# Nonlocal Navier-Stokes problem with a small parameter

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

**2013**:107

**DOI: **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

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