On the Existence and Uniqueness of an R_ν-Generalized Solution to the Stokes Problem with Corner Singularity

Abstract

We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an $R_{\nu }$-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to construct numerical methods for finding an approximate solution without loss of accuracy. In the paper, the existence and uniqueness of an $R_{\nu }$-generalized solution in weighted sets is proved.

1. Introduction

Boundary value problems with singularity play an important role in fracture mechanics [1,2]. The singularity can be caused both by the degeneracy of the coefficients and the right-hand sides of the equation (see [3,4,5,6]) and by the presence of re-entrant corners on the boundary of a polygonal domain (see, [7,8,9,10]). The singularity of the differential problem affects the accuracy of finding an approximate solution numerically, since the error depends on the regularity of the solution [11]. An efficient numerical method with a special mesh refinement to the boundary for the problems with degeneracy of the solution on the entire boundary of the domain has been developed. The method allows us to find a solution without a loss of accuracy at a rate of $\mathcal{O}\left(h\right)$ with respect to the grid step h in the norm of the weighted Sobolev space [12,13]. Several numerical methods for problems with a singularity caused by the presence of re-entrant corners at the boundary have been developed. The methods make it possible to reduce the influence of the singularity on the accuracy of finding an approximate solution. An overview of such approaches is given in [14]. Let us select from them the weighted finite element method (FEM), which is based on the definition of an $R_{\nu }$-generalized solution that takes into account the asymptotics of the solution behavior [15,16,17,18] in the vicinity of the singularity point and introducing special weighted basis functions [19,20]. This allows us to find approximate solutions without a loss of accuracy.

The solution $(\mathbf{w},p)$ of the Stokes problem in a domain with a re-entrant corner in polar coordinates $(r,\phi )$ is a linear combination of singular components and a regular one. The singular ones of the components of vector function $\mathbf{w}$ and function p have asymptotic behaviors $r^{{\lambda }_i}$ and $r^{{\lambda }_{i-1}}$ respectively, where ${\lambda }_i$ are eigenvalues of the Stokes operator satisfying the following equation in the case of homogeneous Dirichlet boundary conditions.

$${\lambda }_i^2{sin}^2\omega -{sin}^2\left({\lambda }_i\omega \right)=0,{\lambda }_i\in \mathbb{C},{\lambda }_i\ne 0.$$

In particular, if the re-entrant corner $\omega$ is equal to $\frac{3\pi }{2},$ then the smallest positive eigenvalue that characterizes the behavior of the solution in a neighborhood of the re-entrant corner is approximately equal to $0.544483.$ In this case, the solution (components of the velocity field) belongs to space $W_2^{1+0.544483-\epsilon }(\Omega )$, where $\epsilon$ is an arbitrary positive number. According to the principle of consistent estimates [11], the classical FEM allows one to find an approximate solution at a rate not higher than $\mathcal{O}\left(h^{0.544}\right)$. We have developed a weighted FEM for the Stokes problem [21,22], which allows us to find an approximate solution with a rate $\mathcal{O}\left(h\right)$ independently on the value of the re-entrant corner at the boundary. The approach is based on the introduction of the concept of an $R_{\nu }$-generalized solution. The existence and uniqueness of which in weighted sets is studied in this paper. The solution proposed by us is determined in the nonsymmetric mixed variational formulation of the problem: find a pair of functions $(\mathbf{w},p)$ such that the following identities:

$$a(\mathbf{w},\mathbf{v})+b_1(\mathbf{v},p)=l\left(\mathbf{v}\right),$$

(1)

$$b_2(\mathbf{w},q)=0$$

(2)

hold, for arbitrary pairs of functions $(\mathbf{v},q)$. Moreover, $a(\mathbf{w},\mathbf{v})$ is not in symmetric bilinear form, and forms $b_1(.,.)$ and $b_2(.,.)$ are not equal to each other and do not coincide with the standard form $b(.,.)$ (see [23]). The question of existence and uniqueness in the abstract mixed variational formulation (1), (2) was previously studied in [24,25] and generalized [26] for the presence of bilinear form $c(p,q)\ne 0$ in Equation (2). In the present study, the existence and uniqueness of the $R_{\nu }$-generalized solution in weighted sets is proved based on auxiliary statements (see [27,28]).

The paper has the following structure. In Section 2, the Stokes problem is posed, and the necessary notation has been introduced. The concept of an $R_{\nu }$-generalized solution is presented in a nonsymmetric mixed variational formulation of the problem. In Section 3, relying on the proved auxiliary assertions, we establish the existence and uniqueness of the $R_{\nu }$-generalized solution in weighted sets. An estimate for the norms of the solution is obtained using the norm of the right-hand side function. Conclusions and useful remarks are made in Section 4.

2. Problem Statement

Let ${\mathbf{R}}^2$ be the two-dimensional Euclidean space a d $\mathbf{x}=(x_1,x_2)$ be its arbitrary element. Denote by $\Omega \subset {\mathbf{R}}^2$ a non-convex polygonal domain with one re-entrant corner $\omega \in (\pi ,2\pi )$ on boundary $\partial \Omega$ with a vertex at origin $\mathcal{O}=(0,0).$ Let $\overline{\Omega }=\Omega \cup \partial \Omega$ be its closure.

We consider the Stokes problem: find the velocity of the fluid $\mathbf{w}=\mathbf{w}\left(\mathbf{x}\right)=(w_1\left(\mathbf{x}\right),$$w_2\left(\mathbf{x}\right))$ and pressure $p=p\left(\mathbf{x}\right),$ which satisfy the system of differential equations and boundary conditions.

$$-▵\mathbf{w}+\nabla p=\mathbf{f},\mathrm{in}\Omega ,$$

(3)

$$\mathrm{div}\mathbf{w}=0,\mathrm{in}\Omega ,$$

(4)

$$\mathbf{w}=\mathbf{0},\mathrm{on}\partial \Omega .$$

(5)

Due to the fact that, on boundary $\partial \Omega$, there is a corner $\omega$ greater than $\pi ,$ although the solution $(\mathbf{w},p)$ of the Stokes problem (3)–(5) is analytic in $\Omega \backslash (0,0),$ but $\nabla \mathbf{w}$ and p are singular at the origin. In particular, $\mathbf{w}\notin {\mathbf{W}}_2^2(\Omega )$ and $p\notin W_2^1(\Omega )$. As a result, it is necessary to determine the $R_{\nu }$-generalized solution of problem (3)–(5) in special sets. The main idea behind this solution is based on the introduction of the weighted function $\rho \left(\mathbf{x}\right)$ satisfying the following conditions in $\overline{\Omega }$: $\rho \left(\mathbf{x}\right)={(x_1^2+x_2^2)}^{1/2}$ if $\mathbf{x}\in {\Omega }_{\delta }$ and $\rho \left(\mathbf{x}\right)=\delta$; otherwise, we have the variational formulation of the problem to some degree $\nu .$ Here and later, let ${\Omega }_{\delta }$ be the intersection of a circle with a radius $\delta ,\delta >0,$ and center at origin $\mathcal{O}=(0,0)$ with closure $\overline{\Omega }.$ The degree $\nu$ of the weighted function $\rho \left(\mathbf{x}\right)$ depends on function $\mathbf{f}=\mathbf{f}\left(\mathbf{x}\right)$ of the right-hand side of Equation (3) and the value of re-entrant corner $\omega$ on $\partial \Omega$. The presence of function ${\rho }^{\nu }\left(\mathbf{x}\right)$ in the variational formulation suppresses the singularity and allows $R_{\nu }$-generalized solution components ${\mathbf{w}}_{\nu }$ and $p_{\nu }$ to belong to sets of the weighted spaces ${\mathbf{W}}_{2,\nu }^2(\Omega )$ and $W_{2,\nu }^1(\Omega )$, respectively (see below for definitions). Let us introduce the necessary spaces and sets of functions. For more details, see [28].

Let $D^{l}z\left(\mathbf{x}\right)=\frac{{\partial }^{\left|l\right|}z\left(\mathbf{x}\right)}{\partial x_1^{l_1}\partial x_2^{l_2}}$, where $l=(l_1,l_2),$$\left|l\right|=l_1+l_2,l_i$ are non-negative integers, $i\in \{1,2\}$ and $d\mathbf{x}=d{x}_{1}d{x}_2$.

Denote by $L_{2,\beta }(\Omega )$ and $W_{2,\beta }^k(\Omega )$ the spaces of functions $z\left(\mathbf{x}\right)$ with bounded norms:

$${\parallel z\parallel }_{L_{2,\beta }(\Omega )}=(\underset{\Omega }{\int }{\rho }^{2\beta }\left(\mathbf{x}\right)z^2\left(\mathbf{x}\right)d\mathbf{x}{)}^{1/2}$$

(6)

and

$${\parallel z\parallel }_{W_{2,\beta }^k(\Omega )}=({\parallel z\parallel }_{L_{2,\beta }(\Omega )}^2+\sum _{1\le \left|l\right|\le k}\underset{\Omega }{\int }{\rho }^{2\beta }\left(\mathbf{x}\right){\left|D^{l}z\left(\mathbf{x}\right)\right|}^{2}d\mathbf{x}{)}^{1/2}$$

(7)

respectively. Let ${\left|z\right|}_{W_{2,\beta }^k(\Omega )}={\left({\displaystyle \sum _{\left|l\right|=k}}\underset{\Omega }{\int }{\rho }^{2\beta }\left(\mathbf{x}\right){\left|D^{l}z\left(\mathbf{x}\right)\right|}^{2}d\mathbf{x}\right)}^{1/2}$ be the seminorm of $W_{2,\beta }^k(\Omega )$. Denote by $W_{2,\beta }^{k,0}(\Omega )$ the closure with respect to the norm (7) of the set of infinitely differentiable compactly supported functions in $\Omega$.

Next, we define the conditions where functions $z\left(\mathbf{x}\right)$ obey the following:

$$0<C_1\le {\parallel z\parallel }_{L_{2,\beta }(\Omega \backslash {\Omega }_{\delta })},$$

(8)

$$\left|z\left(\mathbf{x}\right)\right|\le C_2{\delta }^{\beta -\tau }{\rho }^{\tau -\beta }\left(\mathbf{x}\right),\mathbf{x}\in {\Omega }_{\delta },$$

(9)

$$|D^{1}z\left(\mathbf{x}\right)|\le C_2{\delta }^{\beta -\tau }{\rho }^{\tau -\beta -1}\left(\mathbf{x}\right),\mathbf{x}\in {\Omega }_{\delta },$$

(10)

where $C_2$ is a positive constant, and $\tau$ is a small positive parameter independent from $\beta ,\delta$ and $z\left(\mathbf{x}\right).$

Denote by $L_{2,\beta }(\Omega ,\delta )$ the set of functions $z\left(\mathbf{x}\right)$ from the space $L_{2,\beta }(\Omega ),$ satisfying conditions (8) and (9) with bounded norm (6). Define its subset $L_{2,\beta }^0(\Omega ,\delta )=\{z\left(\mathbf{x}\right)\in L_{2,\beta }(\Omega ,\delta ):\parallel {\rho }^{\beta }z{\parallel }_{L_1(\Omega )}=0\}$ with limited norm (6).

Let $W_{2,\beta }^1(\Omega ,\delta )$ be the set of functions $z\left(\mathbf{x}\right)$ from the space $W_{2,\beta }^1(\Omega )$ satisfying conditions (8)–(10) with limited norm (7). Denote by $W_{2,\beta }^{1,0}(\Omega ,\delta )$ the set of functions from the space $W_{2,\beta }^{1,0}(\Omega )$ satisfying conditions (8)–(10) with bounded norm (7). Moreover, the linear combination of functions from $L_{2,\beta }(\Omega ,\delta )$ ($W_{2,\beta }^{1,0}(\Omega ,\delta )$) also belongs to set $L_{2,\beta }(\Omega ,\delta )$ ($W_{2,\beta }^{1,0}(\Omega ,\delta )$).

We will highlight in bold the sets of vector functions; that is, ${\mathbf{L}}_{2,\beta }(\Omega ,\delta )=\{\mathbf{z}\left(\mathbf{x}\right)=(z_1\left(\mathbf{x}\right),z_2\left(\mathbf{x}\right)):z_i\left(\mathbf{x}\right)\in L_{2,\beta }(\Omega ,\delta )\}$ with bounded vector norm ${\parallel \mathbf{z}\parallel }_{{\mathbf{L}}_{2,\beta }(\Omega )}={\left(\parallel z_1{\parallel }_{L_{2,\beta }(\Omega )}^2+{\parallel z_2\parallel }_{L_{2,\beta }(\Omega )}^2\right)}^{1/2}.$ A similar selection takes place for sets of vector functions ${\mathbf{W}}_{2,\beta }^1(\Omega ,\delta )$ (${\mathbf{W}}_{2,\beta }^{1,0}(\Omega ,\delta )$) with vector norm (7).

Next, we define the bilinear and linear forms as follows.

$$a(\mathbf{u},\mathbf{v})=\underset{\Omega }{\int }\left[\nabla \mathbf{u}:\nabla \left({\rho }^{2\nu }\mathbf{v}\right)\right]d\mathbf{x},$$

(11)

$$b_1(\mathbf{v},s)=-\underset{\Omega }{\int }sdiv\left({\rho }^{2\nu }\mathbf{v}\right)d\mathbf{x},b_2(\mathbf{u},q)=-\underset{\Omega }{\int }\left({\rho }^{2\nu }q\right)div\mathbf{u}d\mathbf{x},$$

$$l\left(\mathbf{v}\right)=\underset{\Omega }{\int }\mathbf{f}·\left({\rho }^{2\nu }\mathbf{v}\right)d\mathbf{x}.$$

Definition 1.

The pair $({\mathbf{w}}_{\nu },p_{\nu })\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )\times L_{2,\nu }^0(\Omega ,\delta )$ is called an $R_{\nu }$-generalized solution of problem (3)–(5) if for all pairs $(\mathbf{z},g)\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )\times L_{2,\nu }^0(\Omega ,\delta )$, the integral identities of the following:

$$a({\mathbf{w}}_{\nu },\mathbf{z})+b_1(\mathbf{z},p_{\nu })=l\left(\mathbf{z}\right),b_2({\mathbf{w}}_{\nu },g)=0$$

(12)

hold, where $\mathbf{f}\in {\mathbf{L}}_{2,\gamma }(\Omega ,\delta ),\nu \ge \gamma \ge 0.$

In contrast to the classical variational formulation of the Stokes problem (3)–(5), bilinear forms $b_1(·,·)$ and $b_2(·,·)$ in Equation (12) are not equal to each other and standard bilinear form $b(·,·)$ (see [23]). Thus, the variational statement (12) is a nonsymmetric one. Guided by this fact, in order to prove the existence and uniqueness $R_{\nu }$-generalized solution of the problem, we need to use the following sets of functions (see [28]).

$$K_1=\{\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta ):\forall s\in L_{2,\nu }^0(\Omega ,\delta ),b_1(\mathbf{v},s)=-\underset{\Omega }{\int }s\mathrm{div}\left({\rho }^{2\nu }\mathbf{v}\right)d\mathbf{x}=0\}.$$

$$K_2=\{\mathbf{u}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta ):\forall q\in L_{2,\nu }^0(\Omega ,\delta ),b_2(\mathbf{u},q)=-\underset{\Omega }{\int }\left({\rho }^{2\nu }q\right)\mathrm{div}\mathbf{u}d\mathbf{x}=0\}.$$

In particular, the second one must contain the first component of the $R_{\nu }$-generalized solution, namely ${\mathbf{w}}_{\nu }.$

3. Existence and Uniqueness of an R_ν-Generalized Solution

3.1. Supplementary Statements

Lemma 1.

(Friedrichs’s inequality). For any $z\in W_{2,0}^{1,0}(\Omega ,\delta )$, the inequality of the following:

$${\parallel z\parallel }_{L_{2,0}(\Omega )}\le C_3{\parallel \nabla z\parallel }_{L_{2,0}(\Omega )}$$

(13)

holds, where $C_3$ is a positive constant that does not depend on z.

Lemma 2 ([28]).

For any $z\in L_{2,\nu }(\Omega )$ satisfying the conditions (8) and (9), the inequality of the following:

$$\underset{{\Omega }_{\delta }}{\int }{\rho }^{2(\nu -1)}{z}^{2}d\mathbf{x}\le C_4^2{\delta }^{2\nu }{\parallel z\parallel }_{L_{2,\nu }(\Omega )}^2$$

(14)

holds, where $C_4$ is a positive constant equal to $\frac{C_2}{C_1}\sqrt{\frac{{\phi }_1-{\phi }_0}{2\tau }}$, ${\phi }_1-{\phi }_0$ is the value of the corner ω change in polar coordinates.

Lemma 3 ([28]).

Function $z\in W_{2,\nu }^{1,0}(\Omega ,\delta )$ if and only if ${\rho }^{\nu }z\in W_{2,0}^{1,0}(\Omega ,\delta )$ and the following inequalities:

$$\parallel \nabla \left({\rho }^{\nu }z\right){\parallel }_{L_{2,0}(\Omega )}^2\le {2\parallel \nabla z\parallel }_{L_{2,\nu }(\Omega )}^2+2{\nu }^2{C}_4^2{\delta }^{2\nu }{\parallel z\parallel }_{L_{2,\nu }(\Omega )}^2,$$

(15)

$${\left|z\right|}_{W_{2,\nu }^1(\Omega )}^2\le 2\parallel \nabla \left({\rho }^{\nu }z\right){\parallel }_{L_{2,0}(\Omega )}^2+2{\nu }^2{C}_4^2{\delta }^{2\nu }{\parallel {\rho }^{\nu }z\parallel }_{L_{2,0}(\Omega )}^2$$

(16)

hold.

Theorem 1 ([28]).

Let $\nu >0,$ then there exists ${\delta }_0={\delta }_0\left(\nu \right)>0$ such that for any $\delta \in (0,{\delta }_0),$ any function $\mathbf{u}\in K_2,$ represented as $\left(\frac{\partial \psi }{\partial x_2},-\frac{\partial \psi }{\partial x_1}\right),$ function $\mathbf{v}=\left({\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_2},-{\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_1}\right)$ belongs to the set $K_1$, and an estimate of the following:

$${\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_5{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}$$

(17)

holds, where $C_5=max\{\sqrt{2+16{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu +2}{(1+\tau )}^{-1}},\sqrt{3+48{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu }(48C_3^2+1)}\}.$

Theorem 2 ([28]).

There exists ${\epsilon }_1>0,$ such that, for $\nu >0$, there exists ${\delta }_1={\delta }_1({\epsilon }_1,\nu )>0,$ such that for any $\delta \in (0,{\delta }_1),$ any function $\mathbf{u}=\left(\frac{\partial \psi }{\partial x_2},-\frac{\partial \psi }{\partial x_1}\right)\in K_2$ and for $\mathbf{v}\in K_1,$ represented as $\left({\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_2},-{\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_1}\right),$ the inequality of the following:

$$\frac{1}{4}{\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2\le a(\mathbf{u},\mathbf{v})$$

(18)

holds, where ${\delta }_1<{\delta }_0$.

Theorem 3 ([28]).

Let $\nu >0,$ then there exists ${\delta }_2={\delta }_2\left(\nu \right)=min\{{\delta }_0\left(\nu \right),\sqrt{\frac{1+\tau }{8}}\}>0$, such that for any $\delta \in (0,{\delta }_2),$ arbitrary function $\mathbf{v}\in K_1,$ represented as $\left({\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_2},-{\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_1}\right),$ function $\mathbf{u}=\left(\frac{\partial \psi }{\partial x_2},-\frac{\partial \psi }{\partial x_1}\right)$ belongs to the set $K_2$, and an estimate of the following:

$${\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_6{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}$$

(19)

holds, where a constant $C_6$ is equal to $\sqrt{4+48{\nu }^2{C}_4^2{\delta }^{2\nu }(48C_3^2+1)}.$

Theorem 4 ([28]).

There exists ${\epsilon }_2>0,$ such that for $\nu >0$ there exists ${\delta }_3={\delta }_3({\epsilon }_2,\nu )>0,$ such that for arbitrary $\delta \in (0,min\{{\delta }_2,{\delta }_3\}),$ any function $\mathbf{v}=\left({\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_2},-{\rho }^{-2\nu }\frac{\partial \left({\rho }^{2\nu }\psi \right)}{\partial x_1}\right)\in K_1$ and for $\mathbf{u}\in K_2,$ represented as $\left(\frac{\partial \psi }{\partial x_2},-\frac{\partial \psi }{\partial x_1}\right),$ the inequality of the following:

$$\frac{1}{4}{\parallel \nabla \left({\rho }^{\nu }\mathbf{v}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2\le a(\mathbf{u},\mathbf{v})$$

(20)

holds.

Let us formulate statements that define the $LBB$-conditions of the bilinear forms $b_1(\mathbf{v},s)$ and $b_2(\mathbf{u},q),$ $\mathbf{v},\mathbf{u}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )$ and $s,q\in L_{2,\nu }^0(\Omega ,\delta ).$

Theorem 5.

Let $\nu >0;$ then there exists ${\delta }_4={\delta }_4\left(\nu \right)>0,$ such that for any $\delta \in (0,{\delta }_4)$, the inequalities of the following:

$$\forall s\in L_{2,\nu }^0(\Omega ,\delta ):\underset{\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )}{sup}\frac{b_1(\mathbf{v},s)}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel s\parallel }_{L_{2,\nu }(\Omega )}}\ge {\beta }_1>0,$$

(21)

$$\forall q\in L_{2,\nu }^0(\Omega ,\delta ):\underset{\mathbf{u}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )}{sup}\frac{b_2(\mathbf{u},q)}{{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel q\parallel }_{L_{2,\nu }(\Omega )}}\ge {\beta }_2>0$$

(22)

hold, where ${\beta }_1$ and ${\beta }_2$ constants that do not depend on $\mathbf{v},s$ and $\mathbf{u},q$, respectively.

Proof. 

Inequality (22) is proved in [27]. The inequality (21) is established in the same way. □

3.2. Relationship between Functions $\mathbf{v}\in K_1$ and $\mathbf{u}\in K_2$ and The Nonsymmetric Bilinear form $a(\mathbf{u},\mathbf{v}):$ Continuity of Bilinear and Linear Forms

Let us prove a theorem relating the norms of the functions $\mathbf{v}\in K_1$ and $\mathbf{u}\in K_2$ in space ${\mathbf{W}}_{2,\nu }^{1,0}(\Omega )$ with a nonsymmetric bilinear form $a(\mathbf{u},\mathbf{v})$ (see Equation (11)).

Theorem 6.

There exist ${\epsilon }_1>0$ and ${\epsilon }_2>0$ such that, for $\nu >0,$ there exists ${\delta }_5=min\{{\delta }_0,{\delta }_1,{\delta }_2,{\delta }_3\}>0$, such that for arbitrary $\delta \in (0,{\delta }_5)$, the following inequalities:

$$\forall \mathbf{u}\in K_2:\underset{\mathbf{v}\in K_1}{sup}\frac{a(\mathbf{u},\mathbf{v})}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}\ge \alpha {\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )},$$

(23)

$$\forall \mathbf{v}\in K_1:\underset{\mathbf{u}\in K_2}{sup}\frac{a(\mathbf{u},\mathbf{v})}{{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}\ge \alpha {\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}$$

(24)

hold, where a constant $\alpha =min\{\frac{1}{C_7},\frac{1}{C_8}\}$ and $C_7=4C_5(2+(1+2{\nu }^2{C}_4^2{\delta }^{2\nu })C_3^2),$ $C_8=4C_6(2+(1+2{\nu }^2{(1+2\nu )}^2{C}_4^2{\delta }^{2\nu })C_3^2).$

Proof. 

1. For an arbitrary function $\mathbf{u}\in K_2$ defined in the formulations of Theorem 1, we apply estimate (16) of Lemma 3 combined with inequality (13) of Lemma 1, and we have the following.

$${\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2={\parallel \mathbf{u}\parallel }_{{\mathbf{L}}_{2,\nu }(\Omega )}^2+{\left|\mathbf{u}\right|}_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2\le \parallel {\rho }^{\nu }{\mathbf{u}\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2+2{\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2+$$

$$+2{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel {\rho }^{\nu }{\mathbf{u}\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2=2\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right){\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2+(1+2{\nu }^2{C}_4^2{\delta }^{2\nu }){\parallel {\rho }^{\nu }\mathbf{u}\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2\le$$

$$\le 2\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right){\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2+(1+2{\nu }^2{C}_4^2{\delta }^{2\nu })C_3^2{\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2,$$

thus

$$\begin{array}{c}\hfill {\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2\le (2+(1+2{\nu }^2{C}_4^2{\delta }^{2\nu })C_3^2){\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2.\end{array}$$

(25)

Applying Theorems 1 and 2 and their estimates (17) and (18), respectively, to Equation (25), for the function $\mathbf{v}\in K_1,$ defined in the formulation of Theorem 1, we obtain a sequence of inequalities:

$${\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_5{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2\le$$

$$\le 4C_5(2+(1+2{\nu }^2{C}_4^2{\delta }^{2\nu })C_3^2){\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2\le C_{7}a(\mathbf{u},\mathbf{v}),$$

such that the following is the case:

$$\begin{array}{c}\hfill {\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_{7}a(\mathbf{u},\mathbf{v}),\end{array}$$

(26)

where $C_7$ is a constant equal to $4C_5(2+(1+2{\nu }^2{C}_4^2{\delta }^{2\nu })C_3^2).$

2. For an arbitrary function $\mathbf{v}\in K_1,$ defined in the formulation of Theorem 3, by analogy with item 1, we conclude the following.

$$\begin{array}{c}\hfill {\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2\le (2+(1+2{\nu }^2{(1+2\nu )}^2{C}_4^2{\delta }^{2\nu })C_3^2){\parallel \nabla \left({\rho }^{\nu }\mathbf{v}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2.\end{array}$$

(27)

Applying Theorems 3 and 4 and their estimates (19) and (20), respectively, to Equation (27), for the function $\mathbf{u}\in K_2,$ defined in the formulation of Theorem 3, we obtain a sequence of inequalities:

$${\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_6{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2\le$$

$$\le 4C_6(2+(1+2{\nu }^2{(1+2\nu )}^2{C}_4^2{\delta }^{2\nu })C_3^2){\parallel \nabla \left({\rho }^{\nu }\mathbf{u}\right)\parallel }_{{\mathbf{L}}_{2,0}(\Omega )}^2\le C_{8}a(\mathbf{u},\mathbf{v}),$$

such that

$$\begin{array}{c}\hfill {\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_{8}a(\mathbf{u},\mathbf{v}),\end{array}$$

(28)

where $C_8$ is a constant equal to $4C_6(2+(1+2{\nu }^2{(1+2\nu )}^2{C}_4^2{\delta }^{2\nu })C_3^2).$

3. We perform the following sequence of reasoning. Divide both parts (26) and (28) into positive quantities $C_7{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}$ and $C_8{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}$, respectively. We define the least upper bound over all functions $\mathbf{v}\in K_1$ and $\mathbf{u}\in K_2$ in the first and second inequalities, respectively. Let $\alpha =min\{\frac{1}{C_7},\frac{1}{C_8}\}.$ Hence, we obtain estimates (23) and (24).

Theorem 6 is proved. □

Let us prove the continuity of the bilinear forms $a(·,·), b_1(·,·), b_2(·,·)$ and the linear form $l(·).$

Theorem 7.

For $\nu >0$ the following statements hold:

(1) 

For any functions $\mathbf{u},\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta ):$

$$\begin{array}{c}\hfill a(\mathbf{u},\mathbf{v})\le C_9{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )},\end{array}$$

(29)

where $C_9=8(1+{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu })(1+{\nu }^2{C}_4^2{\delta }^{2\nu });$

(2) 

For arbitrary functions $s\in L_{2,\nu }^0(\Omega ,\delta )$ and $\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta ):$

$$\begin{array}{c}\hfill b_1(\mathbf{v},s)\le C_{10}{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel s\parallel }_{L_{2,\nu }(\Omega )},\end{array}$$

(30)

where $C_{10}=2\sqrt{2+3{\nu }^2{C}_4^2{\delta }^{2\nu }};$

(3) 

For any functions $q\in L_{2,\nu }^0(\Omega ,\delta )$ and $\mathbf{u}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta ):$

$$\begin{array}{c}\hfill b_2(\mathbf{u},q)\le C_{11}{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel q\parallel }_{L_{2,\nu }(\Omega )},\end{array}$$

(31)

where $C_{11}=\sqrt{2};$

(4) 

For arbitrary functions $\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )$ and $\mathbf{f}\in {\mathbf{L}}_{2,\gamma }(\Omega ,\delta ),\nu \ge \gamma \ge 0:$

$$\begin{array}{c}\hfill l\left(\mathbf{v}\right)\le C_{12}{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )},\end{array}$$

(32)

where $C_{12}=\sqrt{2}{C}_{13},$ and $C_{13}$ are constants in the estimate of norms for functions under embedding $L_{2,\gamma }(\Omega ,\delta )$ to $L_{2,\nu }(\Omega ,\delta )$.

Proof. 

(1) Consider arbitrary functions $\mathbf{u},\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )$ and the following:

$${\left[a(\mathbf{u},\mathbf{v})\right]}^2={\left[\underset{\Omega }{\int }\sum _{i,j=1}^2\frac{\partial u_i}{\partial x_j}\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_j}d\mathbf{x}\right]}^2,$$

then

$${\left[a(\mathbf{u},\mathbf{v})\right]}^2\le 2{\left[\underset{\Omega }{\int }\sum _{j=1}^2\frac{\partial u_1}{\partial x_j}\frac{\partial \left({\rho }^{2\nu }{v}_1\right)}{\partial x_j}d\mathbf{x}\right]}^2+2{\left[\underset{\Omega }{\int }\sum _{j=1}^2\frac{\partial u_2}{\partial x_j}\frac{\partial \left({\rho }^{2\nu }{v}_2\right)}{\partial x_j}d\mathbf{x}\right]}^2\le$$

$$\le 4\sum _{i,j=1}^2{\left[\underset{\Omega }{\int }\frac{\partial u_i}{\partial x_j}\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_j}d\mathbf{x}\right]}^2=4\sum _{i,j=1}^2{S}_{i,j},$$

such that

$$\begin{array}{c}\hfill {\left[a(\mathbf{u},\mathbf{v})\right]}^2\le 4\sum _{i,j=1}^2{S}_{i,j},\end{array}$$

(33)

where $S_{i,j}$ has the following form.

$$\begin{array}{c}\hfill S_{i,j}={\left[\underset{\Omega }{\int }\frac{\partial u_i}{\partial x_j}\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_j}d\mathbf{x}\right]}^2.\end{array}$$

(34)

Insofar as we have the following:

$$\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}={\rho }^{\nu }\left(\frac{\partial u_i}{\partial x_j}\right)+\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)u_i,$$

then

$$\begin{array}{c}\hfill \frac{\partial u_i}{\partial x_j}={\rho }^{-\nu }\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}-{\rho }^{-\nu }\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)u_i,\end{array}$$

(35)

and

$$\begin{array}{c}\hfill \frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_j}=\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)\left({\rho }^{\nu }{v}_i\right)+{\rho }^{\nu }\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}.\end{array}$$

(36)

Let us substitute the right-hand parts (35) and (36) instead of their left-hand parts to Equation (34); then, by applying the Cauchy–Schwarz inequality, we have the following.

$$S_{i,j}={\left[\underset{\Omega }{\int }\left({\rho }^{-\nu }\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}-{\rho }^{-\nu }\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)u_i\right)\left(\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)\left({\rho }^{\nu }{v}_i\right)+{\rho }^{\nu }\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)d\mathbf{x}\right]}^2=$$

$$=[\underset{\Omega }{\int }\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}\right)\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{v}_i\right)d\mathbf{x}+\underset{\Omega }{\int }\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}\right)\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)d\mathbf{x}-\underset{\Omega }{\int }\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{u}_i\right)\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{v}_i\right)d\mathbf{x}-$$

$$-\underset{\Omega }{\int }\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{u}_i\right)\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)d\mathbf{x}{]}^2\le 4[{\left(\underset{\Omega }{\int }\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}\right)\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{v}_i\right)d\mathbf{x}\right)}^2+$$

$$+{\left(\underset{\Omega }{\int }\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}\right)\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)d\mathbf{x}\right)}^2+$$

$$+{\left(\underset{\Omega }{\int }\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{u}_i\right)\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{v}_i\right)d\mathbf{x}\right)}^2+{\left(\underset{\Omega }{\int }\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}{u}_i\right)\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)d\mathbf{x}\right)}^2]\le$$

$$\le 4[\left(\underset{\Omega }{\int }{\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}\right)}^{2}d\mathbf{x}\right)\left(\underset{\Omega }{\int }{\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)}^2{v}_i^{2}d\mathbf{x}\right)+\left(\underset{\Omega }{\int }{\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_j}\right)}^{2}d\mathbf{x}\right)\left(\underset{\Omega }{\int }{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)}^{2}d\mathbf{x}\right)+$$

$$+\left(\underset{\Omega }{\int }{\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)}^2{u}_i^{2}d\mathbf{x}\right)\left(\underset{\Omega }{\int }{\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)}^2{v}_i^{2}d\mathbf{x}\right)+\left(\underset{\Omega }{\int }{\left(\frac{\partial {\rho }^{\nu }}{\partial x_j}\right)}^2{u}_i^{2}d\mathbf{x}\right)\left(\underset{\Omega }{\int }{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_j}\right)}^{2}d\mathbf{x}\right)].$$

Let us summarize $S_{i,1}$ and $S_{i,2}$ using inequality $a_1{b}_1+a_2{b}_2\le (a_1+a_2)(b_1+b_2)$, $a_k,b_k\ge 0;$ we conclude the following.

$$\sum _{j=1}^2{S}_{i,j}\le 4[\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial {\rho }^{\nu }}{\partial x_1}\right)}^2+{\left(\frac{\partial {\rho }^{\nu }}{\partial x_2}\right)}^2\right]v_i^{2}d\mathbf{x}\right)·\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_1}\right)}^2+{\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_2}\right)}^2\right]d\mathbf{x}\right)+$$

$$+\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_1}\right)}^2+{\left(\frac{\partial \left({\rho }^{\nu }{u}_i\right)}{\partial x_2}\right)}^2\right]d\mathbf{x}\right)·\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_1}\right)}^2+{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_2}\right)}^2\right]d\mathbf{x}\right)+$$

$$+\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial {\rho }^{\nu }}{\partial x_1}\right)}^2+{\left(\frac{\partial {\rho }^{\nu }}{\partial x_2}\right)}^2\right]u_i^{2}d\mathbf{x}\right)·\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial {\rho }^{\nu }}{\partial x_1}\right)}^2+{\left(\frac{\partial {\rho }^{\nu }}{\partial x_2}\right)}^2\right]v_i^{2}d\mathbf{x}\right)+$$

$$+\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial {\rho }^{\nu }}{\partial x_1}\right)}^2+{\left(\frac{\partial {\rho }^{\nu }}{\partial x_2}\right)}^2\right]u_i^{2}d\mathbf{x}\right)·\left(\underset{\Omega }{\int }\left[{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_1}\right)}^2+{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_2}\right)}^2\right]d\mathbf{x}\right)].$$

For an arbitrary $\beta$, we have the following.

$$\frac{\partial {\rho }^{\beta }}{\partial x_i}=\left\{\begin{array}{c}\beta {\rho }^{\beta -2}{x}_i,\mathbf{x}\in {\Omega }_{\delta },\hfill \\ 0,\mathbf{x}\in \overline{\Omega }\backslash {\Omega }_{\delta },\hfill \end{array}\right.$$

(37)

Apply (37) with $\beta =\nu ;$ then, we have the following.

$${\left(\frac{\partial {\rho }^{\nu }}{\partial x_1}\right)}^2+{\left(\frac{\partial {\rho }^{\nu }}{\partial x_2}\right)}^2=\left\{\begin{array}{c}{\nu }^2{\rho }^{2\nu -2},\mathbf{x}\in {\Omega }_{\delta },\hfill \\ 0,\mathbf{x}\in \overline{\Omega }\backslash {\Omega }_{\delta },\hfill \end{array}\right.$$

(38)

Using the inequality (14) of Lemma 2 for $u_i$ and $v_i$ and the definition of the $L_{2,0}(\Omega )$ norm for $\nabla \left({\rho }^{\nu }{u}_i\right)$ and $\nabla \left({\rho }^{\nu }{v}_i\right),$ we have the following:

$$\sum _{j=1}^2{S}_{i,j}\le 4{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel v_i{\parallel }_{L_{2,\nu }(\Omega )}^2·\parallel \nabla \left({\rho }^{\nu }{u}_i\right){\parallel }_{L_{2,0}(\Omega )}^2+4\parallel \nabla \left({\rho }^{\nu }{u}_i\right){\parallel }_{L_{2,0}(\Omega )}^2·{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2+$$

$$+4{\left[{\nu }^2{C}_4^2{\delta }^{2\nu }\right]}^2\parallel u_i{\parallel }_{L_{2,\nu }(\Omega )}^2·\parallel v_i{\parallel }_{L_{2,\nu }(\Omega )}^2+4{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel u_i{\parallel }_{L_{2,\nu }(\Omega )}^2·{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2=$$

$$=4\left[{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel u_i{\parallel }_{L_{2,\nu }(\Omega )}^2+{\parallel \nabla \left({\rho }^{\nu }{u}_i\right)\parallel }_{L_{2,0}(\Omega )}^2\right]·\left[{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel v_i{\parallel }_{L_{2,\nu }(\Omega )}^2+{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2\right],$$

such that the following is the case.

$$\sum _{j=1}^2{S}_{i,j}\le 4\left[{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel u_i{\parallel }_{L_{2,\nu }(\Omega )}^2+{\parallel \nabla \left({\rho }^{\nu }{u}_i\right)\parallel }_{L_{2,0}(\Omega )}^2\right]\times$$

$$\begin{array}{c}\hfill \times \left[{\nu }^2{C}_4^2{\delta }^{2\nu }\parallel v_i{\parallel }_{L_{2,\nu }(\Omega )}^2+{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2\right].\end{array}$$

(39)

Applying equalities ${\parallel u_i\parallel }_{L_{2,\nu }(\Omega )}={\parallel {\rho }^{\nu }{u}_i\parallel }_{L_{2,0}(\Omega )},$ ${\parallel v_i\parallel }_{L_{2,\nu }(\Omega )}={\parallel {\rho }^{\nu }{v}_i\parallel }_{L_{2,0}(\Omega )}$ and inequality (13) of Lemma 1 for evaluating the right-hand side (39), we derive the following.

$$\begin{array}{c}\hfill \sum _{j=1}^2{S}_{i,j}\le 4{\left(1+{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu }\right)}^2\parallel \nabla \left({\rho }^{\nu }{u}_i\right){\parallel }_{L_{2,0}(\Omega )}^2·{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2.\end{array}$$

(40)

Let us sum the inequalities (40) over variable $i,i=1,2,$ and substitute it into Equation (33). Using inequality $a_1{b}_1+a_2{b}_2\le (a_1+a_2)(b_1+b_2),a_k,b_k\ge 0,$ we have a sequence of inequalities:

$${\left[a(\mathbf{u},\mathbf{v})\right]}^2\le 16{\left(1+{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu }\right)}^2\sum _{i=1}^2\left[\parallel \nabla \left({\rho }^{\nu }{u}_i\right){\parallel }_{L_{2,0}(\Omega )}^2·{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2\right]\le$$

$$\le 16{\left(1+{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu }\right)}^2\left[\sum _{i=1}^2\parallel \nabla \left({\rho }^{\nu }{u}_i\right){\parallel }_{L_{2,0}(\Omega )}^2\right]·\left[\sum _{i=1}^2\parallel \nabla \left({\rho }^{\nu }{v}_i\right){\parallel }_{L_{2,0}(\Omega )}^2\right],$$

so that the following is the case.

$$\begin{array}{c}\hfill {\left[a(\mathbf{u},\mathbf{v})\right]}^2\le 16{\left(1+{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu }\right)}^2\left[\sum _{i=1}^2\parallel \nabla \left({\rho }^{\nu }{u}_i\right){\parallel }_{L_{2,0}(\Omega )}^2\right]·\left[\sum _{i=1}^2\parallel \nabla \left({\rho }^{\nu }{v}_i\right){\parallel }_{L_{2,0}(\Omega )}^2\right].\end{array}$$

(41)

Due to inequality (15) of Lemma 3 for the factors of the right-hand side (41), we derive the following.

$${\left[a(\mathbf{u},\mathbf{v})\right]}^2\le 64{\left(1+{\nu }^2{C}_3^2{C}_4^2{\delta }^{2\nu }\right)}^2{\left(1+{\nu }^2{C}_4^2{\delta }^{2\nu }\right)}^2{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2.$$

It remains to take the square root of both sides of the last inequality. The estimate (29) is proven.

(2) Consider arbitrary functions $s\in L_{2,\nu }^0(\Omega ,\delta ),$$\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )$ and the following.

$$\begin{array}{c}\hfill {\left[b_1(\mathbf{v},s)\right]}^2={\left[\underset{\Omega }{\int }s(\frac{\partial \left({\rho }^{2\nu }{v}_1\right)}{\partial x_1}+\frac{\partial \left({\rho }^{2\nu }{v}_2\right)}{\partial x_2})d\mathbf{x}\right]}^2.\end{array}$$

(42)

Insofar as the following is the case:

$$\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_i}={\rho }^{\nu }\left(\frac{\partial {\rho }^{\nu }}{\partial x_i}\right)v_i+{\rho }^{\nu }\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_i},i=1,2,$$

then, using this and the Cauchy–Schwarz inequality, we have a sequence of estimates:

$${\left(\underset{\Omega }{\int }s\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_i}d\mathbf{x}\right)}^2\le 2\left(\underset{\Omega }{\int }{\rho }^{2\nu }{s}^{2}d\mathbf{x}\right)·\left(\underset{\Omega }{\int }{\left(\frac{\partial {\rho }^{\nu }}{\partial x_i}\right)}^2{v}_i^{2}d\mathbf{x}\right)+2\left(\underset{\Omega }{\int }{\rho }^{2\nu }{s}^{2}d\mathbf{x}\right)\times$$

$$\times \left(\underset{\Omega }{\int }{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_i}\right)}^{2}d\mathbf{x}\right)\le 2{\parallel s\parallel }_{L_{2,\nu }(\Omega )}^2·\left(\underset{\Omega }{\int }[\sum _{i=1}^2{\left(\frac{\partial {\rho }^{\nu }}{\partial x_i}\right)}^2]v_i^{2}d\mathbf{x}+\underset{\Omega }{\int }\sum _{i=1}^2{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_i}\right)}^{2}d\mathbf{x}\right),$$

such that the following is the case.

$$\begin{array}{c}\hfill {\left(\underset{\Omega }{\int }s\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_i}d\mathbf{x}\right)}^2\le 2{\parallel s\parallel }_{L_{2,\nu }(\Omega )}^2\left(\underset{\Omega }{\int }\left[\sum _{i=1}^2{\left(\frac{\partial {\rho }^{\nu }}{\partial x_i}\right)}^2\right]v_i^{2}d\mathbf{x}+\underset{\Omega }{\int }\sum _{i=1}^2{\left(\frac{\partial \left({\rho }^{\nu }{v}_i\right)}{\partial x_i}\right)}^{2}d\mathbf{x}\right).\end{array}$$

(43)

Using representation (38) and inequality (14) of Lemma 2, we conclude the following.

$$\underset{\Omega }{\int }\left[\sum _{i=1}^2{\left(\frac{\partial {\rho }^{\nu }}{\partial x_i}\right)}^2\right]v_i^{2}d\mathbf{x}={\nu }^2\underset{{\Omega }_{\delta }}{\int }{\rho }^{2(\nu -1)}{v}_i^{2}d\mathbf{x}\le {\nu }^2{C}_4^2{\delta }^{2\nu }{\parallel v_i\parallel }_{L_{2,\nu }(\Omega )}^2.$$

Applying this to estimate the first term of the second factor on the right-hand side (43), we have the following.

$$\begin{array}{c}\hfill {\left(\underset{\Omega }{\int }s\frac{\partial \left({\rho }^{2\nu }{v}_i\right)}{\partial x_i}d\mathbf{x}\right)}^2\le 2{\parallel s\parallel }_{L_{2,\nu }(\Omega )}^2·\left({\nu }^2{C}_4^2{\delta }^{2\nu }\parallel v_i{\parallel }_{L_{2,\nu }(\Omega )}^2+{\parallel \nabla \left({\rho }^{\nu }{v}_i\right)\parallel }_{L_{2,0}(\Omega )}^2\right).\end{array}$$

(44)

Let us use inequality (15) of Lemma 3 for the last term of the second factor on the right-hand side (44); then, we estimate value ${\left[b_1(\mathbf{v},s)\right]}^2$ in Equation (42):

$${\left[b_1(\mathbf{v},s)\right]}^2\le 2\left[{\left(\underset{\Omega }{\int }s\frac{\partial \left({\rho }^{2\nu }{v}_1\right)}{\partial x_1}d\mathbf{x}\right)}^2+{\left(\underset{\Omega }{\int }s\frac{\partial \left({\rho }^{2\nu }{v}_2\right)}{\partial x_2}d\mathbf{x}\right)}^2\right]\le$$

$$\le {4\parallel s\parallel }_{L_{2,\nu }(\Omega )}^2·\left({\nu }^2{C}_4^2{\delta }^{2\nu }{\parallel \mathbf{v}\parallel }_{{\mathbf{L}}_{2,\nu }(\Omega )}^2+{2\left|\mathbf{v}\right|}_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2+2{\nu }^2{C}_4^2{\delta }^{2\nu }{\parallel \mathbf{v}\parallel }_{{\mathbf{L}}_{2,\nu }(\Omega )}^2\right)\le$$

$$\le 4\left(2+3{\nu }^2{C}_4^2{\delta }^{2\nu }\right){\parallel s\parallel }_{L_{2,\nu }(\Omega )}^2·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2,$$

such that the following is the case.

$${\left[b_1(\mathbf{v},s)\right]}^2\le 4\left(2+3{\nu }^2{C}_4^2{\delta }^{2\nu }\right){\parallel s\parallel }_{L_{2,\nu }(\Omega )}^2·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2.$$

It remains to take the square root of both parts of the last inequality. The estimate (30) is set.

(3) Consider arbitrary functions $q\in L_{2,\nu }^0(\Omega ,\delta ),\mathbf{u}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )$ and the following.

$$\begin{array}{c}\hfill {\left[b_2(\mathbf{u},q)\right]}^2={\left[\underset{\Omega }{\int }{\rho }^{2\nu }q\left(\frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}\right)d\mathbf{x}\right]}^2.\end{array}$$

(45)

Using the Cauchy–Schwarz inequality, we have the following.

$${\left(\underset{\Omega }{\int }\left({\rho }^{\nu }q\right)\left({\rho }^{\nu }\left(\frac{\partial u_i}{\partial x_i}\right)\right)d\mathbf{x}\right)}^2\le \left(\underset{\Omega }{\int }{\rho }^{2\nu }{q}^{2}d\mathbf{x}\right)·\left(\underset{\Omega }{\int }{\rho }^{2\nu }{\left(\frac{\partial u_i}{\partial x_i}\right)}^{2}d\mathbf{x}\right)\le$$

$$\le {\parallel q\parallel }_{L_{2,\nu }(\Omega )}^2·\left(\underset{\Omega }{\int }{\rho }^{2\nu }\left[{\left(\frac{\partial u_i}{\partial x_1}\right)}^2+{\left(\frac{\partial u_i}{\partial x_2}\right)}^2\right]d\mathbf{x}\right)\le {\parallel q\parallel }_{L_{2,\nu }(\Omega )}^2·{\parallel u_i\parallel }_{W_{2,\nu }^1(\Omega )}^2,$$

Applying this, to evaluate the right-hand side (45), we conclude the following:

$${\left[b_2(\mathbf{u},q)\right]}^2\le 2\sum _{i=1}^2{\left[\underset{\Omega }{\int }{\rho }^{2\nu }q\left(\frac{\partial u_i}{\partial x_i}\right)d\mathbf{x}\right]}^2\le {2\parallel q\parallel }_{L_{2,\nu }(\Omega )}^2·{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2,$$

then the following is obtained.

$${\left[b_2(\mathbf{u},q)\right]}^2\le {2\parallel q\parallel }_{L_{2,\nu }(\Omega )}^2·{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2.$$

It remains to take the square root of both parts of the last estimate. Inequality (31) is established.

(4) Consider arbitrary functions $\mathbf{v}\in {\mathbf{W}}_{2,\nu }^1(\Omega ,\delta ),$ $\mathbf{f}\in {\mathbf{L}}_{2,\gamma }(\Omega ,\delta ),\nu \ge \gamma \ge 0$ and the following.

$$\begin{array}{c}\hfill {\left[l\left(\mathbf{v}\right)\right]}^2={\left[\sum _{i=1}^2\underset{\Omega }{\int }{\rho }^{2\nu }{f}_i{v}_{i}d\mathbf{x}\right]}^2\le 2\sum _{i=1}^2{\left[\underset{\Omega }{\int }{\rho }^{2\nu }{f}_i{v}_{i}d\mathbf{x}\right]}^2.\end{array}$$

(46)

For $i=1,2,$ we have the following:

$${\left[\underset{\Omega }{\int }{\rho }^{2\nu }{f}_i{v}_{i}d\mathbf{x}\right]}^2\le \parallel f_i{\parallel }_{L_{2,\nu }(\Omega )}^2·\parallel v_i{\parallel }_{L_{2,\nu }(\Omega )}^2\le C_{13}^2\parallel f_i{\parallel }_{L_{2,\gamma }(\Omega )}^2·{\parallel v_i\parallel }_{W_{2,\nu }^1(\Omega )}^2,$$

then

$${\left[l\left(\mathbf{v}\right)\right]}^2\le 2C_{13}^2\sum _{i=1}^2\left[\parallel f_i{\parallel }_{L_{2,\gamma }(\Omega )}^2·{\parallel v_i\parallel }_{W_{2,\nu }^1(\Omega )}^2\right]\le$$

$$\le 2C_{13}^2\left[\sum _{i=1}^2\parallel f_i{\parallel }_{L_{2,\gamma }(\Omega )}^2\right]·\left[\sum _{i=1}^2\parallel v_i{\parallel }_{W_{2,\nu }^1(\Omega )}^2\right]=2C_{13}^2{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}^2·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2,$$

such that the following is obtained.

$${\left[l\left(\mathbf{v}\right)\right]}^2\le 2C_{13}^2{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}^2·{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}^2.$$

It remains to take the square root of both parts of the last inequality. The estimate (32) is set.

Theorem 7 is proved. □

3.3. Existence and Uniqueness Theorem for $R_{\nu }$-Generalized Solution

Let us prove a theorem on the existence and uniqueness of an $R_{\nu }$-generalized solution $({\mathbf{w}}_{\nu },p_{\nu })\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )\times L_{2,\nu }^0(\Omega ,\delta )$ of the Stokes problem (3)–(5) in a nonsymmetric variational formulation (12).

Theorem 8.

Let the following conditions be satisfied for $\nu >0$:

(1) 

The bilinear forms $a(\mathbf{u},\mathbf{v}), b_1(\mathbf{v},s), b_2(\mathbf{u},q)$ and the linear form $l\left(\mathbf{v}\right)$ are continuous for arbitrary functions $\mathbf{u},\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta ),$ $s, q\in L_{2,\nu }^0(\Omega ,\delta )$ and $\mathbf{f}\in {\mathbf{L}}_{2,\gamma }(\Omega ,\delta ), \nu \ge \gamma \ge 0$, and we have inequalities (29)–(32);

(2) 

There exists ${\delta }_6=min\{{\delta }_4,{\delta }_5\}>0,$ such that for any $\delta \in (0,{\delta }_6)$, the estimates (21)–(24) of the following:

$$\forall s\in L_{2,\nu }^0(\Omega ,\delta ):\underset{\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )}{sup}\frac{b_1(\mathbf{v},s)}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel s\parallel }_{L_{2,\nu }(\Omega )}}\ge {\beta }_1>0,$$

$$\forall q\in L_{2,\nu }^0(\Omega ,\delta ):\underset{\mathbf{u}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )}{sup}\frac{b_2(\mathbf{u},q)}{{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel q\parallel }_{L_{2,\nu }(\Omega )}}\ge {\beta }_2>0$$

$$\forall \mathbf{u}\in K_2:\underset{\mathbf{v}\in K_1}{sup}\frac{a(\mathbf{u},\mathbf{v})}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}\ge \alpha {\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )},$$

$$\forall \mathbf{v}\in K_1:\underset{\mathbf{u}\in K_2}{sup}\frac{a(\mathbf{u},\mathbf{v})}{{\parallel \mathbf{u}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}\ge \alpha {\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}$$

hold; then, for an arbitrary function $\mathbf{f}\in {\mathbf{L}}_{2,\gamma }(\Omega ,\delta ), \gamma \le \nu ,$ there is a unique $R_{\nu }$-generalized solution $({\mathbf{w}}_{\nu },p_{\nu })\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )\times L_{2,\nu }^0(\Omega ,\delta )$ of the Stokes problem (3)–(5) in statement (12). There is a positive constant $C_{14}=\frac{C_{12}}{\alpha }\left(1+\frac{\alpha +C_9}{{\beta }_1}\right)$ that satisfies the following inequality.

$$\begin{array}{c}\hfill {\parallel {\mathbf{w}}_{\nu }\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}+\parallel p_{\nu }{\parallel }_{L_{2,\nu }(\Omega )}\le C_{14}{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}.\end{array}$$

(47)

Proof. 

Due to the fulfillment of inequalities (21)–(24), we can apply Theorem 2.1 in combination with Remark 2.1 (see [24]), which was proved for an abstract mixed problem in a nonsymmetric formulation, to our variational formulation (12). Then, there is a unique $R_{\nu }$-generalized solution $({\mathbf{w}}_{\nu },p_{\nu })\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )\times L_{2,\nu }^0(\Omega ,\delta ).$

Let us prove estimate (47).

• Consider an arbitrary function $\mathbf{v}\in K_1$ and then $b_1(\mathbf{v},p_{\nu })=0$ and $a({\mathbf{w}}_{\nu },\mathbf{v})=l\left(\mathbf{v}\right).$ Using sequentially estimates (23) $({\mathbf{w}}_{\nu }\in K_2)$ and (32), we conclude the following:

  $$\alpha \parallel {\mathbf{w}}_{\nu }{\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le \underset{\mathbf{v}\in K_1}{sup}\frac{a({\mathbf{w}}_{\nu },\mathbf{v})}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}=\underset{\mathbf{v}\in K_1}{sup}\frac{l\left(\mathbf{v}\right)}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}\le$$

  $$\le \underset{\mathbf{v}\in K_1}{sup}\frac{C_{12}{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}·{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}=C_{12}{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )},$$

  such that the following is the case.

  $$\begin{array}{c}\hfill {\parallel {\mathbf{w}}_{\nu }\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le \frac{C_{12}}{\alpha }{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}.\end{array}$$

  (48)
• Consider an arbitrary function $\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )$. Using sequentially the estimates (21), (29), (32) and (48), we derive the following:

  $${\beta }_1{\parallel p_{\nu }\parallel }_{L_{2,\nu }(\Omega )}\le \underset{\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )}{sup}\frac{b_1(\mathbf{v},p_{\nu })}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}=\underset{\mathbf{v}\in {\mathbf{W}}_{2,\nu }^{1,0}(\Omega ,\delta )}{sup}\frac{l\left(\mathbf{v}\right)-a({\mathbf{w}}_{\nu },\mathbf{v})}{{\parallel \mathbf{v}\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}}\le$$

  $$\le C_{12}{\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}+C_9\parallel {\mathbf{w}}_{\nu }{\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}\le C_{12}\left(1+\frac{C_9}{\alpha }\right){\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )},$$

  then we have the following.

  $$\begin{array}{c}\hfill {\parallel p_{\nu }\parallel }_{L_{2,\nu }(\Omega )}\le \frac{C_{12}}{{\beta }_1}\left(1+\frac{C_9}{\alpha }\right){\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}.\end{array}$$

  (49)
• Summarize the right-hand and left-hand parts of inequalities (48) and (49), respectively; we have

  $${\parallel {\mathbf{w}}_{\nu }\parallel }_{{\mathbf{W}}_{2,\nu }^1(\Omega )}+\parallel p_{\nu }{\parallel }_{L_{2,\nu }(\Omega )}\le \frac{C_{12}}{\alpha }\left(1+\frac{\alpha +C_9}{{\beta }_1}\right){\parallel \mathbf{f}\parallel }_{{\mathbf{L}}_{2,\gamma }(\Omega )}.$$

Theorem 8 is proved. □

4. Conclusions

In the presented paper, the concept of an $R_{\nu }$-generalized solution of the Stokes problem with a homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary is introduced. The variational formulation of the problem is nonsymmetric. The existence and uniqueness of the $R_{\nu }$-generalized solution of the Stokes problem in weighted sets is established. An estimate of its norm in terms of the norm of the right-hand side function is obtained.

The results of the paper can be extended to the case of a polygonal domain with several re-entrant corners on its boundary. Apply the approach to solve other fluid dynamics problems with corner singularity. In particular, for linearized Navier–Stokes equations in rotation and convective forms, see [29,30]. To perform this, we need to prove estimates (23) and (24) of Theorem 6 and (29) of Theorem 7 that correspond to these problems. Consider cases of other boundary conditions that are different from the homogeneous Dirichlet condition, which is a no-slip behavior of the fluid on the fixed walls (for example, Neumann, Robin and periodic or mixed boundary conditions, including inhomogeneous ones). An established fact of the existence and uniqueness of an $R_{\nu }$-generalized solution in weighted sets is that, firstly, it will make it possible to create efficient numerical methods without a loss of accuracy for hydrodynamic problems with a corner singularity. Secondly, it will help us in obtaining a priori estimates for the convergence rate of an approximate $R_{\nu }$-generalized solution to an exact one in the norms of weighted spaces. Thirdly, it will promote the determination of the optimal sets of parameters $\nu$ and $\delta$ to achieve the required order of accuracy, as well as finding the body of optimal parameters depending on the value of the re-entrant corner. In particular, this was established for the Lame system in a domain with a re-entrant corner equal to $2\pi ,$ which is a mathematical model for the crack problem (see [14,20]).