Stokes operators with variable coefficients and applications

Abstract

The stationary and instationary Stokes problems with variable coefficients in abstract $L^p$ spaces are considered. The problems contain abstract operators and nonlocal boundary conditions. The well-posedness of these problems is derived.

MSC:35Q30, 76D05, 34G10, 35J25.

1 Introduction

We consider the initial and boundary value problem (BVP) for the following Stokes type equation with variable coefficients:

$$\frac{\partial u}{\partial t}+\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x_k^2}+A(x)u+\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}+\mathrm{\nabla }\phi =f(x,t),divu=0,$$

(1.1)

$$\sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0})+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb})]=0,k=1,2,\dots ,n,j=1,2,$$

(1.2)

$$u(x,0)=a(x),x\in G,t\in (0,T),$$

(1.3)

where

$$\begin{array}{c}x=(x_1,x_2,\dots ,x_n)\in G=\prod _{k=1}^n(0,b_k),m_{kj}\in \{0,1\},\hfill \\ G_{k0}=(x_1,x_2,\dots ,x_{k-1},0,x_{k+1},\dots ,x_n),\hfill \\ G_{kb}=(x_1,x_2,\dots ,x_{k-1},b_k,x_{k+1},\dots ,x_n),\hfill \end{array}$$

$A=A(x)$ and $A_k=A_k(x)$ are linear operators in a Banach space E, ${\alpha }_{kji}$, ${\beta }_{kji}$ are complex numbers, and $a_k$ are complex-valued functions. Here $f=(f_1(x,t),f_2(x,t),\dots ,f_n(x,t))$ represents a given, a denotes the initial data and

$$u=(u_1(x,t),u_2(x,t),\dots ,u_n(x,t)),\phi =\phi (x,t)$$

represent the unknown functions. Moreover, $u_j(x,t)$, $f_j(x,t)$, and $\phi (x,t)$ are E-valued functions. This problem is characterized by the presence of abstract operator functions and complex-valued variable coefficients in the principal part. Moreover, boundary conditions are nonlocal, generally. The existence, uniqueness, and coercive estimates of maximal regular solution of problem (1.1)-(1.2) are obtained. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing E and operators A, $A_k$ we can obtain maximal regularity properties for numerous class of Stokes type problems with variable coefficients. For $E=\mathbb{C}$, $a_k(x)\equiv -1$, $A=\varkappa >0$ problem (1.1)-(1.2) is reduced to nonlocal Stokes problem

$$\begin{array}{r}\frac{\partial u}{\partial t}-\mathrm{\Delta }u+\varkappa u+\mathrm{\nabla }\phi =f(x,t),divu=0,\\ \sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0},t)+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb},t)]=0,\\ u(x,0)=a(x),x\in G,t\in (0,T),k=1,2,\dots ,n,j=1,2,\end{array}$$

(1.4)

where ℂ is the set of complex numbers. Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems were extensively studied, e.g., in [1–10]. There is an extensive literature on the solvability of the initial value problems (IVPs) for the Stokes equation (see, e.g., [1, 2, 10] and further papers cited there). Solonnikov [8] proved that for every $f\in L^p(\mathrm{\Omega }\times (0,T);R^3)=B(p)$, $p\in (1,\mathrm{\infty })$ the instationary Stokes problem

$$\begin{array}{r}\frac{\partial u}{\partial t}-\mathrm{\Delta }u+\mathrm{\nabla }\phi =f(x,t),divu=0,u{|}_{\partial \mathrm{\Omega }}=0,\\ u(x,0)=0,x\in \mathrm{\Omega },t\in (0,T)\end{array}$$

(1.5)

has a unique solution $(u,\mathrm{\nabla }\phi )$ so that

$${\parallel \frac{\partial u}{\partial t}\parallel }_{B(p)}+{\parallel {\mathrm{\nabla }}^{2}u\parallel }_{B(p)}+{\parallel \mathrm{\nabla }\phi \parallel }_{B(p,q)}\le C{\parallel f\parallel }_{B(p,q)}.$$

Then Giga and Sohr [2] improved the result of Sollonikov for spaces with different exponents in space and time, i.e., they proved that for every $f\in L^p(0,T;{(L^q(\mathrm{\Omega }))}^n)$ there is a unique solution $(u,\mathrm{\nabla }\phi )$ of problem (1.5) so that

$${\parallel \frac{\partial u}{\partial t}\parallel }_{B(p,q)}+{\parallel {\mathrm{\nabla }}^{2}u\parallel }_{B(p,q)}+{\parallel \mathrm{\nabla }\phi \parallel }_{B(p,q)}\le C{\parallel f\parallel }_{B(p,q)},$$

(1.6)

where

$$B(p,q)=L^p(0,T;{(L^q(\mathrm{\Omega }))}^n),p,q\in (1,\mathrm{\infty }).$$

Moreover, the estimate obtained was global in time, i.e., the constant $C=C(\mathrm{\Omega },p,q)$ is independent of T and f. To derive the global $L^p$-$L^q$ estimates (1.6), Giga and Sohr used abstract parabolic semigroup theory in UMD spaces. The estimate (1.6) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem (see, e.g., [4]). Consider first at all, the stationary version of problem (1.1)-(1.3), i.e., consider the abstract Stokes problem

$$\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x_k^2}+(A(x)+\lambda )u+\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}+\mathrm{\nabla }\phi =f(x),divu=0,x\in G,$$

(1.7)

$$\sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0})+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb})]=0,k=1,2,\dots ,n,j=1,2,$$

(1.8)

λ is a complex number. By applying the corresponding projection transformation P, (1.7)-(1.8) can be reduced to the following problem:

$$P\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x_k^2}+P(A(x)+\lambda )u+P\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}=f(x),x\in G,$$

(1.9)

$$L_{kj}u=\sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0})+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb})]=0,k=1,2,\dots ,n,j=1,2.$$

(1.10)

Let $O_q$ denote the operator generated by problem (1.9)-(1.10), i.e., let $O_q$ be a Stokes operator in the E-valued solenoidal space $L_{\sigma }^q(G;E)$ defined by

$$\begin{array}{c}D(O_q)={\left(W_{\sigma }^{2,q}(G;E(A),E)\right)}^n=\{u\in {\left(W^{2,q}(G;E(A),E)\right)}^n,L_{kj}u=0,divu=0\},\hfill \\ O_{q}u=P\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x^2}u+PAu+P\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}.\hfill \end{array}$$

We prove that $O_q$ is a positive operator and $-O_q$ is a generator of an analytic semigroup in $L_{\sigma }^q(G;E)$. In other words, we consider the instationary Stokes problem (1.1)-(1.3) and prove the well-posedness of this problem. We prove that there is a unique solution $(u,\mathrm{\nabla }\phi )$ of problem (1.1)-(1.3) for $f\in {(L^p(0,T;X_q))}^n$, $a\in Y_{p,q}$, and the following estimate holds:

$${\parallel \frac{\partial u}{\partial t}\parallel }_{L^{p,q}}+\sum _{k=1}^n{\parallel \frac{{\partial }^{2}u}{\partial x_k^2}\parallel }_{L^{p,q}}+{\parallel Au\parallel }_{L^{p,q}}+{\parallel \mathrm{\nabla }\phi \parallel }_{L^{p,q}}\le C({\parallel f\parallel }_{L^{p,q}}+{\parallel a\parallel }_{Y_{p,q}}),$$

(1.11)

where $X_q$ is the class of E-valued $L^q$-spaces and $Y_{p,q}$ is a corresponding interpolation space. The estimate (1.11) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem. Finally, we give some application of this abstract Stokes problem to anisotropic Stokes equations and systems of equations. Note that the abstract Stokes problem with constant coefficients was studied in [11].

2 Definitions and background

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

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

The Banach space E is called an UMD space if the Hilbert operator

$$(Hf)(x)=\underset{\epsilon \to 0}{lim}{\int }_{|x-y|>\epsilon }\frac{f(y)}{x-y}dy$$

is bounded in $L^p(R,E)$, $p\in (1,\mathrm{\infty })$ (see, e.g., [12]). UMD spaces include e.g. $L^p$, $l_p$ spaces, and Lorentz spaces $L_{pq}$, $p,q\in (1,\mathrm{\infty })$.

Let

$$\begin{array}{c}{S}_{\psi }=\{\lambda \in \mathbb{C},|arg\lambda |\le \psi ,0\le \psi <\pi \},\hfill \\ S_{\psi ,\varkappa }=\{\lambda \in S_{\psi },|\lambda |>\varkappa >0\}.\hfill \end{array}$$

A linear operator 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, and $B(E)$ is the space of bounded linear operators in E. It is known [[13], §1.15.1] that there exist fractional powers $A^{\theta }$ of a positive operator A. Let $E(A^{\theta })$ denote the space $D(A^{\theta })$ with norm

$${\parallel u\parallel }_{E(A^{\theta })}={({\parallel u\parallel }^p+{\parallel A^{\theta }u\parallel }^p)}^{\frac{1}{p}},1\le p<\mathrm{\infty },0<\theta <\mathrm{\infty }.$$

Let $E_1$ and $E_2$ be two Banach spaces. By ${(E_1,E_2)}_{\theta ,p}$, $0<\theta <1$, $1\le p\le \mathrm{\infty }$, will be denoted the interpolation spaces obtained from $\{E_1,E_2\}$ by the K-method [[13], §1.3.2].

Let ℕ denote the set of natural numbers. A set $\mathrm{\Phi }\subset B(E_1,E_2)$ is called R-bounded (see, e.g., [14]) if there is a positive constant C such that for all $T_1,T_2,\dots ,T_m\in \mathrm{\Phi }$ and $u_1,u_2,\dots ,u_m\in E_1$, $m\in \mathbb{N}$,

$${\int }_{\mathrm{\Omega }}{\parallel \sum _{j=1}^m{r}_j(y)T_j{u}_j\parallel }_{E_2}dy\le C{\int }_{\mathrm{\Omega }}{\parallel \sum _{j=1}^m{r}_j(y)u_j\parallel }_{E_1}dy,$$

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 a R-bound of the collection Φ and denoted by $R(\mathrm{\Phi })$.

A set ${\mathrm{\Phi }}_h\subset B(E_1,E_2)$ is called uniform R-bounded in h, if there is a constant C independent on $h\in \sigma \subset \mathbb{R}$ such that

$${\int }_{\mathrm{\Omega }}{\parallel \sum _{j=1}^m{r}_j(y)T_j(h)u_j\parallel }_{E_2}dy\le C{\int }_{\mathrm{\Omega }}{\parallel \sum _{j=1}^m{r}_j(y)u_j\parallel }_{E_1}dy$$

for all $T_1(h),T_2(h),\dots ,T_m(h)\in {\mathrm{\Phi }}_h$ and $u_1,u_2,\dots ,u_m\in E_1$, $m\in \mathbb{N}$. It is implied that ${sup}_{h\in Q}R({\mathrm{\Phi }}_h)\le C$.

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 with respect to t with bound $M>0$ if $D(A(t))$ is independent on t, $D(A(t))$ is dense in E and $\parallel {(A(t)+\lambda )}^{-1}\parallel \le \frac{M}{1+|\lambda |}$ for all $\lambda \in S_{\psi }$, $0\le \psi <\pi$, where M does not depend on t and λ.

Let $E_0$ and E be two Banach spaces and $E_0$ continuously and densely embedded into E. Let Ω be a domain in $R^n$ and m is a positive integer. $W^{m,p}(\mathrm{\Omega };E_0,E)$ denotes the space of all functions $u\in L^p(\mathrm{\Omega };E_0)$ that have generalized derivatives $\frac{{\partial }^{m}u}{\partial x_k^m}\in L^p(\mathrm{\Omega };E)$ with the norm

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

For $n=1$, $\mathrm{\Omega }=(a,b)$, $a,b\in R$ the space $W^{m,p}(\mathrm{\Omega };E_0,E)$ will be denoted by $W^{m,p}(a,b;E_0,E)$. For $E_0=E$ the space $W^{m,p}(\mathrm{\Omega };E_0,E)$ is denoted by $W^{m,p}(\mathrm{\Omega };E)$.

Let $L^{s,p}(\mathrm{\Omega };E)$, $-\mathrm{\infty }<s<\mathrm{\infty }$, denote the E-valued Liouville space of order s such that $L^{0,p}(\mathrm{\Omega };E)=L^p(\mathrm{\Omega };E)$. It is known that if E is a UMD space, then $L^{m,p}(\mathrm{\Omega };E)=W^{m,p}(\mathrm{\Omega };E)$ for positive integer m (see, e.g., [[15], §15].

Let $L^{s,p}(\mathrm{\Omega };E_0,E)$ denote a Liouville-Lions type space, i.e.,

$$\begin{array}{rl}{L}^{s,p}(\mathrm{\Omega };E_0,E)=& \{u\in L^{s,p}(\mathrm{\Omega };E)\cap L^q(\mathrm{\Omega };E_0),\\ {\parallel u\parallel }_{L^{s,p}(\mathrm{\Omega };E_0,E)}={\parallel u\parallel }_{L^p(\mathrm{\Omega };E_0)}+{\parallel u\parallel }_{L^{s,p}(\mathrm{\Omega };E)}<\mathrm{\infty }\}.\end{array}$$

$L_{\sigma }^q(\mathrm{\Omega };E)$ denote the E-valued solenoidal space, i.e., the closure of ${(C_{0\sigma }^{\mathrm{\infty }}(\mathrm{\Omega };E))}^n$ in ${(L^q(\mathrm{\Omega };E))}^n$, where

$$C_{0\sigma }^{\mathrm{\infty }}(\mathrm{\Omega };E)=\{u\in C_0^{\mathrm{\infty }}(\mathrm{\Omega };E),divu=0\}.$$

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 $C_{\alpha }$.

The embedding theorems in vector-valued spaces play a key role in the theory of DOEs. For estimating lower order derivatives we use the following embedding theorems from [16].

Theorem A ₁ Suppose the following conditions are satisfied:

• (1) E is a UMD space and A is an R-positive operator in E;

• (2) $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$ and m is a positive integer such that $\varkappa ={\sum }_{k=1}^n\frac{|\alpha |}{m}\le 1$, $0\le \mu \le 1-\varkappa$, $1<p<\mathrm{\infty }$, $0<h\le h_0$, $h_0$ is a fixed positive number;

• (3) $\mathrm{\Omega }\subset R^n$ is a region such that there exists a bounded linear extension operator from $W^{m,p}(\mathrm{\Omega };E(A),E)$ to $W^{m,p}(R^n;E(A),E)$.

Then the embedding $D^{\alpha }{W}^{m,p}(\mathrm{\Omega };E(A),E)\subset L^p(\mathrm{\Omega };E(A^{1-\varkappa -\mu }))$ is continuous and for all $u\in W^{m,p}(\mathrm{\Omega };E(A),E)$ the following uniform estimate holds:

$${\parallel D^{\alpha }u\parallel }_{L^p(\mathrm{\Omega };E(A^{1-\varkappa -\mu }))}\le h^{\mu }{\parallel u\parallel }_{W^{m,p}(\mathrm{\Omega };E(A),E)}+h^{-(1-\mu )}{\parallel u\parallel }_{L^p(\mathrm{\Omega };E)}.$$

Remark 2.1 If $\mathrm{\Omega }\subset R^n$ is a region satisfying the strong l-horn condition (see [[4], §7]), $E=R$, $A=I$, then for $p\in (1,\mathrm{\infty })$ there exists a bounded linear extension operator from $W^{m,p}(\mathrm{\Omega })=W^{m,p}(\mathrm{\Omega };R,R)$ to $W^{m,p}(R^n)=W^{m,p}(R^n;R,R)$.

Theorem A ₂ Suppose all conditions of Theorem  A₁ are satisfied and $0<\mu \le 1-\varkappa$. Moreover, let Ω be a bounded region and $A^{-1}\in {\sigma }_{\mathrm{\infty }}(E)$. Then the embedding

$$D^{\alpha }{W}^{m,p}(\mathrm{\Omega };E(A),E)\subset L^p(\mathrm{\Omega };E\left(A^{1-\varkappa -\mu }\right))$$

is compact.

Theorem A ₃ Suppose all conditions of Theorem  A₁ satisfied and $0<\mu \le 1-\varkappa$. Then the embedding

$$D^{\alpha }{W}^{m,p}(\mathrm{\Omega };E(A),E)\subset L^p(\mathrm{\Omega };{(E(A),E)}_{\varkappa ,p})$$

is continuous and there exists a positive constant $C_{\mu }$ such that for all $u\in W_p^l(\mathrm{\Omega };E(A),E)$ the uniform estimate holds

$${\parallel D^{\alpha }u\parallel }_{L^p(\mathrm{\Omega };{(E(A),E)}_{\varkappa ,p})}\le C_{\mu }[h^{\mu }{\parallel u\parallel }_{W^{m,p}(\mathrm{\Omega };E(A),E)}+h^{-(1-\mu )}{\parallel u\parallel }_{L^p(\mathrm{\Omega };E)}].$$

From [[17], Theorem 2.1] we obtain the following.

Theorem A ₄ Let E be a Banach space, A be a φ-positive operator in E with bound M, $0\le \phi <\pi$. 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, φ, m, α and p) such that for every $u\in {(E,E(A^m))}_{\frac{\alpha }{m}-\frac{1}{2mp},p}$ and $\lambda \in S_{\phi }$,

$${\int }_0^{\mathrm{\infty }}{\parallel A_{\lambda }^{\alpha }{e}^{-x{A}_{\lambda }^{\frac{1}{2}}}u\parallel }^{p}dx\le C[{\parallel u\parallel }_{{(E,E(A^m))}_{\frac{\alpha }{m}-\frac{1}{2mp},p}}^p+|\lambda {|}^{\alpha p-\frac{1}{2}}{\parallel u\parallel }_E^p].$$

3 The stationary Stokes system with variable coefficients

In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem (1.7)-(1.8).

First at all, we consider the BVP for the variable coefficient differential operator equation (DOE)

$$\begin{array}{r}\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x_k^2}+(A(x)+\lambda )u+\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}=f(x),x\in G,\\ \sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0})+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb})]=0,k=1,2,\dots ,n,j=1,2,\end{array}$$

(3.1)

where $A(x)$ and $A_k(x)$ are linear operators in a Banach space E, $a_k$ are complex-valued functions, and λ is a complex parameter.

Maximal regularity properties for DOEs studied, e.g., in [1, 11, 12, 14, 16–24]. Nonlocal BVPs for PDE were studied in [25].

Let ${\omega }_{ki}={\omega }_{ki}(x)$, $i=1,2$, be roots of the equations

$$a_k(x){\omega }^2=1,k=1,2,\dots ,n.$$

Let ${\alpha }_{kj}={\alpha }_{kj{m}_k}$ and ${\beta }_{kj}={\beta }_{kj{m}_k}$ and

$${\eta }_k(x)=\left|\begin{array}{cc}{\alpha }_{k1}{(-{\omega }_{k1})}^{m_{k1}}& {\beta }_{k1}{\omega }_{k1}^{m_{k1}}\\ {\alpha }_{k2}{(-{\omega }_{k1})}^{m_{k2}}& {\beta }_{k2}{\omega }_{k2}^{m_{k2}}\end{array}\right|.$$

Condition 3.1 Assume:

• (1) E is a UMD space and $A(x)$ is a uniformly R-positive operator in E for $\phi \in [0,\pi )$;

• (2) $a_k(x)\in C(\overline{G})$, $a_k(G_{i0})=a_k(G_{ib})$, $a_k\ne 0$, $a_k\in S({\phi }_0)\cap \mathbb{C}/{\mathbb{R}}_{-}$ for all $x\in G$, $\phi +{\phi }_0<\pi$;

• (3) $A(x)A^{-1}(\overline{x})\in C(\overline{G};B(E))$, $A(G_{i0})=A(G_{ib})$, $A_i(x)A^{-(\frac{1}{2}-\nu )}(x)\in C^{(m)}(\overline{G};B(E))$, $0<\nu <\frac{1}{2}$;

• (4) $|{\alpha }_{kj{m}_j}|+|{\beta }_{kj{m}_j}|>0$, ${\eta }_k(x)\ne 0$, $k,i=1,2,\dots ,n$, $j=1,2$, $p\in (1,\mathrm{\infty })$.

Remark 3.1 Let $a_k=-b_k(x)$, where $b_k$ are real-valued positive functions and. Then Condition 3.1 is satisfied.

Remark 3.2 The conditions $a_k(G_{i0})=a_k(G_{ib})$, $A(G_{i0})=A(G_{ib})$ are given due to the nonlocality of the boundary conditions. For local boundary conditions these assumptions are not required.

From [[19], Theorem 4.1] we have the following.

Theorem 3.1 Suppose Condition 3.1 is satisfied. Then problem (3.1) has a unique solution $u\in W^{2,p}(G;E(A),E)$ for $f\in L^p(G;E)$ and for sufficiently large $\lambda \in S_{\phi }$. Moreover, the following coercive uniform estimate holds:

$$\sum _{k=1}^n\sum _{i=0}^2|\lambda {|}^{1-\frac{i}{2}}{\parallel \frac{{\partial }^{i}u}{\partial x_k^i}\parallel }_{L^q(G;E)}+{\parallel Au\parallel }_{L^q(G;E)}\le C{\parallel f\parallel }_{L^q(G;E)}.$$

Consider the differential operator $Q=Q_q$ in $L^q(G;E)$ generated by problem (1.7)-(1.8), i.e.,

$$D(Q)=W^{2,q}(G;E(A),E,L_{kj}),Qu=\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x^2}+A(x)+\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}.$$

Let $B_q=B(L^q(G;E))$. From Theorem 3.1 we obtain the following.

Result 3.1 For $\lambda \in S_{\psi ,\varkappa }$ there is a resolvent ${(Q+\lambda )}^{-1}$ and the following uniform coercive estimate holds:

$$\sum _{k=1}^n\sum _{i=0}^2|\lambda {|}^{1-\frac{i}{2}}{\parallel \frac{{\partial }^i}{\partial x_k^i}{(Q+\lambda )}^{-1}\parallel }_{B_q}+{\parallel A{(Q+\lambda )}^{-1}\parallel }_{B_q}\le C.$$

Let E be a Banach space and $X_q=X_q(G)={(L^q(G;E))}^n$ denote the class of E-valued system of function $f=(f_1(x),f_2(x),\dots ,f_n(x))$ with norm

$${\parallel f\parallel }_{X_q}={\left(\sum _{i=1}^n{\parallel f_i\parallel }_{L^q(G;E)}^q\right)}^{\frac{1}{q}},q\in (1,\mathrm{\infty }),$$

$X_{q\sigma }=L_{\sigma }^q(G;E)$ denote the E-valued solenoidal space and A be a positive operator in E. The spaces ${(L^{s,q}(G;E))}^n$, ${(L^{s,q}(G;E(A),E))}^n$ will be denoted by $X_q^s(G)$ and $X_q^s(G,A)$.

Consider the problem

$$\begin{array}{r}\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x_k^2}+(A(x)+\lambda )u+\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}=f(x),x\in G,\\ \sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0})+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb})]=0,k=1,2,\dots ,n,j=1,2,\end{array}$$

(3.2)

where $f=(f_1(x),f_2(x),\dots ,f_n(x))$, $A(x)$ and $A_k(x)$ are linear operators in a Banach space E, $a_k$ are complex-valued functions, and λ is a complex parameter. From Theorem 3.1 we obtain the following result.

Result 3.2 Suppose Condition 3.1 is satisfied. Then problem (3.2) has a unique solution $u\in X_q^2(G,A)$ for $f\in X_q$ and for sufficiently large $\lambda \in S_{\phi }$. Moreover, the following coercive uniform estimate holds:

$$\sum _{k=1}^n\sum _{i=0}^2|\lambda {|}^{1-\frac{i}{2}}{\parallel \frac{{\partial }^{i}u}{\partial x_k^i}\parallel }_{X_q}+{\parallel Au\parallel }_{X_q}\le C{\parallel f\parallel }_{X_q}.$$

Consider the differential operator $B=B_q$ in $X_q$ generated by problem (3.2), for $\lambda =0$, i.e.,

$$D(B)=X_q^2(G,A),Bu=\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x^2}+A(x)+\sum _{k=1}^n{A}_k(x)\frac{\partial u}{\partial x_k}.$$

From Result 3.2 we obtain the following uniform coercive estimate:

$$\sum _{k=1}^n\sum _{i=0}^2|\lambda {|}^{1-\frac{i}{2}}{\parallel \frac{{\partial }^i}{\partial x_k^i}{(B+\lambda )}^{-1}\parallel }_{B(X_q)}+{\parallel A{(B+\lambda )}^{-1}\parallel }_{B(X_q)}\le C.$$

(3.3)

Consider the space

$$\begin{array}{c}{Y}_q(A)=\{u\in X_q(E(A)),divu\in L^q(G;E)\},\hfill \\ {\parallel u\parallel }_{Y_q(A)}={({\parallel u\parallel }_{X_q(E(A))}^q+{\parallel divu\parallel }_{L^q(G;E)}^q)}^{\frac{1}{q}}.\hfill \end{array}$$

$Y_q$ becomes a Banach space with this norm.

It is known that (see, e.g., [3, 5]) the vector field $u\in {(L^q(G))}^n$ has a Helmholtz decomposition. In the following theorem we generalize this result for the E-valued function space $X_q$.

Theorem 3.2 Let E be an UMD space and $q\in (1,\mathrm{\infty })$. Then $u\in X_q$ has a Helmholtz decomposition, i.e., there exists a linear bounded projection operator $P_q$ from $X_q$ onto $X_{q\sigma }$ with null space $N(P_q)=\{\mathrm{\nabla }\phi \in X_q:\phi \in L_{loc}^q(G;E)\}$. In particular, all $u\in X_q$ have a unique decomposition $u=u_0+\mathrm{\nabla }\phi$ with $u_0\in X_{q\sigma }$, $u_0=P_{q}u$ so that

$${\parallel \mathrm{\nabla }\phi \parallel }_{X_q}+{\parallel u_0\parallel }_{X_q}\le C{\parallel u\parallel }_{X_q}.$$

(3.4)

Moreover, ${(X_{q\sigma })}^{\ast }=X_{q^{\prime }\sigma }$, where $\frac{1}{q}+\frac{1}{q^{\prime }}=1$.

For proving Theorem 3.2 we need some lemmas.

Consider the equation

$$-\mathrm{\Delta }u+A(x)u=f(x),x\in G.$$

(3.5)

Lemma 3.1 Let E be an UMD space, A a R-positive operator in E and $q\in (1,\mathrm{\infty })$. Then, for $f\in X_q^{-1}$, problem (3.2) has a unique solution $u\in X_q^1(A)$ and the following coercive estimate holds:

$${\parallel u\parallel }_{X_q^1(G)}+{\parallel Au\parallel }_{X_q(G)}\le C{\parallel f\parallel }_{X_q^{-1}(G)}.$$

(3.6)

Proof Consider the problem

$$-\mathrm{\Delta }u+Au=\tilde{f}(x),x\in R^n,$$

(3.7)

where $\tilde{f}(x)$ is an extension of the function $f(x)$ on $R^n$. Then, by using the Fourier inversion formula, operator-valued multiplier theorems in $L^q$ spaces, and by reasoning as in [[17], Theorem 3.2] we see that problem (3.7) has a unique solution $\tilde{u}\in X_q^1(R^n,A)$ for $f\in X_q^{-1}(R^n;E)$ and the following coercive estimate holds:

$${\parallel \tilde{u}\parallel }_{X_q^1(R^n)}+{\parallel A\tilde{u}\parallel }_{X_q(R^n)}\le C{\parallel \tilde{f}\parallel }_{X_q^{-1}(R^n)}.$$

This fact implies that the function u which is a restriction of $\tilde{u}$ on G is a solution of problem (3.5). The estimate (3.6) is obtained from the above estimate.

Let $\nu =({\nu }_1,{\nu }_2,\dots ,{\nu }_n)$ be a unit normal to the boundary Γ of the domain G and $f_{\nu }$ is a normal component of $f=(f_1,f_2,\dots ,f_n)\in X_q(G)$ on Γ, i.e.,

$$f_{\nu }=\left(\sum _{k=1}^n{\nu }_k{f}_k\right){|}_{\mathrm{\Gamma }}.$$

Here and hereafter $E^{\ast }$ will denoted the conjugate of E, and $(,\cdot )$ (resp. $〈,\cdot 〉$) denotes the duality pairing of functions on G (resp. Γ). □

By reasoning as in [[3], Lemma 2] we get the following.

Lemma 3.2 $C_0^{\mathrm{\infty }}(G;E)$ is dense in $Y_q(A)$.

Proposition 3.1 There exists a unique bounded linear operator $u\to u_{\nu }$ from $Y_q(A)$, $q\in (1,\mathrm{\infty })$ onto

$$W_q(\mathrm{\Gamma })=W^{-1/q,q}(\mathrm{\Gamma },{(E\left(A^{\ast }\right),E^{\ast })}_{q,1/q^{\prime }},E^{\ast })$$

such that

$$(u_{\nu },\upsilon {|}_{\mathrm{\Gamma }})=(divu,\upsilon )+(u,\mathrm{\nabla }\upsilon ),\upsilon \in W^{1,q^{\prime }}(G,E(A),E)$$

and the following estimate holds:

$${\parallel u_{\nu }\parallel }_{W_q(\mathrm{\Gamma })}\le C({\parallel u\parallel }_{X_q}+{\parallel divu\parallel }_{L^q(G;E)}),$$

(3.8)

where

$$\frac{1}{q}+\frac{1}{q^{\prime }}=1.$$

Proof For $u\in Y_q(A)$ consider the linear form

$$T_u(\upsilon )=(divu,\mathrm{\Phi })+(u,\mathrm{\nabla }\mathrm{\Phi }),\mathrm{\Phi }\in W^{q^{\prime },1}(G,E(A),E),{\mathrm{\Phi }}_{\nu }=\upsilon .$$

(3.9)

By virtue of the trace theorem in $W^{1,q}(0,a;E(A),E)$, the interpolation of intersection and dual spaces (see, e.g., [[13], §1.8.2, 1.12.1, 1.11.2]) and by a localization argument we obtain the result that the operator $\mathrm{\Phi }\to {\mathrm{\Phi }}_{\nu }$ is a bounded linear and surjective from $W^{1,q^{\prime }}(G;E(A),E)$ onto

$$Z(\mathrm{\Gamma })=W^{1-1/q^{\prime },q^{\prime }}(\mathrm{\Gamma },{(E(A),E)}_{q^{\prime },1/q^{\prime }},E).$$

Hence, we can find for each $\upsilon \in Z(\mathrm{\Gamma })$ an element $\mathrm{\Phi }\in W^{1,q^{\prime }}(G;E(A),E)$ so that

$${\mathrm{\Phi }}_{\nu }=\upsilon ,{\parallel \mathrm{\Phi }\parallel }_{W^{1,q^{\prime }}(G;E(A),E)}\le C{\parallel \upsilon \parallel }_{Z(\mathrm{\Gamma })}.$$

Therefore, from (3.9) we get

$$\begin{array}{rl}|T_u(\upsilon )|& \le ({\parallel u\parallel }_{X_q}+{\parallel divu\parallel }_{L^q(G;E)}){\parallel \mathrm{\Phi }\parallel }_{W^{1,q^{\prime }}(G;E(A),E)}\\ \le C({\parallel u\parallel }_{X_q}+{\parallel divu\parallel }_{L^q(G;E)}){\parallel \upsilon \parallel }_{Z(\mathrm{\Gamma })}.\end{array}$$

This implies the existence of an element

$$u_{\nu }\in {(Z(\mathrm{\Gamma }))}^{\ast }=W^{-1/q,q}(\mathrm{\Gamma },{(E\left(A^{\ast }\right),E^{\ast })}_{q,1/q^{\prime }},E^{\ast })$$

such that

$$〈u_{\nu },\upsilon 〉=T_u(\upsilon )\text{for }\upsilon \in Z(\mathrm{\Gamma })$$

and

$${\parallel u_{\nu }\parallel }_{{(Z(\mathrm{\Gamma }))}^{\ast }}\le C({\parallel u\parallel }_{X_q}+{\parallel divu\parallel }_{L^q(G;E)}).$$

Thus, we have proved the existence of the operator $u\to u_{\nu }$. The uniqueness follows from Lemma 3.2. □

Proposition 3.1 implies the following.

Result 3.3 Assume the conditions of Proposition 3.1 are satisfied. Then

$$X_{q\sigma }\subset X_{q\nu }=\{u\in X_q,divu=0,u_{\nu }=0\}$$

and $X_{q\nu }$ is a closed subspace of $X_q$.

Let $f\in X_q$ and $f=(f_1(x),f_2(x),\dots ,f_n(x))$. Consider the following problem:

$$\begin{array}{r}-\mathrm{\Delta }u+Au=divf(x),x\in G,\\ \frac{\partial u}{\partial x_k}(G_{k0})=f_k(G_{k0}),\frac{\partial u}{\partial x_k}(G_{kb})=f_k(G_{kb}),k=1,2,\dots ,n.\end{array}$$

(3.10)

Lemma 3.3 Let E be an UMD space, A a R-positive operator in E and $q\in (1,\mathrm{\infty })$. Then, for $f\in X_q$, problem (3.10) has a unique solution $u\in W^{1,q}(G;E(A),E)$ and the coercive estimate holds

$${\parallel u\parallel }_{W^{1,q}(G;E)}+{\parallel Au\parallel }_{L^q(G;E)}\le C{\parallel divf\parallel }_{W^{-1,q}(G;E)}+{\parallel f_{\nu }\parallel }_{W_q(\mathrm{\Gamma })}.$$

(3.11)

Proof Consider the equation

$$-\mathrm{\Delta }\upsilon +A\upsilon =divf(x),x\in G.$$

(3.12)

By Lemma 3.1, problem (3.12) has a unique solution $\upsilon \in L^{1,q}(G;E(A),E)$ for $f\in X_q$ and the following estimate holds:

$${\parallel \upsilon \parallel }_{L^{1,q}(G;E)}+{\parallel A\upsilon \parallel }_{L^q(G;E)}\le C{\parallel divf\parallel }_{L^{-1,q}(G;E)}.$$

(3.13)

Consider now the BVP

$$\begin{array}{r}-\mathrm{\Delta }w+Aw=0,x\in G,\\ \frac{\partial w}{\partial x_k}(G_{k0})=f_k(G_{k0})-\frac{\partial \upsilon }{\partial x_k}(G_{k0}),\\ \frac{\partial w}{\partial x_k}(G_{kb})=f_k(G_{kb})-\frac{\partial \upsilon }{\partial x_k}(G_{kb}),k=1,2,\dots ,n.\end{array}$$

(3.14)

By using Theorem 3.1, Result 3.3, and Proposition 3.1 we conclude that problem (3.14) for $f\in X_q$ has a unique solution $w\in W^{1,q}(G;E(A),E)$ and the following coercive estimate holds:

$${\parallel w\parallel }_{W^{1,q}(G;E)}+{\parallel A^{\frac{1}{2}}w\parallel }_{L^q(G;E)}\le C{\parallel f_{\nu }\parallel }_{W_q(\mathrm{\Gamma })}.$$

(3.15)

Then we conclude that problem (3.10) has a unique solution $\phi (x)=\upsilon (x)+w(x)$ and (3.13), (3.15) imply the estimate (3.11). □

Result 3.4 For the case of $A=\varkappa >0$ we obtain from (3.9), (3.11), and (3.12) that the problem

$$\begin{array}{r}(-\mathrm{\Delta }+\varkappa )w=0,x\in G,\\ \frac{\partial w}{\partial x_k}(G_{k0})=f_k(G_{k0})-\frac{\partial \upsilon }{\partial x_k}(G_{k0}),\\ \frac{\partial w}{\partial x_k}(G_{kb})=f_k(G_{kb})-\frac{\partial \upsilon }{\partial x_k}(G_{kb}),k=1,2,\dots ,n\end{array}$$

(3.16)

has a unique solution $w\in W^{1,q}(G;E)$ for $f\in X_q$ and the following estimate holds:

$${\parallel w\parallel }_{W^{1,q}(G;E)}\le C{\parallel f_{\nu }-\frac{\partial \upsilon }{\partial \nu }\parallel }_{W^{-1/q,q}(\mathrm{\Gamma },E^{\ast })}.$$

(3.17)

Result 3.5 For the case of $A=\varkappa >0$ we obtain from Lemma 3.1 and from (3.9), (3.10) that the problem

$$(-\mathrm{\Delta }+\varkappa )\upsilon =divf(x),x\in G$$

(3.18)

has a unique solution $\upsilon \in W^{1,q}(G;E)$ for $f\in X_q$ and the following estimate holds:

$${\parallel \upsilon \parallel }_{W^{1,q}(G;E)}\le C{\parallel divf\parallel }_{X_q^{-1}}.$$

(3.19)

By (3.9), $div(f-\mathrm{\nabla }\upsilon )=0$. So, by (3.14) and Proposition 3.1 we get

$${\parallel w\parallel }_{W^{1,q}(G;E)}\le C{\parallel f\parallel }_{X_q}.$$

(3.20)

From Results 3.4, 3.5, and estimate (3.17) we obtain

Result 3.6 The problem

$$\begin{array}{r}(-\mathrm{\Delta }+\varkappa )w=divf(x),x\in G,\\ \frac{\partial w}{\partial x_k}(G_{k0})=f_k(G_{k0})-\frac{\partial \upsilon }{\partial x_k}(G_{k0}),\\ \frac{\partial w}{\partial x_k}(G_{kb})=f_k(G_{kb})-\frac{\partial \upsilon }{\partial x_k}(G_{kb}),k=1,2,\dots ,n\end{array}$$

(3.21)

has a unique solution $u=w+\upsilon \in W^{1,q}(G;E)$ for $f\in X_q$ and the estimate holds

$${\parallel u\parallel }_{W^{1,q}(G;E)}\le C{\parallel f\parallel }_{X_q}.$$

(3.22)

Consider the operator $P=P_q$ defined by

$$D(P)=X_q,Pf=f-grad(\upsilon +w),$$

where w and υ are solutions of problems (3.18), (3.13), respectively. It is clear that we have the following.

Lemma 3.4 Let E be an UMD space and $q\in (1,\mathrm{\infty })$. Then $P_q{X}_q$ is a closed subspace of $X_q$.

Lemma 3.5 Let E be an UMD space and $q\in (1,\mathrm{\infty })$. Then the operator $P_q$ is a bounded linear operator in $X_q$ and $Pf=f$ if $divf(x)=0$.

Proof The linearity of the operator P is clear by construction. Moreover, by Result 3.6 we have

$${\parallel Pf\parallel }_{X_q}\le {\parallel f\parallel }_{X_q}+{\parallel gradu\parallel }_{X_q}\le C{\parallel f\parallel }_{X_q}.$$

(3.23)

If $divf(x)=0$ then by Result 3.3 we get $\upsilon =0$. Moreover, by the estimate (3.20) we obtain $w=0$, i.e., $Pf=f$. □

Lemma 3.6 Assume E is an UMD space and $q\in (1,\mathrm{\infty })$. Then the conjugate of $P_q$ is defined as $P_q^{\ast }=P_{q^{\prime }}$, $\frac{1}{q}+\frac{1}{q^{\prime }}=1$ and is bounded linear in ${(L^{q^{\prime }}(G;E^{\ast }))}^n$.

Proof It is known (see, e.g., [13, 18]) that the dual space of $L^q(G;E)$ is $L^{q^{\prime }}(G;E^{\ast })$. Since $C_0^{\mathrm{\infty }}(G;E^{\ast })$ is dense in $L^{q^{\prime }}(G;E^{\ast })$ we have only to show $P_q^{\ast }\phi =P_{q^{\prime }}\phi$ for any $\phi \in {(C_0^{\mathrm{\infty }}(G;E^{\ast }))}^n$. But this is deriving by reasoning as in [[3], Lemma 5]. Moreover, by Lemma 3.5 the dual operator $P_q^{\ast }$ is a bounded linear in $L^{q^{\prime }}(G;E^{\ast })$. □

Let

$$\begin{array}{c}{W}_q=\{\mathrm{\nabla }\phi :\phi \in W^{q,1}(G;E)\},\hfill \\ {(P_q{X}_q)}^{\perp }=\{f\in {\left(L^{q^{\prime }}(G;E^{\ast })\right)}^n,〈f,\upsilon 〉=0\text{ for any }\upsilon \in P_q{X}_q\}.\hfill \end{array}$$

From Lemmas 3.5, 3.6 we obtain the following.

Result 3.7 Assume E is an UMD space and $q\in (1,\mathrm{\infty })$. Then any element $f\in X_q$ uniquely can be expressed as a sum of elements of $P_q{X}_q$ and $W_q$.

In a similar way as Lemmas 6, 7 of [3] we obtain, respectively, the following.

Lemma 3.7 Assume E is an UMD space and $q\in (1,\mathrm{\infty })$. Then

$${(P_q{X}_q)}^{\perp }=W_{q^{\prime }},\frac{1}{q}+\frac{1}{q^{\prime }}=1.$$

Lemma 3.8 Assume E is an UMD space and $q\in (1,\mathrm{\infty })$. Then

$$X_{q\sigma }^{\perp }=W_{q^{\prime }},\frac{1}{q}+\frac{1}{q^{\prime }}=1.$$

Now we are ready to prove Theorem 3.2.

Proof of Theorem 3.2 From Lemmas 3.7, 3.8 we get $X_{q\sigma }={(P_q{X}_q)}^{\perp }$. Then, by construction of $P_q$, we have $X_q=X_{q\sigma }\oplus W_q$. By Lemmas 3.3, 3.5, we obtain the estimate (3.4). Moreover, by Lemma 3.4, $W_q$ is a close subspace of $X_q$. Then it is known that the dual space of the quotient space $X_q/W_q$ is $W_q^{\perp }$. In view of first assertion we have $X_q/W_q=X_{q\sigma }$ and by Lemma 3.8 we obtain the second assertion. □

Theorem 3.3 Let Condition 3.1 hold. Then problem (1.7)-(1.8) has a unique solution $u\in X_q^2$ for $f\in X_q$, $\phi \in W^{1,q}(G;E)$, $\lambda \in S_{\psi ,\varkappa }$, and the following coercive uniform estimate holds:

$$\sum _{k=1}^n\sum _{i=0}^2{\lambda }^{1-\frac{i}{2}}{\parallel \frac{{\partial }^{i}u}{\partial x_k^i}\parallel }_{X_q}+{\parallel Au\parallel }_{X_q}+{\parallel \mathrm{\nabla }\phi \parallel }_{X_q}\le C{\parallel f\parallel }_{X_q}.$$

(3.24)

Proof By virtue of Result 3.2, we find that problem (3.2) has a unique solution $u\in X_q^2(G,A)$ for $f\in X_q$ and for sufficiently large $\lambda \in S_{\phi }$. Moreover, the following coercive uniform estimate holds:

$$\sum _{k=1}^n\sum _{i=0}^2|\lambda {|}^{1-\frac{i}{2}}{\parallel \frac{{\partial }^{i}u}{\partial x_k^i}\parallel }_{X_q}+{\parallel Au\parallel }_{X_q}\le C{\parallel f\parallel }_{X_q}.$$

By applying the operator $P_q$ to problem (1.7)-(1.8) we get the Stokes problem (1.9)-(1.10). It is clear that

$$D(O_q)=D(B)\cap X_{q\sigma },$$

where $O_q$ is the Stokes operator and B is a operator generated by problem (3.2) for $\lambda =0$. Then by Theorem 3.2 we obtain the assertion. □

Result 3.8 From Result 3.2 we find that $O=O_q$ is a positive operator in $X_q$ and −O generate a bounded holomorphic semigroup $S(t)=exp(-Ot)$ for $t>0$.

In a similar way as in [6] we show the following.

Proposition 3.2 The following estimate holds:

$$\parallel O^{\alpha }S(t)\parallel \le C{t}^{-\alpha }$$

for $\alpha \ge 0$ and $t>0$.

Proof From the estimate (3.3) we see that the operator O is positive in $X_q$, i.e., for $\lambda \in S_{\psi ,\varkappa }$, $0<\psi <\pi$ the following estimate holds:

$$\parallel {(O+\lambda )}^{-1}\parallel \le M|\lambda {|}^{-1},$$

where the constant M is independent of λ. Then, by using the Danford integral and operator calculus (see, e.g., in [12]), we obtain the assertion. □

4 Well-posedness of instationary Stokes problems with variable coefficients

Let

$$B(p,q)=L^p(0,T;X_q),D(p,q)={(X_q^2(A),X_q)}_{\frac{1}{p},p}.$$

In this section, we will show the well-posedness of problem (1.1)-(1.2).

Theorem 4.1 Then, for $f\in B(p,q)$, $\phi \in L_p(0,T;W^{1,q}(G;E))$, and $a\in D(p,q)$, $p,q\in (1,\mathrm{\infty })$, there is a unique solution $(u,\mathrm{\nabla }\phi )$ of problem (1.1)-(1.2) and the following estimate holds:

$${\parallel \frac{\partial u}{\partial t}\parallel }_{B(p,q)}+\sum _{k=1}^n{\parallel \frac{{\partial }^{2}u}{\partial x_k^2}\parallel }_{B(p,q)}+{\parallel Au\parallel }_{B(p,q)}+{\parallel \mathrm{\nabla }\phi \parallel }_{B(p,q)}\le C({\parallel f\parallel }_{B(p,q)}+{\parallel a\parallel }_{D(p,q)}).$$

(4.1)

Proof Problem (1.1)-(1.2) can be expressed as the following abstract parabolic problem:

$$\frac{du}{dt}+Ou=f(t),u(0)=a.$$

(4.2)

If we put $E=X_q$ then by Proposition 3.2, operator O is positive and generates a bounded holomorphic semigroup in $X_q$. Moreover, by using [[17], Theorem 3.1] we see that the operator O is R-positive in E. Since E is a UMD space, in a similar way as in [[13], Theorem 4.2] we see that for all $f\in L^p(0,T;E)$ and $a\in {(D(O),E)}_{\frac{1}{p},p}$ there is a unique solution $u\in W^{1,p}(0,T,D(O),E)$ of problem (4.2) so that the following estimate holds:

$${\parallel \frac{du}{dt}\parallel }_{L^p(0,T;E)}+{\parallel Ou\parallel }_{L^p(0,T;E)}\le C({\parallel f\parallel }_{L^p(0,T;E)}+{\parallel a\parallel }_{{(D(O),E)}_{\frac{1}{p},p}}).$$

(4.3)

 □

From the estimates (3.3) and (4.3) we obtain the assertion.

Remark 4.1 There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1.7)-(1.8) and (1.1)-(1.3) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of A, by virtue of Theorem 3.3 and Theorem 4.1 we can obtain the maximal regularity properties of different class of stationary and instationary Stokes problems, respectively, which occur in numerous physics and engineering problems.

Let us now show some application of Theorem 3.3 and Theorem 4.1.

5 Application

Consider the stationary Stokes problem

$$\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u(x,y)}{\partial x_k^2}+\sum _{|\beta |\le 2m}{a}_{\beta }(x,y)D_y^{\beta }u(x,y)+\mathrm{\nabla }\phi =f(x,y),divu=0,$$

(5.1)

$$\sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0},y)+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb},y)]=0,y\in \mathrm{\Omega },j=1,2,$$

(5.2)

$$B_{j}u=\sum _{|\beta |\le m_j}{b}_{j\beta }(y)D_y^{\beta }u(x,y){|}_{y\in \partial \mathrm{\Omega }}=0,x\in G,j=1,2,\dots ,m,$$

(5.3)

where $f=(f_1(x,y,t),f_2(x,y,t),\dots ,f_n(x,y,t))$ represents a given and

$$u=(u_1(x,y),u_2(x,y),\dots ,u_n(x,y)),\phi =\phi (x,y)$$

are unknown functions;

$$\begin{array}{c}x=(x_1,x_2,\dots ,x_n)\in G=\prod _{k=1}^n(0,b_k),y=(y_1,\dots ,y_{\mu })\in \mathrm{\Omega }\subset R^{\mu },\hfill \\ G_{k0}=(x_1,x_2,\dots ,x_{k-1},0,x_{k+1},\dots ,x_n),p\in (1,\mathrm{\infty }),\hfill \\ G_{kb}=(x_1,x_2,\dots ,x_{k-1},b_k,x_{k+1},\dots ,x_n),q\in (1,\mathrm{\infty }),m_{kj}\in \{0,1\},\hfill \end{array}$$

${\alpha }_{kji}$, ${\beta }_{kji}$ are complex numbers, $a_{\alpha }$, $b_{j\beta }$ are complex-valued functions, $D_j=-i\frac{\partial }{\partial y_j}$.

Let $\tilde{\mathrm{\Omega }}=G\times \mathrm{\Omega }$, $\mathbf{p}=(p_1,p)$. Now $L^{\mathbf{p}}(\tilde{\mathrm{\Omega }})$ will denote the space of all p-summable scalar-valued functions with mixed norm i.e., the space of all measurable functions f defined on $\tilde{\mathrm{\Omega }}$, for which

$${\parallel f\parallel }_{L^{\mathbf{p}}(\tilde{\mathrm{\Omega }})}={\left({\int }_G{\left({\int }_{\mathrm{\Omega }}\right|f(x,y){|}^{p_1}dx)}^{\frac{p}{p_1}}dy\right)}^{\frac{1}{p}}<\mathrm{\infty }.$$

Analogously, $W^{m,\mathbf{p}}(\tilde{\mathrm{\Omega }})$ denotes the Sobolev space with corresponding mixed norm.

$X_{\mathbf{p}}={(L^{\mathbf{p}}(\tilde{\mathrm{\Omega }}))}^n$ denotes the class of vector function

$$f=(f_1(x),f_2(x),\dots ,f_n(x))$$

with norm

$${\parallel f\parallel }_{X_{\mathbf{p}}}=\sum _{i=1}^n{\parallel f_i\parallel }_{L^{\mathbf{p}}(G;E)}$$

and $X_{\mathbf{p}}^{2,2m}={(W^{2,2m,\mathbf{p}}(\tilde{\mathrm{\Omega }}))}^n$, where $W^{2,2m,\mathbf{p}}(\tilde{\mathrm{\Omega }})$ is the anisotropic Sobolev space with mixed norm.

From Theorem 3.3 we obtain the following.

Theorem 5.1 Let the following conditions be satisfied:

• (1) Ω is a domain in $R^{\mu }$ with sufficiently smooth boundary ∂ Ω, $a_{\alpha }(x,y)\in C(\overline{\mathrm{\Omega }})$, $a_{\alpha }(G_{i0},y)=a_{\alpha }(G_{ib},y)$ for each $|\alpha |=2m$, $y\in \mathrm{\Omega }$, $a_{\alpha }\in [L_{\mathrm{\infty }}+L_{r_k}](\mathrm{\Omega })$ for each $|\alpha |=k<2m$ with $r_k\ge p_1$, $p_1\in (1,\mathrm{\infty })$ and $2m-k>\frac{l}{r_k}$, ${\nu }_{\alpha }\in L_{\mathrm{\infty }}$;

• (2) $b_{j\beta }\in C^{2m-m_j}(\partial \mathrm{\Omega })$ for each $j,\beta ,m_j<2m$, $p\in (1,\mathrm{\infty })$;

• (3) for $y\in \overline{\mathrm{\Omega }}$, $\xi \in R^{\mu }$, $\eta \in S({\phi }_1)$, ${\phi }_1\in [0,\frac{\pi }{2})$, $|\xi |+|\eta |\ne 0$ let

  $$\eta +\sum _{|\alpha |=2m}{a}_{\alpha }(y){\xi }^{\alpha }\ne 0;$$

• (4) for each $x_0\in \partial \mathrm{\Omega }$ the local BVPs in local coordinates corresponding to $x_0$

  $$\begin{array}{c}\eta +\sum _{|\alpha |=2m}{a}_{\alpha }(x_0,{\xi }^{\prime },D_{\mu })\vartheta (y)=0,y>0,\hfill \\ B_{j0}\vartheta =\sum _{|\beta |=m_j}{b}_{j\beta }(x_0,{\xi }^{\prime },D_{\mu })\vartheta (y){|}_{y=0}=h_j,j=1,2,\dots ,m\hfill \end{array}$$

has a unique solution $\vartheta \in C_0(R_{+})$ for all $h=(h_1,h_2,\dots ,h_m)\in R^m$ and for ${\xi }^{\prime }\in R^{\mu -1}$ with $|{\xi }^{\prime }|+|\eta |\ne 0$;

• (5) $a_k\in C(\overline{G})$, $a_k(G_{i0},y)=a_k(G_{ib},y)$, $a_k\ne 0$, $a_k\in S({\phi }_0)\cap \mathbb{C}/{\mathbb{R}}_{-}$ for all $x\in G$, $\phi +{\phi }_0<\pi$, $y\in \mathrm{\Omega }$, $k,i=1,2,\dots ,n$.

Then, problem (5.1)-(5.3) has a unique solution $u\in X_{\mathbf{p}}^{2,2m}$ for $f\in X_{\mathbf{p}}$, $\phi \in W^{1,\mathbf{p}}(\tilde{\mathrm{\Omega }};E)$, $\lambda \in S_{\psi ,\varkappa ,}$ and the following coercive uniform estimate holds:

$$\sum _{k=1}^n\sum _{i=0}^2|\lambda {|}^{1-\frac{i}{2}}{\parallel \frac{{\partial }^{i}u}{\partial x_k^i}\parallel }_{X_{\mathbf{p}}}+\sum _{|\alpha |=2m}{\parallel D_y^{\alpha }u\parallel }_{X_{\mathbf{p}}}+{\parallel \mathrm{\nabla }\phi \parallel }_{X_{\mathbf{p}}}\le C{\parallel f\parallel }_{X_{\mathbf{p}}}.$$

(5.4)

Proof Let $E=L^{p_1}(\mathrm{\Omega })$. By virtue of [[18], Theorem 4.5.2] $L^{p_1}(\mathrm{\Omega })$ is an UMD space. Consider the operator A which is defined by

$$D(A)=W^{2m,p_1}(\mathrm{\Omega };B_{j}u=0),Au=\sum _{|\beta |\le 2m}{a}_{\beta }(y)D^{\beta }u(y).$$

Problem (5.1)-(5.3) can be rewritten in the form of (1.7)-(1.8) for $A_k=0$, $k=1,2,\dots ,n$, where $u(x)=u(x,\cdot )$ and $f(x)=f(x,\cdot )$ are functions with values in $E=L^{p_1}(\mathrm{\Omega })$. In view of [[14], Theorem 8.2] the problem

$$\begin{array}{r}\eta u(y)+\sum _{|\beta |\le 2m}{a}_{\beta }(x,y)D^{\beta }u(y)=f(y),\\ B_{j}u=\sum _{|\beta |\le m_j}{b}_{j\beta }(y)D^{\beta }u(y)=0,j=1,2,\dots ,m\end{array}$$

(5.5)

has a unique solution for $f\in L^{p_1}(\mathrm{\Omega })$ and arg $\eta \in S({\phi }_1)$, $|\eta |\to \mathrm{\infty }$, and the operator A is R-positive in $L^{p_1}$. Hence, all conditions of Theorem 3.2 are satisfied, i.e., we obtain the assertion.

Consider now the instationary Stokes problem

$$\frac{\partial u}{\partial t}+\sum _{k=1}^n{a}_k(x)\frac{{\partial }^{2}u}{\partial x_k^2}+\sum _{|\beta |\le 2m}{a}_{\beta }(x,y)D_y^{\beta }u+\mathrm{\nabla }\phi =f(x,y,t),divu=0,$$

(5.6)

$$\sum _{i=0}^{m_{kj}}[{\alpha }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{k0},y,t)+{\beta }_{kji}\frac{{\partial }^{i}u}{\partial x_k^i}(G_{kb},y,t)]=0,j=1,2,$$

(5.7)

$$u(x,y,0)=a(x,y),t\in (0,T),x\in G,y\in \mathrm{\Omega },$$

(5.8)

$$B_{j}u=\sum _{|\beta |\le m_j}{b}_{j\beta }(y)D_y^{\beta }u(x,y,t){|}_{y\in \partial \mathrm{\Omega }}=0,x\in G,j=1,2,\dots ,m,$$

(5.9)

where $f(x,y,t)$, $u=(x,y,t)$ are data and solution vector-functions, respectively. □

From Theorem 4.1 and Theorem 5.1 we obtain the following.

Theorem 5.2 Assume all conditions of Theorem  5.1 are satisfied $p,p_1,q\in (1,\mathrm{\infty })$. Then, for $f\in L^q(0,T;X_{\mathbf{p}})=B(p,p_1,q)$, $\phi \in L^q(0,T;W^{1,\mathbf{p}}(\tilde{\mathrm{\Omega }}))$, and $a\in {(X_{\mathbf{p}}^{2,2m},X_{\mathbf{p}})}_{\frac{1}{q},q}=D(p,p_1,q)$, there is a unique solution $(u,\mathrm{\nabla }\phi )$ of problem (5.6)-(5.9) and the following estimate holds:

$$\begin{array}{c}{\parallel \frac{\partial u}{\partial t}\parallel }_{B(p,p_1,q)}+\sum _{k=1}^n{\parallel \frac{{\partial }^{2}u}{\partial x_k^2}\parallel }_{B(p,p_1,q)}+\sum _{|\alpha |=2m}{\parallel D_y^{\alpha }u\parallel }_{B(p,p_1,q)}\hfill \\ +{\parallel \mathrm{\nabla }\phi \parallel }_{B(p,p_1,q)}\le C({\parallel f\parallel }_{B(p,p_1,q)}+{\parallel a\parallel }_{D(p,p_1,q)}).\hfill \end{array}$$