Navier-Stokes problems in half space with parameters

TL;DR

This paper establishes existence, uniqueness, and uniform estimates for solutions of parameter-dependent Navier-Stokes problems in half spaces, including applications to Wentzell-Robin boundary conditions.

Contribution

It provides new results on the solvability and estimates for Navier-Stokes equations with parameters and boundary conditions in half spaces.

Findings

Existence and uniqueness of solutions with parameter dependence

Uniform L^p estimates for solutions

Application to Wentzell-Robin boundary problems

Abstract

The existence, uniqueness and uniformly estimates for solutions of the parameter dependent abstract Navier-Stokes problem on half space are derived. In application the existence, uniqueness and uniformly L^{p} estimates for solution of the Wentzell-Robin type mixed problem for Navier-Stokes equation is established.

Equations

$$\frac{\partial u}{\partial t}-{△}_{\epsilon }u+\left(u.\nabla \right)u+\nabla \phi +Au=f\left(x,t\right),\mathrm{div}u=0,$$

$$\sum _{i=0}^{\nu }{\epsilon }_n^{{\sigma }_i}{\alpha }_i\frac{{\partial }^{i}u}{\partial x_n^i}\left(x^{\prime },0,t\right)=0\text{, }\nu \in \left\{0,1\right\},$$

$$u\left(x,0\right)=a\left(x\right),x\in R_{+}^n,t\in \left(0,T\right),$$

$$R_{+}^n=\left\{x\in R^n,x_n>0,x=\left(x^{\prime },x_n\right),x^{\prime }=\left(x_1,x_2,...,x_{n-1}\right)\right\},$$

$${△}_{\epsilon }u=\sum _{k=1}^n{\epsilon }_k\frac{{\partial }^{2}u}{\partial x_k^2},{\sigma }_i=\frac{1}{2}\left(i+\frac{1}{q}\right)\text{, }q\in \left(1,\infty \right)\text{,}$$

$$u=u_{\epsilon }\left(x,t\right)=\left(u_1\left(x,t\right),u_2\left(x,t\right),...,u_n\left(x,t\right)\right),u_k\left(x,t\right)=u_{k\epsilon }\left(x,t\right)$$

$$\frac{\partial u}{\partial t}-{△}_{\epsilon }u+Au+\nabla \phi =f\left(x,t\right),\mathrm{div}u=0,x\in R_{+}^n,t\in \left(0,T\right),$$

$$\sum _{i=0}^{\nu }{\epsilon }_n^{{\sigma }_i}{\alpha }_i\frac{{\partial }^{i}u}{\partial x_n^i}\left(x^{\prime },0,t\right)=0\text{, }\nu \in \left\{0,1\right\},u\left(x,0\right)=a\left(x\right)$$

$${\Vert \frac{\partial u}{\partial t}\Vert }_{B\left(p,q\right)}+\sum _{k=1}^n{\Vert {\epsilon }_k\frac{{\partial }^{2}u}{\partial x_k^2}\Vert }_{B\left(p,q\right)}+{\Vert Au\Vert }_{B\left(p,q\right)}+{\Vert \nabla \phi \Vert }_{B\left(p,q\right)}\le$$

$$C\left({\Vert f\Vert }_{B\left(p,q\right)}+{\Vert a\Vert }_{B_{p,q}^{2-\frac{2}{p}}}\right)$$

$$t^{\left(1-\frac{n}{q}\right)/2}{u}_{\epsilon }\text{, }{t}^{\left(1-\frac{n}{2q}\right)}\nabla u_{ϵ}\in C\left([0,\infty );L^q\left(R_{+}^n;E\right)\right)\text{ for }n\le q\le \infty .$$

$$\underset{t,{\epsilon }_k}{\sup }{\Vert t^{\left(1-\frac{n}{q}\right)/2}{u}_{\epsilon }\Vert }_{L^q\left(R_{+}^n;E\right)}\le C,\underset{t,{\epsilon }_k}{\sup }{\Vert t^{\left(1-\frac{n}{2q}\right)}\nabla u_{\epsilon }\Vert }_{L^q\left(R_{+}^n;E\right)}\le C,k=1,2,...,n$$

$$D\left(A\right)=\left\{u\in W_{p_1}^2\left(0,1\right),B_{j}u=Au\left(j\right)+\sum _{i=0}^1{\alpha }_{ji}{u}^{\left(i\right)}\left(j\right),j=0,1\right\},$$

$$Au=a{u}^{\left(2\right)}+b{u}^{\left(1\right)}+cu,$$

$$\frac{\partial u}{\partial t}-{△}_{\epsilon }u+\left(u.\nabla \right)u+\nabla \phi +a\frac{{\partial }^{2}u}{\partial y^2}+b\frac{\partial u}{\partial y}+cu=f\left(x,y,t\right),$$

$${\mathrm{div}}_{x}u=0,u=u\left(x,y,t\right),x\in R_{+}^n,$$

$$\sum _{i=0}^{\nu }{\epsilon }_n^{{\sigma }_i}{\alpha }_i\frac{{\partial }^{i}u}{\partial x_n^i}\left(x^{\prime },0,y,t\right)=0\text{, }\nu \in \left\{0,1\right\},x^{\prime }\in R^{n-1}\text{, }y\in \left(0,1\right)$$

$$Au\left(x,j,t\right)+\sum _{i=0}^1{\alpha }_{ji}{u}^{\left(i\right)}\left(x,j,t\right)=0\text{, }u\left(x,0\right)=a\left(x\right).$$

$$\tilde{\Omega }=R_{+}^n\times \left(0,1\right),\mathbf{p}=\left(p_1,p\right).$$

$${\Vert f\Vert }_{L^{\mathbf{p}}\left(\tilde{\Omega }\right)}={\left(\underset{R_{+}^n}{\int }{\left(\underset{0}{\overset{1}{\int }}{\left|f\left(x,y\right)\right|}^{p_1}dy\right)}^{\frac{p}{p_1}}dx\right)}^{\frac{1}{p}}<\infty .$$

$${\Vert f\Vert }_{L^p}={\Vert f\Vert }_{L^p\left(\Omega ;E\right)}={\left(\underset{\Omega }{\int }{\Vert f\left(x\right)\Vert }_E^{p}dx\right)}^{\frac{1}{p}},1\le p<\infty .$$

$${\Vert u\Vert }_{W^{m,p}\left(\Omega ;E_0,E\right)}={\Vert u\Vert }_{L^p\left(\Omega ;E_0\right)}+\sum _{k=1}^n{\Vert \frac{{\partial }^{m}u}{\partial x_k^m}\Vert }_{L^p\left(\Omega ;E\right)}<\infty .$$

$$-{△}_{\epsilon }u+\left(A+\lambda \right)u=f\left(x\right),x\in R_{+}^n,$$

$$\sum _{i=0}^{\nu }{\epsilon }_n^{{\sigma }_i}{\alpha }_i\frac{{\partial }^{i}u}{\partial x_n^i}\left(x^{\prime },0,t\right)=0\text{, }$$

$${△}_{\epsilon }u=\sum _{k=1}^n{\epsilon }_k\frac{{\partial }^{2}u}{\partial x_k^2},{\sigma }_i=\frac{1}{2}\left(i+\frac{1}{q}\right)\text{, }\nu \in \left\{0,1\right\}.$$

$$\sum _{k=1}^n\sum _{i=0}^{m+2}{\epsilon }_k^{\frac{i}{m+2}}{\left|\lambda \right|}^{1-\frac{i}{m+2}}{\Vert \frac{{\partial }^{i}u}{\partial x_k^i}\Vert }_{L^q\left(R_{+}^n;E\right)}+{\Vert Au\Vert }_{L^q\left(R_{+}^n;E\right)}\le C{\Vert f\Vert }_{W^{m,q}\left(R_{+}^n;E\right)}$$

$$D\left(Q_{\epsilon }\right)=W^{2,q}\left(R_{+}^n;L_{1\epsilon }\right)=\left\{u\in W^{2,q}\left(R_{+}^n\right),L_{1\epsilon }u=0\right\}\text{, }$$

$$Q_{\epsilon }u=-{△}_{\epsilon }u+Au.$$

$$\sum _{k=1}^n\sum _{i=0}^2{\left|\lambda \right|}^{1-\frac{i}{2}}{\epsilon }_k^{\frac{i}{2}}{\Vert \frac{{\partial }^i}{\partial x_k^i}{\left(Q_{\epsilon }+\lambda \right)}^{-1}\Vert }_{B\left(L^q\left(R_{+}^n;E\right)\right)}\le C.$$

$$\sum _{k=1}^n\sum _{i=1}^2{\epsilon }_k^{\frac{i}{2}}{\Vert \frac{{\partial }^{i}u}{\partial {\epsilon }_k^i}\Vert }_{L^q\left(R_{+}^n;E\right)}\le \frac{C}{\left|\lambda \right|}{\Vert \left(Q_{\epsilon }+\lambda \right)u\Vert }_{L^q\left(R_{+}^n;E\right)}.$$

Full text

**Navier-Stokes problems in half space with parameters **

Veli.B. Shakhmurov

Department of Mechanical Engineering, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey,

E-mail: veli.sahmurov@okan.edu.tr

Khazar University, Baky Azerbaijan

Abstract

The existence, uniqueness and uniformly $L^{p}$ estimates for solutions of the parameter dependent abstract Navier-Stokes problem on half space are derived. In application the existence, uniqueness and uniformly $L^{\mathbf{p}}$ estimates for solution of the Wentzell-Robin type mixed problem for Navier-Stokes equation is established.

**Key Word:  **Stokes systems, Navier-Stokes equations, Differential equations with small parameters, Semigroups of operators, Boundary value problems, Differential-operator equations, Maximal $L^{p}$ regularity

MSC 2010: 35xx, 35Jxx, 35Kxx, 43Axx, 47Axx

1. Introduction

We will consider the initial boundary value problems (IBVP) for Navier-Stokes equation (NSE) with small parameter

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[TABLE]

[TABLE]

where

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[TABLE]

$\alpha_{i}$ are complex numbers, $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$, $\varepsilon_{k}$ are small positive parameters and $A$ is a linear operator in a Banach space $E$. Here

[TABLE]

and $\varphi=\varphi\left(x,t\right)$ are represent the $E$-valued unknown velocity and pressure like functions, respectively; $f=\left(f_{1}\left(x,t\right),f_{2}\left(x,t\right),...,f_{n}\left(x,t\right)\right)$ and $a$ represent a given $E$-valued external force and the initial velocity. In this work, we show the uniform existence and uniqueness of the stronger local and global solution of the Navier-Stokes problem with small parameter $\left(1.1\right)-\left(1.3\right)$. This problem is characterized by presence abstract operator $A$ and a small parameters $\varepsilon_{k}$ which corresponds to the inverse of Reynolds number $Re$ very large for the Navier-Stokes equations. The regularity properties of Navier-Stokes equations studied in e.g. $\left[4-6\right]$ and $\left[9-15\right]$. Navier-Stokes equations with small viscosity when the boundary is either characteristic or non-characteristic have been well-studied see, e.g. in $\left[\text{9, 11, 21}\right].$ Moreover, regularity properties of differential operator equation (DOE) were investigated e.g. in $\left[\text{1, 2, 16-20, 23}\right].$ Here we consier Navier-Stokes operator equation in a Banach space $E$. Since the Banach space $E$ is arbitrary and $A$ is a possible linear operator, by chousing spaces $E$ and operators $A$ we can obtained existence, uniqueness and $L^{p}$ estimates of solutions for numerous class of Novier-Stokes type problems.

In this paper, firstly we prove that the Stokes problem

[TABLE]

[TABLE]

has a unique solution $\left(u,\nabla\varphi\right)$ for $f\in L^{p}\left(0,T;L^{q}\left(R_{+}^{n};E\right)\right)=B\left(p,q\right),$ $p,q\in\left(1,\infty\right)$ and the following uniform estimate holds

[TABLE]

[TABLE]

with $C=C\left(T,p,q\right)$ independent of $f$ and $\varepsilon.$

Then, by following Kato and Fujita $\left[6,\text{ }10\right]$ method and using the above uniform coercive estimate for Stokes problem we derive a local a priori estimates for solutions of $\left(1.1\right)-\left(1.3\right)$, i.e., we prove that for $\gamma<1$ and $\delta\geq 0$ such that $\frac{n}{2q}-\frac{1}{2}\leq\gamma,$ $-\gamma<\delta<1-\left|\gamma\right|,$ $a\in D\left(O_{\varepsilon q}^{\gamma}\right)$ there is $T_{\ast}\in\left(0,T\right)$ independent of $\varepsilon_{k}\in\left(0\right.,\left.1\right]$ such that $\left\|O_{\varepsilon q}^{-\delta}Pf\left(t\right)\right\|$ is continuous on $\left(0,T\right)$ and satisfies $\left\|O_{\varepsilon q}^{-\delta}Pf\left(t\right)\right\|=o\left(t^{\gamma+\delta-1}\right)$ as $t\rightarrow 0,$ then there is a local solution of $\left(1.1\right)-\left(1.3\right)$ such that $u\in C\left(\left[0,T_{\ast}\right];D\left(O_{\varepsilon q}^{\gamma}\right)\right)$, $u\left(0\right)=a,$ $u\in C\left(\left(0\right.\left.T_{\ast}\right];D\left(O_{\varepsilon q}^{\alpha}\right)\right)$ for some $T_{\ast}>0,\left\|O_{\varepsilon q}^{\alpha}u\left(t\right)\right\|=o\left(t^{\gamma-\alpha}\right)$ as $t\rightarrow 0$ for all $\alpha$ with $\gamma<\alpha<1-\delta$ uniformly in $\varepsilon.$ Moreover, the solution of $\left(1.1\right)-\left(1.3\right)$ is unique if $u\in C\left(\left(0\right.\left.T_{\ast}\right];D\left(O_{\varepsilon q}^{\beta}\right)\right),$ $\left\|O_{\varepsilon q}^{\alpha}u\left(t\right)\right\|=o\left(t^{\gamma-\beta}\right)$ as $t\rightarrow 0$ for some $\beta$ with \beta>\left|\gamma\right|\uniformly in $\varepsilon$. For sufficiently small date we show that, there is a global solution of the problem $\left(1.1\right)-\left(1.3\right)$. Particularly, we prove that there is a $\delta>0$ such that if $\left\|a\right\|_{L^{q}\left(R_{+}^{n};E\right)}<\delta$, then there is a global solution $u_{\varepsilon}$ of $\left(1.1\right)-\left(1.3\right)$ so that

[TABLE]

Moreover, the following uniform estimates hold

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In application we choose $E=L_{p_{1}}\left(\Omega\right)$ and $A$ to be differential operator with generalized Wentzell-Robin boundary condition defined by

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[TABLE]

in $\left(1.1\right)-\left(1.2\right),$ where $\alpha_{ji}$ are complex numbers, $a,$ $b$, $c$ are complex-valued functions. Then, we obtain the following Wentzell-Robin type mixed problem for Novier-Stokes equation

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[TABLE]

[TABLE]

[TABLE]

Note that, the regularity properties of Wentzell-Robin type BVP for elliptic equations were studied e.g. in $\left[\text{7, 8}\right]$ and the references therein. Here

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$L^{\mathbf{p}}\left(\tilde{\Omega}\right)$ denotes the space of all $\mathbf{p}$-summable complex-valued functions with mixed norm i.e., the space of all measurable functions $f$ defined on $\tilde{\Omega}$, for which

[TABLE]

By using the above general abstract result, the existence, uniqueness and uniformly $L^{\mathbf{p}}\left(\tilde{\Omega}\right)$ estimates for solution of the problem $\left(1.5\right)-\left(1.7\right)$ is obtained.

Let $E$ be a Banach space and $L^{p}\left(\Omega;E\right)$ denotes the space of strongly measurable $E$-valued functions that are defined on the measurable subset $\Omega\subset R^{n}$ with the norm

[TABLE]

The Banach space $E$ is called an $UMD$-space if the Hilbert operator $\left(Hf\right)\left(x\right)=\lim\limits_{\varepsilon\rightarrow 0}\int\limits_{\left|x-y\right|>\varepsilon}\frac{f\left(y\right)}{x-y}dy$ is bounded in $L^{p}\left(R,E\right),$ $p\in\left(1,\infty\right)$ (see. e.g. $\left[\text{2, \S\ 4}\right]$). $UMD$ spaces include e.g. $L^{p}$, $l^{p}$ spaces and Lorentz spaces $L_{pq},$ $p$, $q\in\left(1,\infty\right)$.

Let $E_{1}$ and $E_{2}$ be two Banach spaces. Let $B\left(E_{1},E_{2}\right)$ denote the space of all bounded linear operators from $E_{1}$ to $E_{2}.$ For $E_{1}=E_{2}=E$ it will be denoted by $B\left(E\right).$

A linear operator $A$ is said to be positive in a Banach space $E$ with bound $M>0$ if $D\left(A\right)$ is dense on $E$ and $\left\|\left(A+\lambda I\right)^{-1}\right\|_{B\left(E\right)}\leq M\left(1+\left|\lambda\right|\right)^{-1}$ for any $\lambda\in\left(-\infty\right.,\left.0\right]$ where $I$ is the identity operator in $E$ (see e.g $\left[\text{22, \S 1.15.1}\right]$).

The positive operator $A$ is said to be $R$-positive in a Banach space $E$ if the set $L_{A}=\left\{\xi\left(A+\xi\right)^{-1}\text{: }\xi\in\left(-\infty\right.,\left.0\right]\right\}$, is $R$-bounded (see $\left[\text{2, \S\ 4}\right]$).

The operator $A\left(s\right)$ is said to be positive in $E$ uniformly with respect to papameter $s$ with bound $M>0$ if $D\left(A\left(s\right)\right)$ is independent on $s$, $D\left(A\left(s\right)\right)$ is dense in $E$ and $\left\|\left(A\left(s\right)+\lambda\right)^{-1}\right\|\leq\frac{M}{1+\left|\lambda\right|}$ for all $\lambda\in S_{\psi},0\leq\psi<\pi$, where $M$ does not depend on $s$ and $\lambda.$

Assume $E_{0}$ and $E$ are two Banach spaces and $E_{0}$ is continuously and densely embeds into $E$. Here $\Omega$ is a measurable set in $R^{n}$ and $m$ is a positive integer. Let $W^{m,p}\left(\Omega;E_{0},E\right)$ denote the space of all functions $u\in L^{p}\left(\Omega;E_{0}\right)$ that have the generalized derivatives $\frac{\partial^{m}u}{\partial x_{k}^{m}}\in L^{p}\left(\Omega;E\right)$ with the norm

[TABLE]

2. Regularity properties of solutions for DOEs with parameters

In this section, we consider the boundary value problem (BVP) for the elliptic DOE with small parameters in half-space. We will derive the maximal regularity properties of the following problem

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[TABLE]

where $A$ is a linear operator in $E$, $\alpha_{i}$ are complex numbers, $\varepsilon_{k}$ are positive and $\lambda$ is a complex parameters and

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By virtue of $\left[\text{19,Theorem 2.2}\right]$ we have

**Theorem 2.1. **Let $E$ be a UMD space space and $A$ is an $R$-positive operator in $E$. Assume $m$ is a nonnegative number, $q\in\left(1,\infty\right),$ $\alpha_{\nu}\neq 0,$ $0<t_{k}\leq 1,$ $k=1,2,...,n$. Then for all $f\in W^{m,q}\left(R_{+}^{n};E\right)$, $\lambda\in S_{\psi,\varkappa}$ and sufficiently large $\varkappa>0$ problem $\left(2.1\right)-\left(2.2\right)$ has a unique solution $u$ that belongs to $W^{2+m,q}\left(R_{+}^{n};E\left(A\right),E\right)$ and the following coercive uniform estimate holds

[TABLE]

with $C=C\left(q,A\right)$ independent of $\varepsilon_{1},$ $\varepsilon_{2}$,…,$\varepsilon_{n}$, $\lambda$ and $f.$

Consider the operator $Q_{\varepsilon}$ generated by problem $\left(2.1\right)-\left(2.2\right)$, i.e.,

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[TABLE]

From Theorem 2.1 we obtain the following

Result 2.1. Suppose the conditions of Theorem 2.1 are satisfied. For $\lambda\in S_{\psi,\varkappa}$ there is a resolvent $\left(Q_{\varepsilon}+\lambda\right)^{-1}$ of the operator $Q_{\varepsilon}$ satisfying the following uniform estimate

[TABLE]

It is clear that the solution of the problem $\left(2.1\right)-\left(2.2\right)$ depend on parameters $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$, i.e. $u=u_{\varepsilon}\left(x\right).$ In view of the Theorem 2.1, we derive the properties of the solutions $\left(2.1\right)-\left(2.2\right).$ Particularly, by resoning as $\left[\text{19,Theorem 2.2}\right]$ we show the following:

Corollary 2.1. Let all conditions of the Theorem 2.1. hold. Then, the solution of $\left(2.1\right)-\left(2.2\right)$ satisfies the following uniform estimate

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From Theorem 2.1 we obtain the following

Result 2.2. For $\lambda\in S_{\psi,\varkappa}$ there is a resolvent $\left(Q_{\varepsilon}+\lambda\right)^{-1}$ of the operator $Q_{\varepsilon}$ satisfying the following uniform estimate

[TABLE]

3. Initial-boundary value problems for Stokes system with small parameters

Consider the following BVP for the stationary Stoces equation with parameter

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The function

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satisfying the equation $\left(3.1\right)$ a.e. on $R_{+}^{n}$ is called the stronger solution of the problem $\left(3.1\right)-\left(3.2\right).$

Let $W^{s,q}\left(R_{+}^{n};E\right)$, $0<s<\infty$ be the $E-$valued Sobolev space of order $s$ such that W^{q,0}\left(R_{+}^{n},E\right)=L^{q}\left(R_{+}^{n};E\right).\For $q\in\left(1,\infty\right)$ let $X_{q}=L_{\sigma}^{q}\left(R_{+}^{n},E\right)$ denote the closure of $C_{0\sigma}^{\infty}\left(R_{+}^{n};E\right)$ in $L^{p}\left(R_{+}^{n};E\right),$ where

[TABLE]

By virtue of $\left[19\right],$ vector field $u\in L^{q}\left(R_{+}^{n};E\right)$ has a Helmholtz decomposition, i.e. all $u\in L^{q}\left(R_{+}^{n};E\right)$ can be uniquely decomposed as $u=u_{0}+\nabla\varphi$ with $u_{0}\in L_{\sigma}^{q}\left(R_{+}^{n};E\right)$, $u_{0}=P_{q}u,$where $P_{q}=P$ is a projection operator from $L^{q}\left(R_{+}^{n};E\right)$ to $L_{\sigma}^{q}\left(R_{+}^{n};E\right)$ and $\varphi\in L_{loc}^{q}\left(R_{+}^{n};E\right),$ $\nabla\varphi\in L^{q}\left(R_{+}^{n};E\right)$ so that

[TABLE]

with $C$ independent of $u$, where $B$ is an open ball in $R^{n}$ and $\left\|u\right\|_{p}$ denotes the norm of $u$ in $L^{q}\left(R_{+}^{n};E\right).$

Then the problem $\left(3.1\right)-\left(3.2\right)$ can be reduced to the following BVP

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[TABLE]

Consider the parameter dependent Stokes operator $O_{\varepsilon}=$ $O_{\varepsilon,q}$ generated by problem $\left(3.3\right)-\left(3.4\right)$, i.e.,

[TABLE]

From the Rezult 2.2 we get that the operator $O_{\varepsilon}$ is positive and generates a bounded holomorphic semigroup $S_{\varepsilon}\left(t\right)=\exp\left(-O_{\varepsilon}t\right)$ for $t>0.$

In a similar way as in $\left[6\right]$ we show

**Proposition 3.1. **The following estimate holds

[TABLE]

uniformly in $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$ for $\alpha\geq 0$ and $t>0.$

**Proof. **From Result 2.2 we obtain that the operator $O_{\varepsilon}$ is uniformly positive in $L_{q}\left(R_{+}^{n};E\right)$, i.e. for $\lambda\in\in\left(-\infty\right.,\left.0\right]$ the following uniform estimate holds

[TABLE]

where the constant $M$ is independent of $\lambda$ and $\varepsilon.$ Then, by using Danford integral and operator calculus as in $\left[6\right]$ we obtain the assertion.

From $\left[19\right]$ we obtain the following result

Theorem 3.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. Then for every $f\in L^{p}\left(0,T;L^{q}\left(R_{+}^{n};E\right)\right)=B\left(p,q\right)$ and $a\in B_{p,q}^{2-\frac{2}{p}},$ $p,q\in\left(1,\infty\right)$ there is a unique solution $\left(u,\nabla\varphi\right)$ of the problem $\left(1.9\right)$ and the following uniform estimate holds

[TABLE]

[TABLE]

with $C=C\left(T,p,q\right)$ independent of $f$ and $\varepsilon.$

4. Existence and Uniqueness for Navier-Stokes equation with parameters

In this section, we study the Navier-Stokes problem $\left(1.1\right)-1.3$ in $X_{q}$. The problem $\left(1.1\right)-\left(1.3\right)$ can be expressed as

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We consider this equation in integral form

[TABLE]

For the proving the main result we need the following lemma which is obtained from $\left[\text{4, Theorem 2}\right].$

Lemma 4.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. For any $0\leq\alpha\leq 1$ the domain $D\left(O_{\varepsilon}^{\alpha}\right)$ is the complex interpolation space $\left[X_{q},D\left(O_{\varepsilon}\right)\right]_{\alpha,}.$

**Lemma 4.2. **Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. For each $k=1,2,...,n$ the operator $u\rightarrow O_{\varepsilon}^{-\frac{1}{2}}P\left(\frac{\partial}{\partial x_{k}}\right)u$ extends uniquely to a uniformly bounded linear operator from $L^{q}\left(R_{+}^{n};E\right)$ to $X_{q}.$

**Proof. **Since $O_{\varepsilon}$ is a positive operator, it has a fractional powers $O_{\varepsilon}^{\alpha}.$ From the Lemma 4.1 It follows that the domain $D\left(O_{\varepsilon}^{\alpha}\right)$ is continuously embedded in $X_{q}\cap H_{q}^{2\alpha}\left(R_{+}^{n};E\left(A\right),E\right)$ for any $\alpha>0$. Then by using the duality argument and due to uniform positivity of $O_{\varepsilon}^{\frac{1}{2}}$ we obtain the following uniformly in $\varepsilon$ estimate holds

[TABLE]

By reasoning as in $\left[3\right]$ we obtain the following

**Lemma 4.3. **Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. Let $0\leq\delta<\frac{1}{2}+\frac{n}{2}\left(1-\frac{1}{q}\right).$ Then the following estimate holds

[TABLE]

uniformly in $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$ with constant $M=M\left(\delta,\theta,q,\sigma\right)$ provided that $\theta>0$, $\sigma>0,$ $\sigma+\delta>\frac{1}{2}$ and

[TABLE]

**Proof. Assume that $0<\nu<\frac{n}{2}\left(1-\frac{1}{q}\right)$. **Since $D\left(O_{\varepsilon}^{\alpha}\right)$ is continuously embedded in $X_{q}\cap H_{q}^{2\alpha}\left(R_{+}^{n};E\left(A\right),E\right)$ and $L^{q^{\prime}}\left(R_{+}^{n};E\right)\cap X_{q^{\prime}}$ is the same as $X_{s^{\prime}},$ by Sobolev imbedding theorem we obtain that the operators

[TABLE]

is bounded, where

[TABLE]

By duality argument then, we get that the operator $u\rightarrow O_{\varepsilon,q}^{-\nu}$ is bounded from $X_{s}$ to $X_{q},$ where

[TABLE]

Consider first the case $\delta>\frac{1}{2}$. Since $P(u,\nabla)\upsilon$ is bilinear in $u,\upsilon$, it suffices to prove the estimate on a dense subspace. Therefore assume that $u$ and $\upsilon$ are smooth. Since div $u=0$, we get

[TABLE]

Taking $\nu=\delta-\frac{1}{2}$and using the uniform boundednes of $O_{\varepsilon,q}^{-\nu},$ from $X_{s}$ to $X_{q}$ and Lemma 4.2 for all $\varepsilon>0$ we obtain

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By assumption we can take $r$ and $\eta$ such that

[TABLE]

Since $D\left(O_{\varepsilon,q}^{\alpha}\right)$ is continuously embedded in $X_{q}\cap H_{q}^{2\alpha}\left(R_{+}^{n};E\left(A\right),E\right),$ then by Sobolev imbedding we get

[TABLE]

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

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Similarly we obtain

[TABLE]

for $\frac{1}{r}+\frac{1}{\eta}=\frac{1}{q}$ and $\delta=0$. The above two estimates show that the map $\upsilon\rightarrow P\left(u,\nabla\right)\upsilon$ is a uniform bounded operator from $D\left(O_{\varepsilon}^{\beta}\right)$ to $D\left(O_{\varepsilon}^{-\frac{1}{2}}\right)$ and from $D\left(O_{\varepsilon}^{\beta+\frac{1}{2}}\right)$ to $X_{q}.$ By using the Lemma 4.1 and the interpolation theory for $0\leq\delta\leq\frac{1}{2}$ we obtain

[TABLE]

By using Lemma 4.3 and iteration argument, by reasoning as in Fujita and Kato $\left[6\right]$ we obtain the following

Theorem 4.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. Let $\gamma<1$ be a real number and $\delta\geq 0$ such that

[TABLE]

Suppose that $a\in D\left(O_{\varepsilon}^{\gamma}\right)$, and that $\left\|O_{\varepsilon}^{-\delta}Pf\left(t\right)\right\|$ is continuous on $\left(0,T\right)$ and satisfies

[TABLE]

Then there is $T_{\ast}\in\left(0,T\right)$ independent of $\varepsilon$ and local solution of $\left(4.1\right)$ such that

$u\in C\left(\left[0,T_{\ast}\right];D\left(O_{\varepsilon}^{\gamma}\right)\right)$, $u\left(0\right)=a,$ $u\in C\left(\left(0\right.\left.T_{\ast}\right];D\left(O_{\varepsilon}^{\alpha}\right)\right)$ for some $T_{\ast}>0,$ $\left\|O_{\varepsilon}^{\alpha}u\left(t\right)\right\|=o\left(t^{\gamma-\alpha}\right)$ as $t\rightarrow 0$ for all $\alpha$ with $\gamma<\alpha<1-\delta$ uniformly in $\varepsilon$. Moreover, the solution of $\left(4.1\right)$ is unique if $u\in C\left(\left(0\right.\left.T_{\ast}\right];D\left(O_{\varepsilon}^{\beta}\right)\right),$ $\left\|O_{\varepsilon}^{\alpha}u\left(t\right)\right\|=o\left(t^{\gamma-\beta}\right)$ as $t\rightarrow 0$ for some $\beta$ with \beta>\left|\gamma\right|\uniformly in $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$.

**Proof. **We introduce the following iteration scheme

[TABLE]

[TABLE]

By estimating the term $u_{0}\left(t\right)$ in $\left(4.3\right)$ and by using the Lemma 4.3 for $\gamma\leq\alpha<1-\delta$ we get

[TABLE]

[TABLE]

uniformly with respect to parameters $\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}$ with

[TABLE]

where $N=\sup\limits_{0<t\leq T}t^{1-\gamma-\delta}\left\|O_{\varepsilon}^{-\delta}Pf\left(t\right)\right\|$ and $B\left(a,b\right)$ is the beta function. Here we suppose $\gamma+\delta>0.$ By induction assume that $u_{m}\left(t\right)$ satisfies the following

[TABLE]

We shall estimate $O_{\varepsilon}^{\alpha}u_{m+1}\left(t\right)$ by using $(5.2).$To estimate the term $\left\|O_{\varepsilon}^{-\delta}Fu_{m}\left(s\right)\right\|$ we suppose

[TABLE]

[TABLE]

so that the numbers $\theta,$ $\sigma,$ $\delta$ satisfy the assumptions of Lemma 4.3. Using Lemma 4.3 and $\left(4.4\right),$ we get

[TABLE]

Therefore, we obtain

[TABLE]

[TABLE]

with

[TABLE]

We get the uniform estimate. So, the remaining part of proof is obtainedthe same as in $\left[\text{3},\text{Theorem 2.3}\right]$.

By reasoning as in $\left[\text{6}\right]$ we obtain

**Lemma 4.4. **Let the parameter dependent operator $A_{\varepsilon}$ be uniform positive in a Banach space $E$ and $\alpha$ be a positive number with $0<\alpha<1$. Then, the following uniform inequality holds

[TABLE]

for all $u\in E.$

**Proposition 4.1. **Let $E$ be a space satisfying a multiplier condition, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. Let $u$ be the solution given by Theorem 4.1. Then $O_{\varepsilon}^{\alpha}u$ for $\gamma<\alpha<1-\delta$ is uniform Hölder continuous on every interval $\left[\eta,T_{\ast}\right]$, $0<\eta<T_{\ast}$ for all parameters $\varepsilon_{k}>0.$

Proof. It suffices to prove the Hölder continuity of $O_{\varepsilon}^{\alpha}\upsilon$, where

[TABLE]

Using the Lemma 4.4 we get the uniform estimate

[TABLE]

Then as a similar way as in $\left[\text{3, Proposition 2.4}\right]$ we obtain the assertion.

Theorem 4.2. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. Assume $Pf:\left(0\right.\left.T_{\ast}\right]\rightarrow X_{q}$ is Hölder continuous on each subinterval $\left[\eta,T_{\ast}\right].$ Then, the solution of $\left(4.2\right)$ given by Theorem 4.1 satisfies equation $\left(4.1\right)$ for all parameters $\varepsilon_{k}>0.$ Moreover, $u\in D\left(O_{\varepsilon}\right)$ for $t\in\left(0\right.\left.T_{\ast}\right]$.

**Proof. **It suffices to show Hölder continuity of $Fu\left(t\right)$ on each interval $\left[\eta,T_{\ast}\right].$ It is clear to see that $u\left(\eta\right)\in X_{q}$ and

[TABLE]

Since $Pf$ is continuous on $\left[\eta,T_{\ast}\right]$ we get

[TABLE]

The uniqueness of $u\left(t\right)$, ensured by Theorem 4.1, implies the following estimates

[TABLE]

[TABLE]

uniformly in $\varepsilon_{k},$ where $\nu=\max\left\{\gamma,0\right\}.$ So, by Proposition 5.1, $O_{\varepsilon}^{\alpha}u\left(t\right)$ is continuous on every subinterval $\left[\eta,T_{\ast}\right].$ Since we can choose $\theta$, $\sigma$ so that

[TABLE]

Lemma 4.2 implies that $Fu\left(t\right)$ is Hölder continuous on every interval $\left[\eta,T_{\ast}\right].$

5. **Regularity properties **

The purposes of this section is to show that the solutions of the equation $\left(1.1\right)$ are smooth if the data are smooth. For simplicity, we assume $Pf=0$. The proof when $Pf\neq 0$ is the same. Consider first all of the Stokes problem $\left(3.3\right)-\left(3.4\right).$

By reasonıng as in $\left[\text{6, Lemma 2.14}\right]$ we obtain

**Lemma 5.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. **Let $f\in C^{\mu}\left(\left[0,T\right];X_{q}\right)$, for some $\mu\in\left(0,1\right).$ Then for every $\eta\in\left(0,\mu\right)$ we have

[TABLE]

In a similar way as Lemma 3.3, 3.6.,3.7 in $\left[\text{3}\right]$ we obtain, respectively:

**Lemma 5.2. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. ** For $u,$ $\upsilon\in W^{m,q}\left(R_{+}^{n};E\left(A\right),E\right),$ $q\in\left(1,\infty\right)$ the following hold:

(1) $Pu\in W^{m,q}\left(R_{+}^{n};E\left(A\right),E\right)\cap X_{q}$ and $\left\|Pu\right\|_{W^{m,q}\left(R_{+}^{n};E\right)}\leq C_{m,q}\left\|u\right\|_{W^{m,q}\left(R_{+}^{n};E\right)};$

(2) for $m>\frac{n}{q}$ there exists a constant $C_{m,q}$ such that

[TABLE]

(3) when $q>n$ we have

[TABLE]

**Lemma 5.3. **Let $E$ be a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. Let $u=u_{\varepsilon}\left(t\right)$ be solution of $\left(4.2\right)$ for $Pf=0,$ then $u\in C^{\mu}\left(\left(0,\right.\left.T\right];D\left(O_{\varepsilon}\right)\right)$ and $\frac{du}{dt}\in C^{\mu}\left(\left(0,\right.\left.T\right];X_{q}\right)$ for $\mu\in\left(0,\frac{1}{2}\right).$ Moreover,

[TABLE]

**Lemma 5.4. **Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. ** **Let $u=u_{\varepsilon}\left(t\right)$ be solution of $\left(4.2\right)$ for $Pf=0,$ then $u\in C^{\mu}\left(\left(0,\right.\left.T\right];D\left(O_{\varepsilon}^{\frac{1}{2}}\right)\right)$ for $\mu\in\left(0,\frac{1}{2}\right).$

Now by reasoning as in $\left[\text{3, Proposition 3.5 }\right]$ we can state the following

**Proposition 5.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. **Let $E$ be Banach algebra, $q>n$ and $a\in X_{q}.$ Suppose that the solution $u=u_{\varepsilon}\left(t\right)$ of $\left(4.2\right)$ for $Pf=0$ given by Theorem 4.1 exists on $\left[0,T\right].$ Then $u\in C^{\infty}\left(R_{+}^{n}\times\left[0,T\right];E\right).$

**Proof. **The solution $u=u_{\varepsilon}\left(t\right)$ of $\left(4.2\right)$ for $Pf=0$ given by Theorem 4.1  is expressed as

[TABLE]

where $Fu=-P\left(u,\nabla\right)u.$ From $\left(5.1\right)$ we get

[TABLE]

[TABLE]

Since $S_{\varepsilon}\left(t-\eta\right)O_{\varepsilon}^{\frac{1}{2}}u\left(\eta\right)\in C^{\infty}\left(\left(\delta,\right.\left.T\right];X_{q}\right)$ and $0<\eta<T,$ we will examining only $\upsilon\left(t\right)$. Integrating by parts, we obtain

[TABLE]

[TABLE]

Moreover, since $u\left(s\right)\in D\left(O_{\varepsilon}\right)$ for all $\varepsilon_{k}>0$, $0<s\leq T$, we have

[TABLE]

where $u\left(s\right)=\left(u_{1}\left(s\right),u_{2}\left(s\right),...,u_{n}\left(s\right)\right),$ $u_{k}=u_{k\varepsilon}.$ Hence, by Lemma 4.1 we get the following uniform estimate

[TABLE]

[TABLE]

This estimates together with Lemma 5.3 shows that

[TABLE]

Lemma 5.1 and Lemma 5.2 now imply that

[TABLE]

Since $D\left(O_{\varepsilon}^{\frac{1}{2}}\right)\subset W^{1,q}\left(R_{+}^{n};E\left(A\right),E\right),$ Corollary 5.1, Lemmas 5.3, 5.4 and the identity $u\left(t\right)=O_{\varepsilon}^{\frac{1}{2}}\left(Fu-\frac{du}{dt}\right)$ imply

[TABLE]

Then the proof will be completed as in $\left[\text{ 3, Proposition 3.5}\right]$ by using the induction.

Now we can state the main result of this section

**Theorem 5.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. **Let $E$ be Banach algebra and $a\in X_{q}.$ Suppose that the solution $u=u_{\varepsilon}\left(t\right)$ of $\left(4.2\right)$ for $PF=0$ given by Theorem 4.1 exists on $\left[0,T\right].$ Then $u\in C^{\infty}\left(R_{+}^{n}\times\left[0,T_{\ast}\right];E\right).$

**Proof. **For $q>n$ the assertion is obtained from the Proposition 5.1. Let us show that the assertion is valid for $1<q\leq n.$ Indeed, the solution $u=u_{\varepsilon}\left(t\right)$ of $\left(5.2\right)$ for $PF=0$ given by Theorem 4.1 satisfies the equation $\left(5.1\right)$ on every subinterval $\left[\eta,T_{\ast}\right],$ $0<\eta<T$. Theorem 4.2 shows that $u_{\varepsilon}\left(\eta\right)\in D\left(O_{\varepsilon}\right).$ Since $0\leq\frac{n}{2q}-\frac{1}{2}\leq\gamma<1$, we have $D\left(O_{\varepsilon}^{\gamma}\right)\subset X_{n}$ so that $D\left(O_{\varepsilon}\right)\subset X_{s}$ for some $s>n.$ By $\left(4.2\right)$ this means that we may assume $q>n$ and $a\in X_{q}$.

6. Existence of global solutions

In this section, we prove the existence and estimate of global solution of the problem $\left(1.1\right)-\left(1.3\right).$ The proofs of these theorems are based on the theory of holomorphic semigroups and fractional powers of generators. We assume for simplicity that $f=0$, although it is not difficult to include nonzero $f$ under appropriate conditions. The main result is the following

**Theorem 6.1. Let $E$ be a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$ ** and $a\in L^{q}\left(R_{+}^{n};R^{n}\right).$ There is a $T>0$ and a unique solution $u=u_{\varepsilon}$ of $\left(1.1\right)-\left(1.3\right)$ so that $t^{\left(1-\frac{n}{q}\right)/2}u\in C\left(\left[0\right.,\left.T\right);L^{q}\left(R_{+}^{n};E\right)\right)$ for $n\leq q\leq\infty$ and $t^{\left(1-\frac{n}{2q}\right)}\nabla u\in C\left(\left[0\right.,\left.T\right);L^{q}\left(R_{+}^{n};E\right)\right)$ for $n\leq q<\infty$. Moreover, the following estimates hold

[TABLE]

**Proof. **The solution $u=u_{\varepsilon}\left(t\right)$ of $\left(4.2\right)$ for $Pf=0$ given by Theorem 4.1  is expressed as

[TABLE]

where,

[TABLE]

By applying the generalized Minkovskii inequality and by Proposition 3.1 we can see that

[TABLE]

By using the above estimate we get

[TABLE]

[TABLE]

Moreover, by using $\left(6.1\right)$, $\left(6.2\right)$ and  by applying the Hölder inequality, we get

[TABLE]

Then ın view of $\left(6.1\right)$-$\left(6.4\right)$ we obtain the following uniform estimate

[TABLE]

[TABLE]

where

[TABLE]

Then solving the equation $\left(6.1\right)$ by successive approximation, starting with $u_{0}=S_{\varepsilon}\left(t\right)a$ we get

[TABLE]

First by reasoning as in $\left[\text{22, Theorem 1}\right]$ and by using $\left(6.3\right)$-$\left(6.5\right)$ we show by induction that $u_{k}=u_{\varepsilon k}$ exists, moreover,

[TABLE]

and for $\delta\in\left(0,1\right)$ the following uniform estimates hold

[TABLE]

By applying $\left(6.3\right)$-$\left(6.5\right)$ for $q=n$ and $p=\frac{n}{\delta}$ we have

[TABLE]

where $C$ is a positive constant. From $\left(6.5\right)$ and $\left(6.7\right)$ for $n\leq p<\infty$ we obtain

[TABLE]

[TABLE]

It follows that $u_{\varepsilon k}\left(t\right)$ converges to a limit function $u_{\varepsilon}$ uniformly with respect to $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$, moreover, $u_{\varepsilon}\in C\left(\left[0\right.,\left.T\right);L^{n}\left(R_{+}^{n};E\right)\right)$ for $p=n$ and $u_{\varepsilon}$ satisfies $\left(6.1\right)$ for $n<p<\infty.$

**Theorem 6.2. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. ** There is a $\mu>0$ such that if $\left\|a\right\|_{L^{q}\left(R_{+}^{n};E\right)}<\mu$, then there is a global solution $u_{\varepsilon}$ of the problem $\left(1.1\right)-\left(1.3\right)$, so that $t^{\left(1-\frac{n}{q}\right)/2}u_{\varepsilon}\in C\left(\left[0\right.,\left.\infty\right);L^{q}\left(R_{+}^{n};E\right)\right)$ for $n\leq q\leq\infty,$ $t^{\left(1-\frac{n}{q}\right)/2}$ and $t^{\left(1-\frac{n}{2q}\right)}\nabla u_{\epsilon}\in C\left(\left[0\right.,\left.\infty\right);L^{q}\left(R_{+}^{n};E\right)\right)$ for $n\leq q<\infty$. Moreover, the following uniform estimates hold

[TABLE]

**Proof. **It is clear to see from proof of Theorem 6.1 that $M_{k}$ and $M_{k}^{{}^{\prime}}$ are bounded by a constant $M$ if $M_{0}\leq\lambda$. By $\left(7.9\right)$ this is true if $\left\|a\right\|_{L^{q}\left(R_{+}^{n};E\right)}$ is sufficiently small. In this case, as in $\left[\text{10}\right]$ we prove that the sequences $t^{\left(1-\delta\right)/2}u_{\varepsilon k}$, $t^{1/2}\nabla u_{\varepsilon k}$ are bounded on $\left(0,\infty\right)$ uniformly in $k$ and $\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}$ i.e.,

[TABLE]

Then $\left(6.11\right)$ is obtained from $\left(6.10\right).$

**Remake 6.1. Let $E$ be a a UMD space, $A$ an $R$-positive operator in $E$, $q\in\left(1,\infty\right)$ and $0<\varepsilon_{k}\leq 1$. **Theorem 6.2 shows that all $L^{p}$ norms of $u_{\varepsilon}\left(t\right)$ decay as $t\rightarrow\infty$ for $p>q$ uniformly in $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right).$

For $p=q$ we obtain the following result

**Theorem 6.3. **Let all conditions of Theorem 6.2 hold. Then $\left\|u_{\varepsilon}\left(t\right)\right\|_{p}\rightarrow 0$ uniformly in $\varepsilon$ as $t\rightarrow\infty$. More precisely, we have

[TABLE]

where, $u_{0\varepsilon}\left(t\right)=S_{\varepsilon}\left(t\right)a$ and $\delta<\min\left\{1,n-\frac{n}{q},\frac{n}{q}-1\right\}.$

7. The Wentzell-Robin type mixed problem for Novier-Stokes equations

Consider the problem $\left(1.5\right)-\left(1.7\right)$. Here, $W^{2,\mathbf{p}}\left(\tilde{\Omega}\right)$ denotes the Sobolev space with corresponding mixed norm

The main aim of this section is to prove the following result:

**Theorem 7.1. **Let $a\in W^{1,\infty}\left(0,1\right)$, $a\left(x\right)\geq\delta>0,$ $b,c\in L^{\infty}\left(0,1\right)$. Suppose the condition 7.1 hold. Let $\gamma<1$ be a real number and $\delta\geq 0$ such that

[TABLE]

Suppose $a\in D\left(O_{\varepsilon}^{\gamma}\right)$ such that $\left\|O_{\varepsilon}^{-\delta}Pf\left(t\right)\right\|$ is continuous on $\left(0,T\right)$ and satisfies

[TABLE]

Then there is $T_{\ast}\in\left(0,T\right)$ independent of $\varepsilon$ and local solution of $\left(4.1\right)$ such that

$u\in C\left(\left[0,T_{\ast}\right];\right)$, $u\left(0\right)=a,$ $u\in C\left(\left(0\right.\left.T_{\ast}\right];D\left(O_{\varepsilon}^{\alpha}\right)\right)$ for some $T_{\ast}>0,$ $\left\|O_{\varepsilon}^{\alpha}u\left(t\right)\right\|=o\left(t^{\gamma-\alpha}\right)$ as $t\rightarrow 0$ for all $\alpha$ with $\gamma<\alpha<1-\delta$ uniformly with respect to $\varepsilon.$ Moreover, the solution of $\left(4.1\right)$ is unique if $u\in C\left(\left(0\right.\left.T_{\ast}\right];D\left(O_{\varepsilon}^{\beta}\right)\right)$, $\left\|O_{\varepsilon}^{\alpha}u\left(t\right)\right\|=o\left(t^{\gamma-\beta}\right)$ as $t\rightarrow 0$ for some $\beta$ with \beta>\left|\gamma\right|\uniformly in $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{n}\right)$.

Then problem $\left(1.5\right)-\left(1.7\right)$ has a unique local strange solution $u\in C^{\left(2\right)}\left(\left[0\right.,\left.T_{0}\right);Y_{\infty}^{2,p}\right)$, where $T_{0}$ is a maximal time interval that is appropriately small relative to $M$. Moreover, if

[TABLE]

then $T_{0}=\infty.$

Proof. Let $E=L^{p_{1}}\left(0,1\right)$. It is known $\left[2\right]$ that $L^{p_{1}}\left(0,1\right)$ is an $UMD$ space for $p_{1}\in\left(1,\infty\right).$ Consider the operator $A$ defined by

[TABLE]

Therefore, the problem $\left(1.7\right)-\left(1.8\right)$ can be rewritten in the form of $\left(1.1\right)-\left(1.3\right)$, where $u\left(x\right)=u\left(x,.\right),$ $f\left(x\right)=f\left(x,.\right)$ are functions with values in $E=L^{p_{1}}\left(0,1\right).$ From $\left[\text{7, 8}\right]$ we get that the operator $A$ generates analytic semigroup in $L^{p_{1}}\left(0,1\right).$ Moreover, we obtain that the operator $A$ is $R$-positive in $L^{p_{1}}.$ Then from Theorem 4.1 we obtain the assertion.

References

H. Amann, Linear and quasi-linear equations,1, Birkhauser, Basel 1995. 2. 2.

R. Denk, M. Hieber, J. Pruss, $R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 2005,166 (788). 3. 3.

Y. Giga, T. Miyakava, Solutions in $L_{r}$ of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 1985, 89, 267-281. 4. 4.

Y. Giga, Domains of fractional powers of the Stokes operator in $L_{r}$ spaces, Arch. Ration. Mech. Anal., 1985, 89, 251-265. 5. 5.

D. Fujiwara and H. Morimoto, An $L_{r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sec. 1977, (I) 24, 685-700. 6. 6.

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I., Arch. Rational Mech. Anal., 1964, 16, 269-315. 7. 7.

A. Favini, G. R. Goldstein, Jerome A. Goldstein and Silvia Romanelli, Degenerate Second Order Differential Operators Generating Analytic Semigroups in $L_{p}$ and $W^{1,p}$, Math. Nachr. 238 (2002), 78 – 102. 8. 8.

V. Keyantuo, M. Warma, The wave equation with Wentzell–Robin boundary conditions on Lp-spaces, J. Differential Equations 229 (2006) 680–697. 9. 9.

M. Hamouda and R. Temam, Some singular perturbation problems related to the Navier-Stokes equations, Advances in deterministic and stochastic analysis, 197-227, World. Sci. Publ., Hackensack, NJ, 2007. 10. 10.

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova, 1962, 32, 243-260. 11. 11.

N. Masmoudi, Examples of singular limits in hydrodynamics. In Evolutionary equations. Vol. III, Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam, 2007, 195–276. 12. 12.

O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1966 13. 13.

J-L. Lions, Mathematical topics in fluid mechanics. V. 1. The Clarendon Press Oxford University Press, New  York, 1996. Incompressible models, Oxford Science Publications. 14. 14.

V. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Sov. Math., 1977(8), 467-529. 15. 15.

P. E. Sobolevskii, Study of Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces, Soviet Math. Dokl. 1964(5), 720-723. 16. 16.

V. B. Shakhmurov, The Cauchy problem for generalized abstract Boussinesq equations, Dynamic systems and applications, 25, (2016),109-122. 17. 17.

V. B. Shakhmurov, Linear and nonlinear abstract equations with parameters, Nonlinear Anal-Theor., 2010, v. 73, 2383-2397. 18. 18.

V. B. Shakhmurov, A. Shahmurova, Nonlinear abstract boundary value problems atmospheric dispersion of pollutants, Nonlinear Anal-Real., 2010 v.11 (2), 932-951. 19. 19.

V. B. Shakhmurov, Stokes equations with small parameters in half plane, Computers and Mathematics with Applications, 2014, 67, 91-115. 20. 20.

V. B. Shakhmurov, Nonlocal Novier-Stokes problem with small parameter, Boundary Value Problems, 2013, v. 2013-107, 1-21. 21. 21.

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1984. 22. 22.

H. Triebel, Interpolation theory. Function spaces. Differential operators, North-Holland, Amsterdam, 1978. 23. 23.

S. Yakubov and Ya. Yakubov, Differential-operator Equations. Ordinary and Partial  Differential Equations , Chapman and Hall /CRC, Boca Raton, 2000.