Parameter-dependent Stokes problems in vector-valued spaces and applications

R E S E A R C H A R T I C L E

Open Access

Parameter-dependent Stokes problems in

vector-valued spaces and applications

Veli B Shakhmurov

Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA, Texas A&M University-Kingsville-2012

*_{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 equations with operator coefficients in abstract function spaces are studied. The problems are considered in the whole space, and equations include small parameters. The uniform separability of these problems is established.

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

Keywords: Stokes systems; Navier-Stokes equations; differential equations with small parameters; semigroups of operators; boundary value problems;

differential-operator equations; maximalLp_regularity

1 Introduction

We consider the initial value problem (IVP) for the following Stokes equation with small parameter:

∂u

∂t –εu+Au+∇ϕ=f(x,t), x∈R

n_{,t∈(,T), (.)}

divu= , u(x, ) =a(x), (.)

whereεu=

n

k=εk∂ _u

∂x_k,Ais a linear operator in a Banach spaceEandεkare a small

pos-itive parameters. Hereu= (u(x,t),u(x,t), . . . ,un(x,t)),ϕ=ϕ(x,t) areE-valued unknown

solutions,f= (f(x,t),f(x,t), . . . ,fn(x,t)) is a given function anda= (a(x),a(x), . . . ,an(x))

is an initial date. This problem is characterized by the presence of an abstract operatorA and small termsεkwhich correspond to the inverse of Reynolds numberRevery large. We

prove that problem (.)-(.) has a unique strong maximal regular solutionuon a time interval [,T] independent ofεk. Forεk= ,E=C,A=b, problem (.)-(.) is reduced to

the Stokes problem

∂u

∂t –u+bu+∇ϕ=f(x,t), divu= , (.)

u(x, ) =a(x), x∈Rn,t∈(,T), (.)

whereCis the set of complex numbers andbis a positive constant.

Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems has been extensively studied,e.g., in [–]. There is an extensive literature on the solvability of the 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 time-dependent 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 Solonnikov for spaces with different expo-nents in space and time,i.e., they proved that forf ∈Lp(,T; (Lq())n) =B(p,q) 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), p,q∈(,∞). (.)

Moreover, the estimate obtained was global in time,i.e., the constantC=C(,p,q) is independent ofT andf. To derive globalLp_–Lq_{estimates (.), Giga and Sohr used the}

abstract parabolic semigroup theory inUMD-type Banach spaces. Estimate (.) allows to study the existence of a solution and regularity properties of the corresponding Navier-Stokes problem (see,e.g., []).

In this paper, first we consider the following differential operator equation (DOE) with small parameters:

–εu+ (A+λ)u=f(x), x∈Rn, (.)

whereAis a linear operator in a Banach spaceE,εk are positive andλis a complex

pa-rameter.

We show that forf ∈Wm,q(Rn;E),λ∈Sψ, problem (.) has a unique solutionu belong-ing toW+m,q_(Rn_{;E(A),E) and the uniform coercive estimate holds}

n

k=

m+

i=

ε

i m+ k |λ|–

i m+∂

i_u

∂xi_k

Lq_(Rn_;E)

+AuLq_(Rn_;E)≤Cf_Wm,q_(Rn_;E),

whereC(q) is independent ofε,ε, . . . ,εn,λandf.

We consider, then, the stationary abstract Stokes problem with small parameters

–εu+Au+∇ϕ=f(x), divu= , x∈Rn, (.)

wheref = (f(x),f(x), . . . ,fn(x)) is data andu= (u(x),u(x), . . . ,un(x)) is a solution. By

problem:

–Pεu+Au=f(x), x∈Rn. (.)

LetOε,qdenote the operator generated by problem (.),i.e.,Oε,qis a Stokes operator in

solenoidal spaceLqσ(Rn;E) defined by

D(Oε,q) =

Wσ,q

Rn;E(A),En=u∈W,qRn;E(A),En,divu= , Oε,qu= –Pεu+Au.

We prove that the operatorOε,qis uniformly positive and also is a generator of an analytic

semigroup inLqσ(Rn;E). Finally, the instationary Stokes problem (.)-(.) is considered and the well-posedness of this problem is derived. In application we show the separability properties of the anisotropic stationary Stokes operator in mixedLp_{spaces and maximal}

regularity properties of infinity system of instationary Stokes equations inLp_spaces.

2 Notations and background

LetEbe a Banach space and letLp_{(;E) denote the space of strongly measurableE-valued}

functions that are defined on the measurable subset⊂Rn_{with the norm}

fLp=f_Lp_(;E)=

f(x)p_Edx



p

, ≤p<∞.

The Banach spaceEis called aUMD-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., []).UMDspaces include,e.g.,Lp,lp spaces

and Lorentz spacesLpq,p,q∈(,∞).

LetEandEbe two Banach spaces.B(E,E) denotes the space of bounded linear

oper-ators fromEintoEendowed with the usual uniform operator topology. ForE=E=E,

it is denoted byB(E). Now (E,E)θ,p,  <θ < , ≤p≤ ∞, denotes interpolation spaces

defined by theKmethod [, §..]. Let

Sψ=

λ∈C,|argλ| ≤ϕ∪ {}, ≤ψ<π.

A linear operatorAis said to beψ-positive in a Banach spaceEwith boundM>  if the domainD(A) is dense onEand(A+λI)–

B(E)≤M( +|λ|)–for anyλ∈Sψ, ≤ψ<π, whereIis the identity operator inE. It is known [, §..] that there exist the fractional powersAθ_{of the positive operatorA. LetE(A}θ_{) denote the spaceD(A}θ_{) with the norm}

u_E(Aθ₎=

up+Aθupp_{, }≤_p<∞_{,  <θ<}∞_.

Ndenotes the set of natural numbers. A setG⊂B(E,E) is calledR-bounded (see,e.g.,

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 smallestC, for which the estimate above holds, is called anR-bound of the collection Gand denoted byR(G).

A set Gh ⊂B(E,E) depending of parameter h∈ Q is called uniform R-bounded

with respect to h if there is a constant C, independent of h ∈ Q such that for all T(h),T(h), . . . ,Tm(h)∈Ghandu,u, . . . ,um∈E,m∈N,

m j=

rj(y)Tj(h)uj

E

dy≤C

m j=

rj(y)uj

E

dy.

It implies thatsup_h∈QR(Gh)≤C.

Theψ-positive operatorAis said to beR-positive in a Banach spaceEif the setLA=

{ξ(A+ξ)–_:ξ∈S

ψ}, ≤ψ<π, isR-bounded.

The operatorA(t) is said to beψ-positive inEuniformly with respect totwith bound M>  ifD(A(t)) is independent oft,D(A(t)) is dense inEand(A(t) +λ)– ≤M( +|λ|)– for allλ∈Sψ, ≤ψ<π, whereMdoes not depend oftandλ.

LetEandEbe two Banach spaces and letEbe continuously and densely embedded

intoE. Letbe a measurable set inRn_{andmbe a positive integer.W}p,m_(;E

,E) denotes

the class of all functionsu∈Lp_(;E

) that have the generalized derivatives∂

m_u ∂xm_k ∈L

p_(;E)

with the norm

uWp,m_(;E,E)=u_Lp_(;E)+

n

k=

∂_∂m_xmu

k

Lp_(;E) <∞.

Forn= ,= (a,b),a,b∈R, the spaceWp,m_(;E

,E) is denoted byWp,m(a,b;E,E).

ForE=Ethe spaceWp,m(;E,E) is denoted byWp,m(;E).

LetHq,s_(Rn_{;E), –∞<s<∞denote anE-valued Liouville space of orders,i.e.,}

Hq,sRn;E=u∈LqRn;E,uHq,s_(Rn_;E)=F– +|ξ| s

_Fu

Lq_(Rn_;E)<∞

,

whereFandF–_{denote the Fourier and inverse Fourier transforms, respectively.}

LetHq,s_(Rn_;E

,E) be a Liouville-Lions type space,i.e.,

Hq,sRn;E,E

=u∈Hq,sRn;E∩LqRn;E

,

uHq,s_(Rn_;E,E)=u_Lq_(Rn_;E)+u_Hq,s_(Rn_;E)<∞.

Forε= (ε,ε, . . . ,εn) we define the parameter-dependent norm inHq,s(Rn;E,E) such

that

u_Hq,s

ε (Rn;E,E)=uLq(Rn;E)+

F–  + n k=

εkξk

s  Fu

Sometimes we use one and the same symbolCwithout distinction 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.

3 Boundary value problems for abstract elliptic equations

In this section, we derive the maximal regularity properties of problem (.).

BVPs for DOEs were studied,e.g., in [, , –]. For references, see,e.g., []. From [, Theorem .] we have the following result.

Theorem . Let E be a UMD space and let A be an R-positive operator in E for≤ψ<π. Then problem(.)has a unique solution u∈Wq,_(Rn_{;E(A),E)for f ∈L}q_(Rn_;E)andλ∈S

ψ.

Moreover,the following uniform coercive estimate holds:

n

k= 

i=

ε

i

 k|λ|–

i

∂ i_u

∂xi_k

Lq_(Rn_;E)

+AuLq_(G;E)≤Cf_Lq_(Rn_;E) (.)

with C(q)independent ofε,ε, . . . ,εn,λand f.

Consider the differential operatorQε=QεqinLq(Rn;E) generated by problem (.),i.e.,

D(Qε) =Wq,

Rn, Qεu= –εu+Au.

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

Result . Forλ∈Sψ, there is a resolvent (Qε+λ)–satisfying the uniform estimate

n

k= 

i=

|λ|–i_ε

i

 k

∂i

∂xi_k(Qε+λ)

–

Bq

+A(Qε+λ)–_B q≤C.

Next we show the smoothness of problem (.). The main result is the following.

Theorem . Assume that E is a UMD space,A is an R-positive operator in E,q∈(,∞) and m is a positive integer.

Then,for all f∈Wq,m_(Rn_;E),λ∈S

ψ,problem(.)has a unique solution u that belongs

to Wq,+m_(Rn_{;E(A),E)and the following uniform coercive estimate holds:}

n

k=

m+

i=

ε

i m+ k |λ|–

i m+∂

i_u

∂xi_k

Wq,m_(Rn_;E)

+AuLq_(Rn_;E)≤Cf_Wq,m_(Rn_;E) (.)

with C=C(q,A)independent ofε,ε, . . . ,εn,λand f.

Proof A solution of equation (.) is given by

u(x) =F–L–(λ,ε,ξ)Ff=  (π)n/

Rne

whereL(λ,ε,ξ) =A+n_k=εkξk+λandfˆ(ξ) =Ff. It follows from the expression above that n k= m+ i= ε i m+ k |λ|

–_mi_+∂iu

∂xi k

Lq_(Rn_;E)

+AuLq_(Rn_;E)

= n k= m+ i= ε i m+ k |λ|–

i m+_F–_ξi

kL–(λ,ε,ξ)fˆLq(Rn_;E)

+F–AL–(λ,ε,ξ)ˆf_Lq_(Rn_;E). (.)

It is sufficient to show that the operator-functions

ξ λ(ξ) =AL–(λ,ε,ξ)

 +

n

k=

|ξk|m

–

,

σελ(ξ) =

n k= m+ i= ε i m+ k |λ|–

i m+_ξi

k  + n k=

|ξk|m

–

L–(λ,ε,ξ)

are uniform Fourier multipliers inLq_(Rn_{;E). Actually, due to positivity ofA, we have}

L–(λ,ε,ξ)≤M

 +

n

k=

εkξk+|λ|

–

,

ε,λ(ξ)=AL–(λ,ε,ξ)

 +

n

k=

|ξk|m

– ≤C  + n k=

|ξk|m

–

.

(.)

It is clear to observe that

ξk ∂ ∂ξk

ελ(ξ) = –εkξkAL–(λ,ε,ξ)

=–εkξkL–(λ,ε,ξ)

AL–(λ,ε,ξ), k= , , . . . ,n.

Due toR-positivity of the operatorA, the sets

–εkξkL–(λ,ε,ξ)

:ξ∈Rn\{}, AL–(λ,ε,ξ) :ξ∈Rn\{}

areR-bounded. So, in view of Kahane’s contraction principle and from the product

prop-erties of the collection ofR-bounded operators (see,e.g., [], Lemma ., Proposition .), we obtain

R

ξk ∂ ∂ξk

ελ(ξ):ξ∈Rn\{}

≤Ck.

Namely, theR-bounds of sets{ξk_∂ξ∂

kελ(ξ):ξ∈R

n_{\{}}are independent ofεandλ. Next,}

let us considerσελ(ξ). It is clear to see that

σελ(ξ)_B(E)≤C|λ|

n k= m+ i= ε  m+ k |ξk||λ|–



m+i

 +

n

k=

|ξk|m

–

By using the well-known inequality

n

k=

yαk

k ≤C

 +

n

k=

yl_k

, αk,yk≥,|α| ≤l

foryk= (ε   k|λ|–



|_ξk|_{) andl=m+ , we get the uniform estimate}

n k= m+ i= ε i m+ k |ξk||λ|–

i m+

≤C|λ|

 +

n

k=

|ξk|m

 +|λ|– n

k=

εk|ξk|m+

–

. (.)

From (.) and (.) we have the uniform estimate

σελ(ξ)_B(E)≤C|λ|

 +|λ|– n

k=

εk|ξk|m+

 +

n

k=

|ξk|m

–

L–(λ,ε,ξ)≤C.

Due toR-positivity of the operatorA, the set

|λ|+

n

k=

εkξk

L–(λ,ε,ξ) :ξ∈Rn\{}

isR-bounded. By using this fact, in view of (.) and Kahane’s contraction principle, we

obtain theR-boundedness of the set{σελ(ξ) :ξ∈R\{}}. In a similar way, we obtain the uniform estimates

∂

∂ξk ε,λ(ξ)

B(E)

≤C,

∂

∂ξk σελ(ξ)

B(E)

≤C.

Let

σkελ(ξ) =

ξk ∂ ∂ξk

σελ(ξ) :ξ∈Rn\{}

,

kελ(ξ) =

ξk ∂

∂ξkελ(ξ) :ξ∈R n_\{}

.

By the aid of the estimates above, due toR-positivity of the operatorA, in view of esti-mate (.), by virtue of Kahane’s contraction principle, from the additional and product properties of the collection ofR-bounded operators, forξ,ξ, . . . ,ξμ∈R,u,u, . . . ,uμ∈E and independent symmetric{–, }-valued random variablesrj(y),j= , , . . . ,μ,μ∈N, we

obtain the uniform estimate

μ j=

rj(y)σkελ

ξ(j)uj

E dy ≤C n k= μ j=

σkελ

ξ(j)rj(y)uj

E

dy

≤Csup ε,λ

R ξk

∂

∂ξσkελ(ξ) :ξ∈R\{} μ j=

rj(y)uj

E

The same estimates are obtained forkελ(ξ) in a similar way. Hence, by virtue of [, The-orem .] it follows thatε,λ(ξ) andσελ(ξ) are the uniform collection of multipliers in

Lp_(Rn_{;E). Then, by using equality (.), we obtain the assertion.}

4 The stationary Stokes system with small parameters

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

LetHq,s_(Rn_{;E), –∞<s<∞denote theE-valued Liouville space of orderssuch that}

Hq,_(Rn_{;E) =L}q_(Rn_{;E). It is known that ifEis aUMDspace, thenH}q,m_(Rn_{;E) =W}q,m_(Rn_;E)

for a positive integerm(see,e.g., [, §]). Forq∈(,∞) letXq= (Lq(Rn;E))ndenote the

space of anE-valued system of functionsf = (f(x),f(x), . . . ,fn(x)) with the norm

fXq=

_n

i=

fiq_Lq_(Rn_;E) 

q

.

Xqσ =Lqσ(Rn;E) denotes theE-valued solenoidal space,i.e., closure of (C∞σ(Rn;E))nin (Lq(Rn;E))n, where

C_∞σ

Rn;E=u∈C_∞Rn;E,divu= . Let

Xq,s=

Hq,sRn;En, Xq,s(A) =

Hq,sRn;E(A),En.

LetEbe a Banach space. Consider the space

Yq=

u∈Xq,divu∈Lq

Rn;E,

uYq=

uq_Xq+divuq_Lq_(Rn_;E) 

q_.

Yqbecomes a Banach space with this norm.

It is known that (see,e.g., Fujiwara and Morimoto []) the vector fieldu∈(Lq_(Rn₎₎n_has

a Helmholtz decomposition. In the following theorem we generalize this result for an E-valued function spaceXq.

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

decom-position,i.e.,there exists a bounded linear projection operator Pq from Xqonto Xqσ with the null space N(Pq) ={∇ϕ∈Xq:ϕ∈Lqloc(Rn;E)}.In particular,all u∈Xqhas the unique

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

∇ϕLq_(B;E)+u_X

q≤CuXq, (.)

for any open ball B⊂Rn_{.Moreover, (X}

qσ)∗=X_qσ, _q+ 

q = .

To prove Theorem ., we need some lemmas. Consider the problem

Lemma . Let E be a UMD space, let A be an R-positive operator in E, q ∈(,∞) and– <s<∞. Then, for f ∈Hq,s_(Rn_{;E), λ∈S}

ψ,problem(.) has a unique solution

u∈Hq,+s_(Rn_{;E(A),E)and the following uniform coercive estimate holds:}

u_Hq,s+

ε (Rn;E(A),E)+AuL

q_(Rn_;E)+|λ|u_Lq_(Rn_;E)≤Cf_Hq,s_(Rn_;E). (.)

Proof By using the Fourier transform, we see that estimate (.) is equivalent to the

fol-lowing estimate:

F–

 +

n

k=

εkξk

s+ 

L–(λ,ε,ξ)fˆ

Lq_(Rn_;E)

+F–AL–(λ,ε,ξ)ˆf_Lq_(Rn_;E)+|λ|F–L–(λ,ε,ξ)fˆ_Lq_(Rn_;E)

≤CF– +|ξ|

s

_fˆ

Lq_(Rn_;E). (.)

To prove (.) it is sufficient to show that the operator functions

 +

n

k=

εkξ_k

L–(λ,ε,ξ), A +|ξ|–

s

_L–_(λ,ε,ξ), |_λ|_{ +}|_ξ|–s_L–_(λ,ε,ξ)

are multipliers inLq_(Rn_{;E) uniformly inλandε. This fact is derived as the step in the proof}

of Theorem ..

Now consider the system of equations

–εu+Au+λu=f(x), x∈Rn, (.)

wheref = (f(x),f(x), . . . ,fn(x))∈Xqandu= (u(x),u(x), . . . ,un(x)) is a solution of (.).

We define inXq,sthe following parameter-dependent norm:

uXε,q,s= F–

 +

n

k=

εkξk

s



Fu

Xq <∞.

Lemma . Let E be a UMD space,let A be an R-positive operator in E, – <s<∞and q∈(,∞).Then,for f∈Xq,s,λ∈Sψ,problem(.)has a unique solution u∈Xq,s+and the

following coercive uniform estimate holds:

uXε,q,s++AuXq+|λ|uXq≤CfXq,s. (.)

Proof Problem (.) can be expressed as the following system:

–εuj+Auj+λuj=fj, x∈Rn,j= , , . . . ,n. (.)

By Lemma . we obtain that forfj∈Hq,s(Rn;E),λ∈Sψ, equation (.) has a unique so-lutionu= (u,u, . . . ,un)∈(Xq,s+(A))nand the following uniform coercive estimate holds:

Hence, we get that u= (u,u, . . . ,un) is a unique solution of problem (.) and (.)

implies (.).

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

Lemma . C∞(Rn_{;E)is dense in Y} p.

Consider the problem

–ϕ+Aϕ+λϕ=divf(x), x∈Rn. (.)

From Lemma . we obtain the following results.

Result . LetEbe aUMDspace, letAbe anR-positive operator inE andq∈(,∞). Then, forf ∈Lq(Rn;E),λ∈Sψ, problem (.) has a unique solutionϕ∈Hq,(Rn;E(A),E) and the following coercive uniform estimate holds:

uXε,q,+AuXq+|λ|uXq ≤CdivfXq,–.

Consider the operatorP=Pqdefined by

D(P) =LqRn;E, Pf =f –gradϕ,

whereϕis a solution of problem (.).

Result . LetEbe aUMDspace, letAbe anR-positive operator inE andq∈(,∞). ThenPqXqis a closed subspace ofXq.

Lemma . Let E be a UMD space,let A be an R-positive operator in E and q∈(,∞). Then the operator Pqis a bounded linear operator in Xqand Pf =f ifdivf(x) = .

Proof The linearity of the operatorPis clear by construction. Moreover, by Result . we

have

PfXq≤ fXq+gradϕXq≤CfXq. (.)

Ifdivf(x) = , then by Lemma . we get thatϕ= ,i.e.,Pf =f.

LetE∗denote the dual space ofE.

Lemma . Assume that E is a UMD space and q∈(,∞).Then the conjugate of Pqis

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

Proof It is known (see, e.g., [, ]) that the dual space of Lq_(Rn_{;E) is L}q_(Rn_;E∗_).

Since C_∞(Rn_;E∗_{) is dense in L}q_(Rn_;E∗_{), we only have to show P}∗

qϕ = Pqϕ for any ϕ ∈(C_∞(Rn_;E∗₎₎n_{. But this is derived by reasoning as in [, Lemma ]. Moreover, by}

Let

Gq=

∇ϕ:ϕ∈Wq,Rn;E(A);E, (PqXq)⊥=

f ∈LqRn;E∗n,f,υ=  for anyυ∈PqXq

.

From Lemmas ., . we obtain the following result.

Result . Assume thatEis aUMDspace,Ais anR-positive operator inEandq∈(,∞). Then any elementf ∈Xq uniquely can be expressed as the sum of elements ofPqXqand

Gq.

In a similar way to Lemmas ,  of [] we obtain, respectively, the following lemmas.

Lemma . Assume E is a UMD space and q∈(,∞).Then

(PqXq)⊥=Gq,



q+

 q = .

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

X_q⊥σ =Gq,



q+

 q= .

Now we are ready to prove Theorem ..

Proof of Theorem. From Lemmas ., . we get thatXqσ = (PqXq)⊥. Then, by

con-struction ofPq, we have Xq=Xqσ ⊕Gq. By Lemmas ., ., we obtain estimate (.).

Moreover, by Result .,Gqis a close subspace ofXq. It is known that the dual space of the

quotient spaceXq/GqisG⊥q. By the first assertion we haveXq/Gq=Xqσ, and by Lemma .

we obtain the second assertion.

Theorem . Let E be a UMD space,let A be an R-positive operator in E,q∈(,∞).Then, for f ∈Xq,ϕ∈Xq,,λ∈Sψ,problem(.)has a unique solution u∈Xq,and the uniform

coercive estimate holds

n

k= 

i=

ε

i

 k|λ|

–_i∂iu

∂xi k

Xq

+AuXq+∇ϕXq ≤CfXq (.)

with C=C(q,A)independent ofε,ε, . . . ,εn,λand f.

Proof By applying the operatorPqto equation (.), we get problem (.). It is clear to see

that

D(Qεq) =D(Bε)∩Xqσ,

whereQεqis the Stokes operator andBεis an operator inXqgenerated by problem (.)

forλ= ,i.e.,

Then by Lemma . we obtain the assertion.

Result . From Theorem . we get thatQε=Qεqis a positive operator inXqand it also

generates a bounded holomorphic semigroupSε(t) =exp(–Qεt) fort> .

In a similar way to that in [] we show the following.

Proposition . The following estimate holds

QαεSε(t)≤Ct–α

uniformly inε= (ε,ε, . . . ,εn)forα≥and t> .

Proof From Theorem . we obtain that the operatorQεis uniformly positive inXq,i.e.,

the following estimate holds

(Qε+λ)–≤M|λ|–

forλ∈Sψ,κ,  <ψ<π, where the constantMis independent ofλandε. Hence, by using

Danford integral and operator calculus (see,e.g., in []) we obtain the assertion.

5 Well-posedness of the instationary parameter-dependent Stokes problem In this section, we show the uniform well-posedness of problem (.)-(.).

Theorem . Assume that E is a UMD space,A is an R-positive operator in E and <

εk≤.Then,for f ∈Lp_(,T;X

q) =B(p,q)and a∈(Xq,(A),Xq)

p,p=G(p,q),p,q∈(,∞), there is a unique solution(u,∇ϕ)of problem(.)-(.)and the following uniform estimate holds:

∂_∂u_t

B(p,q)

+

n

k=

εk

∂u

∂x_k

B(p,q)

+AuB(p,q)+∇ϕB(p,q)

≤CfB(p,q)+aG(p,q)

(.)

with C=C(T,p,q)independent of f andε.

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

du

dt +Qεu=f(t), u() =a. (.)

If we putE=Xq, then by Proposition . operatorQεis uniformly positive and generates a bounded holomorphic semigroup inXquniformly inε. Moreover, by using [,

Theo-rem .] we get that the operatorQεisR-positive inEuniformly inε. SinceEis aUMD space, in a similar way to that in [, Theorem .] we obtain that forf ∈Lp_{(,T;E) and}

a∈(D(Qε),E)

p,p, there is a unique solutionu∈W

,p_(,T,D(Q

that the following uniform estimate holds:

du_dt

Lp_(,T;E)

+QεuLp_(,T;E)≤Cf_Lp_(,T;E)+a_(D(Aε),E)



p,p

. (.)

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

Result . It should be noted that ifε=ε=· · ·=εn= , then we obtain maximal

regu-larity properties of an abstract Stokes problem without any parameters in principal part.

Remark . There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (.) and (.) concrete Banach spaces instead ofEand concrete positive differ-ential, pseudo differential operators, or finite, infinite matrices,etc.instead ofA, by virtue of Theorem . and Theorem ., we can obtain the maximal regularity properties of a different class of stationary and instationary Stokes problems which occur in numerous physics and engineering problems.

6 Separability properties of anisotropic Stokes equations

Let⊂Rm_{be an open connected set with compactC}l_{-boundary∂. Consider the BVP}

for the following anisotropic Stokes equations with parameters:

(L+λ)u=εu+∇ϕ+

|α|≤l

aα(y)Dα_yu+λu=f, (.)

div_xu= , x∈Rn,y∈, (.)

Bju=

|β|≤lj

bjβ(y)Dβ_yu(x,y) = , y∈∂,j= , , . . . ,l, (.)

whereaαandbjβare complex-valued functions,

u=u(x,y) =u(x,y),u(x,y), . . . ,un(x,y)

, ϕ=ϕ(x)

are unknown solutions and

f =f(x,y) =f(x,y),f(x,y), . . . ,fn(x,y)

is a given function;

εu=

n

k=

εk ∂u

∂x

k

, Dj= –i

∂ ∂yj

, y= (y, . . . ,ym), x= (x, . . . ,xn),

ε= (ε,ε, . . . ,εn),εkare positive andλis a complex parameter.

IfG=Rn_×,p=(p

,p),Lp(G) will denote the space of allp-summable scalar-valued

functions with mixed norm (see,e.g., [, §],i.e., the space of all measurable functionsf defined onG, for which

fLp_(G)=

Rn

f(x,y)pdy

p p

dx



Analogously, Wp,,l_{(G) denotes the anisotropic Sobolev space with a corresponding}

mixed norm [, §]. LetXp= (Lp(G))n. From Theorem . we obtain the following result.

Theorem . Let the following conditions be satisfied:

() aα∈C(¯)for each|α|= landaα∈[L∞+Lγk]()for each|α|=k< lwithrk≥q

andl–k>m rk;

() bjβ∈Cl–lj(∂)for eachj,βandlj< l,

l

j=bjβ(y)σj= ,for|β|=lj,y∈∂G,where σ= (σ,σ, . . . ,σm)∈Rmis normal to∂;

() fory∈ ¯,ξ∈Rm_{,ν∈S(ϕ),ϕ∈(,π),|ξ|+|ν| = letν+}

|α|=laα(y)ξα= ; () for eachy∈∂,the local BVP in local coordinates corresponding toy

ν+

|α|=l

aα(y)Dαϑ(y) = ,

Bjϑ=

|β|=lj

bjβ(y)Dβu(y) =hj, j= , , . . . ,l,

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

with|ξ|+|ν| = .Then forf∈Xp,λ∈S(ϕ)with sufficiently large|λ|problem

(.)-(.)has a unique solutionubelonging toWp,,l(G;Rn)and the uniform coercive estimate holds

n

k= 

i=

ε

i

 k|λ|

–_i∂iu

∂xi k

Xp

+

|β|=m

Dβ

yuLp_(G)+∇ϕXq≤CfXp.

Proof LetE=Lp_{(). By virtue of [, Theorem .],E is a UMDspace. Consider the}

operatorAinLp() defined by

D(A) =Wp,l(;Bju= ), Au=

|α|≤l

aα(y)Dαu(y).

Problem (.)-(.) can be rewritten in the form (.), where u(x,y) =u(x,·),f(x,y) = f(x,·) are vector-functions with values inE=Lp_{(). By virtue of [, Theorem .] the}

problem

νu(y) +

|α|≤l

aα(y)Dαyu(y) =f(y),

Bju=

|β|≤lj

bjβ(y)Dβyu(y) = , j= , , . . . ,l

has a unique solution forf ∈Lp() and forν∈S(ϕ),|ν| → ∞. Moreover, the operator AisR-positive inLp_{(),i.e., all the conditions of Theorem . hold. So, we obtain the}

7 Infinite system of Stokes equations with small parameters

Consider the IVB for the following infinite system of instationary Stokes equations with small parameters:

∂um ∂t –

n

k=

εk ∂_u

m ∂x

k

+

∞

j=

gjuj+∇ϕm=fm(x,t), (.)

divu= , um(x, ) = , x∈Rn,t∈(,T),m= , , . . . ,∞,

whereεkare positive parameters. Hereum= (um(x,t),um(x,t), . . . ,umn(x,t)),ϕm=ϕm(x,t)

are unknown solutions, andf = (fm(x,t),fm(x,t), . . . ,fmn(x,t)) is a given function. Let

G={gm}, gm> , u={um}, Gu={gmum}, m= , , . . . ,

lq(G) =

u:u∈lσ,ulσ(G)=Gulσ =

_∞

m=

|gmum|σ

 σ

<∞

,  <σ<∞.

Xp,q,σ =Lp(,T;Lq(Rn;lσ)) is a class of functions

f =f(x,t),f(x,t), . . . ,fn(x,t)

with the norm

fXp,q,σ =

_∞

i=

fiσLp_(,T;Lq_(Rn₎₎ 

σ

<∞.

LetX

p,q,σ=Wp,(,T;Lq(Rn;lσ)). From Theorem . we obtain the following.

Theorem . Let <εk≤and gj> .Then for f∈Xp,q,σ,p,q,σ∈(,∞)there is a unique solution(um,∇ϕm)of problem(.)belonging to Xp,q,σ ×Xp,q,σ and the following uniform coercive estimate holds:

∂u

∂t

Xp,q,σ

+

n

k=

εk

∂u

∂x

k

Xp,q,σ

+GuXp,q,σ +∇ϕXp,q,σ≤CfXp,q,σ

with C=C(T,p,q)independent of f andε.

Proof Really, letE=lσ,Aand be an infinite matrix, defined by

A= [gmδjm], m,j= , , . . . ,∞.

It is easy to see that

B(λ) =λ(A+λ)–=λ(gm+λ)–δjm

For allu,u, . . . ,uμ∈lq,λ,λ, . . . ,λμ∈C,λi= –gm,m= , , . . . ,∞and independent

sym-metric{–, }-valued random variablesri(y),j= , , . . . ,μ,μ∈N, we have

μ i=

ri(y)B(λi)ui

σ

lσ

dy≤C

∞ m= μ i=

λi(gm+λi)–ri(y)ui

σ dy ≤sup

m,i

λi(gm+λi)–

σ ∞ m= μ i=

ri(y)ui

σ

dy.

Sincesup_m,i|λi(gm+λi)–|σ≤Cforλi= –gm, from the above we get

μ i=

ri(y)B(λi)ui

σ

lσ

dy≤C

μ i=

ri(y)ui

σ lσ ,

i.e., the operatorAisR-positive inlσ. Therefore, all the conditions of Theorem . hold

and we obtain the assertion.

Competing interests

The author declares that they have no competing interests.

Authors’ contributions All results belong to VS.

Received: 8 March 2013 Accepted: 28 June 2013 Published: 23 July 2013

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doi:10.1186/1687-2770-2013-172