One Navier’s problem for the Brinkman system

Abstract

In this paper we study the Brinkman system and the Darcy-Forchheimer-Brinkman system with the boundary condition of the Navier’s type ${\textbf{u}}_{{\mathbf {\mathcal {T}}}} = {\textbf{g}}_{{\mathbf {\mathcal {T}}}}$, $\rho =h$ on $\partial \Omega$ for a bounded planar domain $\Omega$ with connected boundary. Solutions are looked for in the Sobolev spaces $W^{s+1,q}(\Omega ,{\mathbb R}^2)\times W^{s,q}(\Omega )$ and in the Besov spaces $B_{s+1}^{p,r}(\Omega ,{\mathbb R}^2)\times B_s^{q,r}(\Omega )$. Classical solutions are from the spaces ${\mathcal C}^{k+1,\gamma }(\overline{\Omega },{\mathbb R}^2) \times {\mathcal C}^{k,\gamma }(\overline{\Omega })$. For the Brinkman system we show the unique solvability of the problem. Then we study the Navier problem for the Darcy-Forchheimer-Brinkman system and small boundary conditions.

1 Introduction

For systems of partial differential equations on domains several kinds of Neumann conditions are possible. The system

$$\begin{aligned} -\Delta \textbf{u}+\lambda \textbf{u}+\nabla \rho =\textbf{f}, \ \nabla \cdot \textbf{u}=\chi \quad \text {in}\ \Omega \end{aligned}$$

(1.1)

(called Stokes system for $\lambda =0$ and Brinkman system for $\lambda >0$) has the relevant Neumann conditions

$$\begin{aligned} \frac{\partial \textbf{u}}{\partial \textbf{n}} -\rho \textbf{n} =\textbf{g}\qquad \text {on}\ \partial \Omega \end{aligned}$$

(1.2)

(studied for example in [49, 50, 56]) and

$$\begin{aligned}{}[\nabla \textbf{u}+(\nabla \textbf{u})^T]\textbf{n} -\rho \textbf{n} =\textbf{g}\qquad \text {on}\ \partial \Omega \end{aligned}$$

(1.3)

(studied for example in [18, 38,39,40, 49, 58, 59]). Here $\Omega \subset {{\mathbb {R}}}^m$ is a bounded domain with Lipschitz boundary and $\textbf{n}=\textbf{n}^\Omega$ is the outward unit normal vector of $\Omega$. In ${{\mathbb {R}}}^3$ we have $\Delta \textbf{u}=\nabla (\nabla \cdot \textbf{u})-\nabla \times (\nabla \times {\nabla u})$. This gives another Neumann condition for the Stokes and Brinkman systems

$$\begin{aligned} \textbf{n}\times (\nabla \times \textbf{u})+\rho \textbf{n} =\textbf{g}\qquad \text {on}\ \partial \Omega . \end{aligned}$$

(1.4)

Very interesting boundary value problems are problems of Navier’s type. There are two types of Navier’s problem: (1) It is given the normal part of the Dirichlet condition and the tangential part of the Neumann condition. (2) It is given the tangential part of the Dirichlet condition and the normal part of the Neumann condition. If $\textbf{v}$ is a vector, then $\textbf{v}_\textbf{n}=(\textbf{v}\cdot \textbf{n})\textbf{n}$ denotes the normal part of $\textbf{v}$, and $\textbf{v}_{\tau }=\textbf{v}-\textbf{v}_\textbf{n}$ is the tangential part of $\textbf{v}$. The Navier conditions corresponding to the Neumann condition (1.2) are

$$\begin{aligned} {\textbf{u}}_{\tau } = {\textbf{g}}_{\tau }, \quad [\partial \textbf{u}/\partial \textbf{n}]_\textbf{n}-\rho \textbf{n} =\textbf{h}_\textbf{n} \quad \text {on}\ \partial \Omega \end{aligned}$$

and

$$\begin{aligned} \textbf{u}_\textbf{n}=\textbf{g}_\textbf{n}, \quad [\partial \textbf{u}/\partial \textbf{n}]_\tau =\textbf{h}_\tau \quad \text {on}\ \partial \Omega \end{aligned}$$

(studied in [20] and [54]). The Navier conditions corresponding to the Neumann condition (1.3) are

$$\begin{aligned} {\textbf{u}}_{\tau } = {\textbf{g}}_{\tau } , \quad [(\nabla \textbf{u}+(\nabla \textbf{u})^T)\textbf{n} -\rho \textbf{n}]_\textbf{n} =\textbf{h}_\textbf{n} \quad \text {on}\ \partial \Omega \end{aligned}$$

and

$$\begin{aligned} \textbf{u}_\textbf{n}=\textbf{g}_\textbf{n}, \quad [(\nabla \textbf{u}+(\nabla \textbf{u})^T)\textbf{n} ]_\tau =\textbf{h}_\tau \quad \text {on}\ \partial \Omega \end{aligned}$$

(studied in [3, 4, 7, 13, 19, 26, 27, 29, 37, 40, 44, 45, 55, 57, 61]), [62, 66]). The Navier conditions corresponding to the Neumann condition (1.4) are

$$\begin{aligned} {\textbf{u}}_{\tau } = {\textbf{g}}_{\tau } ,\quad \rho =h\quad \text {on}\ \partial \Omega , \end{aligned}$$

(1.5)

and

$$\begin{aligned} \textbf{u}_\textbf{n}=\textbf{g}_\textbf{n}, \quad \textbf{n}\times (\nabla \times \textbf{u})=\textbf{n}\times \textbf{h}\quad \text {on}\ \partial \Omega . \end{aligned}$$

(1.6)

These problems were studied in three-dimensional domains in [5, 6, 8, 11,12,13,14,15, 17, 21, 22, 25, 43] from the theoretical point of view and in [1, 2, 51] from the numerical point of view. The paper [16] studies the Brinkman system with the Navier condition (1.6) in a planar domain.

We study the Brinkman system with the condition (1.5) in planar domains. The papers [22] and [17] studied the problem with the condition (1.5) for the Stokes system in $H^1(\Omega ,{{\mathbb {R}}}^3)\times H^1(\Omega )$ for $\Omega \subset {{\mathbb {R}}}^3$. The paper [12] is devoted to this problem in $W^{1,p}(\Omega ,{{\mathbb {R}}}^3)\times W^{1,p}(\Omega )$ for $\Omega \subset {{\mathbb {R}}}^3$. The question when the velocity $\textbf{u}\in W^{2,p}(\Omega ,{\mathbb R}^3)$ is answered. The paper [8] studies the Brinkman system with the condition (1.5) in $W^{1,p}(\Omega ,{\mathbb R}^3)\times W^{1,p}(\Omega )$ for $\Omega \subset {{\mathbb {R}}}^3$. The paper [14] treated a very weak solution $(\textbf{u},p)\in L^p(\Omega ,{{\mathbb {R}}}^3) \times L^p(\Omega )$ of the Stokes system with the condition (1.5) in $\Omega \subset {\mathbb R}^3$. The paper [11] studies the Stokes system with the condition (1.5) in an exterior domain $\Omega \subset {{\mathbb {R}}}^3$ in weighted Sobolev spaces. The Stokes system with the condition (1.5) on planar domains was studied from the numerical point of view in [10, 23, 41]. The author studied in [46] the Stokes system with the condition (1.5) on planar domains with connected boundary in Sobolev spaces $W^{t,p}(\Omega ,{{\mathbb {R}}}^2)\times W^{s,q}(\Omega )$, in Besov spaces $B_t^{p,\beta }(\Omega ,{\mathbb R}^2)\times B_s^{q,r}(\Omega )$ and in ${{\mathcal {C}}}^{k+1,\alpha }(\Omega ,{{\mathbb {R}}}^2)\times {{\mathcal {C}}}^{k,\alpha }(\Omega )$.

First we study problems with the boundary condition (1.5) on planar domains with connected boundary. We concern not only the Brinkman system (i.e. $\lambda >0$) but more generally the Stokes resolvent system (i.e. for complex $\lambda$). (Results for the Stokes resolvent system are useful in the study of boundary value problems for non-steady Stokes system.) We show that there exists a unique solution of the problem in the Sobolev spaces $W^{s+1,q}(\Omega ,{{\mathbb {R}}}^2)\times W^{s,q}(\Omega )$ and in the Besov spaces $B_{s+1}^{q,r}(\Omega ,{{\mathbb {R}}}^2)\times B_s^{q,r}(\Omega )$ for $1/q<s<k-1$ and $\partial \Omega \in {{\mathcal {C}}}^{k,1}$, and classical solutions in ${\mathcal C}^{k+1,\gamma }({\overline{\Omega }} ,{{\mathbb {R}}}^2)\times {\mathcal C}^{k,\gamma }({\overline{\Omega }} )$ for $0<\gamma <1$ and $\partial \Omega \in {{\mathcal {C}}}^{k+2,\gamma }$. As an application we study Darcy-Forchheimer-Brinkman system

$$\begin{aligned} -\Delta \textbf{u}+\lambda \textbf{u}+a|\textbf{u}|\textbf{u}+b(\textbf{u}\cdot \nabla )\textbf{u} +\nabla \rho =\textbf{f}, \ \nabla \cdot \textbf{u}=G \quad \text {in}\ \Omega \end{aligned}$$

(1.7)

with the boundary condition (1.5) in the same spaces.

Darcy-Forchheimer-Brinkman system with Navier’s condition has never been studied. But there are some papers concerning Darcy-Forchheimer-Brinkman system (1.7) with Dirichlet condition ( [31, 33, 53]), with the transmission condition ( [9]), the mixed Dirichlet-Neumann problem and the mixed Dirichlet-Robin problem ( [32, 34]).

2 Function spaces

Denote by ${{\mathbb {N}}}$ the set of all positive integers and by ${{\mathbb {N}}}_0$ the set of all non-negative integers.

Let $\Omega \subset {{\mathbb {R}}}^m$ be a domain (i.e. an open connected set). For $k\in {{\mathbb {N}}}_0$ denote by ${\mathcal C}^k({\overline{\Omega }} )$ the set of all $f\in {{\mathcal {C}}}^k(\Omega )$ for which $\partial ^\alpha f$ can be continuously extended onto ${\overline{\Omega }}$ for all $|\alpha |\le k$. The space ${{\mathcal {C}}}^k({\overline{\Omega }} )$ is equipped with the norm

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {C}}}^k({\overline{\Omega }} )}:=\sum _{|\alpha |\le k}\sup _{x\in {\overline{\Omega }} } |\partial ^\alpha f(x)|. \end{aligned}$$

For $k\in {{\mathbb {N}}}_0$ and $0<\beta <1$ we denote

$$\begin{aligned}{} & {} \Vert f\Vert _{{{\mathcal {C}}}^{k,\beta }({\overline{\Omega }} )}:=\sum _{|\alpha |\le k}\sup _{x\in {\overline{\Omega }} } |\partial ^\alpha f(x)|+\sum _{|\alpha |=k}\sup _{x,y\in \overline{\Omega },x\ne y} \frac{|\partial ^\alpha f(x)-\partial ^\alpha f(y)|}{|x-y|^\beta }, \\{} & {} {{\mathcal {C}}}^{k,\beta }({\overline{\Omega }} ):=\{ f\in {\mathcal C}^k({\overline{\Omega }} ); \Vert f\Vert _{{{\mathcal {C}}}^{k,\beta }({\overline{\Omega }} )}<\infty \} . \end{aligned}$$

Let $1<q<\infty$ and $k\in {{\mathbb {N}}}_0$. Denote by $W^{k,q}(\Omega )$ the space of all functions $f\in L^q(\Omega )$ such that $\partial ^\alpha f\in L^q(\Omega )$ in the sense of distributions for each multi-index $\alpha$ with $|\alpha |\le k$. If $s=k+\delta$ with $0<\delta <1$ denote $W^{s,q}(\Omega ):=\{ u\in W^{k,q}(\Omega ); \Vert u\Vert _{W^{s,q}(\Omega )}<\infty \}$ where

$$\begin{aligned} \Vert u\Vert _{W^{s,q}(\Omega )}^q:= \Vert u\Vert _{W^{k,q}(\Omega )}^q +\sum _{|\alpha |=k}\ \int \limits _{\Omega \times \Omega } \frac{|\partial ^\alpha u(x)-\partial ^\alpha u(y)|^q }{|x-y|^{m+q\delta }}\ {\mathrm d}x\ {\mathrm d}y. \end{aligned}$$

Denote by ${{\mathcal {C}}}_c^\infty (\Omega )$ the space of infinitely differentiable functions with compact support in $\Omega$. If $s>0$ denote by $W^{s,q}_0(\Omega )$ the closure of ${\mathcal C}_c^\infty (\Omega )$ in $W^{s,q}(\Omega )$. Put $q'=q/(q-1)$. Then $W^{-s,q'}(\Omega )$ denotes the dual space of $W^{s,q}_0(\Omega )$.

If $s\in {{\mathbb {R}}}^1$ and $1<p,q<\infty$ denote by $B_s^{p,q}({{\mathbb {R}}}^m)$ the Besov space. (For the definition see for example [65].) If k is a non-negative integer and $s=k+\delta$ with $0<\delta <1$ then $u\in B_s^{p,q}({\mathbb R}^2)$ if $u\in W^{k,p}({{\mathbb {R}}}^2)$ and

$$\begin{aligned} \sum _{|\beta |=k}\int _0^\infty \left( \int _{{{\mathbb {R}}}^2}\int _{\{ y\in {{\mathbb {R}}}^3;|x-y|<t\} } \frac{|\partial ^\beta u(x)-\partial ^\beta u(y)|^p}{t^2}\ {\mathrm d}y\ {\mathrm d}x\right) ^{q/p} \frac{{\mathrm d}t}{t^{\delta q+1}}<\infty . \end{aligned}$$

By $B_s^{p,q}(\Omega )$ we denote the space of restrictions of functions from $B_s^{p,q}({{\mathbb {R}}}^2)$ onto $\Omega$. The norm on $B_s^{p,q}(\Omega )$ is defined by

$$\begin{aligned} \Vert u\Vert _{B_s^{p,q}(\Omega )}:=\inf \{ \Vert f\Vert _{B_s^{p,q}({\mathbb R}^2)}; f=u\ \text {on}\ \Omega \} . \end{aligned}$$

Suppose that $\Omega$ is a bounded domain with Lipschitz boundary. If $s>t$ and $1<r<\infty$ then $B_s^{p,q}(\Omega )\hookrightarrow B_t^{p,r}(\Omega )$ (see [64, §4.6.1, Theorem]). If s is not integer then $B_s^{p,p}(\Omega )=W^{s,p}(\Omega )$. (See [24, Theorem 6.7] or [63, Lemma 36.1].)

Let $\varphi$ be a Lipschitz function on ${{\mathbb {R}}}^{m-1}$. Denote by $G:=\{ [x;\varphi (x)];x\in {{\mathbb {R}}}^{m-1}\}$ the graph of $\varphi$. Let X(U) be a space of functions on the set U (i.e ${{\mathcal {C}}}^{s,\alpha }(U)$, $W^{s,p}(U)$ or $B_s^{p,q}(U)$). We say that $f\in X(G)$ if $f(x,\varphi (x))\in X({{\mathbb {R}}}^{m-1})$.

Let $\Omega \subset {{\mathbb {R}}}^m$ be a bounded domain with Lipschitz boundary. Let $z_1,\dots ,z_k\in \partial \Omega$ be such that $\partial \Omega \subset B(z_1;r_1)\cup \dots B(z_k;r_k)$. Let $G_1,\dots ,G_k$ be graph Lipschitz domains such that $\Omega \cap B(z_j;r_j)=G_j\cap B(z_j;r_j)$ for $j=1,\dots ,k$. Choose $\alpha _j\in {{\mathcal {C}}}_c^\infty (B(z_j;r_j))$ such that $\alpha _1 +\dots +\alpha _k =1$ on a neighborhood of $\partial \Omega$. We say that $f\in X(\partial \Omega )$ (where $X=W^{s,p}$ or $X=B_s^{p,q}$) if $f\alpha _j \in X(\partial G_j)$ for $j=1,\dots ,k$.

Let $X(\Omega )$ be a space of functions (here $X=W^{s,q}$, $X=B_s^{q,r}$, ...). We denote $X(\Omega ;{{\mathbb {R}}}^k):=\{ (u_1,\dots ,u_k); u_j \in X(\Omega )\}$, $X(\Omega ;{\mathbb C}):=\{ (u_1+iu_2); u_j \in X(\Omega )\}$, $X(\Omega ;{\mathbb C}^k):=\{ (u_1,\dots ,u_k); u_j \in X(\Omega ;{{\mathbb {C}}})\}$. Similarly for $X(\partial \Omega )$.

Proposition 2.1

Let $\Omega \subset {{\mathbb {R}}}^m$ be a bounded domain. Let $p(0),p(1),q(0),q(1)\in (1,\infty )$ and $-\infty<s(1)<s(0)<\infty$. If $s(0)-m/p(0)>s(1)-m/p(1)$ then the identity is a compact operator from $B_{s(0)}^{p(0),q(0)}(\Omega )$ to $B_{s(1)}^{p(1),q(1)}(\Omega )$.

(See [65, Theorem 1.97].)

Corollary 2.2

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with Lipschitz boundary. Let $1<p<\infty$ and $s<t$. Then the identity is a compact operator from $W^{t,p}(\Omega )$ to $W^{s,p}(\Omega )$.

Proof

If $s\ge 0$ then $W^{t,p}(\Omega )\hookrightarrow W^{s,p}(\Omega )$ by [52, Chap. 2, §5.4, Lemma 5.4]. Let now $t\le 0$. Put $p'=p/(p-1)$. Since $W^{-s,p'}(\Omega )\hookrightarrow W^{-t,p'}(\Omega )$, we infer that $W^{t,p}(\Omega )\hookrightarrow W^{s,p}(\Omega )$. If $s<0<t$ then $W^{t,p}(\Omega )\hookrightarrow L^p(\Omega )\hookrightarrow W^{s,p}(\Omega )$.

Fix $\tau$ and $\theta$ such that $s<\theta<\tau <t$ and $\tau$ and $\theta$ are not integer. Then $W^{t,p}(\Omega )\hookrightarrow W^{\tau ,p}(\Omega )$ and $W^{\theta ,p}(\Omega )\hookrightarrow W^{s,p}(\Omega )$. Since $W^{\tau ,p}(\Omega )=B_\tau ^{p,p}(\Omega )\hookrightarrow B_\theta ^{p,p}(\Omega )=W^{\theta ,p}(\Omega )$ compactly by Proposition 2.1, we infer that the mapping $W^{t,p}(\Omega )\hookrightarrow W^{s,p}(\Omega )$ is compact. $\square$

Lemma 2.3

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with Lipschitz boundary, $1<p,q<\infty$ and $0<s<\infty$. Then $\partial _j :B_s^{p,q}(\Omega )\rightarrow B_{s-1}^{p,q}(\Omega )$ is a bounded operator.

Proof

Choose small positive $\epsilon$ such that $s-\epsilon$ and $s+\epsilon$ are not integer. Then

$$\begin{aligned}{} & {} \partial _j :B_{s+\epsilon }^{p,p}(\Omega )=W^{s+\epsilon ,p}(\Omega )\rightarrow W^{s+\epsilon -1,p}(\Omega )= B_{s+\epsilon -1}^{p,p}(\Omega ), \\{} & {} \partial _j :B_{s-\epsilon }^{p,p}(\Omega )=W^{s-\epsilon ,p}(\Omega )\rightarrow W^{s-\epsilon -1,p}(\Omega )= B_{s-\epsilon -1}^{p,p}(\Omega ) \end{aligned}$$

are bounded operators by [30, Theorem 1.4.4.6]. We now use interpolation. [65, Corollary 1.111] gives

$$\begin{aligned}{} & {} (B_{s-\epsilon }^{p,p}(\Omega ),B_{s+\epsilon }^{p,p}(\Omega ))_{1/2,q}=B_s^{p,q}(\Omega ), \\{} & {} (B_{s-1-\epsilon }^{p,p}(\Omega ),B_{s-1+\epsilon }^{p,p}(\Omega ))_{1/2,q}=B_{s-1}^{p,q}(\Omega ). \end{aligned}$$

Thus $\partial _j :B_s^{p,q}(\Omega )\rightarrow B_{s-1}^{p,q}(\Omega )$ is a bounded operator by [63, Lemma 22.3]. $\square$

3 Brinkman system

We suppose that $\Omega \subset {{\mathbb {R}}}^2$ is a bounded domain with connected boundary. The problem (1.1), (1.5) can be rewritten as

$$\begin{aligned}{} & {} -\Delta \textbf{u}+\lambda \textbf{u}+\nabla \rho =\textbf{F}, \quad \nabla \cdot \textbf{u}=G \quad \text {in}\ \Omega , \end{aligned}$$

(3.1a)

$$\begin{aligned}{} & {} \textbf{u}\cdot {\tau }=g, \quad \rho =h \quad \text {on}\ \partial \Omega . \end{aligned}$$

(3.1b)

We need the following auxiliary lemma:

Lemma 3.1

Let $X_1$, $X_2$, $Y_1$ and $Y_2$ be Banach spaces. Suppose that $X_1$ is a dense subset of $X_2$, $Y_1$ is a dense subset of $Y_2$, $X_1\hookrightarrow X_2$ and $Y_1\hookrightarrow Y_2$. Let $T_j :X_j \rightarrow Y_j$ for $j=1,2$ be bounded linear Fredholm operators with the same index such that $T_1 x=T_2 x$ for all $x\in X_1$. Then $T_1$ is an isomorphism if and only if $T_2$ is an isomorphism.

(See [47, Lemma 1.3.4].)

Theorem 3.2

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with connected boundary of class ${{\mathcal {C}}}^{k,1}$ with $k\in {\mathbb N}$. Let $\lambda \in {{\mathbb {C}}}\setminus (-\infty ,0)$, $1<q<\infty$ and $1/q<s<k-1$ with $s-1/q \not \in {{\mathbb {N}}}$. Suppose that $h\in W^{s-1/q,q}(\partial \Omega ;{{\mathbb {C}}})$, $g\in W^{s+1-1/q,q}(\partial \Omega ;{{\mathbb {C}}})$, $\textbf{F}\in W^{s-1,q}(\Omega ,{{\mathbb {C}}}^2)$ and $G\in W^{s,q}(\Omega ;{{\mathbb {C}}})$. Then there exists a unique solution $(\textbf{u},\rho )\in W^{s+1,q}(\Omega ,{{\mathbb {C}}}^2)\times W^{s,q}(\Omega ;{{\mathbb {C}}})$ of the problem (3.1). Moreover,

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}+\Vert \rho \Vert _{W^{s,q}(\Omega )}\le & {} c \big ( \Vert h\Vert _{W^{s-1/q,q}(\partial \Omega )} +\Vert g\Vert _{W^{s+1-1/q,q}(\partial \Omega )} \\{} & {} +\Vert \textbf{F}\Vert _{W^{s-1,q}(\Omega )}+\Vert G\Vert _{W^{s,q}(\Omega )}\big ) \end{aligned}$$

where c depends only on $\Omega$, $\lambda$, q and s. If $\lambda \ge 0$ and $\textbf{F}$, h, g, G are real, then $(\textbf{u},\rho )\in W^{s+1,q}(\Omega ,{{\mathbb {R}}}^2)\times W^{s,q}(\Omega )$.

Proof

Denote

$$\begin{aligned}{} & {} X^{s,q}:=W^{s+1,q}(\Omega ,{{\mathbb {C}}}^2)\times W^{s,q}(\Omega ,{{\mathbb {C}}}), \\{} & {} Y^{s,q}:=W^{s-1,q}(\Omega ,{{\mathbb {C}}}^2)\times W^{s,q}(\Omega ,{{\mathbb {C}}})\times W^{s+1-1/q,q}(\partial \Omega ,{\mathbb C})\times W^{s-1/q,q}(\partial \Omega ,{{\mathbb {C}}}). \end{aligned}$$

For a complex $\delta$ define the operator $S_\delta$ by

$$\begin{aligned} S_\delta (\textbf{u},\rho ):=[-\Delta \textbf{u}+\delta \textbf{u}-\nabla \rho ,\nabla \cdot \textbf{u}, \textbf{u}\cdot \tau ,\rho ] . \end{aligned}$$

(3.2)

Then $S_0:X^{s,q}\rightarrow Y^{s,q}$ is an isomorphism by [46, Theorem 5.5].

Let now $\lambda \ne 0$. Remark that

$$\begin{aligned} S_\lambda (\textbf{u},\rho )-S_0(\textbf{u},\rho )=[\lambda \textbf{u},0,0,0]. \end{aligned}$$

(3.3)

Since $W^{s+1,q}(\Omega )\hookrightarrow W^{s-1,q}(\Omega )$ compactly by Corollary 2.2, we infer that $S_\lambda :X^{s,q}\rightarrow Y^{s,q}$ is a Fredholm operator with index 0.

Suppose first that $q\ge 2$. Let $(\textbf{u},\rho )\in X^{s,q}$ and $S_\lambda (\textbf{u},\rho )=0$. Since $\nabla \cdot \textbf{u}=0$ we obtain

$$\begin{aligned} \Delta \rho =\nabla \cdot \nabla \rho =\nabla \cdot (\Delta \textbf{u}-\lambda \textbf{u})= \Delta (\nabla \cdot \textbf{u})-\lambda \nabla \cdot \textbf{u}=0\quad \text {in}\ \Omega . \end{aligned}$$

Since $\rho =0$ on $\partial \Omega$ we have $\rho \equiv 0$. (See for example [46, Proposition 7.8].) We now show that there exists $\psi \in L^1(\Omega ;{{\mathbb {C}}})$ such that $\textbf{u}=(\partial _2 \psi ,-\partial _1 \psi )$. Fix $z\in \Omega$. For $x\in \Omega$ choose a piece-wise smooth functions $\Phi _1$ ,$\Phi _2$ on $\langle t_1,t_2\rangle$ such that for $\Phi :=(\Phi _1 ,\Phi _2)$ we have $\Phi :\langle t_1,t_2\rangle \rightarrow \Omega$ and $\Phi (t_1)=z$, $\Phi (t_2)=x$. Put

$$\begin{aligned} \psi (x):=\int _{t_1}^{t_2} [\Phi _2'(t)u_1(\Phi (t))-\Phi _1 '(t)u_2(\Phi (t))]\ {\mathrm d}t. \end{aligned}$$

We show that $\psi$ does not depend on the choice of $\Phi$. Let $\Phi =(\Phi _1 ,\Phi _2): \langle t_1,t_2\rangle \rightarrow \Omega$ and ${\tilde{\Phi }} =({\tilde{\Phi }}_1 ,{\tilde{\Phi }}_2):\langle {\tilde{t}}_1, {\tilde{t}}_2\rangle \rightarrow \Omega$ be two such vector functions. Denote $\Gamma :=\{ \Phi (t); t\in \langle t_1, t_2\rangle \}$ and ${\tilde{\Gamma }} :=\{ {\tilde{\Phi }} (t); t\in \langle {\tilde{t}}_1,{\tilde{t}}_2\rangle \}$. Suppose moreover that $\Gamma \cap \{ {\tilde{\Phi }} (t); t\in ({\tilde{t}}_1,{\tilde{t}}_2)\} =\emptyset$ and ${\tilde{\Gamma }} \cap \{ \Phi (t); t\in (t_1,t_2)\} =\emptyset$. Let $G\subset {\mathbb R}^2$ be a domain with boundary $\Gamma \cup {\tilde{\Gamma }}$. Since $\nabla \cdot \textbf{u}=0$, the divergence theorem gives

$$\begin{aligned} 0= & {} \left| \int _G \nabla \cdot \textbf{u}\ {\mathrm d}y \right| =\left| \int _{\Gamma \cup {\tilde{\Gamma }} }{} \textbf{u}\cdot \textbf{n}^G \ {\mathrm d}\sigma \right| = \Big | \int _{t_1}^{t_2} [\Phi _2'(t)u_1(\Phi (t))-\Phi _1 '(t)u_2(\Phi (t))]\ {\mathrm d}t \\{} & {} -\int _{{\tilde{t}}_1}^{{\tilde{t}}_2} [{\tilde{\Phi }} _2'(t)u_1({\tilde{\Phi }} (t))- {\tilde{\Phi }}_1 '(t)u_2({\tilde{\Phi }} (t))]\ {\mathrm d}t\Big | . \end{aligned}$$

This forces that $\psi$ does not depend on the choice of $\Phi$. Since $u_j\in W^{s+1,q}(\Omega ) \hookrightarrow {\mathcal C}({\overline{\Omega }} )$ by [28, Satz 9.38] and [28, Satz 6.40], we deduce that $\psi \in L^\infty (\Omega ;{\mathbb C}^2)$. Clearly,

$$\begin{aligned}{} & {} \partial _1 \psi (x)=\lim _{\delta \rightarrow 0}\frac{1}{\delta }\int _0^\delta [-u_2(x_1+t,x_2)]\ {\mathrm d}t=-u_2(x), \\{} & {} \partial _1 \psi (x)=\lim _{\delta \rightarrow 0}\frac{1}{\delta }\int _0^\delta [u_1(x_1,x_2+t)]\ {\mathrm d}t=u_1(x). \end{aligned}$$

As $\rho \equiv 0$ we have $-\Delta \textbf{u}+\lambda \textbf{u}=0$. Hence $-\Delta \partial _j \psi +\lambda \partial _j \psi =0$ for $j=1,2$. Since $\nabla (-\Delta \psi +\lambda \psi )=0$, there is a constant $c_1$ such that $-\Delta \psi +\lambda \psi \equiv c_1$. Put $\varphi :=\psi -c_1/\lambda$. Then $-\Delta \varphi +\lambda \varphi \equiv 0$ and $\textbf{u}= (\partial _2 \varphi ,-\partial _1 \varphi )$. Since $\textbf{u}\in W^{s+1,q}(\Omega ,{{\mathbb {C}}}^2)$, we deduce that $\varphi \in W^{s+2,q}(\Omega ;{{\mathbb {C}}})$. As $\partial \varphi /\partial n =\tau \cdot \textbf{u}=0$, Green’s formula forces

$$\begin{aligned} 0=\int _{\partial \Omega }{\overline{\varphi }} \cdot \frac{\partial \varphi }{\partial n}\ {\mathrm d}\sigma =\int _\Omega (\overline{\varphi }\cdot \Delta \varphi +|\nabla \varphi |^2)\ {\mathrm d}x =\int _\Omega (\lambda |\varphi |^2+|\nabla \varphi |^2)\ {\mathrm d}x. \end{aligned}$$

Since $\lambda \in {{\mathbb {C}}}\setminus (-\infty ,0\rangle$, we infer that $\varphi \equiv 0$. Thus $\textbf{u}= (\partial _2 \varphi ,-\partial _1 \varphi )\equiv 0$. As $S_\lambda$ is a Fredholm operator with index 0 from $X^{s,q}$ to $Y^{s,q}$, we infer that $S_\lambda$ is an isomorphism from $X^{s,q}$ to $Y^{s,q}$.

Suppose now that $q<2$. Choose $t\in (s,k-1)$ such that $t\not \in {{\mathbb {N}}}$ and $t-1/2\not \in {{\mathbb {N}}}$. Since $t-2/2 >s-2/q$, $X^{t,2}$ is a dense subset of $X^{s,q}$ and $Y^{t,2}$ is a dense subset of the space $Y^{s,q}$. (See Corollary 2.2 and [28, Satz 6.38].) We have proved that $S_\lambda :X^{t,2}\rightarrow Y^{t,2}$ is an isomorphism. Lemma 3.1 forces that $S_\lambda :X^{s,q}\rightarrow Y^{s,q}$ is an isomorphism, too. $\square$

Theorem 3.3

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with connected boundary of class ${{\mathcal {C}}}^{k+2,\gamma }$, where $k\in {{\mathbb {N}}}$ and $0<\gamma <1$. Let $\lambda \in {\mathbb C}\setminus (-\infty ,0)$, $h\in {{\mathcal {C}}}^{k,\gamma }(\partial \Omega ;{{\mathbb {C}}})$, $g\in {{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega ; {{\mathbb {C}}})$, $\textbf{F}\in {{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} ,{{\mathbb {C}}}^2)$ and $G\in {{\mathcal {C}}}^{k,\gamma }(\overline{\Omega };{{\mathbb {C}}})$. Then there exists a unique solution $(\textbf{u},\rho )\in {{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} ,{{\mathbb {C}}}^2)\times {{\mathcal {C}}}^{k,\gamma }(\overline{\Omega };{{\mathbb {C}}})$ of the problem (3.1). Moreover,

$$\begin{aligned} \Vert \textbf{u}\Vert _{{{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} )}+\Vert \rho \Vert _{{{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )}\le & {} c \big ( \Vert h\Vert _{{{\mathcal {C}}}^{k,\gamma }(\partial \Omega )}+\Vert g\Vert _{{{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega )} \\{} & {} +\Vert \textbf{F}\Vert _{{{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} )}+\Vert G\Vert _{{{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )}\big ) \end{aligned}$$

where c depends only on $\Omega$, $\lambda$, k and $\gamma$. If $\lambda \ge 0$, $\textbf{F}\in {{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} ,{{\mathbb {R}}}^2)$ and h, g, G are real functions, then $(\textbf{u},\rho )\in {{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} , {{\mathbb {R}}}^2)\times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )$.

Proof

For a complex $\delta$ define the operator $S_\delta$ by (3.2). Then

$$\begin{aligned} S_0:{{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} ,{\mathbb R}^2)\times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} ) \rightarrow {{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} ,{{\mathbb {R}}}^2)\times {{\mathcal {C}}}^{k,\gamma }(\overline{\Omega }) \times {{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega )\times {{\mathcal {C}}}^{k,\gamma }(\partial \Omega ) \end{aligned}$$

is an isomorphism by [46, Theorem 6.2]. Remark that

$$\begin{aligned} S_\lambda (\textbf{u},\rho )-S_0(\textbf{u},\rho )=[\lambda \textbf{u},0,0,0]. \end{aligned}$$

Since ${{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} ,{\mathbb C}^2)\hookrightarrow {{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} ,{{\mathbb {C}}}^2)$ compactly (see [28, Satz 2.42] or [42, Remark 1.2.15]), we infer that $S_\lambda$ is a Fredholm operator with index 0 from the space X:=${{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} ,{{\mathbb {C}}}^2)\times {{\mathcal {C}}}^{k,\gamma } (\overline{\Omega };{{\mathbb {C}}})$ to $Y:={{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} ,{{\mathbb {C}}}^2) \times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} ;{{\mathbb {C}}})\times {{\mathcal {C}}}^{k+1,\gamma } (\partial \Omega ;{{\mathbb {C}}})\times {{\mathcal {C}}}^{k,\gamma }(\partial \Omega ;{{\mathbb {C}}})$. If $\lambda >0$ then

$$\begin{aligned} S_\lambda :{{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} ,{\mathbb R}^2)\times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} ) \rightarrow {{\mathcal {C}}}^{k-1,\gamma }(\overline{\Omega },{{\mathbb {R}}}^2)\times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} ) \times {{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega )\times {{\mathcal {C}}}^{k,\gamma }(\partial \Omega ) \end{aligned}$$

is a Fredholm operator with index 0.

Let $(\textbf{u},\rho )\in {{\mathcal {C}}}^{k+1,\gamma }(\overline{\Omega },{{\mathbb {C}}}^2)\times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} ;{{\mathbb {C}}})$ and $S_\lambda (\textbf{u},\rho )=0$. Then $(\textbf{u},\rho )\equiv 0$ by Theorem 3.2. Thus $S_\lambda :X\rightarrow Y$ is an isomorphism. If $\lambda \ge 0$ and $\textbf{F}$, g, h and G are real, then $\textbf{u}$, $\rho$ are real by Theorem 3.2. $\square$

Theorem 3.4

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with connected boundary of class ${{\mathcal {C}}}^{k,1}$ with $k\in {\mathbb N}$. Let $\lambda \in {{\mathbb {C}}}\setminus (-\infty ,0)$, $1<q, r<\infty$ and $1/q<s<k-1$ with $s-1/q\not \in {{\mathbb {N}}}$. If $g\in B_{s+1-1/q}^{q,r}(\partial \Omega ;{{\mathbb {C}}})$, $h\in B_{s-1/q}^{q,r}(\partial \Omega ;{{\mathbb {C}}})$, $\textbf{F}\in B_{s-1}^{q ,r} (\Omega ,{{\mathbb {C}}}^2)$ and $G\in B_s^{q ,r}(\Omega ;{{\mathbb {C}}})$, then there exists a unique solution $(\textbf{u},\rho )\in B_{s+1}^{q,r}(\Omega ,{{\mathbb {C}}}^2)\times B_s^{q,r}(\Omega ;{{\mathbb {C}}})$ of the Navier problem (3.1). Moreover,

$$\begin{aligned} \Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )}+\Vert \rho \Vert _{B_s^{q,r}(\Omega )}\le & {} c \big ( \Vert h\Vert _{B_{s-1/q}^{q,r}(\partial \Omega )}+\Vert g\Vert _{B_{s+1-1/q}^{q,r}(\partial \Omega )} \\{} & {} +\Vert \textbf{F}\Vert _{B_{s-1}^{q,r}(\Omega )}+\Vert G\Vert _{B_s^{q,r}(\Omega )}\big ) \end{aligned}$$

where c depends only on $\Omega$, $\lambda$, q, r and s. If $\lambda \ge 0$ and $\textbf{F}$, h, g, G are real, then $(\textbf{u},\rho )\in B_{s+1}^{q,r}(\Omega ,{{\mathbb {R}}}^2)\times B_s^{q,r}(\Omega )$.

Proof

For a complex $\delta$ define the operator $S_\delta$ by (3.2). $S_\delta$ is a bounded operator from $X:=B_{s+1}^{q,r}(\Omega ,{{\mathbb {C}}}^2)\times B_s^{q,r}(\Omega ;{{\mathbb {C}}})$ to $Y:= B_{s-1}^{q ,r} (\Omega ,{\mathbb C}^2)\times B_s^{q ,r}(\Omega ;{{\mathbb {C}}})\times B_{s+1-1/q}^{q,r}(\partial \Omega ;{{\mathbb {C}}})\times B_{s-1/q}^{q,r}(\partial \Omega ;{{\mathbb {C}}})$ by [36, Chapter VI, Theorem 1] and Lemma 2.3. [46, Theorem 5.4] gives that $S_0$ is an isomorphism from X to Y. Since

$$\begin{aligned} S_\lambda (\textbf{u},\rho )-S_0(\textbf{u},\rho )=[\lambda \textbf{u},0,0,0] \end{aligned}$$

and $B_{s+1}^{q,r}(\Omega )\hookrightarrow B_{s-1}^{q ,r} (\Omega )$ compactly by Proposition 2.1, we infer that $S_\lambda -S_0$ is compact and therefore $S_\lambda$ is a Fredholm operator with index 0 from X to Y.

Let $(\textbf{u},\rho )\in X$ and $S_\lambda (\textbf{u},\rho )=0$. Choose $\tau \in (1/q,s)$ such that $\tau$ and $\tau -1/q$ are not integer. Since $\textbf{u}\in B_{s+1}^{q,r}(\Omega ;{\mathbb C}^2)\subset B_{\tau +1}^{q,q}(\Omega ;{{\mathbb {C}}}^2)=W^{\tau +1,q}(\Omega ;{{\mathbb {C}}}^2)$ and $\rho \in B_s^{q,r}(\Omega ;{{\mathbb {C}}})\subset B_\tau ^{q,q}(\Omega ;{{\mathbb {C}}})=W^{\tau ,q}(\Omega ;{{\mathbb {C}}})$ by Proposition 2.1, Theorem 3.2 gives that $\textbf{u}\equiv 0$, $\rho \equiv 0$. So, $S_\lambda$ is an isomorphism from X to Y. $\square$

4 Darcy-Forchheimer-Brinkman system

This section is devoted to the Darcy-Forchheimer-Brinkman system (1.7) with the boundary condition (3.1b). We begin with proving some auxiliary results.

Lemma 4.1

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with Lipschitz boundary, $1<q,r<\infty$, $1/q<s\le 1$ and $a\in L^q (\Omega )$. Define

$$\begin{aligned} A_a(\textbf{u},\textbf{v}):=a|\textbf{u}|\textbf{v}. \end{aligned}$$

• (1) Then there exists a positive constant C such that $A_a(\textbf{u},\textbf{v})\in W^{s-1,q}(\Omega ;{{\mathbb {R}}}^2)$ for $\textbf{u},\textbf{v}\in W^{s+1,q}(\Omega ;{{\mathbb {R}}}^2)$ and

  $$\begin{aligned}{} & {} \Vert A_a(\textbf{u},\textbf{v})\Vert _{W^{s-1,q}(\Omega )}\le C\Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}\Vert \textbf{v}\Vert _{W^{s+1,q}(\Omega )}, \\{} & {} \Vert A_a(\textbf{u},\textbf{u})-A_a(\textbf{v},\textbf{v})\Vert _{W^{s-1,q}(\Omega )}\le C\Vert \textbf{u}-\textbf{v}\Vert _{W^{s+1,q}(\Omega )} \left( \Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}+\Vert \textbf{v}\Vert _{W^{s+1,q}(\Omega )}\right) . \end{aligned}$$

• (2) If $s<1$ then there exists a positive constant C such that $A_a(\textbf{u},\textbf{v})\in B_{s-1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$ for $\textbf{u},\textbf{v}\in B_{s+1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$ and

  $$\begin{aligned}{} & {} \Vert A_a(\textbf{u},\textbf{v})\Vert _{B_{s-1}^{q,r}(\Omega )}\le C\Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )}\Vert \textbf{v}\Vert _{B_{s+1}^{q,r}(\Omega )}, \\{} & {} \Vert A_a(\textbf{u},\textbf{u})-A_a(\textbf{v},\textbf{v})\Vert _{B_{s-1}^{q,r}(\Omega )}\le C\Vert \textbf{u}-\textbf{v}\Vert _{B_{s+1}^{q,r}(\Omega )} \left( \Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )}+\Vert \textbf{v}\Vert _{B_{s+1}^{q,r}(\Omega )}\right) . \end{aligned}$$

Proof

In the case 1) put $X:=W^{s+1,q}(\Omega ;{{\mathbb {R}}}^2)$, $Y:=W^{s-1,q}(\Omega ;{{\mathbb {R}}}^2)$. In the case 2) put $X:=B_{s+1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$, $Y:=B_{s-1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$. Since $X\hookrightarrow W^{1/q+1,q}(\Omega ;{\mathbb R}^2)=B_{1/q+1}^{q,q}(\Omega ;{{\mathbb {R}}}^2)\hookrightarrow {{\mathcal {C}}}({\overline{\Omega }} ;{{\mathbb {R}}}^2)$ by [52, Chap. 2, §5.4, Lemma 5.4] and [65, Proposition 4.6], there exists a positive constant $C_1$ such that

$$\begin{aligned} \Vert u\Vert _{L^\infty (\Omega )}\le C_1 \Vert u\Vert _X\qquad \forall u\in X. \end{aligned}$$

Thus

$$\begin{aligned}{} & {} \Vert A_a(\textbf{u},\textbf{v})\Vert _{L^q(\Omega )}\le 4C_1^2 \Vert a\Vert _{L^q(\Omega )}\Vert \textbf{u}\Vert _X\Vert \textbf{v}\Vert _X, \\{} & {} \Vert A_a(\textbf{u},\textbf{u})-A_a(\textbf{v},\textbf{v})\Vert _{L^q(\Omega )}\le \Vert A_a(\textbf{u},\textbf{u}-\textbf{v})\Vert _{L^q(\Omega )} +\Vert A_a(\textbf{u},\textbf{v})-A_a(\textbf{v},\textbf{v})\Vert _{L^q(\Omega )} \\{} & {} \quad \le \Vert a\Vert _{L^q(\Omega )}4C_1^2\Vert \textbf{u}\Vert _X\Vert \textbf{u}-\textbf{v}\Vert _X +\Vert a\Vert _{L^q(\Omega )}4C_1^2\Vert \textbf{v}\Vert _X\Vert \textbf{u}-\textbf{v}\Vert _X. \end{aligned}$$

Suppose now that $s<1$. In the case 1) put $Z:=W^{(s-1)/2,q}(\Omega ;{{\mathbb {R}}}^2)$, in the case 2) put $Z:=B_{(s-1)/2}^{q,q}(\Omega ;{{\mathbb {R}}}^2)$. Remark that $L^q_0(\Omega )=L^q(\Omega )$ where $L_s^q(\Omega )$ denotes Bessel spaces. According to [64, §4.6.2, Theorem] there exists a constant $C_2$ such that

$$\begin{aligned} \Vert f\Vert _Z\le C_2\Vert f\Vert _{L^q(\Omega )} \qquad \forall f\in L^q(\Omega ;{{\mathbb {R}}}^2). \end{aligned}$$

According to [64, §4.6.1, Theorem] and [52, Chap. 2, §5.4, Lemma 5.4] there exists a constant $C_3$ such that

$$\begin{aligned} \Vert f\Vert _Y\le C_3\Vert f\Vert _Z \qquad \forall f\in Z. \end{aligned}$$

Hence

$$\begin{aligned}{} & {} \Vert A_a(\textbf{u},\textbf{v})\Vert _Y\le 4C_1^2 C_2C_3\Vert a\Vert _{L^q(\Omega )}\Vert \textbf{u}\Vert _X\Vert \textbf{v}\Vert _X, \\{} & {} \Vert A_a(\textbf{u},\textbf{u})-A_a(\textbf{v},\textbf{v})\Vert _Y \le \Vert a\Vert _{L^q(\Omega )}4C_1^2C_2C_3(\Vert \textbf{u}\Vert _X+\Vert \textbf{v}\Vert _X)\Vert \textbf{u}-\textbf{v}\Vert _X. \end{aligned}$$

$\square$

Lemma 4.2

Let $\Omega \subset {{\mathbb {R}}}^m$ be a bounded domain with Lipschitz boundary. Let $1<p<\infty$ and $0<s<\infty$. Then there exists a bounded linear operator $E:W^{s,p}(\Omega )\rightarrow W^{s,p}({{\mathbb {R}}}^m)$ such that $Ef=f$ on $\Omega$.

Proof

For s integer see [60, Chapter VI, §3, Theorem 5]. If s is not integer then the lemma is a consequence of [63, Lemma 36.1] and [65, Theorem 1.105]. $\square$

Lemma 4.3

Let $\Omega \subset {{\mathbb {R}}}^m$ be a bounded domain with Lipschitz boundary. Let $0<s(1),s(2)<\infty$, $\min (s(1),s(2))\ge s>-\infty$ and $1<p<\infty$. Suppose that $s(1)+s(2)-s>m/p$. Then there exists a positive constant C such that

$$\begin{aligned} \Vert fg\Vert _{W^{s,p}(\Omega )}\le C\Vert f\Vert _{W^{s(1),p}(\Omega )}\Vert g\Vert _{W^{s(2),p}(\Omega )} \end{aligned}$$

for all $f\in W^{s(1),p}(\Omega )$, $g\in W^{s(2),p}(\Omega )$.

Proof

Choose a bounded domain $\omega \subset {{\mathbb {R}}}^m$ with smooth boundary such that ${\overline{\Omega }} \subset \omega$. According to Lemma 4.2 there exists an extension operator $E_j:W^{s(j),p}(\Omega )\rightarrow W^{s(j),p}({{\mathbb {R}}}^m)$ and a positive constant $C_1$ such that

$$\begin{aligned} \Vert E_jf\Vert _{W^{s(j),p}({{\mathbb {R}}}^m)}\le C_1\Vert f\Vert _{W^{s(j),p}(\Omega )}\qquad \forall f\in W^{s(j),p}(\Omega ). \end{aligned}$$

Fix $\varphi \in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^m)$ supported in $\omega$ such that $\varphi =1$ on $\Omega$. Then there exists a positive constant $C_2$ such that

$$\begin{aligned} \Vert \varphi f\Vert _{W^{s(j),p}({{\mathbb {R}}}^m)}\le C_2\Vert f\Vert _{W^{s(j),p}({{\mathbb {R}}}^m)} \qquad \forall f\in W^{s(j),p}({{\mathbb {R}}}^m). \end{aligned}$$

We can choose an m-dimensional smooth closed manifold M such that ${\overline{\omega }} \subset M$. According to [35, Lemma 28] there exists a positive constant $C_3$ such that

$$\begin{aligned} \Vert fg\Vert _{W^{s,p}(M)}\le C_3\Vert f\Vert _{W^{s(1),p}(M)}\Vert g\Vert _{W^{s(2),p}(M)} \end{aligned}$$

for all $f\in W^{s(1),p}(M)$ and $g\in W^{s(2),p}(M)$. Define $Ef:=f$ in $\omega$, $Ef:=0$ in $M\setminus \omega$. If $f\in W^{s(j),p}({{\mathbb {R}}}^m)$ then $E(\varphi f)\in W^{s(j),p}(M)$. Therefore

$$\begin{aligned}{} & {} \Vert fg\Vert _{W^{s,p}(\Omega )}\le \Vert \{ E(\varphi E_1f)\}\{ E(\varphi E_2g)\} \Vert _{W^{s,p}(M)} \\{} & {} \quad \le C_3\Vert E(\varphi E_1f)\Vert _{W^{s(1),p}(M)} \Vert E(\varphi E_2g)\Vert _{W^{s(2),p}(M)} \\{} & {} \quad \le C_3 C_1^2 C_2^2\Vert f\Vert _{W^{s(1),p}(\Omega )}\Vert g\Vert _{W^{s(2),p}(\Omega )} \end{aligned}$$

for all $f\in W^{s(1),p}(\Omega )$, $g\in W^{s(2),p}(\Omega )$. $\square$

Lemma 4.4

Let $\Omega \subset {{\mathbb {R}}}^m$ be a bounded domain with Lipschitz boundary. Let $0<s(1),s(2)<\infty$, $\min (s(1),s(2))>s>-\infty$ and $1<p,q<\infty$. Suppose that $s(1)+s(2)-s>m/p$. Then there exists a positive constant C such that

$$\begin{aligned} \Vert fg\Vert _{B_s^{p,q}(\Omega )}\le C\Vert f\Vert _{B_{s(1)}^{p,q}(\Omega )}\Vert g\Vert _{B_{s(2)}^{p,q}(\Omega )} \end{aligned}$$

for all $f\in B_{s(1)}^{p,q}(\Omega )$, $g\in B_{s(2)}^{p,q}(\Omega )$.

Proof

Choose $\epsilon \in (0,\infty )$ such that $s(j)-\epsilon >s$, $s(1)+s(2)-s-3\epsilon >m/p$ and $s(j)-\epsilon$, $s-\epsilon$, $s(j)+\epsilon$, $s+\epsilon$ are not integer. According to Lemma 4.3 there exists a positive constant $C_1$ such that

$$\begin{aligned} \Vert fg\Vert _{W^{t,p}(\Omega )}\le C_1\Vert f\Vert _{W^{t(1),p}(\Omega )}\Vert g\Vert _{W^{t(2),p}(\Omega )} \end{aligned}$$

for all $f\in W^{t(1),p}(\Omega )$, $g\in W^{t(2),p}(\Omega )$ with $t(j)\in \{ s(j)-\epsilon , s(j)+\epsilon \}$, $t\in \{ s-\epsilon ,s+\epsilon \}$. We use the real interpolation.

$$\begin{aligned}{} & {} (W^{s(j)-\epsilon ,p}(\Omega ),W^{s(j)+\epsilon ,p}(\Omega ))_{1/2,q}\nonumber \\{} & {} \quad =(B_{s(j)-\epsilon }^{p,p}(\Omega ),B_{s(j)+\epsilon }^{p,p}(\Omega ))_{1/2,q}=B_{s(j)}^{p,q}(\Omega ), \end{aligned}$$

(4.1)

$$\begin{aligned}{} & {} (W^{s-\epsilon ,p}(\Omega ),W^{s+\epsilon ,p}(\Omega ))_{1/2,q}=B_s^{p,q}(\Omega ) \end{aligned}$$

(4.2)

by [64, §4.3.1, Theorem 2]. Fix $g\in W^{t(2),p}(\Omega )$ and define $G_g(f):=fg$. According to (4.1), (4.2) and [63, Lemma 22.3]

$$\begin{aligned} \Vert G_g(f)\Vert _{B_s^{p,q}(\Omega )}\le C_1 \Vert g\Vert _{W^{t(2),p}(\Omega )}\Vert f\Vert _{B_{s(1)}^{p,q}(\Omega )}. \end{aligned}$$

We now use the real interpolation with respect to g. According to (4.1), (4.2) and [63, Lemma 22.3]

$$\begin{aligned} \Vert fg\Vert _{B_s^{p,q}(\Omega )}\le C_1 \Vert f\Vert _{B_{s(1)}^{p,q}(\Omega )}\Vert g\Vert _{B_{s(2)}^{p,q}(\Omega )}. \end{aligned}$$

$\square$

Lemma 4.5

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with Lipschitz boundary. Let $1<q,r<\infty$ and $1/q<s<\infty$. For a given function b define

$$\begin{aligned} B_b(\textbf{u},\textbf{v}):=b(\textbf{u}\cdot \nabla )\textbf{v}. \end{aligned}$$

• (1) If $b\in W^{s,q}(\Omega )$ then there exists a positive constant C such that $B_b(\textbf{u},\textbf{v})\in W^{s-1,q}(\Omega ;{{\mathbb {R}}}^2)$ for $\textbf{u},\textbf{v}\in W^{s+1,q}(\Omega ;{{\mathbb {R}}}^2)$ and

  $$\begin{aligned}{} & {} \Vert B_b(\textbf{u},\textbf{v})\Vert _{W^{s-1,q}(\Omega )}\le C\Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}\Vert \textbf{v}\Vert _{W^{s+1,q}(\Omega )}, \\{} & {} \Vert B_b(\textbf{u},\textbf{u})-B_b(\textbf{v},\textbf{v})\Vert _{W^{s-1,q}(\Omega )}\nonumber \\{} & {} \quad \le C\Vert \textbf{u}-\textbf{v}\Vert _{W^{s+1,q}(\Omega )} \left( \Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}+\Vert \textbf{v}\Vert _{W^{s+1,q}(\Omega )}\right) . \end{aligned}$$

• (2) If $b\in B_s^{q,r}(\Omega )$ then there exists a positive constant C such that $B_b(\textbf{u},\textbf{v}) \in B_{s-1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$ for $\textbf{u},\textbf{v}\in B_{s+1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$ and

  $$\begin{aligned}{} & {} \Vert B_b(\textbf{u},\textbf{v})\Vert _{B_{s-1}^{q,r}(\Omega )}\le C\Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )} \Vert \textbf{v}\Vert _{B_{s+1}^{q,r}(\Omega )}, \\{} & {} \Vert B_b(\textbf{u},\textbf{u})-B_b(\textbf{v},\textbf{v})\Vert _{B_{s-1}^{q,r}(\Omega )}\le C\Vert \textbf{u}-\textbf{v}\Vert _{B_{s+1}^{q,r}(\Omega )} \left( \Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )}+\Vert \textbf{v}\Vert _{B_{s+1}^{q,r}(\Omega )}\right) . \end{aligned}$$

Proof

Choose $\epsilon \in (0,1-1/q)$. In the case (1) put $X:=W^{s+1,q}(\Omega )$, $Y:=W^{s,q}(\Omega )$, $W:=W^{s-\epsilon ,q}(\Omega )$, $Z:=W^{s-1,q}(\Omega )$, ${\mathcal X}:=W^{s+1,q}(\Omega ;{{\mathbb {R}}}^2)$, ${\mathcal W}:=W^{s-\epsilon ,q}(\Omega ;{{\mathbb {R}}}^2)$, ${\mathcal Z}:=W^{s-1,q}(\Omega ;{{\mathbb {R}}}^2)$. In the case (2) put $X:=B_{s+1}^{q,r}(\Omega )$, $Y:=B_s^{q,r}(\Omega )$, $W:=B_{s-\epsilon }^{q,r}(\Omega )$, $Z:=B_{s-1}^{q,r}(\Omega )$, ${{\mathcal {X}}}:=B_{s+1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$, ${{\mathcal {W}}}:=B_{s-\epsilon }^{q,r} (\Omega ;{{\mathbb {R}}}^2)$, ${{\mathcal {Z}}}:=B_{s-1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$. Since $(s+1)+s-(s-\epsilon )=s+1-\epsilon >1/q+1-(1-1/q)=2/q$, Lemma 4.3 and Lemma 4.4 force that there exists a positive constant $C_1$ such that

$$\begin{aligned} \Vert fg\Vert _W\le C_1\Vert f\Vert _X\Vert g\Vert _Y \qquad \forall f\in X, g\in Y. \end{aligned}$$

Since $s+(s-\epsilon )-(s-1)=s+1-\epsilon >2/q$, Lemma 4.3 and Lemma 4.4 give that there exists a positive constant $C_2$ such that

$$\begin{aligned} \Vert fg\Vert _Z\le C_2\Vert f\Vert _Y\Vert g\Vert _W \qquad \forall f\in Y, g\in W. \end{aligned}$$

If $\textbf{u},\textbf{v}\in {{\mathcal {X}}}$ then

$$\begin{aligned} \Vert (\textbf{u}\cdot \nabla )\textbf{v}\Vert _{{\mathcal {W}}}\le 2C_1\Vert \textbf{u}\Vert _{{\mathcal {X}}}\Vert \textbf{v}\Vert _{{\mathcal {X}}}. \end{aligned}$$

So,

$$\begin{aligned}{} & {} \Vert B_b(\textbf{u},\textbf{v})\Vert _{{\mathcal {Z}}}\le 2C_2 \Vert b\Vert _Y\Vert (\textbf{u}\cdot \nabla )\textbf{v}\Vert _{{\mathcal {W}}} \le 4C_1C_2 \Vert b\Vert _Y\Vert \textbf{u}\Vert _{{\mathcal {X}}}\Vert \textbf{v}\Vert _{{\mathcal {X}}}, \\{} & {} \Vert B_b(\textbf{u},\textbf{u})-B_b(\textbf{v},\textbf{v})\Vert _{{\mathcal {Z}}}\le \Vert B_b(\textbf{u}-\textbf{v},\textbf{u})\Vert _{{\mathcal {Z}}}+\Vert B_b(\textbf{v},\textbf{u}-\textbf{v})\Vert _{{\mathcal {Z}}} \\{} & {} \quad \le 4C_1C_2\Vert b\Vert _Y\Vert \textbf{u}-\textbf{v}\Vert _{{\mathcal {X}}}\left( \Vert \textbf{u}\Vert _{{\mathcal {X}}}+\Vert \textbf{v}\Vert _{{\mathcal {X}}}\right) . \end{aligned}$$

$\square$

Theorem 4.6

Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded domain with connected Lipschitz boundary and $0\le \lambda <\infty$.

• (1) Let $\partial \Omega$ be of class ${{\mathcal {C}}}^{k+2,\gamma }$ where $k\in {{\mathbb {N}}}$ and $0<\gamma <1$. Let $a\in {{\mathcal {C}}}^{0,\gamma }({\overline{\Omega }} )$ and $b\in {{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} )$. If $k\ne 1$ suppose that $a=0$. Then there exist $\delta ,\epsilon , C\in (0,\infty )$ such that the following holds: If $h\in {\mathcal C}^{k,\gamma }(\partial \Omega )$, $g\in {{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega )$, $\textbf{f}\in {{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} ;{{\mathbb {R}}}^2)$ and $G \in {\mathcal C}^{k,\gamma }({\overline{\Omega }} )$ satisfy

  $$\begin{aligned} \Vert h\Vert _{{{\mathcal {C}}}^{k,\gamma }(\partial \Omega )}+\Vert g\Vert _{{{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega )}+ \Vert \textbf{f}\Vert _{{{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} )}+\Vert G \Vert _{{{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )}<\delta , \end{aligned}$$

  then there exists a unique solution $(\textbf{u},\rho )\in {\mathcal C}^{k+1,\gamma }({\overline{\Omega }} ; {{\mathbb {R}}}^2)\times {{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )$ of (1.7), (3.1b) satisfying $\Vert \textbf{u}\Vert _{{{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} )}<\epsilon$. Moreover

  $$\begin{aligned}{} & {} \Vert \textbf{u}\Vert _{{{\mathcal {C}}}^{k+1,\gamma }({\overline{\Omega }} )}+\Vert \rho \Vert _{{{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )} \\{} & {} \quad \le C\left( \Vert h\Vert _{{{\mathcal {C}}}^{k,\gamma }(\partial \Omega )}+\Vert g\Vert _{{{\mathcal {C}}}^{k+1,\gamma }(\partial \Omega )}+ \Vert \textbf{f}\Vert _{{{\mathcal {C}}}^{k-1,\gamma }({\overline{\Omega }} )}+\Vert G \Vert _{{{\mathcal {C}}}^{k,\gamma }({\overline{\Omega }} )}\right) . \end{aligned}$$

• (2) Suppose that $\partial \Omega$ is of class ${{\mathcal {C}}}^{k,1}$ with $k\in {{\mathbb {N}}}$. Let $1<q<\infty$ and $1/q<s<k-1$ with $s-1/q \not \in {{\mathbb {N}}}$. Let $a\in L^q (\Omega )$ and $b\in W^{s,q}(\Omega )$. If $s>1$ suppose that $a\equiv 0$. Then there exist $\delta ,\epsilon , C\in (0,\infty )$ such that the following holds: If $h\in W^{s-1/q,q}(\partial \Omega )$, $g\in W^{s+1-1/q,q}(\partial \Omega )$, $\textbf{f}\in W^{s-1,q}(\Omega ;{{\mathbb {R}}}^2)$ and $G \in W^{s,q}(\Omega )$ satisfy

  $$\begin{aligned} \Vert h\Vert _{W^{s-1/q,q}(\partial \Omega )}+\Vert g\Vert _{W^{s+1-1/q,q}(\partial \Omega )}+ \Vert \textbf{f}\Vert _{W^{s-1,q}(\Omega )}+\Vert G \Vert _{W^{s,q}(\Omega )}<\delta , \end{aligned}$$

  then there exists a unique solution $(\textbf{u},\rho )\in W^{s+1,q}(\Omega ;{{\mathbb {R}}}^2)\times W^{s,q}(\Omega )$ of (1.7), (3.1b) satisfying $\Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}<\epsilon$. Moreover,

  $$\begin{aligned}{} & {} \Vert \textbf{u}\Vert _{W^{s+1,q}(\Omega )}+\Vert \rho \Vert _{W^{s,q}(\Omega )} \\{} & {} \quad \le C\left( \Vert h\Vert _{W^{s-1/q,q}(\partial \Omega )} +\Vert g\Vert _{W^{s+1-1/q,q}(\partial \Omega )}+ \Vert \textbf{f}\Vert _{W^{s-1,q}(\Omega )}+\Vert G \Vert _{W^{s,q}(\Omega )}\right) . \end{aligned}$$

• (3) Suppose that $\partial \Omega$ is of class ${{\mathcal {C}}}^{k,1}$ with $k\in {{\mathbb {N}}}$. Let $1<q,r<\infty$ and $1/q<s<k-1$ with $s-1/q \not \in {{\mathbb {N}}}$. Let $a\in L^q (\Omega )$ and $b\in B_s^{q,r}(\Omega )$. If $s\ge 1$ suppose that $a\equiv 0$. Then there exist $\delta ,\epsilon , C\in (0,\infty )$ such that the following holds: If $h\in B_{s-1/q}^{q,r}(\partial \Omega )$, $g\in B_{s+1-1/q}^{q,r}(\partial \Omega )$, $\textbf{f}\in B_{s-1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$ and $G \in B_s^{q,r}(\Omega )$ satisfy

  $$\begin{aligned} \Vert h\Vert _{B_{s-1/q}^{q,r}(\partial \Omega )}+\Vert g\Vert _{B_{s+1-1/q}^{q,r}(\partial \Omega )}+ \Vert \textbf{f}\Vert _{B_{s-1}^{q,r}(\Omega )}+\Vert G \Vert _{B_s^{q,r}(\Omega )}<\delta , \end{aligned}$$

  then there exists a unique solution $(\textbf{u},\rho )\in B_{s+1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)\times B_s^{q,r}(\Omega )$ of (1.7), (3.1b) satisfying $\Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )}<\epsilon$. Moreover,

  $$\begin{aligned}{} & {} \Vert \textbf{u}\Vert _{B_{s+1}^{q,r}(\Omega )}+\Vert \rho \Vert _{B_s^{q,r}(\Omega )} \\{} & {} \quad \le C\left( \Vert h\Vert _{B_{s-1/q}^{q,r}(\partial \Omega )} +\Vert g\Vert _{B_{s+1-1/q}^{q,r}(\partial \Omega )}+ \Vert \textbf{f}\Vert _{B_{s-1}^{q,r}(\Omega )}+\Vert G \Vert _{B_s^{q,r}(\Omega )}\right) . \end{aligned}$$

Proof

In the case 1) put $U:={{\mathcal {C}}}^{k+1,\gamma }(\overline{\Omega };{{\mathbb {R}}}^2)$, $V:={{\mathcal {C}}}^{k,\gamma } ({\overline{\Omega }} )$, $W:={{\mathcal {C}}}^{k-1}({\overline{\Omega }} ;{{\mathbb {R}}}^2)$, $Y:={{\mathcal {C}}}^{k+1,\gamma } (\partial \Omega )$, $Z:= {{\mathcal {C}}}^{k,\gamma }(\partial \Omega )$. In the case 2) put $U:=W^{s+1,q}(\Omega ;{{\mathbb {R}}}^2)$, $V:=W^{s,q}(\Omega )$, $W:=W^{s-1,q}(\Omega ;{{\mathbb {R}}}^2)$, $Y:=W^{s+1-1/q,q}(\partial \Omega )$, $Z:=W^{s-1/q,q}(\partial \Omega )$. In the case 3) put $U:=B_{s+1}^{q,r}(\Omega ;{{\mathbb {R}}}^2)$, $V:=B_s^{q,r}(\Omega )$, $W:=B_{s-1}^{q,r}(\Omega ; {{\mathbb {R}}}^2)$, $Y:=B_{s+1-1/q}^{q,r}(\partial \Omega )$, $Z:=B_{s-1/q}^{q,r}(\partial \Omega )$.

Define

$$\begin{aligned} L(\textbf{u}):=a|\textbf{u}|\textbf{u}+b(\textbf{u}\cdot \nabla )\textbf{u}. \end{aligned}$$

According to [48, Lemma 3.1], [48, Lemma 3.2], Lemma 4.1 and Lemma 4.5 there exists a constant $C_1$ such that

$$\begin{aligned}{} & {} \Vert L\textbf{u}\Vert _W\le C_1\Vert \textbf{u}\Vert _U^2, \\{} & {} \Vert L\textbf{u}-L\textbf{v}\Vert _W\le C_1\Vert \textbf{u}-\textbf{v}\Vert _U\left( \Vert \textbf{u}\Vert _U+\Vert \textbf{v}\Vert _U\right) . \end{aligned}$$

By Theorem 3.2, Theorem 3.3 and Theorem 3.4 there exists a constant $C_2$ such that the following holds: For each $h\in Z$, $g\in Y$, $\textbf{F}\in W$ and $G\in V$ there is a unique solution $(\textbf{u},\rho )\in U\times V$ of the problem (3.1) and

$$\begin{aligned} \Vert \textbf{u}\Vert _U+\Vert \rho \Vert _V\le C_2 \big ( \Vert h\Vert _Z+\Vert g\Vert _Y+\Vert \textbf{F}\Vert _W+\Vert G\Vert _V\big ) . \end{aligned}$$

Put

$$\begin{aligned} \epsilon :=\frac{1}{4(C_1+1)(C_2+1)}, \qquad \delta :=\frac{\epsilon }{2(C_2+1)} . \end{aligned}$$

If $(\textbf{u},\rho )\in U\times V$ is a solution of (1.7), (3.1b) with $\Vert \textbf{u}\Vert _U<\epsilon$ and $(\tilde{\textbf{u}},{\tilde{\rho }} )\in U\times V$ is a solution of

$$\begin{aligned}{} & {} -\Delta \tilde{\textbf{u}}+\lambda \tilde{\textbf{u}}+a|\tilde{\textbf{u}}|\tilde{\textbf{u}} +b(\tilde{\textbf{u}}\cdot \nabla )\tilde{\textbf{u}}+\nabla {\tilde{\rho }}=\tilde{\textbf{f}}, \ \nabla \cdot \tilde{\textbf{u}}={\tilde{G}} \quad \text {in}\ \Omega \\{} & {} \tilde{\textbf{u}}\cdot {\tau }={\tilde{g}}, \quad \tilde{\rho }={\tilde{h}} \quad \text {on}\ \partial \Omega \end{aligned}$$

with $\Vert \tilde{\textbf{u}}\Vert _U<\epsilon$, then

$$\begin{aligned}{} & {} \Vert \textbf{u}-\tilde{\textbf{u}}\Vert _U+\Vert \rho -{\tilde{\rho }} \Vert _V\le C_2 \big ( \Vert h-{\tilde{h}}\Vert _Z+\Vert g-{\tilde{g}}\Vert _Y\\ {}{} & {} \qquad +\Vert \textbf{f}-\tilde{\textbf{f}}\Vert _W+\Vert G-{\tilde{G}}\Vert _V+\Vert L\textbf{u}-L\tilde{\textbf{u}}\Vert _W\big ) \\{} & {} \quad \le C_2 \big ( \Vert h-{\tilde{h}}\Vert _Z+\Vert g-{\tilde{g}}\Vert _Y+\Vert \textbf{f}-\tilde{\textbf{f}}\Vert _W +\Vert G-{\tilde{G}}\Vert _V+2\epsilon C_1\Vert \textbf{u}-\tilde{\textbf{u}}\Vert _U\big ) . \end{aligned}$$

Since $2C_1C_2 \epsilon <1/2$ we get subtracting $2\epsilon C_1C_2\Vert \textbf{u}-\tilde{\textbf{u}}\Vert _U$ from the both sides

$$\begin{aligned} \Vert \textbf{u}-\tilde{\textbf{u}}\Vert _U+\Vert \rho -{\tilde{\rho }} \Vert _V\le 2C_2 \big ( \Vert h-{\tilde{h}}\Vert _Z+\Vert g-{\tilde{g}}\Vert _Y +\Vert \textbf{f}-\tilde{\textbf{f}}\Vert _W+\Vert G-{\tilde{G}}\Vert _V\big ) . \end{aligned}$$

Therefore a solution of (1.7), (3.1b) satisfying $\Vert \textbf{u}\Vert _U<\epsilon$ is unique. For $\tilde{\textbf{u}}\equiv 0$, ${\tilde{\rho }} \equiv 0$, $\tilde{\textbf{f}}\equiv 0$, ${\tilde{g}}\equiv 0$, ${\tilde{G}}\equiv 0$ and ${\tilde{h}}\equiv 0$ we obtain the estimate

$$\begin{aligned} \Vert \textbf{u}\Vert _U+\Vert \rho \Vert _V\le 2C_2 \big ( \Vert h\Vert _Z+\Vert g\Vert _Y+\Vert \textbf{f}\Vert _W+\Vert G\Vert _V\big ) . \end{aligned}$$

Denote $X:=\{ \textbf{v}\in U;\Vert \textbf{v}\Vert _U\le \epsilon \}$. Choose $h\in Z$, $g\in Y$, $\textbf{f}\in W$ and $G \in V$ satisfying

$$\begin{aligned} \Vert h\Vert _Z+\Vert g\Vert _Y+\Vert \textbf{f}\Vert _W+\Vert G \Vert _V<\delta . \end{aligned}$$

For a fixed $\textbf{v}\in X$ there exists a unique solution $(\textbf{u}^\textbf{v},\rho ^\textbf{v})\in U\times V$ of (3.1) with $\textbf{F}=\textbf{f}-L\textbf{v}$. (See Theorem 3.2, Theorem 3.3 and Theorem 3.4.) Remark that $(\textbf{u},\rho )$ is a solution of (1.7), (3.1b) if $(\textbf{u},\rho )$ is a solution of (3.1) with $\textbf{F}=\textbf{f}-L\textbf{u}$. We have

$$\begin{aligned} \Vert \textbf{u}^\textbf{v}\Vert _U\le C_2 \big ( \Vert h\Vert _Z+\Vert g\Vert _Y+\Vert \textbf{f}\Vert _W+\Vert G \Vert _V+\Vert L\textbf{v}\Vert _W \big ) \le C_2 \delta +C_2C_1 \epsilon ^2 . \end{aligned}$$

Since $C_2 \delta +C_2 C_12 \epsilon ^2 <\epsilon$ we deduce that $\textbf{u}^\textbf{v}\in X$. If $\textbf{w}\in X$ then

$$\begin{aligned} \Vert \textbf{u}^\textbf{v}-\textbf{u}^\textbf{w}\Vert _U\le C_2\Vert L\textbf{v}-L\textbf{w}\Vert _W\le C_2 C_1 2\epsilon \Vert \textbf{v}-\textbf{w}\Vert _U. \end{aligned}$$

Since $C_2 C_1 2\epsilon <1$, Banach’s fixed point theorem ( [28, Satz 1.24]) gives that there exists $\textbf{v}\in X$ such that $\textbf{u}^\textbf{v}=\textbf{v}$. So, $(\textbf{u}^\textbf{v},\rho ^\textbf{v})$ is a solution of (1.7), (3.1b) in $U\times V$ satisfying $\Vert \textbf{v}\Vert _U<\epsilon$. $\square$