# Nonlocal Navier-Stokes problem with a small parameter

- Veli B Shakhmurov
^{1, 2}Email author

**2013**:107

https://doi.org/10.1186/1687-2770-2013-107

© Shakhmurov; licensee Springer. 2013

**Received: **5 March 2013

**Accepted: **12 April 2013

**Published: **26 April 2013

## 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 uniformly ${L}^{p}$ estimates 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 maximal ${L}_{p}$ regularity## 1 Introduction

*ε*is a small positive parameter,

*a*denotes the initial velocity. This problem is characterized by nonlocality of boundary conditions and by presence of a small term

*ε*which corresponds to the inverse of Reynolds number

*Re*very large for the Navier-Stokes equations. 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.*, [1–6]. In the present work, we established a uniform time of existence and estimates for solutions of problem (1.1)-(1.3). It is clear that for $\epsilon =1$, choosing the boundary conditions locally and ${m}_{kj}=0$, problem (1.1)-(1.3) is reduced to the classical Navier-Stokes problem

*e.g.*, in [1–3, 5, 7–33]. There is extensive literature on the solvability of the initial value problem for the Navier-Stokes equation ( see,

*e.g.*, [25] for further papers cited there ). Hopf [20] proved the existence of a global weak solution of (1.4) using the Faedo-Galerkin approximation and an energy inequality. Another approach to problem (1.4) is to use semigroup theory. Kato and Fujita [18, 22, 34] and Sobolevskii [27] transformed equation (1.4) into an evolution equation in the Hilbert space ${L}^{2}$. They proved the existence of a unique global strong solution for any square-summable initial velocity when $n=2$. On the other hand, when $n=3$ 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 corresponding to the Stokes problem; see, for example, Solonnikov [26] and Heywood [21]. Afterward, Giga and Sohr [13] improved this result in two directions. First, they generalized 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 (boundary value problem) BVP for the following differential operator equation (DOE) with small parameters:

*A*is a linear operator in a Banach space

*E*, ${\alpha}_{kji}$, ${\beta}_{kji}$ are complex numbers, ${\epsilon}_{k}$ are positive and

*λ*is a complex parameter. We show that problem (1.5) has a unique solution $u\in {W}^{2,q}(G;E(A),E)$ for $f\in {W}^{m,q}(G;E)$ and $\lambda \in {S}_{\psi ,\varkappa}$ with sufficiently large $\varkappa >0$, and the following coercive uniform estimate holds:

with $C(q)$ independent of ${\epsilon}_{1}$, ${\epsilon}_{2},\dots ,{\epsilon}_{n}$, *λ* and *f*.

*e.g.*, [33, 35]) for abstract parabolic equations (1.8), we obtain that for every $f\in {L}^{p}(0,T;{L}^{q}(\mathrm{\Omega};{R}^{n}))=B(p,q)$, $p,q\in (1,\mathrm{\infty})$, there is a unique solution $(u,\mathrm{\nabla}\phi )$ of problem (1.8) and the following uniform estimate holds:

with $C=C(T,\mathrm{\Omega},p,q)$ independent of *f* and *ε*. Afterwords, by using the above uniform coercive estimate, we derive local uniform existence and uniform *a priori* estimates of a solution of problem (1.1)-(1.3).

Modern analysis methods, particularly abstract harmonic analysis, the operator theory, the interpolation of Banach spaces, the theory of semigroups of linear operators, embedding and trace theorems in vector-valued Sobolev-Lions spaces are the main tools implemented to carry out the analysis.

## 2 Notations, definitions and background

*E*be a Banach space and ${L}^{p}(\mathrm{\Omega};E)$ denotes the space of strongly measurable

*E*-valued functions that are defined on the measurable subset $\mathrm{\Omega}\subset {R}^{n}$ with the norm

*E*is called a UMD-space if the Hilbert operator

is bounded in ${L}_{p}(R,E)$, $p\in (1,\mathrm{\infty})$ (see, *e.g.*, [36]). UMD spaces include, *e.g.*, ${L}_{p}$, ${l}_{p}$ spaces and Lorentz spaces ${L}_{pq}$, $p,q\in (1,\mathrm{\infty})$.

*A*is said to be

*ψ*-positive in a Banach space

*E*with bound $M>0$ if $D(A)$ is dense on

*E*and ${\parallel {(A+\lambda I)}^{-1}\parallel}_{B(E)}\le M{(1+|\lambda |)}^{-1}$ for any $\lambda \in {S}_{\psi}$, $0\le \psi <\pi $, where

*I*is the identity operator in

*E*, $B(E)$ is the space of bounded linear operators in

*E*. It is known [[30], §1.15.1] that there exist the fractional powers ${A}^{\theta}$ of a positive operator

*A*. Let $E({A}^{\theta})$ denote the space $D({A}^{\theta})$ with the norm

*R*-bounded (see,

*e.g.*, [36]) if there is a positive constant

*C*such that for all ${T}_{1},{T}_{2},\dots ,{T}_{m}\in G$ and ${u}_{1},{u}_{2},\dots ,{u}_{m}\in {E}_{1}$, $m\in \mathbb{N}$,

where $\{{r}_{j}\}$ is a sequence of independent symmetric $\{-1,1\}$-valued random variables on Ω. The smallest *C* for which the above estimate holds is called an *R*-bound of the collection *G* and denoted by $R(G)$.

*R*-bounded if there is a constant

*C*independent of $h\in Q$ such that for all ${T}_{1}(h),{T}_{2}(h),\dots ,{T}_{m}(h)\in {G}_{h}$ and ${u}_{1,}{u}_{2},\dots ,{u}_{m}\in {E}_{1}$, $m\in \mathbb{N}$,

which implies that ${sup}_{h\in Q}R({G}_{h})\le C$.

The *ψ*-positive operator *A* is said to be *R*-positive in a Banach space *E* if the set ${L}_{A}=\{\xi {(A+\xi )}^{-1}:\xi \in {S}_{\psi}\}$, $0\le \psi <\pi $, is *R*-bounded.

The operator $A(t)$ is said to be *ψ*-positive in *E* uniformly in $t\in \sigma $ with bound $M>0$ if $D(A(t))$ is independent of *t*, $D(A(t))$ is dense in *E* and $\parallel {(A(t)+\lambda )}^{-1}\parallel \le M{(1+|\lambda |)}^{-1}$ for all $\lambda \in {S}_{\psi}$, $0\le \psi <\pi $, where *M* does not depend on *t* and *λ*.

*E*be two Banach spaces, and let ${E}_{0}$ be continuously and densely embedded into

*E*. Let Ω be a measurable set in ${R}^{n}$ and

*m*be a positive integer. Let ${W}^{m,p}(\mathrm{\Omega};{E}_{0},E)$ denote the space consisting of all functions $u\in {L}^{p}(\mathrm{\Omega};{E}_{0})$ that have the generalized derivatives $\frac{{\partial}^{m}u}{\partial {x}_{k}^{m}}\in {L}^{p}(\mathrm{\Omega};E)$, with the norm

For $n=1$, $\mathrm{\Omega}=(a,b)$, $a,b\in \mathbb{N}$, the space ${W}^{m,p}(\mathrm{\Omega};{E}_{0},E)$ will be denoted by ${W}^{m,p}(a,b;{E}_{0},E)$.

Sometimes we use one and the same symbol *C* without 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, say *α*, we write ${C}_{\alpha}$.

## 3 Boundary value problems for abstract elliptic equations

In this section, we consider problem (1.5). We derive the maximal regularity properties of this problem.

It should be noted that BVPs for DOEs were studied, *e.g.*, in [35–38] and [6, 26, 27, 39–43]. For references, see [35, 43]. Let ${\alpha}_{kj}={\alpha}_{kj{m}_{k}}$ and ${\beta}_{kj}={\beta}_{kj{m}_{k}}$. First, we prove the following theorem.

**Theorem 3.1**

*Let the following conditions be satisfied*:

- (1)
*E**is a*UMD*space and**A**is an**R*-*positive operator in**E**for*$0\le \psi <\pi $; - (2)
$q\in (1,\mathrm{\infty})$, ${\eta}_{k}={(-1)}^{{m}_{1}}{\alpha}_{k1}{\beta}_{k2}-{(-1)}^{{m}_{2}}{\alpha}_{k2}{\beta}_{k1}\ne 0$, $0<{t}_{k}\le 1$, $k=1,2,\dots ,n$.

*Then problem*(1.5)

*has a unique solution*$u\in {W}^{2,q}(G;E(A),E)$

*for*$f\in {L}^{q}(G;E)$

*and*$\lambda \in {S}_{\psi ,\varkappa}$

*with sufficiently large*$\varkappa >0$.

*Moreover*,

*the following coercive uniform estimate holds*:

*with*$C(q)$*independent of*${\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{n}$, *λ* *and* *f*.

*Proof*Let us consider the BVP

*i.e.*,

*R*-positive in

*F*. Then, by applying again [[40], Theorem 3.2], we get that for $f\in {L}^{q}(0,{b}_{1};F)$, $\lambda \in {S}_{\psi ,\varkappa}$ and sufficiently large $\varkappa >0$, problem (3.5) has a unique solution $u\in {W}^{2,q}(0,{b}_{1};D(B),F)$, and the following coercive uniform estimate holds:

Further, by continuing this process *n*-times, we obtain the assertion.

From Theorem 3.1 we obtain the following. □

**Corollary 3.1**

*Let*$0<{\epsilon}_{k}\le 1$, ${(-1)}^{{m}_{k1}}{\alpha}_{k1}{\beta}_{k2}-{(-1)}^{{m}_{k2}}{\alpha}_{k2}{\beta}_{k1}\ne 0$.

*For*$f\in {L}^{q}(G;{R}^{n})$, $q\in (1,\mathrm{\infty})$

*and for*$\lambda \in {S}_{\psi ,\varkappa}$

*with sufficiently large*$\varkappa >0$,

*there is a unique solution*

*u*

*of problem*(1.5)

*and the following uniform coercive estimate holds*:

*with*$C=C(q)$*independent of* *f*, ${\epsilon}_{k}$*and* *λ*.

*Proof* Let us put $E={R}^{n}$ and $A=\varkappa >0$ in Theorem 3.1. It is known that the operator $A=\varkappa >0$ is *R*-positive in ${R}^{n}$ (see, *e.g.*, [36]). So, the estimate (3.1) implies Corollary 3.1.

From Theorem 3.1 we obtain the following. □

**Result 3.1**For $\lambda \in {S}_{\psi ,\varkappa}$, there is a resolvent ${({Q}_{\epsilon}+\lambda )}^{-1}$ of the operator ${Q}_{\epsilon}$ satisfying the following uniform estimate:

It is clear that the solution *u* of problem (1.5) depends on parameters $\epsilon =({\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{n})$, *i.e.*, $u=u(x,\epsilon )$. In view of Theorem 3.1, we established estimates for the solution of (1.5) uniformly in ${\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{n}$.

## 4 Regularity properties of solutions for DOEs with parameters

In this section, we show the separability properties of problem (1.5) in Sobolev spaces ${W}^{m,q}(G;E)$. The main result is the following theorem.

**Theorem 4.1**

*Let the following conditions be satisfied*:

- (1)
*E**is a*UMD*space and**A**is an**R*-*positive operator in**E*; - (2)
*m**is a positive integer*$q\in (1,\mathrm{\infty})$, $0<{t}_{k}\le 1$,*and*${\eta}_{k}={(-1)}^{{m}_{1}}{\alpha}_{k1}{\beta}_{k2}-{(-1)}^{{m}_{2}}{\alpha}_{k2}{\beta}_{k1}\ne 0,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,n.$

with $C=C(q,A)$ independent of ${\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{n}$, *λ* and *f*.

where ${\sigma}_{i}=\frac{i}{2}+\frac{1}{2p}$, ${m}_{k}\in \{0,m+1\}$, ${\alpha}_{ki}$, ${\beta}_{ki}$ are complex numbers, *t* is positive, *λ* is a complex parameter and *A* is a linear operator in *E*. Let ${A}_{\lambda}=A+\lambda I$.

To prove the main result, we need the following result in [[37], Theorem 2.1].

**Theorem A**

*Let*

*E*

*be a*UMD

*space*,

*A*

*be a*

*ψ*-

*positive operator in*

*E*

*with bound*

*M*, $0\le \psi <\pi $.

*Let*

*m*

*be a positive integer*, $1<p<\mathrm{\infty}$,

*and*$\alpha \in (\frac{1}{2p},\frac{1}{2p}+m)$.

*Then*,

*for*$\lambda \in {S}_{\phi}$,

*an operator*$-{A}_{\lambda}^{\frac{1}{2}}$

*generates a semigroup*${e}^{-x{A}_{\lambda}^{\frac{1}{2}}}$

*which is holomorphic for*$x>0$.

*Moreover*,

*there exists a positive constant*

*C*(

*depending only on*

*M*,

*ψ*,

*m*,

*α*

*and*

*p*)

*such that for every*$u\in {(E,E({A}^{m}))}_{\frac{\alpha}{m}-\frac{1}{2mp},p}$

*and*$\lambda \in {S}_{\psi}$,

In a similar way as in [[43], §1.8.2, Theorem 2], we obtain the following lemma.

**Lemma 4.1**

*Let*

*m*

*and*

*j*

*be integer numbers*, $0\le j\le m-1$, ${\theta}_{j}=\frac{pj+1}{pm}$, $0<t\le 1$, ${x}_{0}\in [0,b]$.

*Then*,

*for*$u\in {W}_{p}^{m}(0,b;{E}_{0},E)$,

*the transformation*$u\to {u}^{(j)}({x}_{0})$

*is bounded linear from*${W}_{p}^{m}(0,b;{E}_{0},E)$

*onto*${({E}_{0},E)}_{{\theta}_{j},p}$

*and the following inequality holds*:

*Let*

**Lemma 4.2**

*Let*

*A*

*be an*

*R*-

*positive operator in a*UMD

*space*

*E*

*and*

*Then problem*(4.3)

*has a unique solution*$u\in {W}^{m+2,p}(0,1;E(A),E)$

*for*${f}_{k}\in {E}_{k}$, $p\in (1,\mathrm{\infty})$, $\lambda \in {S}_{\psi}$,

*and the coercive uniform estimate holds*

*Proof*In a similar way as in [[40], Theorem 3.1], we obtain the representation of the solution of (4.3)

*A*, we obtain from (4.5)

Then from (4.6)-(4.9) we obtain (4.4). □

Now we can represent a more general result for nonhomogeneous problem (4.2).

**Theorem 4.2**

*Assume that the following conditions are satisfied*:

- (1)
*E**is a*UMD*space and**A**is an**R*-*positive operator in**E*; - (2)
$\eta ={(-1)}^{{m}_{1}}{\alpha}_{1}{\beta}_{2}-{(-1)}^{{m}_{2}}{\alpha}_{2}{\beta}_{1}\ne 0$, ${\theta}_{k}=\frac{{m}_{k}}{m+2}+\frac{1}{2p}$, $k=1,2$, $0<t\le 1$, $p\in (1,\mathrm{\infty})$.

*Then the operator*$u\to \{({L}_{t}+\lambda )u,{L}_{1t}u,{L}_{2t}u\}$

*is an isomorphism from*${W}^{m+2,p}(0,1;E(A),E)$

*onto*${W}^{m,p}(0,1;E)\times {E}_{1}\times {E}_{2}$

*for*$\lambda \in {S}_{\psi ,\varkappa}$

*with large enough*$\varkappa >0$.

*Moreover*,

*the uniform coercive estimate holds*

*Proof*The uniqueness of a solution of problem (4.2) is obtained from Lemma 4.2. Let us define

*A*, we have

*R*-positivity of the operator

*A*, the sets

*R*-bounded. Then, in view of the Kahane contraction principle, from the product properties of the collection of

*R*-bounded operators (see,

*e.g.*, [36] Lemma 3.5, Proposition 3.4), we obtain

*u*on $(0,1)$. The estimate (4.16) implies that ${u}_{1}\in {W}^{m+2,p}(0,1;E(A),E)$. By virtue of Lemma 4.1, we get

Finally, from (4.18) and (4.20) we obtain (4.10). □

Now, we can prove the main result of this section.

*Proof of Theorem 4.1*Let ${G}_{2}=(0,{b}_{1})\times (0,{b}_{2})$. It is clear to see that

where ${X}_{0}={W}^{m,q}(0,{b}_{2};E)$ and $X={L}^{q}(0,{b}_{2};E)$.

*X*defined by

*X*are UMD spaces, (see,

*e.g.*, [[35], Theorem 4.5.2]) by virtue of Theorem 4.2, we obtain that problem (4.22) has a unique solution $u\in {W}^{m+2,q}(0,{b}_{1};D({B}_{{\epsilon}_{2}}),X)$ for $f\in {W}^{m,q}(0,{b}_{1};X)$ and $\lambda \in {S}_{\psi ,\varkappa}$ with sufficiently large $\varkappa >0$. Moreover, the coercive uniform estimates holds

From estimates (4.24)-(4.25) we conclude the corresponding claim for problem (4.21). Then, by continuing this process *n*-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 (1.6).

The function $u\in {W}_{\sigma}^{2,q}(G,{L}_{kj\epsilon})=\{u\in {W}^{2,q}(G;{R}^{n}),{L}_{kj\epsilon}u=0,divu=0\}$ satisfying equation (1.6) a.e. on *G* is called the stronger solution of problem (1.6).

*s*such that ${W}^{0,q}(G)={L}^{q}(G)$. For $q\in (1,\mathrm{\infty})$, let ${X}_{q}={L}_{\sigma}^{q}(G)$ denote the closure of ${C}_{0\sigma}^{\mathrm{\infty}}(G)$ in ${L}^{p}(G;{R}^{n})$, where

*e.g.*, Fujiwara and Morimoto [17]) a vector field $u\in {L}^{q}(G;{R}^{n})$ has the Helmholtz decomposition,

*i.e.*, all $u\in {L}^{q}(G;{R}^{n})$ can be uniquely decomposed as $u={u}_{0}+\mathrm{\nabla}\phi $ with ${u}_{0}\in {L}_{\sigma}^{q}(G)$, ${u}_{0}={P}_{q}u$, where ${P}_{q}=P$ is a projection operator from ${L}^{q}(G;{R}^{n})$ to ${L}_{\sigma}^{q}(G)$ and $\phi \in {L}_{\mathrm{loc}}^{q}(\overline{G})$, $\mathrm{\nabla}\phi \in {L}^{q}(G;{R}^{n})$, so that

with *C* independent of *u*, where *B* is an open ball in ${R}^{n}$ and ${\parallel u\parallel}_{p}$ denotes the norm of *u* in ${L}^{q}(G;{R}^{n})$ or ${L}^{q}(G)$.

*i.e.*,

From Corollary 3.1 we get that the operator ${O}_{\epsilon}$ is positive and also is a generator of a bounded holomorphic semigroup ${S}_{\epsilon}(t)=exp(-{O}_{\epsilon}t)$ for $t>0$.

In a similar way as in [18], we show the following.

**Proposition 5.1**

*The following estimate holds*:

*uniformly in*$\epsilon =({\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{n})$*for*$\alpha \ge 0$*and*$t>0$.

*Proof*From Result 3.1 we obtain that the operator ${O}_{\epsilon}$ is uniformly positive in ${L}_{q}(G;{R}^{n})$,

*i.e.*, for $\lambda \in {S}_{\psi ,\varkappa}$, $0<\psi <\pi $, the following estimate holds:

where the constant *M* is independent of *λ* and *ε*. Then, by using the Danford integral and operator calculus as in [18], we obtain the assertion. □

Now consider problem (1.7). The main theorem in this section is the following.

**Theorem 5.1**

*Let*$0<{\epsilon}_{k}\le 1$, ${(-1)}^{{m}_{k1}}{\alpha}_{k1}{\beta}_{k2}-{(-1)}^{{m}_{k2}}{\alpha}_{k2}{\beta}_{k1}\ne 0$

*and*$p,q\in (1,\mathrm{\infty})$.

*Then there is a unique solution*$(u,\mathrm{\nabla}\phi )$

*of problem*(1.7)

*for*$f\in {L}^{p}(0,T;{L}^{q}(G;{R}^{n}))=B(p,q)$

*and*$a\in {B}_{p,q}^{2-\frac{2}{p}}$.

*Moreover*,

*the following uniform estimate holds*:

*with*$C=C(T,G,p,q)$*independent of* *f* *and* *ε*.

*Proof*Problem (1.7) can be expressed as the following abstract parabolic problem with a small parameter:

*R*-positive in

*E*. Since

*E*is a UMD space, in a similar way as in [[33], Theorem 4.2], we obtain that for $f\in {L}^{p}(0,T;E)$ and $a\in {(D({O}_{\epsilon}),E)}_{\frac{1}{p},p}$, there is a unique solution $u\in {W}^{1,p}(0,T,D({O}_{\epsilon}),E)$ of problem (5.3) so that the following uniform estimate holds:

uniformly in $\epsilon =({\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{n})$. □

## 6 Existence and uniqueness for the Navier-Stokes equation with parameters

To prove the main result, we need the following result which are obtained in a similar way as in [[11], Theorem 2].

**Lemma 6.1** *For any*$0\le \alpha \le 1$, *the domain*$D({O}_{\epsilon}^{\alpha})$*is the complex interpolation space*${[{X}_{q},D({O}_{\epsilon})]}_{\alpha}$.

**Lemma 6.2** *For each*$k=1,2,\dots ,n$, *the operator*$u\to {O}_{\epsilon}^{-\frac{1}{2}}P(\frac{\partial}{\partial {x}_{k}})u$*extends uniquely to a uniformly bounded linear operator from*${L}^{q}(G;{R}^{n})$*to*${X}_{q}$.

*Proof*Since ${O}_{\epsilon}$ is a positive operator, it has fractional powers ${O}_{\epsilon}^{\alpha}$. From Lemma 6.1, it follows that the domain $D({O}_{\epsilon}^{\alpha})$ is continuously embedded in ${X}_{q}\cap {H}_{q}^{2\alpha}(G;{R}^{n})$ for any $\alpha >0$, where ${H}_{q}^{2\alpha}(G;{R}^{n})$ is the vector-valued Bessel space. Then, by using the duality argument and due to uniform positivity of ${O}_{\epsilon}^{\frac{1}{2}}$, we obtain the following uniformly in

*ε*estimate:

By reasoning as in [12], we obtain the following. □

**Lemma 6.3**

*Let*$0\le \delta <\frac{1}{2}+\frac{n}{2}(1-\frac{1}{q})$.

*Then the following estimate holds*:

*uniformly in*

*ε*

*with some constant*$M=M(\delta ,\theta ,q,\sigma )$

*provided that*$\theta >0$, $\sigma >0$, $\sigma +\delta >\frac{1}{2}$

*and*

*Proof*Assume that $0<\nu <\frac{n}{2}(1-\frac{1}{q})$. Since $D({O}_{\epsilon}^{\alpha})$ is continuously embedded in ${X}_{q}\cap {H}_{q}^{2\alpha}(G;{R}^{n})$, and since ${L}^{{q}^{\mathrm{\prime}}}(G;{R}^{n})\cap {X}_{{q}^{\mathrm{\prime}}}$ is the same as ${X}_{{s}^{\mathrm{\prime}}}$, by the Sobolev embedding theorem, we obtain that the operator

*u*,

*υ*, it suffices to prove the estimate on a dense subspace. Therefore, assume that

*u*and

*υ*are smooth. Since $divu=0$, we get

*r*and

*η*such that

*i.e.*, we have the required result for $\delta >\frac{1}{2}$. In particular, we get the following uniform estimate:

By using Lemma 6.3 and the iteration argument, by reasoning as in Fujita and Kato [18], we obtain the following. □

**Theorem 6.1**

*Let*$0<{\epsilon}_{k}\le 1$, ${(-1)}^{{m}_{k1}}{\alpha}_{k1}{\beta}_{k2}-{(-1)}^{{m}_{k2}}{\alpha}_{k2}{\beta}_{k1}\ne 0$.

*Let*$\gamma <1$

*be a real number and*$\delta \ge 0$

*such that*

*Suppose that*$a\in D({O}_{\epsilon}^{\gamma})$,

*and that*$\parallel {O}_{\epsilon}^{-\delta}Pf(t)\parallel $

*is continuous on*$(0,T)$

*and satisfies*

*Then there is*${T}_{\ast}\in (0,T)$

*independent of*

*ε*

*and a local solution of*(6.2)

*such that*

- (1)
$u\in C([0,{T}_{\ast}];D({O}_{\epsilon}^{\gamma}))$, $u(0)=a$;

- (2)
$u\in C((0{T}_{\ast}];D({O}_{\epsilon}^{\alpha}))$

*for some*${T}_{\ast}>0$; - (3)
$\parallel {O}_{\epsilon}^{\alpha}u(t)\parallel =o({t}^{\gamma -\alpha})$

*as*$t\to 0$*for all**α**with*$\gamma <\alpha <1-\delta $*uniformly with respect to the parameter**ε*.

*Moreover*,

*the solution of*(5.2)

*is unique if*

- (4)
$u\in C((0{T}_{\ast}];D({O}_{\epsilon}^{\beta}))$;

- (5)
$\parallel {O}_{\epsilon}^{\alpha}u(t)\parallel =o({t}^{\gamma -\beta})$

*as*$t\to 0$*for some**β**with*$\beta >|\gamma |$*uniformly in**ε*.

*ε*with

*θ*,

*σ*,

*δ*satisfy the assumptions of Lemma 6.3. Using Lemma 6.3 and (6.5), we get the following uniform estimate:

Since we get the uniform estimates with respect to the parameter *ε*, the remaining part of the proof is the same as in [[12], Theorem 2.3], so this part is omitted. □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

## References

- Beirâo da Veiga H: Vorticity and regularity for flows under the Navier boundary condition.
*Commun. Pure Appl. Anal.*2006, 5(4):907-918.MATHMathSciNetView ArticleGoogle Scholar - Hamouda M, Temam R:
*Some Singular Perturbation Problems Related to the Navier-Stokes Equations, Advances in Deterministic and Stochastic Analysis*. World Scientific, Hackensack; 2007:197-227.Google Scholar - Iftimie D, Planas G: Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions.
*Nonlinearity*2006, 19(4):899-918. 10.1088/0951-7715/19/4/007MATHMathSciNetView ArticleGoogle Scholar - Lions J-L 1. In
*Mathematical Topics in Fluid Mechanics*. The Clarendon Press Oxford University Press, New York; 1996. Incompressible models, Oxford Science PublicationsGoogle Scholar - Temam R, Wang X: Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case.
*J. Differ. Equ.*2002, 179(2):647-686. 10.1006/jdeq.2001.4038MATHMathSciNetView ArticleGoogle Scholar - Xiao Y, Xin Z: On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition.
*Commun. Pure Appl. Math.*2007, 60(7):1027-1055. 10.1002/cpa.20187MATHMathSciNetView ArticleGoogle Scholar - Amann H: On the strong solvability of the Navier-Stokes equations.
*J. Math. Fluid Mech.*2000, 2: 16-98. 10.1007/s000210050018MATHMathSciNetView ArticleGoogle Scholar - Caffarelli L, Kohn R, Nirenberg L: Partial regularity of suitable weak solutions of the Navier-Stokes equations.
*Commun. Pure Appl. Math.*1982, 35: 771-831. 10.1002/cpa.3160350604MATHMathSciNetView ArticleGoogle Scholar - Cannone M: A generalization of a theorem by Kato on Navier-Stokes equations.
*Rev. Mat. Iberoam.*1997, 13(3):515-541.MATHMathSciNetView ArticleGoogle Scholar - Desch W, Hieber M, Prüss J:${L}_{p}$-theory of the Stokes equation in a half-space.
*J. Evol. Equ.*2001., 2001: Article ID 1Google Scholar - Giga Y:Domains of fractional powers of the Stokes operator in ${L}_{r}$ spaces.
*Arch. Ration. Mech. Anal.*1985, 89: 251-265. 10.1007/BF00276874MATHMathSciNetView ArticleGoogle Scholar - Giga Y, Miyakava T:Solutions in ${L}_{r}$ of the Navier-Stokes initial value problem.
*Arch. Ration. Mech. Anal.*1985, 89: 267-281. 10.1007/BF00276875MATHView ArticleGoogle Scholar - Giga Y, Sohr H:Abstract ${L}_{p}$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains.
*J. Funct. Anal.*1991, 102: 72-94. 10.1016/0022-1236(91)90136-SMATHMathSciNetView ArticleGoogle Scholar - Giga Y:Solutions for semilinear parabolic equation ${L}^{p}$ and regularity of weak solutions of the Navier-Stokes systems.
*J. Differ. Equ.*1986, 61: 186-212.MATHMathSciNetView ArticleGoogle Scholar - Galdi GP:
*An Introduction to the Mathematical Theory of the Navier-Stokes Equations I: Linearized Steady Problems*. 2nd edition. Springer, Berlin; 1998.Google Scholar - Fefferman C, Constantin P: Direction of vorticity and the problem of global regularity for the 3-d Navier-Stokes equations.
*Indiana Univ. Math. J.*1993, 42: 775-789. 10.1512/iumj.1993.42.42034MATHMathSciNetView ArticleGoogle Scholar - Fujiwara D, Morimoto H:An ${L}_{r}$-theorem of the Helmholtz decomposition of vector fields.
*J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math.*1977, 24: 685-700.MATHMathSciNetGoogle Scholar - Fujita H, Kato T: On the Navier-Stokes initial value problem I.
*Arch. Ration. Mech. Anal.*1964, 16: 269-315. 10.1007/BF00276188MATHMathSciNetView ArticleGoogle Scholar - Fabes EB, Lewas JE, Riviere NM: Boundary value problems for the Navier-Stokes equations.
*Am. J. Math.*1977, 99: 626-668. 10.2307/2373933MATHView ArticleGoogle Scholar - Hopf E: Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen.
*Math. Nachr.*1950-51, 4: 213-231. 10.1002/mana.3210040121MATHMathSciNetView ArticleGoogle Scholar - Heywood JG: The Navier-Stokes equations: on the existence, regularity and decay of solutions.
*Indiana Univ. Math. J.*1980, 29: 639-681. 10.1512/iumj.1980.29.29048MATHMathSciNetView ArticleGoogle Scholar - Kato T, Fujita H: On the nonstationary Navier-Stokes system.
*Rend. Semin. Mat. Univ. Padova*1962, 32: 243-260.MATHMathSciNetGoogle Scholar - Masmoudi N: Examples of singular limits in hydrodynamics. Handb. Differ. Equ. III. In
*Evolutionary Equations*. Elsevier, Amsterdam; 2007:195-276.Google Scholar - Leray J: Sur le mouvement d’un liquide visqueux emplissant l’espace.
*Acta Math.*1934, 63: 193-248. 10.1007/BF02547354MATHMathSciNetView ArticleGoogle Scholar - Ladyzhenskaya OA:
*The Mathematical Theory of Viscous Incompressible Flow*. Gordon and Breach, New York; 1969.MATHGoogle Scholar - Solonnikov V: Estimates for solutions of nonstationary Navier-Stokes equations.
*J. Sov. Math.*1977, 8: 467-529. 10.1007/BF01084616MATHView ArticleGoogle Scholar - Sobolevskii PE: Study of Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces.
*Sov. Math. Dokl.*1964, 5: 720-723.Google Scholar - Shakhmurov VB: Separable anisotropic differential operators and applications.
*J. Math. Anal. Appl.*2006, 327(2):1182-1201.MathSciNetView ArticleGoogle Scholar - Teman R:
*Navier-Stokes Equations*. North-Holland, Amsterdam; 1984.Google Scholar - Triebel H:
*Interpolation Theory. Function Spaces. Differential Operators*. North-Holland, Amsterdam; 1978.Google Scholar - Wiegner M: Navier-Stokes equations a neverending challenge.
*Jahresber. Dtsch. Math.-Ver.*1999, 101: 1-25.MATHMathSciNetGoogle Scholar - Weissler FB:The Navier-Stokes initial value problem in ${L}_{p}$.
*Arch. Ration. Mech. Anal.*1980, 74: 219-230. 10.1007/BF00280539MATHMathSciNetView ArticleGoogle Scholar - Weis L:Operator-valued Fourier multiplier theorems and maximal ${L}_{p}$ regularity.
*Math. Ann.*2001, 319: 735-758. 10.1007/PL00004457MATHMathSciNetView ArticleGoogle Scholar - Kato T:Strong ${L}_{p}$-solutions of the Navier-Stokes equation in ${R}^{m}$, with applications to weak solutions.
*Math. Z.*1984, 187: 471-480. 10.1007/BF01174182MATHMathSciNetView ArticleGoogle Scholar - Amann H 1. In
*Linear and Quasi-Linear Equations*. Birkhäuser, Basel; 1995.Google Scholar - Denk R, Hieber M, Pruss J:
*R*-boundedness, Fourier multipliers and problems of elliptic and parabolic type.*Mem. Am. Math. Soc.*2005., 166: Article ID 788Google Scholar - Dore C, Yakubov S: Semigroup estimates and non coercive boundary value problems.
*Semigroup Forum*2000, 60: 93-121. 10.1007/s002330010005MATHMathSciNetView ArticleGoogle Scholar - Lunardi A:
*Analytic Semigroups and Optimal Regularity in Parabolic Problems*. Birkhäuser, Basel; 2003.Google Scholar - Shakhmurov VB: Nonlinear abstract boundary value problems in vector-valued function spaces and applications.
*Nonlinear Anal., Theory Methods Appl.*2006, 67(3):745-762.MathSciNetView ArticleGoogle Scholar - Shakhmurov VB: Linear and nonlinear abstract equations with parameters.
*Nonlinear Anal., Theory Methods Appl.*2010, 73: 2383-2397. 10.1016/j.na.2010.06.004MATHMathSciNetView ArticleGoogle Scholar - Shakhmurov VB, Shahmurova A: Nonlinear abstract boundary value problems atmospheric dispersion of pollutants.
*Nonlinear Anal., Real World Appl.*2010, 11(2):932-951. 10.1016/j.nonrwa.2009.01.037MATHMathSciNetView ArticleGoogle Scholar - Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations.
*J. Math. Anal. Appl.*2004, 292(2):605-620. 10.1016/j.jmaa.2003.12.032MATHMathSciNetView ArticleGoogle Scholar - Yakubov S, Yakubov Ya:
*Differential-Operator Equations. Ordinary and Partial Differential Equations*. Chapman and Hall/CRC, Boca Raton; 2000.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.