A),E⊂Lp_;E(A),E

κ,p

is continuous and there exists a positive constant Cμsuch that for all u∈W_pl(;E(A),E)

the uniform estimate holds

Dαu_Lp_{(;(E(A),E)κ,p)}≤Cμ

hμuWm,p_(;E(A),E)+h–(–μ)uLp_(;E).

From [, Theorem .] we obtain the following.

Theorem A Let E be a Banach space,A be aϕ-positive operator in E with bound M, ≤ϕ<π.Let m be a positive integer,  <p<∞andα∈(__p,__p +m).Then,forλ∈Sϕ,

an operator–A  

λ generates a semigroup e–xA

 

λ which is holomorphic for x> .Moreover, there exists a positive constant C(depending only on M,ϕ,m,αand p)such that for every u∈(E,E(Am))α

m–mp ,pandλ∈Sϕ,

∞



Aα λe

–xA

 

λupdx≤Cup (E,E(Am₎₎

α

m– mp,p

+|λ|αp–_up

E

.

3 The stationary Stokes system with variable coefficients

In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem (.)-(.).

First at all, we consider the BVP for the variable coefficient differential operator equation (DOE)

n

k= ak(x)

∂u

∂x_k +

A(x) +λu+

n

k= Ak(x)

∂u

∂xk

=f(x), x∈G,

mkj

i=

αkji

∂iu

∂xi_k(Gk) +βkji

∂iu

∂xi_k(Gkb)

= , k= , , . . . ,n,j= , ,

(.)

where A(x) and Ak(x) are linear operators in a Banach spaceE,ak are complex-valued

functions, andλis a complex parameter.

Maximal regularity properties for DOEs studied,e.g., in [, , , , –]. Nonlocal BVPs for PDE were studied in [].

Letωki=ωki(x),i= , , be roots of the equations

Letαkj=αkjmkandβkj=βkjmk and

ηk(x) =

αk(–ωk)mk βkω

mk

k

αk(–ωk)mk βkωmkk

.

Condition . Assume:

() Eis a UMD space andA(x)is a uniformlyR-positive operator inEforϕ∈[,π); () ak(x)∈C(G¯),ak(Gi) =ak(Gib),ak= ,ak∈S(ϕ)∩C/R–for allx∈G,ϕ+ϕ<π;

() A(x)A–_(x¯)∈C(G¯_;B(E)),A(G

i) =A(Gib),Ai(x)A–(



–ν)(x)∈C(m)(G¯;B(E)), <_ν<  ;

() |αkjmj|+|βkjmj|> ,ηk(x)= ,k,i= , , . . . ,n,j= , ,p∈(,∞).

Remark . Letak= –bk(x), wherebkare real-valued positive functions and. Then

Con-dition . is satisfied.

Remark . The conditionsak(Gi) =ak(Gib),A(Gi) =A(Gib) are given due to the

non-locality of the boundary conditions. For local boundary conditions these assumptions are not required.

From [, Theorem .] we have the following.

Theorem . Suppose Condition.is satisfied.Then problem(.)has a unique solu-tion u∈W,p_{(G;E(A),E)for f ∈L}p_{(G;E)and for sufficiently largeλ∈S}

ϕ.Moreover,the

following coercive uniform estimate holds:

n

k= 

i=

|λ|–i∂

i_u

∂xi_k

Lq_(G;E)

+AuLq_(G;E)≤CfLq_(G;E).

Consider the differential operator Q=Qq in Lq(G;E)generated by problem(.)-(.),

i.e.,

D(Q) =W,qG;E(A),E,Lkj

, Qu=

n

k= ak(x)

∂_u

∂x +A(x) +

n

k= Ak(x)

∂u

∂xk

.

LetBq=B(Lq(G;E)). From Theorem . we obtain the following.

Result . Forλ∈Sψ,κthere is a resolvent (Q+λ)–and the following uniform coercive

estimate holds:

n

k= 

i=

|λ|–i ∂

i

∂xi k

(Q+λ)–

Bq

+A(Q+λ)–_B

q≤C.

LetEbe a Banach space andXq=Xq(G) = (Lq(G;E))ndenote the class ofE-valued system

of functionf= (f(x),f(x), . . . ,fn(x)) with norm

fXq=

_n

i=

fiq_Lq_(G;E)



q

Xqσ =Lqσ(G;E) denote theE-valued solenoidal space andAbe a positive operator inE.

The spaces (Ls,q_(G;E))n_{, (L}s,q_(G;E(A),E))n_{will be denoted byX}s

q(G) andXqs(G,A).

Consider the problem

n

k= ak(x)

∂_u

∂x

k

+A(x) +λu+

n

k= Ak(x)

∂u

∂xk

=f(x), x∈G,

mkj

i=

αkji

∂i_u ∂xi

k

(Gk) +βkji ∂i_u ∂xi k

(Gkb)

= , k= , , . . . ,n,j= , ,

(.)

wheref= (f(x),f(x), . . . ,fn(x)),A(x) andAk(x) are linear operators in a Banach spaceE,ak

are complex-valued functions, andλis a complex parameter. From Theorem . we obtain the following result.

Result . Suppose Condition . is satisfied. Then problem (.) has a unique solution u∈X_q(G,A) forf ∈Xqand for sufficiently largeλ∈Sϕ. Moreover, the following coercive

uniform estimate holds:

n

k= 

i=

|λ|–i∂

i_u

∂xi_k

Xq

+AuXq≤CfXq.

Consider the differential operatorB=BqinXqgenerated by problem (.), forλ= ,i.e.,

D(B) =X_q(G,A), Bu=

n

k= ak(x)

∂u

∂x +A(x) +

n

k= Ak(x)

∂u

∂xk

.

From Result . we obtain the following uniform coercive estimate:

n

k= 

i=

|λ|–i ∂

i

∂xi k

(B+λ)–

B(Xq)

+A(B+λ)–_B(X

q)≤C. (.)

Consider the space

Yq(A) = u∈Xq

E(A),divu∈Lq(G;E),

uYq(A)=

uq_Xq(E(A))+divuq_Lq_(G;E)



q_.

Yqbecomes a Banach space with this norm.

It is known that (see,e.g., [, ]) the vector fieldu∈(Lq_(G))n_{has a Helmholtz}

decomposi-tion. In the following theorem we generalize this result for theE-valued function spaceXq.

Theorem . Let E be an UMD space and q∈(,∞).Then u∈Xqhas a Helmholtz

de-composition,i.e.,there exists a linear bounded projection operator Pq from Xq onto Xqσ

with null space N(Pq) ={∇ϕ∈Xq:ϕ∈Llocq (G;E)}.In particular,all u∈Xqhave a unique

decomposition u=u+∇ϕwith u∈Xqσ,u=Pqu so that

∇ϕX_q+uXq≤CuXq. (.)

For proving Theorem . we need some lemmas. Consider the equation

–u+A(x)u=f(x), x∈G. (.)

Lemma . Let E be an UMD space,A a R-positive operator in E and q∈(,∞).Then,for f ∈X–

q ,problem(.)has a unique solution u∈Xq(A)and the following coercive estimate

holds:

u_X

q(G)+AuXq(G)≤CfXq–(G). (.)

Proof Consider the problem

–u+Au=f˜(x), x∈Rn, (.)

where f˜(x) is an extension of the function f(x) on Rn_{. Then, by using the Fourier}

in-version formula, operator-valued multiplier theorems inLq _{spaces, and by reasoning as}

in [, Theorem .] we see that problem (.) has a unique solutionu˜∈X

q(Rn,A) for

f ∈X–

q (Rn;E) and the following coercive estimate holds:

˜u_X

q(Rn)+Au˜Xq(Rn)≤C˜fX–q (Rn).

This fact implies that the functionuwhich is a restriction of u˜ onGis a solution of problem (.). The estimate (.) is obtained from the above estimate.

Letν= (ν,ν, . . . ,νn) be a unit normal to the boundaryof the domainGandfνis a

normal component off = (f,f, . . . ,fn)∈Xq(G) on,i.e.,

fν=

_n

k=

νkfk

.

Here and hereafterE∗ will denoted the conjugate ofE, and (,·) (resp.,·) denotes the

duality pairing of functions onG(resp.).

By reasoning as in [, Lemma ] we get the following.

Lemma . C_∞(G;E)is dense in Yq(A).

Proposition . There exists a unique bounded linear operator u→uν from Yq(A),q∈

(,∞)onto

Wq() =W–/q,q

,EA∗,E∗_q,/q,E∗

such that

(uν,υ|) = (divu,υ) + (u,∇υ), υ∈W,q

and the following estimate holds:

uνWq()≤C

uXq+divuLq(G;E)

, (.)

where

 q+

 q= .

Proof Foru∈Yq(A) consider the linear form

Tu(υ) = (divu,) + (u,∇), ∈Wq _,

G,E(A),E, ν=υ. (.)

By virtue of the trace theorem inW,q_{(,a;E(A),E), the interpolation of intersection and}

dual spaces (see,e.g., [, §.., .., ..]) and by a localization argument we obtain the result that the operator→νis a bounded linear and surjective fromW,q

(G;E(A),E) onto

Z() =W–/q,q,E(A),E_q,/q,E

.

Hence, we can find for eachυ∈Z() an element∈W,q_{(G;E(A),E) so that}

ν=υ,_W,q_(G;E(A),E)≤CυZ().

Therefore, from (.) we get

Tu(υ)≤

uXq+divuLq(G;E)

_W,q_(G;E(A),E)

≤CuXq+divuLq(G;E)

υZ().

This implies the existence of an element

uν∈

Z()∗=W–/q,q,EA∗,E∗_q,/q,E∗

such that

uν,υ=Tu(υ) forυ∈Z()

and

uν(Z())∗≤C

uXq+divuLq(G;E)

.

Thus, we have proved the existence of the operator u→uν. The uniqueness follows

from Lemma ..

Result . Assume the conditions of Proposition . are satisfied. Then

Xqσ ⊂Xqν={u∈Xq,divu= ,uν= }

andXqνis a closed subspace ofXq.

Letf ∈Xqandf = (f(x),f(x), . . . ,fn(x)). Consider the following problem:

–u+Au=divf(x), x∈G,

∂u

∂xk

(Gk) =fk(Gk),

∂u

∂xk

(Gkb) =fk(Gkb), k= , , . . . ,n. (.)

Lemma . Let E be an UMD space,A a R-positive operator in E and q∈(,∞).Then,for f ∈Xq,problem(.)has a unique solution u∈W,q(G;E(A),E)and the coercive estimate

holds

uW,q_(G;E)+AuLq_(G;E)≤CdivfW–,q_(G;E)+fνW_q(). (.)

Proof Consider the equation

–υ+Aυ=divf(x), x∈G. (.)

By Lemma ., problem (.) has a unique solutionυ∈L,q_{(G;E(A),E) forf ∈X} qand

the following estimate holds:

υL,q_(G;E)+AυLq_(G;E)≤CdivfL–,q_(G;E). (.)

Consider now the BVP

–w+Aw= , x∈G,

∂w

∂xk

(Gk) =fk(Gk) –

∂υ ∂xk

(Gk), (.)

∂w

∂xk

(Gkb) =fk(Gkb) – ∂υ ∂xk

(Gkb), k= , , . . . ,n.

By using Theorem ., Result ., and Proposition . we conclude that problem (.) forf ∈Xqhas a unique solutionw∈W,q(G;E(A),E) and the following coercive estimate

holds:

wW,q_(G;E)+A

 _w

Lq(G;E)≤CfνWq(). (.)

Then we conclude that problem (.) has a unique solutionϕ(x) =υ(x)+w(x) and (.),

(.) imply the estimate (.).

Result . For the case ofA=κ>  we obtain from (.), (.), and (.) that the prob-lem

∂w

∂xk

(Gk) =fk(Gk) –

∂υ ∂xk

(Gk), (.)

∂w

∂xk

(Gkb) =fk(Gkb) – ∂υ ∂xk

(Gkb), k= , , . . . ,n

has a unique solutionw∈W,q_{(G;E) forf∈X}

qand the following estimate holds:

w_W,q_(G;E)≤C

fν–

∂υ ∂ν

W–/q,q(,E∗)

. (.)

Result . For the case ofA=κ>  we obtain from Lemma . and from (.), (.) that the problem

(–+κ)υ=divf(x), x∈G (.)

has a unique solutionυ∈W,q_{(G;E) forf∈X}

qand the following estimate holds:

υW,q_(G;E)≤Cdivf_X–

q . (.)

By (.),div(f–∇υ) = . So, by (.) and Proposition . we get

wW,q_(G;E)≤CfX_q. (.)

From Results ., ., and estimate (.) we obtain

Result . The problem

(–+κ)w=divf(x), x∈G,

∂w

∂xk

(Gk) =fk(Gk) –

∂υ ∂xk

(Gk), (.)

∂w

∂xk

(Gkb) =fk(Gkb) – ∂υ ∂xk

(Gkb), k= , , . . . ,n

has a unique solutionu=w+υ∈W,q_{(G;E) forf ∈X}

qand the estimate holds

u_W,q_(G;E)≤CfX_q. (.)

Consider the operatorP=Pqdefined by

D(P) =Xq, Pf =f–grad(υ+w),

wherewandυare solutions of problems (.), (.), respectively. It is clear that we have the following.

Lemma . Let E be an UMD space and q∈(,∞).Then PqXqis a closed subspace of Xq.

Lemma . Let E be an UMD space and q∈(,∞).Then the operator Pq is a bounded

Proof The linearity of the operatorPis clear by construction. Moreover, by Result . we have

PfXq≤ fXq+graduXq≤CfXq. (.)

Ifdivf(x) =  then by Result . we getυ= . Moreover, by the estimate (.) we obtain

w= ,i.e.,Pf =f.

Lemma . Assume E is an UMD space and q∈(,∞).Then the conjugate of Pqis defined

as P_q∗=Pq, _q+_q = and is bounded linear in(Lq(G;E∗))n.

Proof It is known (see,e.g., [, ]) that the dual space ofLq_{(G;E) isL}q_(G;E∗_{). Since}

C_∞(G;E∗) is dense inLq(G;E∗) we have only to showP∗_qϕ=Pqϕfor anyϕ∈(C∞(G;E∗))n. But this is deriving by reasoning as in [, Lemma ]. Moreover, by Lemma . the dual operatorP∗_qis a bounded linear inLq_(G;E∗_).

Let

Wq= ∇ϕ:ϕ∈Wq,(G;E)

,

(PqXq)⊥= f ∈

LqG;E∗n,f,υ=  for anyυ∈PqXq

.

From Lemmas ., . we obtain the following.

Result . AssumeEis an UMD space andq∈(,∞). Then any elementf ∈Xquniquely

can be expressed as a sum of elements ofPqXqandWq.

In a similar way as Lemmas ,  of [] we obtain, respectively, the following.

Lemma . Assume E is an UMD space and q∈(,∞).Then

(PqXq)⊥=Wq,

 q+

 q = .

Lemma . Assume E is an UMD space and q∈(,∞).Then

X_qσ⊥ =Wq,

 q+

 q= .

Now we are ready to prove Theorem ..

Proof of Theorem. From Lemmas ., . we getXqσ = (PqXq)⊥. Then, by construction

ofPq, we haveXq=Xqσ ⊕Wq. By Lemmas ., ., we obtain the estimate (.).

More-over, by Lemma .,Wqis a close subspace ofXq. Then it is known that the dual space of

the quotient spaceXq/Wq isWq⊥. In view of first assertion we haveXq/Wq=Xqσ and by

Theorem . Let Condition.hold.Then problem(.)-(.)has a unique solution u∈ X

qfor f ∈Xq,ϕ∈W,q(G;E),λ∈Sψ,κ,and the following coercive uniform estimate holds:

n

k= 

i=

λ–i∂

i_u

∂xi k

Xq

+AuXq+∇ϕXq≤CfXq. (.)

Proof By virtue of Result ., we find that problem (.) has a unique solutionu∈X

q(G,A)

forf ∈Xqand for sufficiently largeλ∈Sϕ. Moreover, the following coercive uniform

esti-mate holds:

n

k= 

i=

|λ|–i∂

i_u

∂xi_k

Xq

+AuXq≤CfXq.

By applying the operatorPqto problem (.)-(.) we get the Stokes problem (.)-(.).

It is clear that

D(Oq) =D(B)∩Xqσ,

whereOqis the Stokes operator andBis a operator generated by problem (.) forλ= .

Then by Theorem . we obtain the assertion.

Result . From Result . we find thatO=Oqis a positive operator inXqand –O

gen-erate a bounded holomorphic semigroupS(t) =exp(–Ot) fort> .

In a similar way as in [] we show the following.

Proposition . The following estimate holds:

OαS(t)≤Ct–α

forα≥and t> .

Proof From the estimate (.) we see that the operatorOis positive inXq,i.e., forλ∈Sψ,κ,  <ψ<πthe following estimate holds:

(O+λ)–≤M|λ|–,

where the constantMis independent ofλ. Then, by using the Danford integral and oper-ator calculus (see,e.g., in []), we obtain the assertion.

4 Well-posedness of instationary Stokes problems with variable coefficients

Let

B(p,q) =Lp(,T;Xq), D(p,q) =

X_q(A),Xq



p,p.

Theorem . Then,for f ∈B(p,q),ϕ∈Lp(,T;W,q(G;E)),and a∈D(p,q),p,q∈(,∞),

there is a unique solution(u,∇ϕ)of problem(.)-(.)and the following estimate holds:

∂u

∂t

B(p,q) +

n

k=

∂u

∂x

k

B(p,q)

+AuB(p,q)+∇ϕB(p,q)≤C

fB(p,q)+aD(p,q)

. (.)

Proof Problem (.)-(.) can be expressed as the following abstract parabolic problem:

du

dt +Ou=f(t), u() =a. (.)

If we putE=Xqthen by Proposition ., operatorOis positive and generates a bounded

holomorphic semigroup inXq. Moreover, by using [, Theorem .] we see that the

op-eratorOisR-positive in E. SinceE is a UMD space, in a similar way as in [, Theo-rem .] we see that for allf ∈Lp_{(,T;E) anda∈(D(O),E)}

p,p there is a unique solution

u∈W,p_{(,T,D(O),E) of problem (.) so that the following estimate holds:}

du_dt

Lp_(,T;E)

+OuLp_(,T;E)≤CfLp_(,T;E)+a_(D(O),E)



p,p

. (.)

From the estimates (.) and (.) we obtain the assertion.

Remark . There are a lot of positive operators in concrete Banach spaces. Therefore,

putting in (.)-(.) and (.)-(.) concrete Banach spaces instead ofEand concrete posi-tive differential, pseudo differential operators, or finite, infinite matrices, etc. instead ofA, by virtue of Theorem . and Theorem . we can obtain the maximal regularity proper-ties of different class of stationary and instationary Stokes problems, respectively, which occur in numerous physics and engineering problems.

Let us now show some application of Theorem . and Theorem ..

5 Application

Consider the stationary Stokes problem

n

k= ak(x)

∂u(x,y)

∂x_k +

|β|≤m

aβ(x,y)Dβ_yu(x,y) +∇ϕ=f(x,y), divu= , (.)

mkj

i=

αkji

∂iu

∂xi k

(Gk,y) +βkji ∂iu

∂xi k

(Gkb,y)

= , y∈,j= , , (.)

Bju=

|β|≤mj

bjβ(y)Dβyu(x,y)_y∈∂= , x∈G,j= , , . . . ,m, (.)

wheref = (f(x,y,t),f(x,y,t), . . . ,fn(x,y,t)) represents a given and

u=u(x,y),u(x,y), . . . ,un(x,y)

are unknown functions;

x= (x,x, . . . ,xn)∈G= n

k=

(,bk), y= (y, . . . ,yμ)∈⊂Rμ,

Gk= (x,x, . . . ,xk–, ,xk+, . . . ,xn), p∈(,∞),

Gkb= (x,x, . . . ,xk–,bk,xk+, . . . ,xn), q∈(,∞),mkj∈ {, },

αkji,βkjiare complex numbers,aα,bjβare complex-valued functions,Dj= –i_∂y∂_j.

Let˜ =G×,p=(p,p). NowLp₍˜_{) will denote the space of allp-summable}

scalar-valued functions with mixed normi.e., the space of all measurable functionsf defined on˜, for which

f_Lp₍˜₎=

G

f(x,y)pdx

p

p_

dy



p

<∞.

Analogously,Wm,p₍˜_{) denotes the Sobolev space with corresponding mixed norm.}

Xp= (Lp(˜))ndenotes the class of vector function

f =f(x),f(x), . . . ,fn(x)

with norm

fXp= n

i=

fiLp_(G;E)

andXp,m= (W,m,p(˜))n, whereW,m,p(˜) is the anisotropic Sobolev space with mixed

norm.

From Theorem . we obtain the following.

Theorem . Let the following conditions be satisfied:

() is a domain inRμ_{with sufficiently smooth boundary∂,a}

α(x,y)∈C(¯),

aα(Gi,y) =aα(Gib,y)for each|α|= m,y∈,aα∈[L∞+Lrk]()for each

|α|=k< mwithrk≥p,p∈(,∞)andm–k>_rl_k,να∈L∞; () bjβ∈Cm–mj(∂)for eachj,β,mj< m,p∈(,∞);

() fory∈ ¯,ξ∈Rμ_,η∈S(ϕ

),ϕ∈[,π_),|ξ|+|η| = let

η+

|α|=m

aα(y)ξα= ;

() for eachx∈∂the local BVPs in local coordinates corresponding tox

η+

|α|=m

aα

x,ξ,Dμ

ϑ(y) = , y> ,

Bjϑ=

|β|=mj

bjβ

x,ξ,Dμ

ϑ(y)

has a unique solutionϑ∈C(R+)for allh= (h,h, . . . ,hm)∈Rmand forξ∈Rμ–

with|ξ|+|η| = ;

() ak∈C(G¯),ak(Gi,y) =ak(Gib,y),ak= ,ak∈S(ϕ)∩C/R–for allx∈G,ϕ+ϕ<π, y∈,k,i= , , . . . ,n.

Then,problem(.)-(.)has a unique solution u∈X,m

p for f ∈Xp, ϕ∈W,p(˜;E), λ∈Sψ,κ,and the following coercive uniform estimate holds:

n

k= 

i=

|λ|–i∂

i_u

∂xi_k

Xp

+

|α|=m _Dα

yuXp+∇ϕXp≤CfXp. (.)

Proof LetE=Lp_{(). By virtue of [, Theorem ..]L}p_{() is an UMD space. Consider} the operatorAwhich is defined by

D(A) =Wm,p_(;B

ju= ), Au=

|β|≤m

aβ(y)Dβu(y).

Problem (.)-(.) can be rewritten in the form of (.)-(.) forAk= ,k= , , . . . ,n,

whereu(x) =u(x,·) andf(x) =f(x,·) are functions with values inE=Lp_{(). In view of [,} Theorem .] the problem

ηu(y) +

|β|≤m

aβ(x,y)Dβu(y) =f(y),

Bju=

|β|≤mj

bjβ(y)Dβu(y) = , j= , , . . . ,m

(.)

has a unique solution forf ∈Lp_{() and argη}∈_S(ϕ

), |η| → ∞, and the operatorAis R-positive inLp_{. Hence, all conditions of Theorem . are satisfied,i.e., we obtain the} assertion.

Consider now the instationary Stokes problem

∂u

∂t +

n

k= ak(x)

∂_u

∂x

k

+

|β|≤m

aβ(x,y)Dβyu+∇ϕ=f(x,y,t), divu= , (.)

mkj

i=

αkji

∂iu

∂xi_k(Gk,y,t) +βkji

∂iu

∂xi_k(Gkb,y,t)

= , j= , , (.)

u(x,y, ) =a(x,y), t∈(,T),x∈G,y∈, (.)

Bju=

|β|≤mj

bjβ(y)Dβyu(x,y,t)_y∈∂= , x∈G,j= , , . . . ,m, (.)

wheref(x,y,t),u= (x,y,t) are data and solution vector-functions, respectively.

From Theorem . and Theorem . we obtain the following.

Theorem . Assume all conditions of Theorem.are satisfied p,p,q∈(,∞).Then, for f ∈Lq_(,T;X

p) =B(p,p,q),ϕ∈Lq(,T;W,p(˜)),and a∈(Xp,m,Xp)

there is a unique solution(u,∇ϕ)of problem(.)-(.)and the following estimate holds:

∂_∂u_t

B(p,p,q) +

n

k=

∂_∂_xu

k

B(p,p,q)

+

|α|=m

Dα_yu_B(p,p,q)

+∇ϕB(p,p,q)≤C

fB(p,p,q)+aD(p,p,q)

.

Competing interests

The author declares that they have no competing interests.

Received: 30 September 2013 Accepted: 21 April 2014 Published:02 May 2014

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