Generalized Navier–Stokes Equations with Non-Homogeneous Boundary Conditions

Abstract

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solutions. The global-in-time existence and uniqueness of a small-data strong solution is proved. For the proof of this result, we propose a new approach. Our approach is based on the operator treatment of the problem with the consequent application of a theorem on the local unique solvability of an operator equation involving an isomorphism between Banach spaces with continuously Fréchet differentiable perturbations.

1. Introduction

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^d$$(d=2,3)$, and fix $T>0$. In this paper, we are concerned with the generalized Navier–Stokes equations, which describe the unsteady motion of an incompressible fluid in $\Omega$:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t\mathit{u}+\delta \mathbf{div}(\mathit{u}\otimes \mathit{u})-{\mu }_0\Delta \mathit{u}-{\mu }_1\Delta {\partial }_t\mathit{u}-{\int }_0^t\eta (s,t)\Delta \mathit{u}(·,s)ds+\nabla p=\mathit{f}}\hfill & \mathrm{in} Q_T,\hfill \\ \nabla ·\mathit{u}=0\hfill & \mathrm{in} Q_T,\hfill \end{array}\right.$$

(1)

where $Q_T\stackrel{\mathrm{def}}{=}\Omega \times (0,T)$, $\mathit{u}$ is the flow velocity, p is the pressure, $\eta$ is the memory kernel, $\mathit{f}$ is the external forces field, ${\mu }_1$ and ${\mu }_0$ are non-negative material constants, $\delta \in \{0,1\}$, the symbol ⊗ denotes the tensor product of vectors, $\mathit{x}\otimes \mathit{y}\stackrel{\mathrm{def}}{=}{\left(x_i{y}_j\right)}_{i,j=1}^d$ for any $\mathit{x},\mathit{y}\in {\mathbb{R}}^d$, the symbol ∇ stands for the gradient with respect to the spatial variables $x_1,\dots ,x_d$, the differential operators $\Delta$ and div are defined as follows:

$$\Delta \mathit{w}\stackrel{\mathrm{def}}{=}(\nabla ·\nabla )\mathit{w}=\sum _{i=1}^d\frac{{\partial }^2\mathit{w}}{\partial x_i^2},\mathbf{div}\mathit{\tau }\stackrel{\mathrm{def}}{=}\left(\sum _{i=1}^d\frac{\partial {\tau }_{i1}}{\partial x_i},\dots ,\sum _{i=1}^d\frac{\partial {\tau }_{id}}{\partial x_i}\right),$$

for a vector-valued function $\mathit{w}:Q_T\to {\mathbb{R}}^d$ and a matrix-valued function $\mathit{\tau }:Q_T\to {\mathbb{R}}^{d\times d}$.

System (1) arises in studying many important models for dynamics of Newtonian and non-Newtonian fluids, including fluids with memory. Indeed, (1) reduces to

• The Euler equations (inviscid fluids) when $\delta =1$, ${\mu }_0=0$, ${\mu }_1=0$, $\eta \equiv 0$ (see [1,2]);
• The classical Navier–Stokes equations (Newtonian fluids) when $\delta =1$, ${\mu }_0>0$, ${\mu }_1=0$, $\eta \equiv 0$ (see [3,4]);
• The unsteady Stokes equations (the creeping flow of a viscous fluid) when $\delta =0$, ${\mu }_0>0$, ${\mu }_1=0$, $\eta \equiv 0$ (see [3,5]);
• The Navier–Stokes–Voigt equations (Kelvin–Voigt type viscoelastic fluids) when $\delta =1$, ${\mu }_0>0$, ${\mu }_1>0$, $\eta \equiv 0$ (see [6,7,8,9]);
• The simplified Jeffreys model (Jeffreys–Oldroyd type viscoelastic fluids) when $\delta =1$, ${\mu }_0>0$, ${\mu }_1=0$, $\eta (s,t)\equiv a{e}^{b(s-t)}$ (see [10,11,12,13]);
• The Oskolkov integro-differential system (generalized Kelvin–Voigt viscoelastic fluids) when $\delta =1$, ${\mu }_0>0$, ${\mu }_1>0$, $\eta \not\equiv 0$ (see [14,15]).

We supplement the generalized Navier–Stokes system (1) with the non-homogeneous Dirichlet boundary condition on ${\Gamma }_T\stackrel{\mathrm{def}}{=}\Gamma \times (0,T)$

$$\mathit{u}=\mathit{\phi }\mathrm{on} {\Gamma }_T$$

(2)

and the initial condition

$${\mathit{u}|}_{t=0}={\mathit{u}}_0\mathrm{in} \Omega .$$

(3)

Despite the large number of works devoted to the analysis of well-posedness of model (1) and its particular cases (see the monographs and the articles mentioned above as well as references therein), the solvability of the non-homogeneous problem (1)–(3) is a completely open problem. The reason is that non-zero Dirichlet boundary data produce major difficulties in the mathematical handling of nonlinear motion equations, especially in the case of 3D flows in a domain with boundary that has several connected components [16]. However, inflow–outflow problems demand careful scrutiny because these are not merely of academic interest but have important consequences with regard to technological applications, for example, in the modeling of fluid flows through channels and pipeline networks with complex geometry.

The main aim of the present work is to study the existence and uniqueness of a strong solution to system (1)–(3) with the boundary data $\mathit{\phi }$ from a fractional Sobolev space.

The paper is organized as follows. In the next section, we give basic notations and function spaces. In Section 3, we introduce the concept of strong solutions and formulate our main result—the existence and uniqueness theorem for problem (1)–(3) in the strong formulation, under smallness assumptions on the data $\eta$, $\mathit{f}$, $\mathit{\phi }$ and ${\mathit{u}}_0$ (Theorem 1). Section 4 is devoted to obtaining auxiliary results, which are needed to prove Theorem 1. Here, we present an abstract theorem on the local unique solvability of an operator equation that contains an isomorphism of Banach spaces with continuously Fréchet differentiable perturbations (Theorem 2). The proof of this theorem is based on the inverse function theorem for nonlinear mappings between Banach spaces. Moreover, in Section 4, we introduce a suitable boundary trace operator and construct its right inverse operator (Lemma 1 on the existence of a divergence-free lifting). Further, using these operators, we study some properties of the operators that describe the linear part of the generalized Navier–Stokes equations (Lemma 2 and Corollary 1). Finally, Section 5 deals with the operator treatment of problem (1)–(3) and the proof of Theorem 1.

2. Preliminaries: Notation and Function Spaces

Let $\mathit{x},\mathit{y}\in {\mathbb{R}}^d$ and $\sigma ,\tau \in {\mathbb{R}}^{d\times d}$. We shall use the standard notation for the scalar product and the Euclidean norm, respectively:

$$\mathit{x}·\mathit{y}\stackrel{\mathrm{def}}{=}\sum _{i=1}^d{x}_i{y}_i,\left|\mathit{x}\right|\stackrel{\mathrm{def}}{=}{\left(\sum _{i=1}^d{x}_i^2\right)}^{1/2},\mathit{\sigma }:\mathit{\tau }\stackrel{\mathrm{def}}{=}\sum _{i,j=1}^d{\sigma }_{ij}{\tau }_{ij},\left|\mathit{\sigma }\right|\stackrel{\mathrm{def}}{=}{\left(\sum _{i,j=1}^d{\sigma }_{ij}^2\right)}^{1/2}.$$

Let $\mathit{X}$ and $\mathit{Y}$ be normed linear spaces. By $\mathcal{L}(\mathit{X},\mathit{Y})$ we denote the space of all continuous linear operators from $\mathit{X}$ into $\mathit{Y}$. Recall that $\mathit{X}$ and $\mathit{Y}$ are called isomorphic normed spaces if there exists a linear bijection $\mathit{A}:\mathit{X}\to \mathit{Y}$ such that $\mathit{A}\in \mathcal{L}(\mathit{X},\mathit{Y})$ and ${\mathit{A}}^{-1}\in \mathcal{L}(\mathit{Y},\mathit{X})$. The operator $\mathit{A}$ is called an isomorphism. By $\mathrm{Isom}(\mathit{X},\mathit{Y})$ denote the set of all isomorphisms from $\mathit{X}$ onto $\mathit{Y}$. If $\mathit{X}=\mathit{Y}$, we write $\mathcal{L}\left(\mathit{X}\right)$ and $\mathrm{Isom}\left(\mathit{X}\right)$ instead of $\mathcal{L}(\mathit{X},\mathit{X})$ and $\mathrm{Isom}(\mathit{X},\mathit{X})$, respectively.

Throughout this paper we shall use the following spaces (see [4]):

• $C_c^{\infty }\left({\mathbb{R}}^d\right)\stackrel{\mathrm{def}}{=}\left\{\alpha \in C^{\infty }\left({\mathbb{R}}^d\right):\mathrm{supp}\left(\alpha \right)\mathrm{is} \mathrm{compact}\right\}$;
• $C_c^{\infty }\left(\overline{\Omega }\right)\stackrel{\mathrm{def}}{=}\left\{{\beta |}_{\overline{\Omega }}:\beta \in C_c^{\infty }\left({\mathbb{R}}^d\right)\right\}$;
• the Lebesgue space $L^r(\Omega )$, $r\ge 1$;
• the Sobolev space $H^m(\Omega )\stackrel{\mathrm{def}}{=}{W}^{m,2}(\Omega )$, $m\in \mathbb{N}$;
• the fractional Sobolev space $H^{m-1/2}(\Gamma )$, for boundary traces of functions from $H^m(\Omega )$.

For the corresponding spaces of vector-valued functions, we use the same notations, but the first letter is bold. For example,

$${\mathit{L}}^r(\Omega )\stackrel{\mathrm{def}}{=}{L}^r{(\Omega )}^d,{\mathit{H}}^m(\Omega )\stackrel{\mathrm{def}}{=}{H}^m{(\Omega )}^d,{\mathit{H}}^{m-1/2}(\Gamma )\stackrel{\mathrm{def}}{=}{H}^{m-1/2}{(\Gamma )}^d.$$

By ${\gamma }_{\Gamma }$ denote the boundary trace operator (for details, see [4], Chap. III), which is a surjective bounded linear operator from ${\mathit{H}}^m(\Omega )$ into ${\mathit{H}}^{m-1/2}(\Gamma )$. Note that ${\gamma }_{\Gamma }{\mathit{v}=\mathit{v}|}_{\Gamma }$ for any vector function $\mathit{v}\in {\mathit{C}}_c^{\infty }\left(\overline{\Omega }\right)$.

Let us introduce the subspace of ${\mathit{H}}^m(\Omega )$ of divergence-free vector functions:

$${\mathit{H}}_{\sigma }^m(\Omega )\stackrel{\mathrm{def}}{=}\{\mathit{v}\in {\mathit{H}}^m(\Omega ):\nabla ·\mathit{v}=0\mathrm{in} \Omega \},m\in \mathbb{N},$$

and the subspace of ${\mathit{H}}^{m-1/2}(\Gamma )$ of vector functions that satisfy the condition of zero total flux on $\Gamma$:

$${\dot{\mathit{H}}}^{m-1/2}(\Gamma )\stackrel{\mathrm{def}}{=}\left\{\mathsf{\omega }\in {\mathit{H}}^{m-1/2}(\Gamma ):{\int }_{\Gamma }\mathsf{\omega }·\mathbf{n}d\Gamma =0\right\},m\in \mathbb{N},$$

where $\mathbf{n}$ denotes the unit exterior normal vector to the surface $\Gamma$.

For handling fluid flow problems in a domain with impermeable boundary, it is convenient to consider the following spaces:

$$\begin{array}{cc}\hfill \mathcal{V}(\Omega )& \stackrel{\mathrm{def}}{=}\left\{\mathit{v}\in {\mathit{C}}^{\infty }(\Omega )\cap {\mathit{H}}_{\sigma }^1(\Omega ):\mathrm{supp}\left(\mathit{v}\right)\subset \Omega \right\},\hfill \\ \hfill {\mathit{V}}^0(\Omega )& \stackrel{\mathrm{def}}{=}\mathrm{the} \mathrm{closure} \mathrm{of} \mathrm{the} \mathrm{set} \mathcal{V}(\Omega )\mathrm{in} \mathrm{the} \mathrm{space} {\mathit{L}}^2(\Omega ),\hfill \\ \hfill {\mathit{V}}^1(\Omega )& \stackrel{\mathrm{def}}{=}\mathrm{the} \mathrm{closure} \mathrm{of} \mathrm{the} \mathrm{set} \mathcal{V}(\Omega )\mathrm{in} \mathrm{the} \mathrm{space} {\mathit{H}}^1(\Omega ),\hfill \\ \hfill {\mathit{V}}^2(\Omega )& \stackrel{\mathrm{def}}{=}{\mathit{H}}^2(\Omega )\cap {\mathit{V}}^1(\Omega ).\hfill \end{array}$$

Clearly, vector functions from the spaces ${\mathit{V}}^1(\Omega )$ and ${\mathit{V}}^2(\Omega )$ “vanish on $\Gamma$”, that is, ${\gamma }_{\Gamma }\mathit{v}=\mathbf{0}$ for any $\mathit{v}\in {\mathit{V}}^{\ell }(\Omega )$, $\ell =1,2$.

We shall use the Leray (or Helmholtz–Weyl) decomposition for vector functions from the Lebesgue space ${\mathit{L}}^2(\Omega )$ into a divergence-free part and a gradient part (see [4], Chap. IV):

$${\mathit{L}}^2(\Omega )={\mathit{V}}^0(\Omega )\oplus \nabla H^1(\Omega ),$$

where the symbol ⊕ stands for the orthogonal sum and the subspace $\nabla H^1(\Omega )$ is defined as follows:

$$\nabla H^1(\Omega )\stackrel{\mathrm{def}}{=}\{\nabla q:q\in H^1(\Omega )\}.$$

The orthogonal projection from ${\mathit{L}}^2(\Omega )$ into ${\mathit{V}}^0(\Omega )$ is known as the Leray projection and denoted as $\mathbb{P}$.

We introduce the equivalence relation “∼” on the Sobolev space $H^1(\Omega )$ by stating that

$$\mathit{\phi }\sim \mathit{\psi }⟺\mathit{\phi }-\mathit{\psi }=\mathrm{const}.$$

As usual, $H^1(\Omega )/\mathbb{R}$ denotes the quotient of $H^1(\Omega )$ by $\mathbb{R}$.

For a function $\xi \in H^1(\Omega )$, we set

$$\overline{\xi }\stackrel{\mathrm{def}}{=}\{\zeta \in H^1(\Omega ):\zeta \sim \xi \}\in H^1(\Omega )/\mathbb{R}$$

and define the gradient and the norm of the class $\overline{\xi }$ as follows:

$$\nabla \overline{\xi }\stackrel{\mathrm{def}}{=}\nabla \xi ,\parallel \overline{\xi }{\parallel }_{H^1(\Omega )/\mathbb{R}}\stackrel{\mathrm{def}}{=}{\parallel \nabla \overline{\xi }\parallel }_{L^2(\Omega )}.$$

Using Proposition 1.2 from the monograph [3] (see Chap. I, § 1), it is easy to show that the norm ${\parallel ·\parallel }_{H^1(\Omega )/\mathbb{R}}$ is well defined.

For handling time-dependent problems, it is convenient to use spaces of functions defined on a time interval with values in a functional space (for example, in a Lebesgue space or a Sobolev space). For a Banach space $\mathit{E}$, by $\mathit{C}\left(\right[0,T];\mathit{E})$ denote the space of continuous functions from $[0,T]$ into $\mathit{E}$. This space is equipped with the max-norm:

$${\parallel \mathit{w}\parallel }_{\mathit{C}\left(\right[0,T];\mathit{E})}\stackrel{\mathrm{def}}{=}\underset{t\in [0,T]}{max}{\parallel \mathit{w}\left(t\right)\parallel }_{\mathit{E}}.$$

Moreover, we shall also use the space of continuously differentiable functions

$${\mathit{C}}^1([0,T];\mathit{E})\stackrel{\mathrm{def}}{=}\left\{\mathit{w}:[0,T]\to \mathit{E}:\mathit{w}\in \mathit{C}([0,T];\mathit{E}),{\mathit{w}}^{\prime }\in \mathit{C}([0,T];\mathit{E})\right\}$$

with the norm

$${\parallel \mathit{w}\parallel }_{{\mathit{C}}^1([0,T];\mathit{E})}\stackrel{\mathrm{def}}{=}{\parallel \mathit{w}\parallel }_{\mathit{C}\left(\right[0,T];\mathit{E})}+{\parallel {\mathit{w}}^{\prime }\parallel }_{\mathit{C}\left(\right[0,T];\mathit{E})},$$

where the symbol ${}^{\prime }$ stands for the derivative with respect to t.

3. Strong Solutions and Main Result

Suppose that

$$\eta \in C([0,T]\times [0,T];\mathbb{R}),\mathit{f}\in \mathit{C}([0,T];{\mathit{L}}^2(\Omega )),\mathit{\phi }\in {\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma )),{\mathit{u}}_0\in {\mathit{H}}_{\sigma }^2(\Omega ).$$

(4)

Definition 1

(Strong solution). We shall say that a pair $(\mathit{u},\overline{p})$ is a strong solution of initial boundary value problem (1)–(3) if $\mathit{u}\in {\mathit{C}}^1([0,T];{\mathit{H}}_{\sigma }^2(\Omega ))$, $\overline{p}\in C([0,T];H^1(\Omega )/\mathbb{R})$, and the functions $\mathit{u}$ and $\overline{p}$ satisfy the following system

$$\left\{\begin{array}{cc}{\displaystyle {\mathit{u}}^{\prime }+\delta \mathbf{div}(\mathit{u}\otimes \mathit{u})-{\mu }_0\Delta \mathit{u}-{\mu }_1\Delta {\mathit{u}}^{\prime }-{\int }_0^t\eta (s,t)\Delta \mathit{u}(·,s)ds+\nabla \overline{p}=\mathit{f}}\hfill & in Q_T,\hfill \\ {\gamma }_{\Gamma }\mathit{u}=\mathit{\phi }\hfill & on {\Gamma }_T,\hfill \\ \mathit{u}\left(0\right)={\mathit{u}}_0\hfill & in \Omega .\hfill \end{array}\right.$$

Theorem 1

(Main result). Let $\Omega \subset {\mathbb{R}}^d$ be a bounded domain of class $C^2$, and fix $T>0$. Assume that ${\mu }_0>0$, ${\mu }_1>0$, and (4) holds. Then there exist positive constants ${\epsilon }_1(Q_T,{\mu }_0,{\mu }_1)$ and ${\epsilon }_2(Q_T,{\mu }_0,{\mu }_1,\eta )$ such that, for any functions η, $\mathit{f}$, φ and ${\mathit{u}}_0$ satisfying the following conditions:

$$\begin{array}{cc}\hfill & {\parallel \eta \parallel }_{C\left(\right[0,T]\times [0,T];\mathbb{R})}\le {\epsilon }_1(Q_T,{\mu }_0,{\mu }_1),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\parallel \mathit{f}\parallel }_{\mathit{C}([0,T];{\mathit{L}}^2(\Omega ))}+{\parallel \mathit{\phi }\parallel }_{{\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma ))}+{\parallel {\mathit{u}}_0\parallel }_{{\mathit{H}}_{\sigma }^2(\Omega )}\le {\epsilon }_2(Q_T,{\mu }_0,{\mu }_1,\eta ),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & {\gamma }_{\Gamma }{\mathit{u}}_0=\mathit{\phi }\left(0\right)\left(the compatibility condition\right),\hfill \end{array}$$

(7)

problem (1)–(3) has a unique strong solution $(\mathit{u},\overline{p})$ in an open neighborhood $\mathcal{U}(Q_T,{\mu }_0,{\mu }_1,\eta )$ of the zero vector function $(\mathbf{0},\overline{0})$ in the space ${\mathit{C}}^1([0,T];{\mathit{H}}_{\sigma }^2(\Omega ))\times C([0,T];H^1(\Omega )/\mathbb{R})$.

Moreover, if $\delta =0$, then problem (1)–(3) is uniquely solvable in the class of strong solutions without restriction (6) on the data $\mathit{f}$, φ and ${\mathit{u}}_0$, provided that inequality (5) and the compatibility condition (7) hold.

The proof of this theorem is given in Section 5.

Remark 1.

The unique solvability of the generalized Navier–Stokes system (1) under the zero Dirichlet boundary condition ${\mathit{u}|}_{{\Gamma }_T}=\mathbf{0}$ is established by Oskolkov [15]. Specifically, assuming that

$$\begin{array}{c}\Gamma \in C^{2,\alpha },{\mu }_1>0,{\mathit{u}}_0\in {\mathit{C}}^{2,\alpha }\left(\overline{\Omega }\right)\cap {\mathit{V}}^1(\Omega ),0<\alpha <1,\\ \mathit{f}\in {\mathit{L}}^{\infty }(0,T;{\mathit{C}}^{\alpha }\left(\overline{\Omega }\right)),{\partial }_t\mathit{f}\in {\mathit{L}}^2\left(Q_T\right),\eta (t,s)\equiv k(t-s),k\in C^1[0,T],\end{array}$$

he proved that the corresponding initial boundary value problem has a unique solution $(\mathit{u},p)$ such that

$$\mathit{u}\in {\mathit{W}}^{1,\infty }(0,T;{\mathit{C}}^{2,\alpha }\left(\overline{\Omega }\right)\cap {\mathit{V}}^1(\Omega )),\frac{\partial p}{\partial x_i}\in L^{\infty }(0,T;C^{\alpha }\left(\overline{\Omega }\right)),i=1,\dots ,d.$$

4. Auxiliary Results

4.1. Abstract Theorem on Local Unique Solvability

Following ideas from [17], we shall establish the next result on the local unique solvability of operator equations in Banach spaces.

Theorem 2.

Let $\mathit{E}$ and $\mathit{F}$ be isomorphic real Banach spaces. Assume that

(i)

the operator ${\mathit{A}}_1:\mathit{E}\to \mathit{F}$ belongs to $\mathrm{Isom}(\mathit{E},\mathit{F});$

(ii)

the operator ${\mathit{A}}_2:\mathit{E}\to \mathit{F}$ belongs to $\mathcal{L}(\mathit{E},\mathit{F})$ and satisfies the following inequality

$$\parallel {\mathit{A}}_2{\parallel }_{\mathcal{L}(\mathit{E},\mathit{F})}<\frac{1}{\parallel {\mathit{A}}_1^{-1}{\parallel }_{\mathcal{L}(\mathit{F},\mathit{E})}};$$

(8)

(iii)

the operator $\mathit{B}:\mathit{E}\to \mathit{F}$ is a continuously Fréchet differentiable mapping such that $\mathit{B}\left(\mathbf{0}\right)=\mathbf{0}$ and the Fréchet derivative $\mathit{D}\mathit{B}\left(\mathbf{0}\right)$ is the zero operator.

Then there exist a positive number ϵ and an open neighborhood $\mathcal{U}$ of the zero element in the space $\mathit{E}$ such that, for any element $\mathit{h}$ from the ball

$$\mathfrak{B}(\mathbf{0},ϵ)\stackrel{\mathrm{def}}{=}{\{\mathit{q}\in \mathit{F}:\parallel \mathit{q}\parallel }_{\mathit{F}}<ϵ\},$$

the operator equation

$${\mathit{A}}_1\mathit{w}+{\mathit{A}}_2\mathit{w}+\mathit{B}\left(\mathit{w}\right)=\mathit{h}$$

(9)

has a unique solution $\mathit{w}={\mathit{w}}_{\mathit{h}}$ in the set $\mathcal{U}$.

Proof. 

Let us consider the operator $\mathbf{\Phi }:\mathit{E}\to \mathit{E}$ defined as follows:

$$\mathbf{\Phi }\stackrel{\mathrm{def}}{=}{\mathit{A}}_1^{-1}\circ {\mathit{A}}_2+{\mathit{A}}_1^{-1}\circ \mathit{B},$$

where ${\mathbf{Id}}_{\mathit{E}}$ is the identity operator in the space $\mathit{E}$.

Applying the operator ${\mathit{A}}_1^{-1}$ to both sides of (9), we arrive at the following equation

$$({\mathbf{Id}}_{\mathit{E}}+\mathbf{\Phi })\left(\mathit{w}\right)={\mathit{A}}_1^{-1}\mathit{h}.$$

(10)

It is obvious that (9) and (10) are equivalent. Therefore, we can focus on (10).

Since $\mathit{B}\left(\mathbf{0}\right)=\mathbf{0}$, we have

$$({\mathbf{Id}}_{\mathit{E}}+\mathbf{\Phi })\left(\mathbf{0}\right)=\mathbf{0}.$$

(11)

Moreover, using inequality (8) and condition (iii), we derive

$$\begin{array}{cc}\hfill {\parallel \mathit{D}\mathbf{\Phi }\left(\mathbf{0}\right)\parallel }_{\mathcal{L}\left(\mathit{E}\right)}& =\parallel {\mathit{A}}_1^{-1}\circ {\mathit{A}}_2+{\mathit{A}}_1^{-1}{\circ \mathit{D}\mathit{B}\left(\mathbf{0}\right)\parallel }_{\mathcal{L}\left(\mathit{E}\right)}\hfill \\ \hfill & =\parallel {\mathit{A}}_1^{-1}\circ {\mathit{A}}_2{\parallel }_{\mathcal{L}\left(\mathit{E}\right)}\hfill \\ \hfill & \le \parallel {\mathit{A}}_1^{-1}{\parallel }_{\mathcal{L}(\mathit{F},\mathit{E})}{\parallel {\mathit{A}}_2\parallel }_{\mathcal{L}(\mathit{E},\mathit{F})}\hfill \\ \hfill & <1.\hfill \end{array}$$

From this estimate it follows that

$$({\mathbf{Id}}_{\mathit{E}}+\mathit{D}\mathbf{\Phi }\left(\mathbf{0}\right))\in \mathrm{Isom}\left(\mathit{E}\right).$$

(12)

Taking into account (11) and (12), by the inverse function theorem (see, e.g., [18], Theorem 10.4) we deduce that there exist open neighborhoods ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ of the zero element in the space $\mathit{E}$ such that the mapping $({\mathbf{Id}}_{\mathit{E}}+\mathbf{\Phi }){|}_{{\mathcal{U}}_1}:{\mathcal{U}}_1\to {\mathcal{U}}_2$ is bijective.

Let $ϵ$ be a positive number such that

$${\mathit{A}}_1^{-1}\mathit{h}\in {\mathcal{U}}_2,\forall \mathit{h}\in \mathfrak{B}(\mathbf{0},ϵ).$$

Then, in order to prove Theorem 2, it suffices to take $\mathcal{U}={\mathcal{U}}_1$. Indeed, for any $\mathit{h}\in \mathfrak{B}(\mathbf{0},ϵ)$, the element ${\mathit{w}}_{\mathit{h}}$ given by the explicit formula

$${\mathit{w}}_{\mathit{h}}\stackrel{\mathrm{def}}{=}[{({\mathbf{Id}}_{\mathit{E}}+\mathbf{\Phi })}^{-1}\circ {\mathit{A}}_1^{-1}]\mathit{h}$$

is the unique solution of (10) in the set $\mathcal{U}$. This concludes the proof. □

4.2. Trace and Lifting Operators

For time-dependent vector functions, we introduce the trace operator ${\mathsf{{\rm Y}}}_{\Gamma }$ as follows:

$$\begin{array}{cc}\hfill & {\mathsf{{\rm Y}}}_{\Gamma }:{\mathit{C}}^1([0,T];{\mathit{H}}_{\sigma }^2(\Omega ))\to {\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma )),\hfill \\ \hfill & \left[{\mathsf{{\rm Y}}}_{\Gamma }\mathit{u}\right]\left(t\right)\stackrel{\mathrm{def}}{=}{\gamma }_{\Gamma }\left[\mathit{u}\left(t\right)\right],\mathrm{for} \mathrm{any} \mathit{u}\in {\mathit{C}}^1([0,T];{\mathit{H}}_{\sigma }^2(\Omega ))\mathrm{and} t\in [0,T].\hfill \end{array}$$

Let us show that the operator ${\mathsf{{\rm Y}}}_{\Gamma }$ has a continuous right inverse.

Lemma 1

(Existence of a divergence-free lifting). Suppose $\Gamma \in C^2;$ then there exists a continuous linear operator

$${\mathsf{\Lambda }}_{\Omega }:{\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma ))\to {\mathit{C}}^1([0,T];{\mathit{H}}_{\sigma }^2(\Omega ))$$

(which is called a divergence-free lifting operator) such that ${\mathsf{{\rm Y}}}_{\Gamma }\circ {\mathsf{\Lambda }}_{\Omega }=\mathbf{Id}$, where $\mathbf{Id}$ is the identity operator in ${\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma ))$. In other words,

$${\gamma }_{\Gamma }\left[{\mathsf{\Lambda }}_{\Omega }\mathit{\psi }\left(t\right)\right]=\mathit{\psi }\left(t\right),for any \mathit{\psi }\in {\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma ))and t\in [0,T].$$

Proof. 

For an arbitrary vector function $\mathsf{\zeta }\in {\dot{\mathit{H}}}^{3/2}(\Gamma )$, consider the boundary value problem

$$\left\{\begin{array}{cc}\mathbb{P}\Delta \mathit{v}=\mathbf{0}\hfill & \mathrm{in} \Omega ,\hfill \\ \nabla ·\mathit{v}=0\hfill & \mathrm{in} \Omega ,\hfill \\ \mathit{v}=\mathsf{\zeta }\hfill & \mathrm{on} \Gamma ,\hfill \end{array}\right.$$

(13)

where $\mathbb{P}:{\mathit{L}}^2(\Omega )\to {\mathit{V}}^0(\Omega )$ is the Leray projection. From the well-known results on the unique solvability of the steady-state Stokes equations with non-homogeneous Dirichlet boundary conditions (see, e.g., [3], Chap. I) it follows that there exists a unique vector function $\mathit{v}\in {\mathit{H}}_{\sigma }^2(\Omega )$ satisfying system (13). Moreover, the following estimate holds

$${\parallel \mathit{v}\parallel }_{{\mathit{H}}_{\sigma }^2(\Omega )}\le K_0(\Omega ){\parallel \mathsf{\zeta }\parallel }_{{\dot{\mathit{H}}}^{3/2}(\Gamma )},$$

(14)

with some positive constant $K_0(\Omega )$.

Let $\mathbb{S}:{\dot{\mathit{H}}}^{3/2}(\Gamma )\to {\mathit{H}}_{\sigma }^2(\Omega )$ be the data-to-solution mapping for the Stokes problem (13), that is, $\mathbb{S}\mathsf{\zeta }\stackrel{\mathrm{def}}{=}\mathit{v}$. Taking into account estimate (14), we obtain

$$\mathbb{S}\in \mathcal{L}({\dot{\mathit{H}}}^{3/2}(\Gamma ),{\mathit{H}}_{\sigma }^2(\Omega )).$$

Let us define the divergence-free lifting operator ${\mathsf{\Lambda }}_{\Omega }$ as follows:

$$\left[{\mathsf{\Lambda }}_{\Omega }\mathit{\psi }\right]\left(t\right)\stackrel{\mathrm{def}}{=}\mathbb{S}\left[\mathit{\psi }\left(t\right)\right],\mathrm{for} \mathrm{any} \mathit{\psi }\in {\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma ))\mathrm{and} t\in [0,T].$$

Clearly, this operator satisfies the equality ${\mathsf{{\rm Y}}}_{\Gamma }\circ {\mathsf{\Lambda }}_{\Omega }=\mathbf{Id}$. Thus, Lemma 1 is proved. □

4.3. Operators Related to the Linear Part of the Generalized Navier–Stokes Equations

For the description of the linear part of system (1), we shall use the linear operators ${\mathbb{A}}_1$ and ${\mathbb{A}}_2$ defined as follows:

$$\begin{array}{cc}\hfill & {\mathbb{A}}_1:\mathit{E}\left(Q_T\right)\to \mathit{F}\left(Q_T\right),{\mathbb{A}}_1(\mathit{u},\overline{p})\stackrel{\mathrm{def}}{=}\left({\mathit{u}}^{\prime }-{\mu }_0\Delta \mathit{u}-{\mu }_1\Delta {\mathit{u}}^{\prime }+\nabla \overline{p},{\mathsf{{\rm Y}}}_{\Gamma }\mathit{u},\mathit{u}\left(0\right)\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\mathbb{A}}_2:\mathit{E}\left(Q_T\right)\to \mathit{F}\left(Q_T\right),{\mathbb{A}}_2(\mathit{u},\overline{p})\stackrel{\mathrm{def}}{=}\left(-{\int }_0^t\eta (s,t)\Delta \mathit{u}(·,s)ds,\mathbf{0},\mathbf{0}\right),\hfill \end{array}$$

(16)

where

$$\begin{array}{cc}\hfill & \mathit{E}\left(Q_T\right)\stackrel{\mathrm{def}}{=}{\mathit{C}}^1([0,T];{\mathit{H}}_{\sigma }^2(\Omega ))\times C([0,T];H^1(\Omega )/\mathbb{R}),\hfill \\ \hfill & \mathit{F}\left(Q_T\right)\stackrel{\mathrm{def}}{=}\left\{(\mathit{g},\mathit{\varphi },\mathit{a})\in \mathit{C}([0,T];{\mathit{L}}^2(\Omega ))\times {\mathit{C}}^1([0,T];{\dot{\mathit{H}}}^{3/2}(\Gamma ))\times {\mathit{H}}_{\sigma }^2(\Omega ):\mathit{\varphi }\left(0\right)={\gamma }_{\Gamma }\mathit{a}\right\}.\hfill \end{array}$$

Lemma 2.

Suppose $\Gamma \in C^2$, ${\mu }_0>0$ and ${\mu }_1>0;$ then

$${\mathbb{A}}_1\in \mathrm{Isom}(\mathit{E}\left(Q_T\right),\mathit{F}\left(Q_T\right)).$$

Proof. 

The proof of this lemma is derived in three steps.

Step 1: Continuity. It is easily proved that there exists a positive constant $K(Q_T,{\mu }_0,{\mu }_1)$ such that

$$\parallel {\mathbb{A}}_1(\mathit{u},\overline{p}){\parallel }_{\mathit{F}\left(Q_T\right)}\le K(Q_T,{\mu }_1,{\mu }_0){\parallel (\mathit{u},\overline{p})\parallel }_{\mathit{E}\left(Q_T\right)},$$

for any pair $(\mathit{u},\overline{p})\in \mathit{E}\left(Q_T\right)$. Hence, we have

$${\mathbb{A}}_1\in \mathcal{L}(\mathit{E}\left(Q_T\right),\mathit{F}\left(Q_T\right)).$$

Step 2: Injectivity. Let us show that the operator ${\mathbb{A}}_1$ is injective. Consider pairs $({\mathit{u}}_1,{\overline{p}}_1)$ and $({\mathit{u}}_2,{\overline{p}}_2)$ belonging to the space $\mathit{E}\left(Q_T\right)$ such that

$${\mathbb{A}}_1({\mathit{u}}_1,{\overline{p}}_1)={\mathbb{A}}_1({\mathit{u}}_2,{\overline{p}}_2).$$

Setting

$$(\tilde{\mathit{u}},\tilde{p})\stackrel{\mathrm{def}}{=}({\mathit{u}}_1-{\mathit{u}}_2,{\overline{p}}_1-{\overline{p}}_2),$$

we obtain

$${\mathbb{A}}_1(\tilde{\mathit{u}},\tilde{p})=(\mathbf{0},\mathbf{0},\mathbf{0}).$$

From this equality, it follows that

$${\tilde{\mathit{u}}}^{\prime }-{\mu }_0\Delta \tilde{\mathit{u}}-{\mu }_1\Delta {\tilde{\mathit{u}}}^{\prime }+\nabla \tilde{p}=\mathbf{0}\mathrm{in} Q_T.$$

(17)

Let us multiply both sides of Equality (17) by $\tilde{\mathit{u}}$ and integrate over the domain $\Omega$:

$${\int }_{\Omega }{\tilde{\mathit{u}}}^{\prime }·\tilde{\mathit{u}}\mathit{dx}-{\mu }_0{\int }_{\Omega }\Delta \tilde{\mathit{u}}·\tilde{\mathit{u}}\mathit{dx}-{\mu }_1{\int }_{\Omega }\Delta {\tilde{\mathit{u}}}^{\prime }·\tilde{\mathit{u}}\mathit{dx}+{\int }_{\Omega }\nabla \tilde{p}·\tilde{\mathit{u}}\mathit{dx}=0,\forall t\in [0,T].$$

(18)

Using the relations $\nabla ·\tilde{\mathit{u}}=0$ in $Q_T$ and ${\gamma }_{\Gamma }\tilde{\mathit{u}}=\mathbf{0}$ on ${\Gamma }_T$, by integration by parts we derive

$$\begin{array}{cc}\hfill {\int }_{\Omega }\Delta \tilde{\mathit{u}}·\tilde{\mathit{u}}\mathit{dx}& =\sum _{i,j=1}^d{\int }_{\Omega }\frac{{\partial }^2{\tilde{u}}_i}{\partial x_j^2}{\tilde{u}}_i\mathit{dx}\hfill \\ \hfill & =\sum _{i,j=1}^d{\int }_{\Gamma }\underset{=0}{\underbrace{\frac{\partial {\tilde{u}}_i}{\partial x_j}{\tilde{u}}_i{n}_j}}d\Gamma -\sum _{i,j=1}^d{\int }_{\Omega }{\left(\frac{\partial {\tilde{u}}_i}{\partial x_j}\right)}^2\mathit{dx}\hfill \\ \hfill & =-{\int }_{\Omega }{|\nabla \tilde{\mathit{u}}|}^2\mathit{dx},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\int }_{\Omega }\Delta {\tilde{\mathit{u}}}^{\prime }·\tilde{\mathit{u}}\mathit{dx}& =\sum _{i,j=1}^d{\int }_{\Omega }\frac{{\partial }^2{\tilde{u}}_i^{\prime }}{\partial x_j^2}{\tilde{u}}_i\mathit{dx}\hfill \\ \hfill & =\sum _{i,j=1}^d{\int }_{\Gamma }\underset{=0}{\underbrace{\frac{\partial {\tilde{u}}_i^{\prime }}{\partial x_j}{\tilde{u}}_i{n}_j}}d\Gamma -\sum _{i,j=1}^d{\int }_{\Omega }\frac{\partial {\tilde{u}}_i^{\prime }}{\partial x_j}\frac{\partial {\tilde{u}}_i}{\partial x_j}\mathit{dx}\hfill \\ \hfill & =-{\int }_{\Omega }\nabla {\tilde{\mathit{u}}}^{\prime }:\nabla \tilde{\mathit{u}}\mathit{dx},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\int }_{\Omega }\nabla \tilde{p}·\tilde{\mathit{u}}\mathit{dx}& =\sum _{i=1}^d{\int }_{\Omega }\frac{\partial (p_1-p_2)}{\partial x_i}{\tilde{u}}_i\mathit{dx}\hfill \\ \hfill & =\sum _{i=1}^d{\int }_{\Gamma }\underset{=0}{\underbrace{(p_1-p_2){\tilde{u}}_i{n}_i}}d\Gamma -\sum _{i=1}^d{\int }_{\Omega }(p_1-p_2)\frac{\partial {\tilde{u}}_i}{\partial x_i}\mathit{dx}\hfill \\ \hfill & ={\int }_{\Omega }(p_1-p_2)\underset{=0}{\underbrace{\nabla ·\tilde{\mathit{u}}}}\mathit{dx}\hfill \\ \hfill & =0.\hfill \end{array}$$

(21)

Substituting (19)–(21) into equality (18), we obtain

$${\int }_{\Omega }{\tilde{\mathit{u}}}^{\prime }·\tilde{\mathit{u}}\mathit{dx}+{\mu }_0{\int }_{\Omega }{|\nabla \tilde{\mathit{u}}|}^2\mathit{dx}+{\mu }_1{\int }_{\Omega }\nabla {\tilde{\mathit{u}}}^{\prime }:\nabla \tilde{\mathit{u}}\mathit{dx}=0,\forall t\in [0,T].$$

(22)

Taking into account the equalities

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\tilde{\mathit{u}}}^{\prime }·\tilde{\mathit{u}}\mathit{dx}=\frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{\left|\tilde{\mathit{u}}\right|}^2\mathit{dx},\forall t\in [0,T],\hfill \\ \hfill & {\int }_{\Omega }\nabla {\tilde{\mathit{u}}}^{\prime }:\nabla \tilde{\mathit{u}}\mathit{dx}=\frac{1}{2}\frac{d}{dt}{\int }_{\Omega }{|\nabla \tilde{\mathit{u}}|}^2\mathit{dx},\forall t\in [0,T],\hfill \end{array}$$

we derive from (22) the following relation

$$\frac{d}{dt}{\int }_{\Omega }\left(|\tilde{\mathit{u}}{|}^2+{\mu }_1{|\nabla \tilde{\mathit{u}}|}^2\right)\mathit{dx}+2{\mu }_0{\int }_{\Omega }{|\nabla \tilde{\mathit{u}}|}^2\mathit{dx}=0,\forall t\in [0,T].$$

(23)

Let us integrate both sides of equality (23) with respect to t from 0 to s. Since $\tilde{\mathit{u}}\left(0\right)=\mathbf{0}$, we find

$${\int }_{\Omega }\left(|\tilde{\mathit{u}}{\left(s\right)|}^2+{\mu }_1{|\nabla \tilde{\mathit{u}}\left(s\right)|}^2\right)\mathit{dx}+2{\mu }_0{\int }_0^s{\int }_{\Omega }{|\nabla \tilde{\mathit{u}}\left(t\right)|}^2\mathit{dx}dt=0,\forall s\in [0,T],$$

whence $\tilde{\mathit{u}}\left(s\right)=\mathbf{0}$ for any $s\in [0,T]$. This means that ${\mathit{u}}_1={\mathit{u}}_2$.

Moreover, from equality $\tilde{\mathit{u}}=\mathbf{0}$ and (17) it follows that $\nabla \tilde{p}=\mathbf{0}$. The last equality implies that $\tilde{p}=\overline{0}$, and hence, ${\overline{p}}_1={\overline{p}}_2$. Thus, we arrive at the equality $({\mathit{u}}_1,{\overline{p}}_1)=({\mathit{u}}_2,{\overline{p}}_2)$.

Step 3: Surjectivity. In view of the bounded inverse theorem for linear operators in Banach spaces (see [18], Theorem 8.34), we must now only prove that ${\mathbb{A}}_1$ is a surjective operator.

Take an arbitrary triple $(\mathit{g},\mathit{\varphi },\mathit{a})$ from the space $\mathit{F}\left(Q_T\right)$. Let us show the solvability of the operator equation

$${\mathbb{A}}_1(\mathit{u},\overline{p})=(\mathit{g},\mathit{\varphi },\mathit{a}).$$

(24)

Note that this equation is equivalent to the following system

$$\left\{\begin{array}{cc}{\mathit{u}}^{\prime }-{\mu }_0\Delta \mathit{u}-{\mu }_1\Delta {\mathit{u}}^{\prime }+\nabla \overline{p}=\mathit{g}\hfill & \mathrm{in} Q_T,\hfill \\ \nabla ·\mathit{u}=0\hfill & \mathrm{in} Q_T,\hfill \\ \mathit{u}=\mathit{\varphi }\hfill & \mathrm{on} {\Gamma }_T,\hfill \\ \mathit{u}\left(0\right)=\mathit{a}\hfill & \mathrm{in} \Omega .\hfill \end{array}\right.$$

(25)

If we represent the unknown vector function $\mathit{u}$ as the sum

$$\mathit{u}=\mathit{v}+{\mathsf{\Lambda }}_{\Omega }\mathit{\varphi },$$

where $\mathit{v}$ is a new unknown vector function, then the initial boundary value problem (25) is reduced to the system

$$\left\{\begin{array}{cc}{\mathit{v}}^{\prime }-{\mu }_0\Delta \mathit{v}-{\mu }_1\Delta {\mathit{v}}^{\prime }+\nabla \overline{p}=\widehat{\mathit{g}}\hfill & \mathrm{in} Q_T,\hfill \\ \nabla ·\mathit{v}=0\hfill & \mathrm{in} Q_T,\hfill \\ \mathit{v}=\mathbf{0}\hfill & \mathrm{on} {\Gamma }_T,\hfill \\ \mathit{v}\left(0\right)=\widehat{\mathit{a}}\hfill & \mathrm{in} \Omega ,\hfill \end{array}\right.$$

(26)

with

$$\begin{array}{cc}\hfill \widehat{\mathit{g}}& \stackrel{\mathrm{def}}{=}\mathit{g}-{\mathsf{\Lambda }}_{\Omega }{\mathit{\varphi }}^{\prime }+{\mu }_0\Delta \left({\mathsf{\Lambda }}_{\Omega }\mathit{\varphi }\right)+{\mu }_1\Delta \left({\mathsf{\Lambda }}_{\Omega }{\mathit{\varphi }}^{\prime }\right)\in \mathit{C}([0,T];{\mathit{L}}^2(\Omega )),\hfill \\ \hfill \widehat{\mathit{a}}& \stackrel{\mathrm{def}}{=}\mathit{a}-\left[{\mathsf{\Lambda }}_{\Omega }\mathit{\varphi }\right]\left(0\right)\in {\mathit{V}}^2(\Omega ).\hfill \end{array}$$

Using techniques similar to the ones in the proof of Theorem 1 from [8], it can be shown that problem (26) has a unique solution $(\mathit{v},\overline{p})$ in the space $\mathit{E}\left(Q_T\right)$. Consequently, equation (24) is solvable. This completes the proof of Lemma 2. □

Corollary 1.

Suppose $\Gamma \in C^2$, ${\mu }_1>0$, ${\mu }_1>0$, and $\eta \in C\left(\right[0,T]\times [0,T];\mathbb{R});$ then

$$({\mathbb{A}}_1+{\mathbb{A}}_2)\in \mathrm{Isom}(\mathit{E}\left(Q_T\right),\mathit{F}\left(Q_T\right)),$$

provided that the norm ${\parallel \eta \parallel }_{C\left(\right[0,T]\times [0,T];\mathbb{R})}$ is sufficiently small.

5. Proof of Main Result

We are now in a position to prove Theorem 1. First of all, we give the operator treatment of the considered initial boundary value problem. Let us introduce the nonlinear operator $\mathbb{B}$ as follows:

$$\mathbb{B}:\mathit{E}\left(Q_T\right)\to \mathit{F}\left(Q_T\right),\mathbb{B}(\mathit{u},\overline{p})\stackrel{\mathrm{def}}{=}\left(\mathbf{div}(\mathit{u}\otimes \mathit{u}),\mathbf{0},\mathbf{0}\right).$$

Clearly, problem (1)–(3) in the strong formulation (see Definition 1) is equivalent to the operator equation

$${\mathbb{A}}_1(\mathit{u},\overline{p})+{\mathbb{A}}_2(\mathit{u},\overline{p})+\delta \mathbb{B}(\mathit{u},\overline{p})=(\mathit{f},\mathit{\phi },{\mathit{u}}_0),$$

with the linear operators ${\mathbb{A}}_1$ and ${\mathbb{A}}_2$ defined in (15) and (16), respectively.

Consider the case $\delta =1$. Since $\Gamma \in C^2$, ${\mu }_0>0$ and ${\mu }_1>0$, the operator ${\mathbb{A}}_1$ is an isomorphism (see Lemma 2). Moreover, it is not hard to show that there exists a positive constant ${\epsilon }_1(Q_T,{\mu }_0,{\mu }_1)$ such that if inequality (5) holds, then

$$\parallel {\mathbb{A}}_2{\parallel }_{\mathcal{L}(\mathit{E}\left(Q_T\right),\mathit{F}\left(Q_T\right))}<\frac{1}{\parallel {\mathbb{A}}_1^{-1}{\parallel }_{\mathcal{L}(\mathit{F}\left(Q_T\right),\mathit{E}\left(Q_T\right))}}.$$

Further, note that the operator $\mathbb{B}$ is continuously Fréchet differentiable and

$$\left[\mathit{D}\mathbb{B}(\mathit{u},\overline{p})\right](\mathit{w},\overline{q})=\left(\mathbf{div}(\mathit{w}\otimes \mathit{u})+\mathbf{div}(\mathit{u}\otimes \mathit{w}),\mathbf{0},\mathbf{0}\right),$$

for any pairs $(\mathit{u},\overline{p})$ and $(\mathit{w},\overline{q})$ belonging to the space $\mathit{E}\left(Q_T\right)$. Hence, the Fréchet derivative $\mathit{D}\mathbb{B}(\mathbf{0},\overline{0}):\mathit{E}\left(Q_T\right)\to \mathit{F}\left(Q_T\right)$ is the zero operator.

Then, applying Theorem 2, we deduce that, under the small data conditions (5) and (6) and the compatibility condition (7), problem (1)–(3) has a unique strong solution $(\mathit{u},\overline{p})$ in some open neighborhood $\mathcal{U}(Q_T,{\mu }_0,{\mu }_1,\eta )$ of the zero function $(\mathbf{0},\overline{0})$ in the space $\mathit{E}\left(Q_T\right)$.

Finally, let us consider the case when $\delta =0$. In this case, the unique solvability of problem (1)–(3) straightforwardly follows from Corollary 1, without restrictions on the size of the data $\mathit{f}$, $\mathit{\phi }$, and ${\mathit{u}}_0$, but provided that (5) and (7) hold. Thus, the proof of Theorem 1 is complete.

6. Conclusions

In the present paper, we have proved the existence and uniqueness of a global-in-time strong solution to the generalized Navier–Stokes system (1) under a non-homogeneous Dirichlet boundary condition, for small data. The proof of this result is based on the application of an abstract theorem on the local unique solvability of an operator equation involving an isomorphism between Banach spaces with continuously Fréchet differentiable perturbations. Note that the proposed approach is quite universal and can be applied to the analysis of solutions of other types, for example, classical or weak solutions. In general, one can deduce that qualitative properties (such as existence, uniqueness, regularity) of small-data solutions to the nonlinear system (1)–(3) (the case $\delta =1$) are determined by the corresponding properties of solutions of the related linearized system (the case $\delta =0$). An important direction for future investigations is the analysis of well-posedness of non-homogeneous boundary value problems for the flow model (1) without the smallness assumption for boundary fluxes.