R E S E A R C H
Open Access
Nonlocal Navier-Stokes problem with a small
parameter
Veli B Shakhmurov
**Correspondence:
[email protected] Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul, Turkey Institute of Mathematics and Mechanics, Azerbaijan National Akademy of Science, Baku, Azerbaijan
Abstract
Initial nonlocal boundary value problems for a Navier-Stokes equation with a small parameter is considered. The uniform maximal regularity properties of the
corresponding stationary Stokes operator, well-posedness of a nonstationary Stokes problem and the existence, uniqueness and uniformlyLpestimates for the solution of the Navier-Stokes problem are established.
MSC: 35Q30; 76D05; 34G10; 35J25
Keywords: Stokes operators; Navier-Stokes equations; differential equations with small parameters; semigroups of operators; boundary value problems;
differential-operator equations; maximalLpregularity
1 Introduction
Consider the following Navier-Stokes problem with a parameter:
∂u
∂t –εu+ (u· ∇)u+∇ϕ=f(x,t), divu= ,x∈G,t∈(,T), (.)
Lkjεu= mkj
i=
εσi
αkji
∂iu
∂xik(Gk) +βkji ∂iu ∂xik(Gkb)
= , (.)
k= , , . . . ,n,j= , ,u(x, ) =a(x), (.)
where
εu=ε
n
k=
∂u
∂x
k
, σi=
i+
q
, q∈(,∞),
G=
n
k=
(,bk), Gk= (x,x, . . . ,xk–, ,xk+, . . . ,xn),
Gkb= (x,x, . . . ,xk–,bk,xk+, . . . ,xn), ≤mkj≤,
αkji,βkjiare complex numbers,εis a small positive parameter,
u=u(x,t),u(x,t), . . . ,un(x,t), ϕ=ϕ(x,t)
represent the unknown velocity and pressure, respectively,
f =f(x,t),f(x,t), . . . ,fn(x,t)
represents a given external force andadenotes the initial velocity. This problem is char-acterized by nonlocality of boundary conditions and by presence of a small termεwhich corresponds to the inverse of Reynolds numberRevery large for the Navier-Stokes equa-tions. From both the theoretical and computational points of view, singularly perturbed problems and asymptotic behavior of the Navier-Stokes equations with small viscosity when the boundary is either characteristic or non-characteristic have been well studied; see,e.g., [–]. In the present work, we established a uniform time of existence and esti-mates for solutions of problem (.)-(.). It is clear that forε= , choosing the boundary conditions locally andmkj= , problem (.)-(.) is reduced to the classical Navier-Stokes
problem
∂u
∂t –u+ (u· ∇)u+∇ϕ=f(x,t), divu= ,x∈,t∈(,T), (.) u|∂= , u(x, ) =a(x).
Note that the existence of weak or strong solutions and regularity properties of classical Navier-Stokes problems were extensively studied,e.g., in [–, , –]. There is extensive literature on the solvability of the initial value problem for the Navier-Stokes equation ( see,e.g., [] for further papers cited there ). Hopf [] proved the existence of a global weak solution of (.) using the Faedo-Galerkin approximation and an energy inequality. Another approach to problem (.) is to use semigroup theory. Kato and Fujita [, , ] and Sobolevskii [] transformed equation (.) into an evolution equation in the Hilbert space L. They proved the existence of a unique global strong solution for any square-summable initial velocity whenn= . On the other hand, whenn= they proved the existence of a unique local strong solution if the initial velocity has some regularity. Other contributions in this field have also assumed some regularity of the initial velocity cor-responding to the Stokes problem; see, for example, Solonnikov [] and Heywood []. Afterward, Giga and Sohr [] improved this result in two directions. First, they general-ized the result of Solonnikov for spaces with different exponents in space and time, and the estimate obtained was global in time. Here, first at all, we consider the nonlocal (bound-ary value problem) BVP for the following differential operator equation (DOE) with small parameters:
–εu+ (A+λ)u=f(x), x∈G,
Lkjεu= mkj
i=
εσi
k
αkji
∂iu
∂xik(Gk) +βkji ∂iu ∂xik(Gkb)
= , (.)
k= , , . . . ,n,j= , ,
whereAis a linear operator in a Banach spaceE,αkji,βkjiare complex numbers,εk are
positive andλis a complex parameter. We show that problem (.) has a unique solution
u∈W,q(G;E(A),E) forf ∈Wm,q(G;E) andλ∈Sψ
following coercive uniform estimate holds:
n
k=
m+
i=
ε
i m+
k |λ|–
i
∂
iu
∂xik
Lq(G;E)
+AuLq(G;E)≤CfWm,q(G;E), m≥,
withC(q) independent ofε,ε, . . . ,εn,λandf.
Further, we consider the nonlocal BVP for the stationary Stokes system with small pa-rameters
εu+∇ϕ=f(x,t), divu= ,x∈G,t∈(,T),
Lkjεu= mkj
i=
εσi
k
αkji
∂iu
∂xi k
(Gk) +βkji
∂iu
∂xi k
(Gkb)
= , (.)
k= , , . . . ,n,j= , ,u(x, ) = ,
where
u=u(x,t),u(x,t), . . . ,un(x,t), f =
f(x,t),f(x,t), . . . ,fn(x,t) .
Then we consider the initial nonlocal BVP for the following nonstationary Stokes equation with small parameters:
∂u
∂t –εu+∇ϕ=f(x,t), divu= ,x∈G,t∈(,T),
Lkjεu= mkj
i=
εσi
k
αkji
∂iu
∂xik(Gk) +βkji ∂iu ∂xik(Gkb)
= , (.)
k= , , . . . ,n,j= , ,u(x, ) = .
Problem (.) can be expressed as the abstract parabolic problem with a parameter
du
dt +Oε,qu=f(t), u() = , (.)
where Oε,q is a stationary parameter depending on the Stokes operator in a solenoidal
spaceLqσ(G;Rn) defined by D(Oε,q) =Wσ,q(G,Lkjε) =
u∈W,q(G),divu= ,Lkjεu= ,k= , , . . . ,n,j= ,
,
Oε,qu= –Pεu.
We prove that the operatorOε,qis positive inLq(G;Rn) uniformly with respect to
param-etersε= (ε,ε, . . . ,εn) and also is a generator of a holomorphic semigroup. Then, by using
Lp-maximal regularity theorems (see,e.g., [, ]) for abstract parabolic equations (.),
we obtain that for every f ∈Lp(,T;Lq(;Rn)) =B(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)
withC=C(T,,p,q) independent off andε. Afterwords, by using the above uniform coercive estimate, we derive local uniform existence and uniforma prioriestimates of a solution of problem (.)-(.).
Modern analysis methods, particularly abstract harmonic analysis, the operator theory, the interpolation of Banach spaces, the theory of semigroups of linear operators, embed-ding and trace theorems in vector-valued Sobolev-Lions spaces are the main tools imple-mented to carry out the analysis.
2 Notations, definitions and background
LetEbe a Banach space andLp(;E) denotes the space of strongly measurableE-valued
functions that are defined on the measurable subset⊂Rnwith the norm
fLp=fLp(;E)=
f(x)pEdx
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∈(,∞).
LetCbe the set of complex numbers and
Sψ=λ∈C,|argλ| ≤ϕ∪ {}, ≤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, whereIis the identity operator inE,B(E) is the space of bounded linear operators inE. It is known [,§..] that there exist the fractional powersAθof a positive operatorA. LetE(Aθ)
denote the spaceD(Aθ) with the norm
uE(Aθ)=
up+Aθup p, ≤p<∞, <θ<∞.
Let N denote the set of natural numbers. A set G⊂ B(E,E) is called R-bounded (see, e.g., []) if there is a positive constantC such that for allT,T, . . . ,Tm ∈Gand
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 anR-bound of the collection
A setGh⊂B(E,E) is called uniformR-bounded if there is a constantCindependent ofh∈Qsuch that for allT(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,
which implies thatsuph∈QR(Gh)≤C.
Theψ-positive operatorAis said to beR-positive in a Banach spaceEif the setLA=
{ξ(A+ξ)–:ξ∈Sψ}, ≤ψ<π, isR-bounded.
The operatorA(t) is said to beψ-positive inEuniformly int∈σ with boundM> if
D(A(t)) is independent oft,D(A(t)) is dense inEand(A(t) +λ)– ≤M( +|λ|)–for all
λ∈Sψ, ≤ψ<π, whereMdoes not depend ontandλ.
LetEandEbe two Banach spaces, and letEbe continuously and densely embedded into E. Letbe a measurable set inRnandmbe a positive integer. LetWm,p(;E,E) denote the space consisting of all functionsu∈Lp(;E) that have the generalized deriva-tives ∂mu
∂xmk ∈L
p(;E), with the norm
uWm,p(;E
,E)=uLp(;E)+
n
k=
∂∂xmmu
k
Lp(;E)
<∞.
Forn= , = (a,b), a,b∈N, the spaceWm,p(;E,E) will be denoted byWm,p(a,b;
E,E).
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.
3 Boundary value problems for abstract elliptic equations
In this section, we consider problem (.). We derive the maximal regularity properties of this problem.
It should be noted that BVPs for DOEs were studied,e.g., in [–] and [, , , –]. For references, see [, ]. Let αkj=αkjmk andβkj=βkjmk. First, we prove the
following theorem.
Theorem . Let the following conditions be satisfied:
() Eis aUMDspace andAis anR-positive operator inEfor≤ψ<π; () q∈(,∞),ηk= (–)mαkβk– (–)mαkβk= , <tk≤,k= , , . . . ,n.
Then problem(.)has a unique solution u∈W,q(G;E(A),E)for f ∈Lq(G;E)andλ∈ Sψ,κwith sufficiently largeκ> .Moreover,the following coercive uniform estimate holds:
n
k=
j=
ε
j
k|λ|–
j
∂
ju
∂xjk
Lq(G;E)
+AuLq(G;E)≤CfLq(G;E) (.)
with C(q)independent ofε,ε, . . . ,εn,λand f.
Proof Let us consider the BVP
–ε
∂u ∂x
–ε
∂u ∂x
wherex,x∈G= (,b)×(,b),Lkjεare defined by equalities (.)-(.). For the
inves-tigation (.), we consider the following BVP for ordinary DOE:
Lu= –ε
du
dx
+ (A+λ)u(x) =f(x), y∈(,b), Ljεu= , (.)
where Lk are boundary conditions of type (.) on (,b). By virtue of [, Theo-rem .], we obtain that problem (.) has a unique solutionu∈W,q(,b
;E(A),E) for
f ∈Lq(,b
;E),λ∈Sψ,κ, with sufficiently largeκ> , and the following coercive uniform estimate holds:
j=
ε
j
|λ|– j
u(j)
Lq(,b
;E)+AuL
q(,b
;E)≤CfLq(,b;E). (.)
Since Lq(,b
;Lq(,b;E)) =Lq(G;E), problem (.) can be expressed as the following problem:
–ε
du dx
+ (Bε+λ)u(x) =f(x), Lkεu= , (.)
whereBεis the differential operator inLq(,b
;E) generated by problem (.),i.e.,
D(Bε) =W,q,b;E(A),E,Lkε , Bεu= –ε
du
dx +Au(x).
By virtue of [, Theorem ..],F=Lq(,b
;E)∈UMDprovidedE∈UMD,q∈(,∞). Hence, by virtue of [, Theorem .] and [, Theorem .], the operator Bε is
uni-formlyR-positive inF. Then, by applying again [, Theorem .], we get that forf ∈ Lq(,b
;F),λ∈Sψ,κand sufficiently largeκ> , problem (.) has a unique solutionu∈ W,q(,b
;D(B),F), and the following coercive uniform estimate holds:
j=
ε
j
|λ|– j
u(j)
Lq(,b
;F)+BεuL
q(,b
;F)≤CfLq(,b;F). (.)
The estimate (.) implies the uniform estimate
j=
ε
j
|λ|– j
u(j)
Lq(,b
;F)≤CB(ε)uLq(,b;F). (.)
By using (.) and (.), we obtain that problem (.) has a unique solutionu∈W,q(G;
E(A),E) forf ∈Lq(G
;E),λ∈Sψ,κwith sufficiently largeκ> , and the coercive uniform
estimate holds
k=
j=
ε
j
k|λ|
–j∂ju
∂xjk
Lq(G
;E)
+AuLq(G
;E)≤CfLq(G;E).
Further, by continuing this processn-times, we obtain the assertion.
Corollary . Let <εk≤, (–)mkαkβk– (–)mkαkβk= .For f ∈Lq(G;Rn), q∈ (,∞)and forλ∈Sψ,κwith sufficiently largeκ> ,there is a unique solution u of problem
(.)and the following uniform coercive estimate holds:
n
k=
i= |λ|–iε
i
k
∂iu
∂xik
q
≤Cfq,
with C=C(q)independent of f,εkandλ.
Proof Let us putE=RnandA=κ> in Theorem .. It is known that the operatorA=
κ> isR-positive inRn(see,e.g., []). So, the estimate (.) implies Corollary .. Consider the differential operatorQεgenerated by problem (.),i.e.,
D(Qε) =W,q(G;Lkjε) =
u∈W,q(G),Lkjεu= ,k= , , . . . ,n,j= ,
,
Qεu= –εu+Au.
From Theorem . we obtain the following.
Result . Forλ∈Sψ,κ, there is a resolvent (Qε+λ)–of the operatorQεsatisfying the following uniform estimate:
n
k=
i= |λ|–iε
i
k
∂i
∂xik(Qε+λ)
–
B(Lq(G;E))≤
C.
It is clear that the solutionuof problem (.) depends on parametersε= (ε,ε, . . . ,εn),
i.e.,u=u(x,ε). In view of Theorem ., we established estimates for the solution of (.) uniformly inε,ε, . . . ,εn.
4 Regularity properties of solutions for DOEs with parameters
In this section, we show the separability properties of problem (.) in Sobolev spaces
Wm,q(G;E). The main result is the following theorem.
Theorem . Let the following conditions be satisfied: () Eis aUMDspace andAis anR-positive operator inE; () mis a positive integerq∈(,∞), <tk≤,and
ηk= (–)mαkβk– (–)mαkβk= , k= , , . . . ,n.
Then problem (.)-(.) has a unique solutionu∈W+m,q(G;E(A),E) forf ∈Wm,q(G;
E),λ∈Sψ,κ, with sufficiently largeκ> , and the following coercive uniform estimate holds:
n
k=
m+
i=
ε
i m+
k |λ|–
i m+∂
iu
∂xik
Wm,q(G;E)
+AuLq(G;E)≤CfWm,q(G;E) (.)
Consider first the following nonlocal BVP for an ordinary DOE with a small parameter:
(Lt+λ)u= –tu()(x) + (A+λ)u(x) =f(x), x∈(, ),
Lktu= mk
i=
tσiα
kiu(i)() +βkiu(i)()
=fk, k= , , (.)
whereσi=i+p,mk∈ {,m+ },αki,βkiare complex numbers,tis positive,λis a complex
parameter andAis a linear operator inE. LetAλ=A+λI.
To prove the main result, we need the following result in [, Theorem .].
Theorem A Let E be aUMDspace,A be aψ-positive operator in E with bound M, ≤
ψ <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,ψ,m,αand p)such that for every u∈
(E,E(Am))
α
m–mp ,pandλ∈Sψ,
∞
Aαλe–xA
λupdx≤Cup (E,E(Am))
α m– mp,p
+|λ|αp–up
E
.
In a similar way as in [,§.., Theorem ], we obtain the following lemma.
Lemma . Let m and j be integer numbers, ≤j≤m– ,θj=pjpm+, <t≤,x∈[,b].
Then, for u∈Wm
p (,b;E,E), the transformation u→u(j)(x) is bounded linear from
Wm
p (,b;E,E)onto(E,E)θj,pand the following inequality holds:
tθju(j)(x )(E,E)
θj,p≤C
tu(m)
Lp(,b;E)+uLp(,b;E) .
Consider at first the homogeneous problem of(.)
(Lt+λ)u= –tu()(x) + (A+λ)u(x) = ,
Lktu= mk
i=
tσiα
kiu(i)() +βkiu(i)()
=fk, k= , . (.)
Let
Ek=
E(A),E θ
k,p.
Lemma . Let A be an R-positive operator in aUMDspace E and
<t≤, η= (–)mα
β– (–)mαβ= .
Then problem(.)has a unique solution u∈Wm+,p(, ;E(A),E)for f
k∈Ek,p∈(,∞),
λ∈Sψ,and the coercive uniform estimate holds
m+
i=
tm+i |λ|–mi+u(i)
Lp(,;E)+AuLp(,;E)≤M
k=
fkEk+|λ| –θk
Proof In a similar way as in [, Theorem .], we obtain the representation of the solution of (.)
u(x) =t–pe–xt– A
λC+d(λ,t)+e–(–x)t– A
λC+d(λ,t)A– m
λ f
+t–pe–xt– A
λC+d(λ,t)+e–(–x)t– A
λC+d(λ,t)A– m
λ f, (.)
whereCijanddijare uniformly bounded operators. Then, in view of positivity ofA, we
obtain from (.)
m+
i=
tm+i |λ|–mi+u(i)
Lp(,;E)+AuLp(,;E)
≤Ct–p k= m+ i=
|λ|μiA–
(mk–i) λ
e–(–x)t– A
λ +e–xt– A
λfk
Lp(,;E)
+AA–
mk
λ e–xt
– A
λfk
Lp(,;E)
≤Ct–p, (.)
k= m+ i=
|λ|–iA–(–
mmk
(m+)+i)
λ A
–mmk+
λ e
–xt– A
λf
kLp(,;E)
+AA–
mk m+ λ
e–xt– A
λ +e–(–x)t– A
λfk
Lp(,;E)
≤Ct–p k= A– mk m+ λ
e–xt– A
λ +e–(–x)t– A
λfkpdx
p + AA– mk m+ λ
e–xt– A
λ +e–(–x)t– A
λfkpdx
p
. (.)
By changing the variablext– =yand in view of Theorem A, we obtain
t–p
A– mk m+ λ
e–xt– A
λ +e–(–x)t– A
λfkpdx
p
≤M
k=
fkEk+|λ| –θkf
k
. (.)
By using the estimate (.), by virtue of Theorem A, we get the uniform estimate
k=
t–p
AA–mmk+ λ
e–xt– A
λ +e–(–x)t– A
λfkpdx
p
≤M
k=
fkEk+|λ| –θkf
k
. (.)
Then from (.)-(.) we obtain (.).
Theorem . Assume that the following conditions are satisfied: () Eis aUMDspace andAis anR-positive operator inE; () η= (–)mα
β– (–)mαβ= ,θk=mm+k +p,k= , , <t≤,p∈(,∞).
Then the operator u→ {(Lt+λ)u,Ltu,Ltu}is an isomorphism from Wm+,p(, ;E(A),E)
onto Wm,p(, ;E)×E
×Eforλ∈Sψ,κwith large enoughκ> .Moreover,the uniform coercive estimate holds
m+
j=
tmj+|λ|–
j m+u(j)
Lp(,;E)+AuLp(,;E)
≤C
fWm,p(,;E)+
k=
fkEk+|λ| –θkf
kE
. (.)
Proof The uniqueness of a solution of problem (.) is obtained from Lemma .. Let us define
¯
f(x) =
⎧ ⎨ ⎩
f(x), ifx∈[, ],
, ifx∈/[, ].
We will show that problem (.) has a solutionu∈Wm+,p(, ;E(A),E) forf ∈Wm,p(, ;
E),fk∈Ekandu=u+u, whereuis the restriction on [, ] of the solution of the equation
(Lt+λ)u=f¯(x), x∈R= (–∞,∞), (.)
anduis a solution of the problem
(Lt+λ)u= , Lktu=fk–Lktu. (.)
A solution of equation (.) is given by
u(x) =F–L–(λ,t,ξ)Ff¯= π
∞
∞ e
iξxL–(λ,t,ξ)(Ff¯)(ξ)dξ,
whereL(λ,t,ξ) =A+tξ+λ. It follows from the above expression that
m+
j=
tmj+|λ|–
j m+u(j)
Lp(R;E)+AuLp(R;E)
=
m+
j=
tmj+|λ|–
j
m+F–ξjL–(λ,t,ξ)Ff¯
Lp(R;E)+F–AL–(λ,t,ξ)Ff¯Lp(R;E). (.)
It is sufficient to show that the operator-functions
tλ(ξ) =AL–(λ,t,ξ)
+ξm –,
σtλ(ξ) =
m+
j=
tmj+|λ|–
j
are uniform Fourier multipliers inLp(R;E). Actually, due to the positivity ofA, we have
L–(λ,t,iξ)≤M +tξ+|λ| –, (.)
t,λ(ξ)=A
A+λ+tξ–≤C.
It is clear to observe that
ξ d
dξtλ(ξ) = –tξ
AL–(λ,t,ξ) =–tξL–(λ,t,ξ)AL–(λ,t,ξ).
Due toR-positivity of the operatorA, the sets
–tξA+tξ+λ–, AA+tξ+λ–, ξ∈R\{}
areR-bounded. Then, in view of the Kahane contraction principle, from the product prop-erties of the collection ofR-bounded operators (see,e.g., [] Lemma ., Proposition .), we obtain
R
ξi d
i
dξitλ(ξ) :ξ∈R\{}
≤C,
R
ξi d
i
dξiσtλ(ξ) :ξ∈R\{}
≤C, i= , .
(.)
By [, Theorem .] it follows thatt,λ(ξ) andσtλ(ξ) are the uniform collection of
mul-tipliers in Lp(R;E). Then in view of (.) we obtain that problem (.) has a solution
u∈Wm+,p(R;E(A),E) and the uniform coercive estimate holds m+
j=
tmj+|λ|–
j m+u(j)
Lp(R;E)+AuLp(R;E)≤C¯fLp(R;E). (.)
Letube the restriction ofuon (, ). The estimate (.) implies thatu∈Wm+,p(, ;
E(A),E). By virtue of Lemma ., we get
u(mk) (·)∈
E(A);E θ
k,p, k= , .
Hence,Lktu∈Ek. Thus, by virtue of Lemma ., problem (.) has a unique solution
u(x) that belongs to the spaceWm+,p(, ;E(A),E) and
m+
j=
tmj+|λ|–
j m+u(j)
Lp(,;E)+AuLp(,;E)
≤C
k=
fkEk+|λ| –θkf
kE+tθku(mk)C([,];Ek)+t
θk|λ|–θku
C([,];E)
. (.)
Moreover, from (.) we obtain
m+
j=
tmj+|λ|–
j m+u(j)
Therefore, by Lemma . and by estimate (.), we obtain
tθku(mk)
(·)Ek ≤CuWtm+,p(,;E(A),E)≤CfW
m,p(,;E). (.)
So, in view of Lemma . and estimates (.)-(.), we get
m+
j=
tmj+|λ|–
j m+u(j)
Lp(,;E)+AuLp(,;E)
≤C
fLp(,;E)+
k=
fkEk+|λ| –θkf
kE
. (.)
Finally, from (.) and (.) we obtain (.).
Now, we can prove the main result of this section.
Proof of Theorem. LetG= (,b)×(,b). It is clear to see that
Wm,q(G;E) =Wm,q(,b;X,X) =Wm,q(,b;X)∩Lq(,b;X), whereX=Wm,q(,b;E) andX=Lq(,b;E).
Let us consider the BVP
–ε
∂u
∂x –ε ∂u
∂x + (A+λ)u(x,x) =f(x,x), Lkjεu= , k,j= , , (.)
whereLkjεare defined by equalities (.). Problem (.) can be expressed as the following
BVP for an ordinary DOE:
Lu= –ε
du
dx + (Bε+λ)u(x) =f(x), x∈(,b), Lkεu= , (.)
whereLkεare boundary conditions of type (.),Bε is the operator acting inXandX
defined by
Bεu= –ε
du
dx +Au(x), D(Bε) =W
m+,q,b
;E(A),E,Lkε , W,q,b;E(A),E,Lkε .
SinceXandXareUMDspaces, (see,e.g., [, Theorem ..]) by virtue of Theorem ., we obtain that problem (.) has a unique solutionu∈Wm+,q(,b
;D(Bε),X) forf ∈
Wm,q(,b
;X) andλ∈Sψ,κwith sufficiently largeκ> . Moreover, the coercive uniform
estimates holds
m+
i=
ε
i m+
|λ|– i m+u(i)
Lq(,b
;X)+BεuLq(,b;X)≤CfWm,q(,b;X),
i=
ε
i
|λ|– i
u(i)
Lq(,b
;X)+BεuL
q(,b
;X)≤CfLq(,b;X).
From (.) we obtain that problem (.) has a unique solution
u∈Wm+,qG;E(A),E forWm,q(G;E).
Moreover, the uniform coercive estimates hold
m+
i=
ε
i m+
|λ|– i m+u(i)
Lq(,b
;X)+BεuL
q(,b
;X)≤CfWm,q(G;E). (.)
By applying Theorem . forfk= andE=X, we get the following uniform estimate: m+
j=
ε
i m+
|λ|– i m+u(i)
X+AuX≤CBεuWm,q(,b;E). (.)
From estimates (.)-(.) we conclude the corresponding claim for problem (.).
Then, by continuing this processn-times, we obtain the assertion.
5 Nonlocal initial-boundary value problems for the Stokes system with small parameters
In this section, we show the uniform maximal regularity properties of the nonlocal initial value problem for nonstationary Stokes equations (.).
The functionu∈Wσ,q(G,Lkjε) ={u∈W,q(G;Rn),Lkjεu= ,divu= }satisfying
equa-tion (.) a.e. onGis called the stronger solution of problem (.).
LetWs,q(G), <s<∞be the Sobolev space of orderssuch thatW,q(G) =Lq(G). For
q∈(,∞), letXq=Lqσ(G) denote the closure ofC∞σ(G) inLp(G;Rn), where
C∞σ(G) =
u∈C∞(G),divu= .
It is known that ( see,e.g., Fujiwara and Morimoto []) a vector fieldu∈Lq(G;Rn) has
the Helmholtz decomposition,i.e., allu∈Lq(G;Rn) can be uniquely decomposed asu=
u+∇ϕwithu∈Lqσ(G),u=Pqu, wherePq=Pis a projection operator fromLq(G;Rn)
toLqσ(G) andϕ∈Lqloc(G¯),∇ϕ∈Lq(G;Rn), so that
∇ϕq≤Cuq, ϕLq(G∩B)≤Cuq
withCindependent ofu, whereBis an open ball inRnandu
pdenotes the norm ofuin
Lq(G;Rn) orLq(G).
Then problem (.) can be reduced to the following BVP:
–Pεu+λu=f(x), x∈G,
Lkjεu= mkj
i=
εσik
k
αkji
∂iu ∂xi k
(Gk) +βkji
∂iu ∂xi k
(Gkb)
= , (.)
Consider the parameter-dependent Stokes operatorOε=Oε,qgenerated by problem (.),
i.e.,
D(Oε) =Wσ,q(G;Lkj) n, Oεu= –Pεu.
From Corollary . we get that the operatorOεis positive and also is a generator of a bounded holomorphic semigroupSε(t) =exp(–Oεt) fort> .
In a similar way as in [], we show the following.
Proposition . The following estimate holds:
OαεSε(t)≤Ct– α
uniformly inε= (ε,ε, . . . ,εn)forα≥and t> .
Proof From Result . we obtain that the operatorOεis uniformly positive inLq(G;Rn),
i.e., forλ∈Sψ,κ, <ψ<π, the following estimate holds:
(Oε+λ)–≤ M |λ|,
where the constantMis independent ofλandε. Then, by using the Danford integral and
operator calculus as in [], we obtain the assertion.
Now consider problem (.). The main theorem in this section is the following.
Theorem . Let <εk≤, (–)mkαkβk– (–)mkαkβk= and p,q∈(,∞).Then
there is a unique solution(u,∇ϕ)of problem(.)for f ∈Lp(,T;Lq(G;Rn)) =B(p,q)and a∈B–
p
p,q .Moreover,the following uniform estimate holds:
∂u∂t
B(p,q) +
n
k=
εk
∂u
∂x
k
B(p,q)
+∇ϕB(p,q)≤CfB(p,q)+a
B– p,qp(G)
(.)
with C=C(T,G,p,q)independent of f andε.
Proof Problem (.) can be expressed as the following abstract parabolic problem with a small parameter:
du
dt +Oεu=f(t), u() =a. (.)
If we putE=Lq(G;Rn) andA=κ> in Theorem ., then the Result . implies that
the operator Oε is uniformly positive and generates bounded holomorphic semigroup in Lq(G;Rn) uniformly inε
k. Moreover, by using [, Theorem .] we get that
opera-tor Oε isR-positive in E. Since E is aUMD space, in a similar way as in [, Theo-rem .], we obtain that forf ∈Lp(,T;E) anda∈(D(Oε),E)
p,p, there is a unique solution
u∈W,p(,T,D(Oε),E) of problem (.) so that the following uniform estimate holds:
dudt
Lp(,T;E)
+OεuLp(,T;E)≤CfLp(,T;E)+a(D(Aε),E)
p,p
From (.) for allu∈Wσ,q(G,Lkjε), we get the following estimate:
n
k=
εk
∂∂xu
k
q
≤CAεuq
uniformly inε= (ε,ε, . . . ,εn).
6 Existence and uniqueness for the Navier-Stokes equation with parameters
In this section, we study the Navier-Stokes problem (.)-(.) in the spaceXq. Problem
(.)-(.) can be expressed as
du
dt +Oεu=Fu+Pf, u() = , t> , (.)
where
Fu= –P(u,∇)u.
We consider equation (.) in an integral form
u(t) =Sε(t)a+
t
Sε(t–s)Fu(s) +Pf(s)ds, t> . (.)
To prove the main result, we need the following result which are obtained in a similar way as in [, Theorem ].
Lemma . For any≤α≤,the domain D(Oα
ε)is the complex interpolation space
[Xq,D(Oε)]α.
Lemma . For each k= , , . . . ,n,the operator u→O– ε P(∂∂x
k)u extends uniquely to a
uniformly bounded linear operator from Lq(G;Rn)to X q.
Proof SinceOεis a positive operator, it has fractional powersOα
ε. From Lemma ., it
follows that the domainD(Oα
ε) is continuously embedded inXq∩Hqα(G;Rn) for anyα> ,
whereHα
q (G;Rn) is the vector-valued Bessel space. Then, by using the duality argument
and due to uniform positivity ofO
ε, we obtain the following uniformly inεestimate:
O–
ε P
∂ ∂xk
u
Lq(G;Rn)≤
CuXq. (.)
By reasoning as in [], we obtain the following.
Lemma . Let≤δ<+n( –q).Then the following estimate holds:
εO–εδP(u,∇)υq≤MO θ εuqO
σ εuq
uniformly inεwith some constant M=M(δ,θ,q,σ)provided thatθ> ,σ> ,σ+δ>
and
θ+σ+δ> n q+
Proof Assume that <ν < n( – q). Since D(Oα
ε) is continuously embedded in Xq ∩
Hα
q (G;Rn), and sinceLq
(G;Rn)∩X
q is the same asXs, by the Sobolev embedding
theo-rem, we obtain that the operator
O–ε,νq:Xq→D
Oνε,q →Xs
is bounded, where
s =
q–
ν n,
q+
q= .
By the duality argument then, we get that the operatoru→O–ν
ε,qis bounded fromXsto
Xq, where
s= –
s=
q +
ν n .
Consider first the caseδ>. SinceP(u,∇)υ is bilinear inu,υ, it suffices to prove the estimate on a dense subspace. Therefore, assume thatuandυare smooth. Sincedivu= , we get
(u,∇)υ=
n
k=
∂ ∂xk
(ukυ).
Takingν=δ–
, using the uniform boundedness ofO –ν
ε,qfromXstoXqand Lemma .
for allε> , we obtain
εO–εδP(u,∇)υq=
εO
–ν ε,q
n
k=
P ∂ ∂xk
(ukυ)
q
≤|u||υ|s.
By assumption we can takerandηsuch that
r≥
q–
θ n ,
η≥
q–
σ n ,
r+
η=
s, r> ,η<∞.
SinceD(Oα
ε,q) is continuously embedded inXq∩Hqα(G;Rn), by the Sobolev embedding,
we get
|u||υ|
s≤ urυη≤MO
θ ε,qurO
σ ε,qυη,
i.e., we have the required result forδ>. In particular, we get the following uniform esti-mate:
εO–
ε P(u,∇)υq≤MOθε,qurO
σ
ε,qυη, θ+β≥ n
q, β> .
Similarly, we obtain
εP(u,∇)υ
q≤Curυη≤CO
θ ε,qurO
for r + η = q andδ= . The above two estimates show that the mapυ →P(u,∇)υ is a uniform bounded operator fromD(Oβ
ε) toD(O
–
ε ) and fromD(O β+
ε ) toXq. By using
Lemma . and the interpolation of Banach spaces [,§..] for ≤δ≤, we obtain
εP(u,∇)υq≤COθε,qurO
σ ε,qυη.
By using Lemma . and the iteration argument, by reasoning as in Fujita and Kato [],
we obtain the following.
Theorem . Let <εk≤, (–)mkαkβk– (–)mkαkβk= .Letγ< be a real number
andδ≥such that
n
q–
≤γ, –γ <δ< –|γ|.
Suppose that a∈D(Oγε),and thatOε–δPf(t)is continuous on(,T)and satisfies
O–εδPf(t)=o
tγ+δ– as t→.
Then there is T∗∈(,T)independent ofεand a local solution of(.)such that () u∈C([,T∗];D(Oγε)),u() =a;
() u∈C((T∗];D(Oαε))for someT∗> ;
() Oα
εu(t)=o(t
γ–α)ast→for allαwithγ<α< –δuniformly with respect to the
parameterε.
Moreover,the solution of(.)is unique if
() u∈C((T∗];D(Oβε));
() Oα
εu(t)=o(t
γ–β)ast→for someβwithβ>|γ|uniformly inε.
Proof We introduce the following iteration scheme:
u(t) =Sε(t)a+
t
Sε(t–s)Pf(s)ds, (.)
um+(t) =u(t) +
t
Sε(t–s)Fum(s)ds, m≥.
By estimating the termu(t) in (.) and by using Proposition . forγ ≤α< –δ, we get
Oαεu(t)≤OαεSε(t)a+
t
Oαε+δSε(t–s)O– δ
ε Pf(s)ds
≤OαεSε(t)a+Cα+δ
t
(t–s)–(α+δ)
O–εδPf(s)ds≤Mαt γ–α
uniformly inεwith
Mα= sup
<t≤T,ε>
where N=sup<t≤Tt–γ–δO–δ
ε Pf(t)andB(a,b) is the beta function. Here we suppose γ +δ> . By induction assume thatum(t) satisfies the following estimate:
Oα
εum(t)≤Mαmt
γ–α, γ ≤α< –δ. (.)
We will estimateOα
εum+(t) by using (.). To estimate the termOε–δFum(s), we suppose
θ+σ+δ= +γ, γ <θ< –δ,γ <σ< –δ,θ> ,σ> ,δ+σ> ,
so that the numbersθ,σ,δsatisfy the assumptions of Lemma .. Using Lemma . and (.), we get the following uniform estimate:
O–εδFum(s)≤CMθmMσmsγ+δ–.
Therefore, we obtain
Oα
εum(t)≤Mαtγ–α+Mα+δ
t
(t–s)–(α+δ)O–δ
ε Fum(s)ds
≤Mαm+tγ–α
with
Mαm+=Mα+Mα+δMB( –δ–α,γ +δ)MθmMσm.
Since we get the uniform estimates with respect to the parameterε, the remaining part
of the proof is the same as in [, Theorem .], so this part is omitted.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA Texas A&M University-Kingsville-2012.
Received: 5 March 2013 Accepted: 12 April 2013 Published: 26 April 2013 References
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