Continuous dependence and convergence for a Kelvin–Voigt fluid of order one

Abstract

It is shown that the solution to the boundary - initial value problem for a Kelvin–Voigt fluid of order one depends continuously upon the Kelvin–Voigt parameters, the viscosity, and the viscoelastic coefficients. Convergence of a solution is also shown.

1 Introduction

The equations for a viscoelastic fluid have been increasingly occupying attention. Such fluids occur everywhere in real life and differ from Navier - Stokes fluids in that the stress depends on the history of the velocity gradient. As such, the equations for such fluids present many mathematical challenges, see e.g. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].

A particular class of viscoelastic fluids of interest here are those associated with the names of Kelvin and of Voigt, see e.g. [16,17,18,19,20,21]. Much of the interest in these fluids stems from work of Russian writers in this field and analytical studies of Kelvin–Voigt fluids are contained in [22, 23], with generalizations of these models and analytical results to encompass the non - isothermal situation in [24,25,26]. A lucid account of viscoelastic fluids associated with the names of Maxwell, of Oldroyd, and of Kelvin and Voigt, is contained in [27], where the solution existence question is analysed, see also [28].

Kelvin–Voigt fluids are being increasingly employed in real life applications especially in industrial and engineering contexts. Many of these are reviewed in [19], but we highlight here their employment in viscous dampers in large buildings, cf. [29, 30]. For example, a large viscous damper is utilized in the 1667 feet high tower Taipei 101 in the city of Taipei. This building has been constructed to withstand earthquakes and typhoons and the large viscous damper is essential.

The goal of this article is to analyse continuous dependence of a solution to the equations for a Kelvin–Voigt fluid of order one. Such questions are important and belong to the general area of structural stability. [31], p. 304, pose the problem of what effect does changing the parameters in a differential equation have upon the solution to such an equation. They introduce this as the concept of structural stability. In this article we concentrate on continuous dependence on parameters in the equations for a Kelvin–Voigt fluid of order one. This is continuous dependence on the model itself which is structural stability of the model. We point out that continuous dependence on the model has been the subject of much recent attention in continuum mechanics, see e.g. [13, 16, 32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54].

In the next section we introduce the Kelvin–Voigt equations of order one. The following section establishes continuous dependence on the Kelvin–Voigt parameter, $\lambda$. This is important as this coefficient multiplies the highest derivative term in the equations. After this we establish continuous dependence upon the remaining coefficients in the governing equations.

2 The Kelvin–Voigt equations of order one

Throughout this article we employ standard indicial notation in conjunction with the Einstein summation convention. Hence, partial differentiation with respect to $x_i$ is written as $_{,i}\equiv \partial /\partial x_i$, and $\varDelta$ denotes the Laplacian in ${{\mathbb {R}}}^3$.

Let $v_i(\mathbf{x},t)$ be the velocity, $p(\mathbf{x},t)$ be the pressure, and let $f_i(\mathbf{x},t)$ be the body force at position $\mathbf{x}$ and time t. Now, let $q_i^m(\mathbf{x},t)$, $m=1,\ldots ,L,$ be viscoelastic variables. Then [27] define a hierarchy of viscoelastic fluid models. They write that a Maxwell fluid of order L, $L\in {{\mathbb {N}}}$, satisfies the equations

$$\begin{aligned} \begin{aligned}&v_{i,t}+v_jv_{i,j}-\sum _{m=1}^L\beta ^{(1)}_m\varDelta q^m_i+p_{,i}=f_i,\\&v_{i,i}=0,\\&q^m_{i,t}+\gamma _mq^m_i=v_i\,,\quad m=1,\ldots ,L. \end{aligned} \end{aligned}$$

(1)

An Oldroyd fluid of order L, $L\in {{\mathbb {N}}}$, satisfies the equations

$$\begin{aligned} \begin{aligned}&v_{i,t}+v_jv_{i,j}-\mu ^{(2)}\varDelta v_i-\sum _{m=1}^L\beta ^{(2)}_m\varDelta q^m_i+p_{,i}=f_i,\\&v_{i,i}=0,\\&q^m_{i,t}+\gamma _mq^m_i=v_i\,,\quad m=1,\ldots ,L. \end{aligned} \end{aligned}$$

(2)

A Kelvin–Voigt fluid of order L, $L\in {{\mathbb {N}}}$, satisfies the equations

$$\begin{aligned}&v_{i,t}+v_jv_{i,j}-\lambda \varDelta v_{i,t}-\mu ^{(3)}\varDelta v_i-\sum _{m=1}^L\beta ^{(3)}_m\varDelta q^m_i+p_{,i}=f_i, \nonumber \\&v_{i,i}=0, \nonumber \\&q^m_{i,t}+\gamma _mq^m_i=v_i\,,\quad m=1,\ldots ,L. \end{aligned}$$

(3)

In equations (1) - (3), the coefficients $\beta ^{(k)}_m,\,\,k=1,2,3,\,\,m=1,\ldots ,L,$ $\gamma _m,m=1,\ldots ,L,$ $\mu ^{(2)}$, $\mu ^{(3)}$ and $\lambda$ are positive constants, and it should be noted that there is no sum on m in the terms $\gamma _mq^m_i$.

The nonlinearity in equations (1) - (3) consists of the $v_jv_{i,j}$ term. An equation which has been studied is one for the so called Navier - Stokes - Voigt fluid, cf. the very interesting results on attractors and regularity by [55, 56]. The Navier - Stokes - Voigt equations, which are also known as the Kelvin–Voigt equations of order zero, are

$$\begin{aligned} \begin{aligned}&v_{i,t}+v_jv_{i,j}-\lambda \varDelta v_{i,t}-\mu \varDelta v_i+p_{,i}=f_i\,,\\&v_{i,i}=0. \end{aligned} \end{aligned}$$

(4)

These equations also contain the nonlinear term $v_jv_{i,j}$ and a very interesting article establishing the existence of weak solutions and of strong solutions in appropriate $L^q(\varOmega )$ spaces for $1<q<\infty$ with $\varOmega$ a bounded domain in ${\mathbb {R}}^d$, $d\ge 2$, is due to [57]. These writers raise the question of how equations (4) might be derived from the balance of linear momentum and continuity equations for a viscous incompressible fluid, namely,

$$\begin{aligned} \begin{aligned}&v_{i,t}+v_jv_{i,j}=T_{ji,j}+f_i\,,\\&v_{i,i}=0, \end{aligned} \end{aligned}$$

(5)

where $T_{ij}$ is the symmetric Cauchy stress tensor. They argue that some writers present for equations (4) a constitutive equation of form

$$\begin{aligned} T_{ij}=-p\delta _{ij}+2\mu d_{ij}+2\lambda d_{ij,t}\,, \end{aligned}$$

(6)

where $d_{ij}=(v_{i,j}+v_{j,i})/2$, and they point out that this is not correct since the tensor $d_{ij,t}$ is not objective. Thus, the term $-\lambda \varDelta v_{i,t}$ in (4) should be regarded as a regularization term for the Navier - Stokes equations.

One may use the [57] argument also for equations (3). However, a fully nonlinear model for a Navier - Stokes - Voigt fluid is derived by [58]. This model is now referred to as a Walters fluid, see e.g. [59]. The model of [58] recognizes the fact that $d_{ij,t}$ is not objective as [57] point out and they replace it with an objective derivative of form

$$\begin{aligned} d_{ij}^{\bigtriangledown }=d_{ij,t}+v_kd_{ij,k}-v_{j,k}d_{ik}-v_{i,k}d_{kj}\,, \end{aligned}$$

see also [59, 60]. This requires one to analyse instead of (4) the momentum equation

$$\begin{aligned} \begin{aligned} v_{i,t}+v_jv_{i,j}=&-p_{,i}+\mu \varDelta v_i+\lambda \varDelta v_{i,t}\\&+\lambda [v_k(v_{i,j}+v_{j,i})_{,k}]_{,j}\\&-\lambda [v_{j,k}(v_{i,k}+v_{k,i})]_{,j}\\&-\lambda [v_{i,k}(v_{k,j}+v_{j,k})]_{,j}\,. \end{aligned} \end{aligned}$$

(7)

Here the divergence of $\mathbf{v}$ is also zero. If we employ this argument for a Kelvin–Voigt fluid of order one then instead of equation (3)$_1$ with $L=1$ we should use (7) with a $-\beta \varDelta q_i$ term added to the right hand side.

Of particular interest to the present article are equations (1) - (3) for fluids of order one, i.e. when $L=1$. [13] study continuous dependence and convergence for a linear system arising from the Maxwell equations (1) and we develop linear equations here. In the linear case we write the analogous systems to (1) - (3) of order one as, with $f_i=0$,

Maxwell,

$$\begin{aligned} \begin{aligned}&v_{i,t}-\beta \varDelta q_i+p_{,i}=0,\\&v_{i,i}=0,\\&q_{i,t}+\gamma q_i=v_i\,, \end{aligned} \end{aligned}$$

(8)

Oldroyd,

$$\begin{aligned} \begin{aligned}&v_{i,t}-\mu \varDelta v_i-\beta \varDelta q_i+p_{,i}=0,\\&v_{i,i}=0,\\&q_{i,t}+\gamma q_i=v_i\,, \end{aligned} \end{aligned}$$

(9)

Kelvin–Voigt,

$$\begin{aligned} \begin{aligned}&v_{i,t}-\lambda \varDelta v_{i,t}-\mu \varDelta v_i-\beta \varDelta q_i+p_{,i}=0,\\&v_{i,i}=0,\\&q_{i,t}+\gamma q_i=v_i\,. \end{aligned} \end{aligned}$$

(10)

It is sometimes convenient to eliminate $q_i$ in these equations and derive the following equations for $v_i$ and a generalized pressure, of form,

Maxwell,

$$\begin{aligned} \begin{aligned}&v_{i,tt}-\beta \varDelta v_i+\gamma v_{i,t}=-\phi _{,i}\,,\\&v_{i,i}=0, \end{aligned} \end{aligned}$$

(11)

Oldroyd,

$$\begin{aligned} \begin{aligned}&v_{i,tt}-\mu \varDelta v_{i,t}+\gamma v_{i,t}-(\beta +\mu \gamma )\varDelta v_i=-\phi _{,i}\,,\\&v_{i,i}=0, \end{aligned} \end{aligned}$$

(12)

Kelvin–Voigt,

$$\begin{aligned} \begin{aligned}&v_{i,tt}-\lambda \varDelta v_{i,tt}-(\gamma \lambda +\mu )\varDelta v_{i,t}+\gamma v_{i,t}-(\beta +\mu \gamma )\varDelta v_i=-\phi _{,i}\,,\\&v_{i,i}=0, \end{aligned} \end{aligned}$$

(13)

where $\phi =p_{,t}+\gamma p$.

We observe that formally, as $\lambda \rightarrow 0$ the Kelvin–Voigt system (13) tends to the Oldroyd equations (12). As $\mu \rightarrow 0$ the Oldroyd equations (12) tend to the Maxwell equations (11). Furthermore, if we put $\gamma =\beta =1/\zeta$ and rescale the pressure $\phi$ as $\phi =\psi /\zeta$, then the Maxwell system (11) becomes

$$\begin{aligned} \begin{aligned}&\zeta v_{i,tt}-\varDelta v_i+v_{i,t}=-\psi _{,i}\,,\\&v_{i,i}=0. \end{aligned} \end{aligned}$$

(14)

When $\zeta \rightarrow 0$, equations (14) tend to those of Stokes flow, cf. [13].

In this article we concentrate on equations (10) or (13) and we establish continuous dependence of the solution upon the parameters $\lambda ,\beta ,\gamma$ and $\mu$.

3 Continuous dependence upon the Kelvin–Voigt coefficient $\lambda$

As remarked at the end of the introduction, the Kelvin–Voigt coefficient multiplies the highest derivative term in (10) or (13), and as such it is highly important that the solution depends continuously on changes in this parameter. To establish continuous dependence upon $\lambda$ we let $(u_i,q^1_i,p^1)$ and $(v_i,q^2_i,p^2)$ be solutions to equations (10) for fixed constants $\mu ,\beta$ and $\gamma$, but for $\lambda _1$ and $\lambda _2$, respectively. In each case the equations are defined on a bounded region $\varOmega$ in ${\mathbb R}^3$ with boundary $\varGamma$ which is sufficiently regular to allow application of the divergence theorem. We here restrict attention to a bounded domain in ${\mathbb {R}}^3$ although the methods work for a bounded domain in ${\mathbb {R}}^n$, $n\ge 2$.

Thus, $(u_i,q^1_i,p^1)$ satisfies the boundary - initial value problem

$$\begin{aligned} \begin{aligned}&u_{i,t}-\lambda _1\varDelta u_{i,t}-\mu \varDelta u_i-\beta \varDelta q^1_i=-p^1_{,i}\,,\\&u_{i,i}=0,\\&q^1_{i,t}+\gamma q^1_i=u_i\,, \end{aligned} \end{aligned}$$

(15)

on $\varOmega \times (0,T)$, for some $T>0$, with the boundary and initial conditions,

$$\begin{aligned} \begin{aligned}&u_i=\ell _i(\mathbf{x}),\quad \mathbf{x}\in \varGamma ,\\&u_i(\mathbf{x},0)=f_i(\mathbf{x}),\qquad q^1_i(\mathbf{x},0)=h_i(\mathbf{x}),\qquad \mathbf{x}\in \varOmega , \end{aligned} \end{aligned}$$

(16)

for prescribed functions $f_i,h_i$ and $\ell _i$. In general, one should like $\ell _i(\mathbf{x})$ to also depend on t. The method we employ requires us to allow $\ell _i$ to depend only on $\mathbf{x}$, which was also the case in [13].

The solution $(v_i,q^2_i,p^2)$ satisfies the boundary - initial value problem

$$\begin{aligned} \begin{aligned}&v_{i,t}-\lambda _2\varDelta v_{i,t}-\mu \varDelta v_i-\beta \varDelta q^2_i=-p^2_{,i}\,,\\&v_{i,i}=0,\\&q^2_{i,t}+\gamma q^2_i=v_i\,, \end{aligned} \end{aligned}$$

(17)

on $\varOmega \times (0,T)$, for some $T>0$, with the boundary and initial conditions,

$$\begin{aligned} \begin{aligned}&v_i=\ell _i(\mathbf{x}),\quad \mathbf{x}\in \varGamma ,\\&v_i(\mathbf{x},0)=f_i(\mathbf{x}),\qquad q^2_i(\mathbf{x},0)=h_i(\mathbf{x}),\qquad \mathbf{x}\in \varOmega . \end{aligned} \end{aligned}$$

(18)

Define the difference variables $w_i,q_i,\pi$ and $\lambda$ as

$$\begin{aligned} w_i=u_i-v_i,\quad q_i=q^1_i-q^2_i,\quad \pi =p^1-p^2,\quad \lambda =\lambda _1-\lambda _2. \end{aligned}$$

Then $(w_i,q_i,\pi )$ satsifies the boundary - initial value problem

$$\begin{aligned} \begin{aligned}&w_{i,t}-\lambda \varDelta u_{i,t}-\lambda _2\varDelta w_{i,t}-\mu \varDelta w_i-\beta \varDelta q_i=-\pi _{,i}\,,\\&w_{i,i}=0,\\&q_{i,t}+\gamma q_i=w_i\,, \end{aligned} \end{aligned}$$

(19)

on $\varOmega \times (0,T)$, for some $T>0$, with the boundary and initial conditions,

$$\begin{aligned} \begin{aligned}&w_i=0,\quad \mathbf{x}\in \varGamma ,\\&w_i(\mathbf{x},0)=0,\qquad q_i(\mathbf{x},0)=0,\qquad \mathbf{x}\in \varOmega . \end{aligned} \end{aligned}$$

(20)

Let now $(\cdot ,\cdot )$ and $\Vert \cdot \Vert$ denote the inner product and norm on $L^2(\varOmega )$.

We commence the continuous dependence analysis by multiplying (19)$_1$ by $w_i$ and integrating over $\varOmega$. After use of the boundary conditions one obtains

$$\begin{aligned} \frac{d}{dt}\Bigl (\frac{1}{2}\Vert \mathbf{w}\Vert ^2+\frac{\lambda }{2}\Vert \nabla \mathbf{w}\Vert ^2\Bigr )+\mu \Vert \nabla \mathbf{w}\Vert ^2 =-\lambda (u_{i,tj},w_{i,j})-\beta (w_{i,j},q_{i,j}). \end{aligned}$$

(21)

Next we differentiate equation (19)$_3$ with respect to $x_j$ and t, multiply by $q_{i,j}$ and integrate over $\varOmega$ to find

$$\begin{aligned} \frac{d}{dt}\frac{1}{2}\Vert \nabla \mathbf{q}\Vert ^2+\gamma \Vert \nabla \mathbf{q}\Vert ^2 =(w_{i,j},q_{i,j}). \end{aligned}$$

(22)

Upon forming (21)+$\beta \times$(22) one derives

$$\begin{aligned} \frac{d}{dt}\Bigl (\frac{1}{2}\Vert \mathbf{w}\Vert ^2+&\frac{\lambda _2}{2}\Vert \nabla \mathbf{w}\Vert ^2+\frac{\beta }{2}\Vert \nabla \mathbf{q}\Vert ^2\Bigr ) +\beta \lambda \Vert \nabla \mathbf{q}\Vert ^2+\mu \Vert \nabla \mathbf{w}\Vert ^2 \nonumber \\&=-\lambda (u_{i,tj},w_{i,j})\nonumber \\&\le \frac{\lambda ^2}{2\epsilon }\Vert \nabla \mathbf{u}_{,t}\Vert ^2 +\frac{\epsilon }{2}\Vert \nabla \mathbf{w}\Vert ^2\,, \end{aligned}$$

(23)

for $\epsilon >0$, where we have employed the arithmetic - geometric mean inequality. Pick $\epsilon =\mu$ and integrate over (0, t) to find

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\Vert \mathbf{w}\Vert ^2+\frac{\lambda _2}{2}\Vert \nabla \mathbf{w}\Vert ^2+\frac{\beta }{2}\Vert \nabla \mathbf{q}\Vert ^2 +\beta \gamma \int _0^t\Vert \nabla \mathbf{q}\Vert ^2ds\\&\qquad +\frac{\mu }{2}\int _0^t\Vert \nabla \mathbf{w}\Vert ^2ds\le \frac{\lambda ^2}{2\mu }\int _0^t\Vert \nabla \mathbf{u}_{,s}\Vert ^2ds. \end{aligned} \end{aligned}$$

(24)

We now proceed to bound the right hand side of (24) in terms of data. To do this we differentiate (15)$_{1,3}$ to obtain

$$\begin{aligned} \begin{aligned}&u_{i,tt}-\lambda _1\varDelta u_{i,tt}-\mu \varDelta u_{i,t}-\beta \varDelta q^1_{i,t}=-p^1_{,it}\,,\\&q^1_{i,ttj}+\gamma q^1_{i,tj}=u_{i,tj}\,. \end{aligned} \end{aligned}$$

(25)

To employ (25)$_1$ we additionally prescribe $u_{i,t}(\mathbf{x},0)=g_i(\mathbf{x})$. Multiply (25)$_1$ by $u_{i,t}$ and integrate over $\varOmega$ noting $u_{i,t}=0$ on $\varGamma$. Multiply (25)$_2$ by $q^1_{i,tj}$ and integrate over $\varOmega$. After some manipulation one may obtain

$$\begin{aligned} \frac{d}{dt}\Bigl (\frac{1}{2}\Vert \mathbf{u}_{,t}\Vert ^2+\frac{\lambda _1}{2}\Vert \nabla \mathbf{u}_{,t}\Vert ^2+\frac{\beta }{2}\Vert \nabla \mathbf{q}^1_{,t}\Vert ^2\Bigr ) +\mu \Vert \nabla \mathbf{u}_{,t}\Vert ^2+\beta \gamma \Vert \nabla \mathbf{q}^1_{,t}\Vert ^2=0. \end{aligned}$$

This equation is integrated over (0, t) to find using the initial data

$$\begin{aligned}&\frac{1}{2}\Vert \mathbf{u}_{,t}\Vert ^2+\frac{\lambda _1}{2}\Vert \nabla \mathbf{u}_{,t}\Vert ^2+\frac{\beta }{2}\Vert \nabla \mathbf{q}^1_{,t}\Vert ^2 +\mu \int _0^t\Vert \nabla \mathbf{u}_{,s}\Vert ^2ds\nonumber \\&\qquad +\beta \gamma \int _0^t\Vert \nabla \mathbf{q}^1_{,s}\Vert ^2ds =\frac{1}{2}\Vert \mathbf{g}\Vert ^2+\frac{\lambda _1}{2}\Vert \nabla \mathbf{g}\Vert ^2+\Vert \nabla \mathbf{f}-\gamma \nabla \mathbf{g}\Vert ^2\nonumber \\&\quad =d_0\,, \end{aligned}$$

(26)

where $d_0$ is the data term defined in (26). From this expression one deduces

$$\begin{aligned} \int _0^t\Vert \nabla \mathbf{u}_{,s}\Vert ^2ds\le \frac{1}{\mu }\,d_0\,. \end{aligned}$$

Employ this in (24) to obtain the inequality

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\Vert \mathbf{w}\Vert ^2+\frac{\lambda _2}{2}\Vert \nabla \mathbf{w}\Vert ^2+\frac{\beta }{2}\Vert \nabla \mathbf{q}\Vert ^2 +\beta \gamma \int _0^t\Vert \nabla \mathbf{q}\Vert ^2ds\\&\qquad +\frac{\mu }{2}\int _0^t\Vert \nabla \mathbf{w}\Vert ^2ds\le \frac{d_0}{2\mu ^2}\,\lambda ^2. \end{aligned} \end{aligned}$$

(27)

Inequality (27) establishes continuous dependence of a solution on $\lambda$ in the measures $\Vert \mathbf{w}\Vert$, $\Vert \nabla \mathbf{w}\Vert$, and $\Vert \nabla \mathbf{q}\Vert$.

Remark

If we set $\lambda _2=0$ and put $\lambda _1=\lambda$ then (27) yields a convergence result indicating how $u_i$ converges to $v_i$, a solution to the Oldroyd order one equations.

4 Continuous dependence upon the parameters $\mu ,\beta ,\gamma$

Let now $(u_i,q^1_i,p^1)$ and $(v_i,q^2_i,p^2)$ be solutions to the Kelvin–Voigt order one equations (10) for the same coefficient $\lambda$, but with values $\mu _1,\beta _1,\gamma _1$, and $\mu _2,\beta _2,\gamma _2$, respectively. Thus, these solutions satisfy the boundary - initial value problems

$$\begin{aligned} \begin{aligned}&u_{i,t}-\lambda \varDelta u_{i,t}-\mu _1\varDelta u_i-\beta _1\varDelta q^1_i=-p^1_{,i}\,,\\&u_{i,i}=0,\\&q^1_{i,t}+\gamma _1 q^1_i=u_i\,, \end{aligned} \end{aligned}$$

(28)

on $\varOmega \times (0,T)$, with

$$\begin{aligned} \begin{aligned}&u_i=\ell _i(\mathbf{x}),\quad \mathbf{x}\in \varGamma ,\\&u_i(\mathbf{x},0)=f_i(\mathbf{x}),\qquad q^1_i(\mathbf{x},0)=h_i(\mathbf{x}),\qquad \mathbf{x}\in \varOmega , \end{aligned} \end{aligned}$$

(29)

and

$$\begin{aligned} \begin{aligned}&v_{i,t}-\lambda \varDelta v_{i,t}-\mu _2\varDelta v_i-\beta _2\varDelta q^2_i=-p^2_{,i}\,,\\&v_{i,i}=0,\\&q^2_{i,t}+\gamma _2 q^2_i=v_i\,, \end{aligned} \end{aligned}$$

(30)

on $\varOmega \times (0,T)$, with

$$\begin{aligned} \begin{aligned}&v_i=\ell _i(\mathbf{x}),\quad \mathbf{x}\in \varGamma ,\\&v_i(\mathbf{x},0)=f_i(\mathbf{x}),\qquad q^2_i(\mathbf{x},0)=h_i(\mathbf{x}),\qquad \mathbf{x}\in \varOmega . \end{aligned} \end{aligned}$$

(31)

Define the difference variables by

$$\begin{aligned}&w_i=u_i-v_i,\quad q_i=q^1_i-q^2_i,\quad \pi =p^1-p^2,\\&\mu =\mu _1-\mu _2,\quad \beta =\beta _1-\beta _2\quad \gamma =\gamma _1-\gamma _2.. \end{aligned}$$

One then finds that the difference solution satisfies the boundary - initial value problem

$$\begin{aligned} \begin{aligned}&w_{i,t}-\lambda \varDelta w_{i,t}-\mu \varDelta u_i-\mu _2\varDelta w_i-\beta \varDelta q^1_i-\beta _2\varDelta q_i=-\pi _{,i}\,,\\&w_{i,i}=0,\\&q_{i,t}+\gamma q^1_i+\gamma _2 q_i=w_i\,, \end{aligned} \end{aligned}$$

(32)

on $\varOmega \times (0,T)$, with

$$\begin{aligned} \begin{aligned}&w_i=0,\quad \mathbf{x}\in \varGamma ,\\&w_i(\mathbf{x},0)=0,\qquad q_i(\mathbf{x},0)=0,\qquad \mathbf{x}\in \varOmega . \end{aligned} \end{aligned}$$

(33)

Next, differentiate (32)$_3$ with respect to $x_j$, and multiply the result by $q_{i,j}$ and integrate over $\varOmega$. Multiply (32)$_1$ by $w_i$ and integrate over $\varOmega$. After some integration by parts and use of the boundary conditions one may combine the results to derive

$$\begin{aligned} \frac{d}{dt}\Bigl (\frac{1}{2}\Vert \mathbf{w}\Vert ^2&+\frac{\lambda }{2}\Vert \nabla \mathbf{w}\Vert ^2+\frac{\beta _2}{2}\Vert \nabla \mathbf{q}\Vert ^2\Bigr ) +\mu _2\Vert \nabla \mathbf{w}\Vert ^2+\gamma _2\beta _2\Vert \nabla \mathbf{q}\Vert ^2\\&=-\mu (u_{i,j},w_{i,j})-\beta (q^1_{i,j},w_{i,j})-\beta _2\gamma (q^1_{i,j},q_{i,j}). \end{aligned}$$

We next employ the arithmetic - geometric mean inequality on the right hand side of this equation to see that

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\Bigl (\frac{1}{2}\Vert \mathbf{w}\Vert ^2&+\frac{\lambda }{2}\Vert \nabla \mathbf{w}\Vert ^2+\frac{\beta _2}{2}\Vert \nabla \mathbf{q}\Vert ^2\Bigr ) +\mu _2\Vert \nabla \mathbf{w}\Vert ^2+\gamma _2\beta _2\Vert \nabla \mathbf{q}\Vert ^2\\&\le \frac{2}{\mu _2}\,\mu ^2\Vert \nabla \mathbf{u}\Vert ^2 +\Bigl (\frac{2}{\mu _2}\,\beta ^2+\frac{1}{\beta _2\gamma _2}\,\gamma ^2\Bigr )\Vert \nabla \mathbf{q}^1\Vert ^2. \end{aligned} \end{aligned}$$

(34)

We need to bound the right hand side of (34) in terms of $\mu ^2,\beta ^2$ and $\gamma ^2$. To do this we observe that $u_i$ satsify the equations (13) for $\gamma _1,\mu _1,\beta _1$, and we impose $u_{i,t}(\mathbf{x},0)=g_i(\mathbf{x})$. We multiply the appropriate version of (13)$_1$ by $u_{i,t}$ and integrate over $\varOmega$ to obtain after integration by parts and integration over (0, t),

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Vert \mathbf{u}_{,t}\Vert ^2&+\frac{\lambda }{2}\Vert \nabla \mathbf{u}_{,t}\Vert ^2+\Bigl (\frac{\beta _1+\gamma _1\mu _1}{2}\Bigr )\Vert \nabla \mathbf{u}\Vert ^2\\&+(\gamma _1\lambda +\mu _1)\int _0^t\Vert \nabla \mathbf{u}_{,s}\Vert ^2ds+\gamma _1\int _0^t\Vert \mathbf{u}_{,s}\Vert ^2ds\\&=\frac{1}{2}\Vert \mathbf{g}\Vert ^2+\frac{\lambda }{2}\Vert \nabla \mathbf{q}\Vert ^2+\Bigl (\frac{\beta _1+\gamma _1\mu _1}{2}\Bigr )\Vert \nabla \mathbf{f}\Vert ^2. \end{aligned} \end{aligned}$$

(35)

Furthermore, differentiate (28)$_3$ to find

$$\begin{aligned} q^1_{i,jt}+\gamma _1q^1_{i,j}=u_{i,j}. \end{aligned}$$

Multiply this by $q^1_{i,j}$ and integrate over $\varOmega$ to find after use of the arithmetic - geometric mean inequality

$$\begin{aligned} \frac{d}{dt}\Vert \nabla \mathbf{q}^1\Vert ^2+\gamma _1\Vert \nabla \mathbf{q}^1\Vert ^2\le \frac{1}{\gamma _1}\Vert \nabla \mathbf{u}\Vert ^2. \end{aligned}$$

(36)

Let $d_1$ be the data term

$$\begin{aligned} d_1=\frac{1}{(\beta _1+\gamma _1\mu _1)}\,\Vert \mathbf{g}\Vert ^2+\frac{\lambda }{(\beta _1+\gamma _1\mu _1)}\,\Vert \nabla \mathbf{g}\Vert ^2 +\Vert \nabla \mathbf{f}\Vert ^2. \end{aligned}$$

Then from (35) we deduce

$$\begin{aligned} \Vert \nabla \mathbf{u}\Vert ^2\le d_1\,. \end{aligned}$$

(37)

We employ this estimate in (36) and integrate to obtain

$$\begin{aligned} \Vert \nabla \mathbf{q}^1\Vert ^2\le \frac{1}{\gamma _1^2}d_1+\Vert \nabla \mathbf{g}\Vert ^2\,e^{-\gamma _1t}=d_2\,, \end{aligned}$$

(38)

where $d_2$ is the indicated data term. Upon employment of (37) and (38) in (34) and a further integration we may obtain

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Vert \mathbf{w}\Vert ^2&+\frac{\lambda }{2}\Vert \nabla \mathbf{w}\Vert ^2+\frac{\beta _2}{2}\Vert \nabla \mathbf{q}\Vert ^2\\&+\frac{\mu _2}{2}\int _0^t\Vert \nabla \mathbf{w}\Vert ^2ds+\frac{\gamma _2\beta _2}{2}\int _0^t\Vert \nabla \mathbf{q}\Vert ^2ds \le \,d_3\,, \end{aligned} \end{aligned}$$

(39)

where $d_3$ is the data term (involving $\mu ^2,\beta ^2,\gamma ^2$)

$$\begin{aligned} d_3=\frac{2d_1t}{\mu _2}\,\mu ^2+\Bigl (\frac{2}{\mu _2}\,\beta ^2 +\frac{1}{\beta _2\gamma _2}\,\gamma ^2\Bigr )\Bigl (\frac{d_1t}{\gamma _1^2}+\frac{1}{\gamma _1}\,\Vert \nabla \mathbf{g}\Vert ^2\Bigr )\,. \end{aligned}$$

Inequality (39) demonstrates continuous dependence of the solution upon the parameters $\mu ,\beta$ and $\gamma$.