1 Introduction

The problems regarding mixtures of two or more continua are relatively recent. But these are increasingly involved in construction issues, resistance structures, etc. The great number of studies dedicated to this issue proves the increased importance of mixtures. Among the first impactful results, dedicated to mixtures, those from works [1,2,3,4,5] should be mentioned. So, in [1] the authors constructed, for each constituent, some balance laws of mass, of microinertia moment, of energy, of production of entropy. In a second paper the author expose a constitutive theory for micromorphic mixtures and analysed the dispersion of waves in the plane case.

In the study [2] the authors derived the main equations of a theory of mixtures and, as a particular example, a constitutive theory for a chemically inert mixture of two ideal gases. Iesan proposed in [4] a linear theory for binary mixtures of two elastic solids with pores, considering as independent variables the displacement, the displacement gradients, the volume fractions and volume fraction gradients, for each constituent.

In the book [5] Rajagopal and Tao summarize the main results regarding mixtures, published up to that date, as well as their practical importance.

Some results on other bodies with microstructure can be found in [6,7,8,9,10,11,12,13,14,15,16].

The importance of voids, not necessarily in the context of mixtures, has been highlighted since the first studies dedicated to this topic. The works [17,18,19] are conclusive in this regards. Here, the authors postulated the fact that bodies with pores are endowed of an elastic matrix and material-free interstices.

For instance, in [19] the authors specify the essential difference between the theory of bodies with voids and the theory of classical elasticity, namely that the volume fraction corresponding to the void volume is taken as an independent constitutive variable. Even if at the beginning the mixtures referred to gases and fluids, later they acquired a larger scope for solids. The models for mixtures between solids are more natural, a more flexible description can be used for them, namely the Lagrangian description, while the Eulerian description was used for the mixture models for gases and fluids. Relative to the Cosserat type structure, it is known that the evolution of a Cosserat body is described, in addition to the displacement vector, by another independent vector, known as the rotation vector, which indicates the orientation of each material point of the body. In our work we propose to extend Iesan’s results, [4], who used, for the first time, the Lagrangian description for a linear model of a binary mixture of two elastic solids with voids.

2 Basic equations

To model our mixture, it is used a domain D from the Euclidean space \(R^3\), which is made up of two interpenetrating constituents, both being porous Cosserat type solids. The boundary \({\partial } D\) of D is assumed to be a piecewise regular surface having the outward unit normal \(\textbf{n}\). When there is no likelihood of confusion, the time argument or the spatial argument of a function can be omitted.

To denote the differentiation of a function f with respect to time t, it is used a superposed dot, \(\dot{f}\), and to denote the differentiation of function f with respect to a spatial variable \(x_m\) it is used a subscript preceded by a comma, \(f_j\).

To describe the evolution of such a mixture we will use the following fields: the displacement vector with components \(v_{m}\), the microrotation vector with components \(\varphi _{m}\), the function for volume fraction \(\phi \) - for first constituent and, respectively, the displacement vector with components \(u_{m}\), the microrotation vector with components \(\psi _{m}\), the function for volume fraction \(\chi \) for second constituent.

With the help of the variables above, it can be defined the tensors of strain \(\varepsilon _{mn}\), \(e_{mn}\) and \(\gamma _{mn}\), \(\kappa _{mn}\) as well as the vector \(f_{m}\), the vector \(d_m\), which are defined in the cylinder \([0,\infty )\times {\bar{D}}\), by means of the following geometric equations:

$$\begin{aligned} e_{mn}= & {} v_{n,m}+\varepsilon _{nmk}\left( \varphi _{k}+\psi _{k}\right) ,\; \gamma _{mn}=\varphi _{n,m},\; d_m=v_m-u_m, \nonumber \\ \varepsilon _{mn}= & {} u_{m,n}+\varepsilon _{nmk}\left( \varphi _{k}+\psi _{k}\right) \!,\; \kappa _{mn}=\psi _{n,m}, \;f_m=\varphi _{m}-\psi _{m}. \end{aligned}$$
(1)

According to [4], the basic equations in the theory of homogeneous and anisotropic Cosserat elastic bodies are:

- the motion equations:

$$\begin{aligned} \sigma _{mn,n}-h_m+\rho _1 F_m^{(1)}= & {} \rho _1\ddot{v}_{m}, \nonumber \\ \mu _{mn,n}+\varepsilon _{mnk}\tau _{jk}-p_m+\rho _1 G_m^{(1)}= & {} I_{mn}^{(1)}\ddot{\varphi }_{n}, \nonumber \\ \tau _{mn,n}+h_m+\rho _2 F_m^{(2)}= & {} \rho _2\ddot{u}_{m}, \nonumber \\ m_{mn,n}+\varepsilon _{mnk}\tau _{jk}+p_m+\rho _2 G_m^{(2)}= & {} I_{mn}^{(2)}\ddot{\psi }_{n}; \end{aligned}$$
(2)

- the motion equations of the equilibrated forces:

$$\begin{aligned} \lambda ^{(1)}_{k,k}+g^{(1)}+\rho _1 L^{(1)}= & {} \rho _1 \kappa _1 \ddot{\phi }, \nonumber \\ \lambda ^{(2)}_{k,k}+g^{(2)}+\rho _2 L^{(2)}= & {} \rho _2 \kappa _2 \ddot{\chi }. \end{aligned}$$
(3)

The domain of definition for Eqs. (2) and (3) is the cylinder \((t, x)\in (0,\infty )\times D.\)

It is assumed that our body has at each point, in its state of reference, a center of symmetry, but is otherwise non-isotropic. Thus, the constitutive relations, defined for \((t,x)\in [0,\infty ) \times {\bar{D}}\), are:

$$\begin{aligned} \sigma _{mn}= & {} A_{mnkl}e_{kl}+B_{mnkl}\gamma _{kl}+C_{mnkl}\varepsilon _{kl}+ D_{mnjl}\kappa _{jl}+ \nonumber \\{} & {} +\left( A_{mn}+M_{mn}\right) \phi +\left( B_{mn}+N_{mn}\right) \chi , \nonumber \\ \tau _{mn}= & {} C_{mnkl}e_{kl}+E_{mnkl}\gamma _{kl}+ F_{mnkl}\varepsilon _{kl}+G_{mnjl}\kappa _{jl}+ \nonumber \\{} & {} +M_{mn}\phi +N_{mn}\chi , \nonumber \\ \mu _{mn}= & {} B_{mnkl}e_{kl}+H_{mnkl}\gamma _{kl}+I_{mnkl}\varepsilon _{kl}+ J_{mnjl}\kappa _{jl}+ \nonumber \\{} & {} +\left( C_{mn}+P_{mn}\right) \phi +\left( D_{mn}+Q_{mn}\right) \chi , \nonumber \\ \nu _{mn}= & {} D_{mnkl}e_{kl}+ J_{mnkl}\gamma _{kl}+G_{mnkl}\varepsilon _{kl}+L_{mnjl}\kappa _{jl}+ \nonumber \\{} & {} +P_{mn}\phi +Q_{mn}\chi , \nonumber \\ h_m= & {} a_{mn}d_n+b_{mn}\phi _{,n}+c_{mn}\chi _{,n}, \nonumber \\ p_m= & {} d_{mn}d_n+e_{mn}\phi _{,n}+f_{mn}\chi _{,n}, \nonumber \\ \lambda _{m}^{(1)}= & {} b_{mn}d_{n}+\alpha _{mn}\phi _{,n}+\beta _{mn}\chi _{,n}, \nonumber \\ \lambda _{m}^{(2)}= & {} c_{mn}d_{n}+\beta _{mn}\phi _{,n}+\delta _{mn}\chi _{,n}, \nonumber \\ g^{(1)}= & {} \!-\left( A_{mn}\!+\!M_{mn}\right) e_{mn}\!-\!M_{mn}\gamma _{mn}\!-\xi \phi -\tau \chi \!-\!C\!, \nonumber \\ g^{(2)}= & {} -\left( B_{mn}\!+\!N_{mn}\right) e_{mn}\!-\!N_{mn}\gamma _{mn}\!-\tau \phi -\eta \chi \!-\!D\!. \end{aligned}$$
(4)

Based on the paper [4], the densities of the two constituents \(\varrho _1\) and \(\varrho _2\), the inertia coefficients of the two constituents \(\kappa _1\) and \(\kappa _2\) and the coupling coefficients \(\xi \) and \(\tau \), from above, satisfy the following conditions:

$$\begin{aligned} \varrho _1>0,\;\varrho _2>0,\;\kappa _1>0,\;\kappa _2>0,\; \xi \ge 0,\;\tau \ge 0. \end{aligned}$$
(5)

The meaning of the notations used in the above equations is as follows: the stress tensor with components \(\sigma _{mn}\), the couple stress tensor with components \(\tau _{mn}\) for the first constituent and, respectively, the stress tensor with components \(\mu _{mn}\), the couple stress tensor with components \(\nu _{mn}\) for the second constituent. The body forces for the two constituents are \(F_m^{(1)}\) and \(F_m^{(2)}\), the body couples for the two constituents are \(G_m^{(1)}\) and \(G_m^{(2)}\). The internal body forces in the two constituents are denoted by \(h_m\), respectively \(p_m\), the inertia in the two constituents has the components \(I_{mn}^{(1)}\), respectively \(I_{mn}^{(2)}\). The heat conduction vector has the components \(q_{m}\). We also denote by \(\eta \) the entropy, by \(\theta \) the variation of the temperature, by r the rate of supply of heat and the alternating symbol is \(\varepsilon _{mnk}\).

The elastic coefficients in (4) are constant constitutive characteristics of the body which satisfy the next symmetry relations:

$$\begin{aligned} A_{mnkl}= & {} A_{klmn},\;C_{mnkl}=C_{klmn},\; B_{mnkl}=B_{klmn}, \nonumber \\ D_{mnkl}= & {} D_{klmn},\; E_{mnkl}=E_{klmn},\;G_{mnkl}=G_{klmn}, \nonumber \\ F_{mnkl}= & {} F_{klmn},\; J_{mnkl}=J_{klmn},\; I_{mn}^{(1)}=I_{nm}^{(1)},\;I_{mn}^{(2)}=I_{nm}^{(2)}. \end{aligned}$$
(6)

In order to consider a mixed problem in our context, the following initial values are needed:

$$\begin{aligned} v_m(0,x)= & {} v_m^0(x),\;\dot{v}_m(0,x)=v_m^1(x),\;u_m(0,x)=u_m^0(x),\;\dot{u}_m(0,x)=u_m^1(x),\; \nonumber \\ \varphi _m(0,x)= & {} \varphi _m^0(x),\;\dot{\varphi }_m(0,x)=\varphi _m^1(x),\; \psi _m(0,x)=\psi _m^0(x),\;\dot{\psi }_m(0,x)=\psi _m^1(x), \nonumber \\ \phi (0,x)= & {} \phi ^0(x),\;\dot{\phi }(0,x)=\phi ^1(x),\; \chi (0,x)=\chi ^0(x),\;\dot{\chi }(0,x)=\chi ^1(x),\;\forall x\in D, \end{aligned}$$
(7)

where the values \(v_m^0,\;v_m^1,\;u_m^0,\;u_m^1,\;\varphi _m^0,\;\varphi _m^1,\;\psi _m^0,\;\psi _m^1,\;\chi _m^0,\;\chi _m^1\) are prescribed.

The mixed problem is complete if the following boundary data are added:

$$\begin{aligned} v_m=\bar{v}_m,\; u_m=\bar{u}_m,\; \varphi _m=\bar{\varphi }_m,\; \psi _m=\bar{\psi }_m,\; \phi =\bar{\phi },\;\chi =\bar{\chi }, \;\text{ on }\; (0,T)\times \partial D, \end{aligned}$$
(8)

where the values \(\bar{v}_m,\;\bar{u}_m,\;\bar{\varphi }_m,\;\bar{\psi }_m,\;\bar{\phi },\;\bar{\chi },\;\) are prescribed.

Let \({{\mathcal {P}}}\) denote the mixed problem consisting of the main Eqs. (1–4), the initial data (7) and the boundary values (8).

A solution of the problem \({{\mathcal {P}}}\) is an ordered array \(S=\left\{ v_m, u_m, \varphi _m, \psi _m, \phi , \chi \right\} \), defined for all \((t,x)\in (0,T)\times D\), which satisfies the following regularity conditions:

$$\begin{aligned}{} & {} v_m(t,x)\in C^{2,2}((0,T)\times D),\; u_m(t,x)\in C^{2,2}((0,T)\times D), \nonumber \\{} & {} varphi_m(t,x)\in C^{2,2}((0,T)\times D),\; \psi _m(t,x)\in C^{2,2}((0,T)\times D), \nonumber \\{} & {} \phi (t,x)\in C^{2,2}((0,T)\times D),\; \chi (t,x)\in C^{2,2}((0,T)\times D). \end{aligned}$$

Since the subsequent considerations will be made only in the context of a linear theory, it is natural that the internal energy density E(S), which is associated with the solution \(S=\left\{ v_m, u_m, \varphi _m, \psi _m, \phi , \chi \right\} \), should be a quadratic form, which has the following expression:

$$\begin{aligned} E(S(t))= & {} \frac{1}{2}\left[ \sigma _{mn}e_{mn}+\tau _{mn}\gamma _{mn}+\mu _{mn}\varepsilon _{mn} +\nu _{mn}\kappa _{mn}+ \right. \nonumber \\{} & {} \left. +\lambda ^{(1)}_{m}\phi _{,m} +\lambda ^{(2)}_{m}\chi _{,m} +h_md_m+p_mf_m-\left( g^{(1)}+C\right) \phi -\left( g^{(2)}+D\right) \chi \right] . \end{aligned}$$
(9)

It is easy to notice that:

$$\begin{aligned} \dot{E}(S(t))= & {} \frac{1}{2}\left[ \sigma _{mn}\dot{e}_{mn}+\tau _{mn}\dot{\gamma }_{mn}+\mu _{mn}\dot{\varepsilon }_{kl} +\nu _{mn}\dot{\kappa }_{mn}+ \right. \nonumber \\{} & {} \left. +\lambda ^{(1)}_{m}\dot{\phi }_{,m} +\lambda ^{(2)}_{m}\dot{\chi }_{,m} +h_m\dot{d}_m+p_m\dot{f}_m-\left( g^{(1)}+C\right) \dot{\phi }-\left( g^{(2)}+D\right) \dot{\chi }\right] . \end{aligned}$$
(10)

Furthermore, it is assumed that the quadratic form E(S) is positive definite with regards to \(e_{mn}\), \(\gamma _{mn}\), \(\varepsilon _{mn}\), \(\kappa _{mn}\), \(d_{m}\), \(f_{m}\), \(\phi \), \(\chi \). As such, there exist the strictly positive constants m, M, \(\kappa \) and K such that:

$$\begin{aligned}{} & {} E(S)\ge m\left[ \varepsilon _{mn}\varepsilon _{mn}+\gamma _{mn}\gamma _{mn}+e_{mn}e_{mn}+g_{mn}g_{mn}+\kappa \left( d_{m}d_{m}+ f_{m}f_{m}\right) \right] , \nonumber \\{} & {} E(S)\le M\left[ \varepsilon _{mn}\varepsilon _{mn}+\gamma _{mn}\gamma _{mn}+e_{mn}e_{mn}+g_{mn}g_{mn}+K \left( d_{m}d_{m}+ f_{m}f_{m}\right) \right] . \end{aligned}$$
(11)

3 Preliminary results

In this section we will use some results for a Dirichlet boundary value problem associated to an elliptic equation on the domain D with the border \(\partial D\). More precisely, we will deduce some estimates on the solutions of this type of problem in terms of the boundary values.

Let us consider the following problem regarding the Laplace’s operator \(\Delta \) and the unknown function w:

$$\begin{aligned} \Delta w= & {} 0\;\text{ in } \;D, \nonumber \\ w= & {} w^0\;\text{ on } \;\partial D, \end{aligned}$$
(12)

in which \(w^0\) is given.

The existence of the solution w for problem (12) can be obtained with the help of the results from Fichera’s work [20], see also [21].

In order to obtain our first estimate, we will assume that the surface \(\partial D\) is star-shaped relative to the origin of the coordinate system and this assures that:

there exist the positive constants \(a_0\) and \(b_0\) so that:

$$\begin{aligned} n_jx_j\ge a_0,\; \left| t_jx_j\right| \le b_0\;\text{ on }\;\partial D, \end{aligned}$$
(13)

where \(\textbf{n}=\left( n_j\right) \) is the unit vector normal outward to \(\partial D\) and \(\textbf{t}=\left( t_j\right) \) is the tangent to the surface \(\partial D\).

In the next proposition we deduce an estimate with respect to the solution of the problem (12). To this aim for the \(\nabla w\), the gradient of function w, it is used the next decomposition:

$$\begin{aligned} \nabla w=\left( \frac{\partial w}{\partial n}\right) \textbf{n}+\left( \nabla _t w\right) \textbf{t}, \end{aligned}$$

where \(\partial w/ \partial n\) is the derivative in the direction of normal and \(\nabla _t w\) is the derivative in the direction of tangent.

Proposition 1

.

If w is a solution of the problem (12), then the following estimate takes place:

$$\begin{aligned} 2\int _D \!w_{,m}w_{,m} dV+a_0\int _{\partial D}\!\left( \frac{\partial w}{\partial n}\right) ^2 dA\le 2 \int _{\partial D}\!\left( n_jx_j+\frac{2b_0^2}{a_0}\right) \left( \nabla _t w\right) ^2 dA. \end{aligned}$$
(14)

Proof

If Eq. (12)\(_1\) is multiplied by \(x_m w_m\), the following equality is obtained:

$$\begin{aligned} \int _D \!\!w_{,m}w_{,m} \;dV\!+\!2\int _{\partial D}\!\! n_lx_m w_{,m}w_{,l}\;dA\!-\!\int _{\partial D}\!\! n_lx_l w_{,m}w_{,m}\;dA\!=\! \int _D \!\!x_m w_{,m}\Delta w \;dV\!=0\!, \end{aligned}$$
(15)

from where, considering the decomposition above of gradient, it is deduced:

$$\begin{aligned}{} & {} \int _D w_{,m}w_{,m} dV+ \int _{\partial D} x_m n_m\left( \frac{\partial w}{\partial n}\right) ^2 dA= \nonumber \\= & {} -2\int _{\partial D} x_m t_m\frac{\partial w}{\partial n} \nabla _t w dA+ \int _{\partial D} x_m n_m \left( \nabla _t w\right) ^2 dA. \end{aligned}$$
(16)

In this identity, the conditions (13) are taken into account, the Cauchy-Schwarz inequality and the arithmetic–geometric mean inequality are applied so that the desired estimate (12) is obtained. This concludes the proof of Proposition 1. \(\square \)

Remark

Regarding the function w and its boundary value \(w^0\) from problem (12), we must remember Poincare’s inequality, according to which there are two positive constants \(c_1\) and \(c_2\) such that:

$$\begin{aligned} c_1\int _D w^2 dV\le \int _D w_{,m}w_{,m} dV+c_2\int _{\partial D}\left( w^0\right) ^2 dA. \end{aligned}$$
(17)

Thus, it can be seen that estimates (13) and (17) provide useful bounds for the integrals:

$$\begin{aligned} \int _D w^2 dV,\;\int _D w_{,m}w_{,m} \;dV,\; \int _{\partial D} \left( \frac{\partial w}{\partial n}\right) ^2 dA. \end{aligned}$$

4 Main results

A main concern in this section is to adapt the results from the previous section to obtain the continuous dependence of the solutions relative to the external data of the problem \({{\mathcal {P}}}\), introduced in Sect. 2. First, some identities will be obtained, useful in deducing the proposed result.

Proposition 2

. If \(v_m\) is a solution of the equation (2)\(_1\) and \(\varphi _m\) is a solution of the equation (2)\(_3\), then the following equality occurs:

$$\begin{aligned}{} & {} \int _D\left[ \varrho _1\dot{v}_m\ddot{v}_m+\sigma _{mn}\dot{e}_{mn}+h_m\dot{v}_m\right] dV+ \nonumber \\{} & {} \quad +\int _D\left[ I_{mn}^{(1)}\dot{\varphi }_m\ddot{\varphi }_n+\mu _{mn}\dot{\gamma }_{mn}+\left( \varepsilon _{mnk}\sigma _{nk}-p_m\right) \dot{\varphi }_m\right] dV= \nonumber \\= & {} \!\frac{d}{dt}\int _D\!\!\varrho _1\dot{v}_mV_m dV\!-\!\int _D\!\!\varrho _1\dot{v}_m\dot{V}_m dV\!+\! \frac{d}{dt}\int _D\!\!I_{mn}^{(1)}\dot{\varphi }_m\Phi _n dV\!-\!\int _D\!\!I_{mn}^{(1)}\dot{\varphi }_m\dot{\Phi }_n dV \nonumber \\{} & {} \quad +\int _D\left[ \sigma _{mn}V_{m,n}+h_mV_m+\mu _{mn}\Phi _{m,n}+p_m\Phi _{m}\right] dV+ \nonumber \\{} & {} \quad +\int _D\left[ \varrho _1F_m^{(1)}\left( \dot{v}_m-V_m\right) +\varrho _1G_m^{(1)}\left( \dot{\varphi }_m-\Phi _m\right) \right] dV. \end{aligned}$$
(18)

Proof

We adapt the Dirichlet problem (12) by substituting the function w with \(\dot{v}_m\) and denote the solution of the obtained problem with \(V_m\). Thus, using estimate (14) we get

$$\begin{aligned} 2\int _D \!V_{m,k}V_{m,k} dV+a_0\int _{\partial D}\!\frac{\partial V_m}{\partial n}\frac{\partial V_k}{\partial n} dA\le 2 \int _{\partial D}\!\left( n_jx_j+\frac{2b_0^2}{a_0}\right) \left( \nabla _t \dot{v}_m^0\right) \left( \nabla _t \dot{v}_m^0\right) dA, \end{aligned}$$
(19)

and from (17), with the same substitution, is obtained:

$$\begin{aligned} c_1\int _D V_mV_m dV\le \int _D V_{m,n}V_{m,n} dV+c_2\int _{\partial D}\dot{v}_m^0\dot{v}_m^0 dA. \end{aligned}$$
(20)

From equation (2)\(_1\), multiplied by \(V_m-\dot{v}_m\), it is deduced

$$\begin{aligned} \int _D \left( \sigma _{nm,n}-h_m-\varrho _1\ddot{v}_m+\varrho _1F_m^{(1)}\right) \left( V_m-\dot{v}_m\right) dV=0. \end{aligned}$$
(21)

Here the divergence theorem is applied and it is taken into account that at the boundary \(\partial D\) the condition \(V_m-\dot{v}_m=0\) is fulfilled. Thus, we are led to the equality:

$$\begin{aligned}{} & {} \int _D\left[ \varrho _1\dot{v}_m\ddot{v}_m+\sigma _{mn}\dot{e}_{mn}+h_m\dot{v}_m\right] dV=\!\frac{d}{dt}\int _D\!\!\varrho _1\dot{v}_mV_m dV\!-\!\int _D\!\!\varrho _1\dot{v}_m\dot{V}_m dV\!+\! \nonumber \\{} & {} \qquad +\int _D\left[ \sigma _{mn}V_{m,n}+h_mV_m\right] dV+\int _D\varrho _1F_m^{(1)}\left( \dot{v}_m-V_m\right) dV. \end{aligned}$$
(22)

Now, we adapt the Dirichlet problem (12) by substituting the function w with \(\dot{\varphi }_m\) and denote the solution of the obtained problem with \(\Phi _m\). Thus, using estimate (14) we get

$$\begin{aligned} 2\int _D \!\Phi _{m,k}\Phi _{m,k} dV+a_0\int _{\partial D}\!\frac{\partial \Phi _m}{\partial n}\frac{\partial \Phi _k}{\partial n} dA\le 2 \int _{\partial D}\!\left( n_jx_j+\frac{2b_0^2}{a_0}\right) \left( \nabla _t \dot{\varphi }_m^0\right) \left( \nabla _t \dot{\varphi }_m^0\right) dA. \end{aligned}$$
(23)

By using the same substitution, from (17) one obtains:

$$\begin{aligned} c_1\int _D \Phi _m \Phi _m dV\le \int _D \Phi _{m,n}\Phi _{m,n} dV+c_2\int _{\partial D}\dot{\varphi }_m^0\dot{\varphi }_m^0 dA. \end{aligned}$$
(24)

If we multiply by \(\Phi _m-\dot{\varphi }_m\) the Eq. (2)\(_3\), we are led to:

$$\begin{aligned} \int _D \left( \mu _{nm,n}-p_m-I_{mn}^{(1)}\ddot{\varphi }_n+\varrho _1G_m^{(1)}\right) \left( \Phi _m-\dot{\varphi }_m\right) dV=0, \end{aligned}$$
(25)

so that, if the divergence theorem is applied and it is taken into account that at the boundary \(\partial D\) the condition \(\Phi _m-\dot{\varphi }_m=0\) is fulfilled, we are led to the equality:

$$\begin{aligned}{} & {} \int _D\left[ I_{mn}^{(1)}\dot{\varphi }_m\ddot{\varphi }_n+\mu _{mn}\dot{\gamma }_{mn}+\left( \varepsilon _{mnk}\sigma _{nk}-p_m\right) \dot{\varphi }_m\right] dV= \nonumber \\= & {} \frac{d}{dt}\int _DI_{mn}^{(1)}\dot{\varphi }_m \Phi _n dV-\int _D\!\!I_{mn}^{(1)}\dot{\varphi }_m\dot{\Phi }_n dV+ \nonumber \\{} & {} +\int _D\left[ \mu _{mn}\Phi _{m,n}+p_m\Phi _m\right] dV+\int _D\varrho _1G_m^{(1)}\left( \dot{\varphi }_m-\Phi _m\right) dV. \end{aligned}$$
(26)

Finally, we add relations (22) and (26), term by term, and obtain the desired identity (18), which concludes the proof of Proposition 2. \(\square \)

Proposition 3

. If \(u_m\) is a solution of the Eq. (2)\(_2\) and \(\psi _m\) is a solution of the Eq. (2)\(_4\), then the following equality occurs:

$$\begin{aligned}{} & {} \int _D\left[ \varrho _2\dot{u}_m\ddot{u}_m+\tau _{mn}\dot{\varepsilon }_{mn}+h_m\dot{u}_m\right] dV+ \nonumber \\{} & {} \quad +\int _D\left[ I_{mn}^{(2)}\dot{\psi }_m\ddot{\psi }_n+\nu _{mn}\dot{\kappa }_{mn}+\left( \varepsilon _{mnk}\tau _{nk}-p_m\right) \dot{\psi }_m\right] dV= \nonumber \\= & {} \!\frac{d}{dt}\int _D\!\!\varrho _2\dot{u}_mU_m dV\!-\!\int _D\!\!\varrho _2\dot{u}_m\dot{U}_m dV\!+\! \frac{d}{dt}\int _D\!\!I_{mn}^{(2)}\dot{\psi }_m\Psi _n dV\!-\!\int _D\!\!I_{mn}^{(2)}\dot{\psi }_m\dot{\Psi }_n dV \nonumber \\{} & {} +\int _D\left[ \tau _{mn}U_{m,n}+h_mU_m+\nu _{mn}\Psi _{m,n}+p_m\Psi _{m}\right] dV+ \hspace{2.5cm} \nonumber \\{} & {} +\int _D\left[ \varrho _2F_m^{(2)}\left( \dot{u}_m-U_m\right) +I_{mn}^{(2)}G_m^{(2)}\left( \dot{\psi }_m-\Psi _m\right) \right] dV. \end{aligned}$$
(27)

Proof

We adapt the Dirichlet problem (12) by substituting the function w with \(\dot{u}_m\) and denote the solution of the obtained problem with \(U_m\).

Then, the Dirichlet problem (12) is adapted by substituting the function w with \(\dot{\psi }_m\) and denote the solution of the obtained problem with \(\Psi _m\).

The same steps and the same calculations are then used as in the proof of Proposition 2 and the identity (27) is obtained. \(\square \)

Proposition 4

. If \(\phi \) is a solution of the equation (3)\(_1\) then the following equality occurs:

$$\begin{aligned}{} & {} \int _D\left[ \varrho _1\kappa _1\dot{\phi }\ddot{\phi }+\lambda _{m}^{(1)}\dot{\phi }_{,m}-\left( g^{(1)}+C\right) \dot{\phi }\right] dV= \frac{d}{dt}\int _D\!\!\varrho _1\kappa _1\dot{\phi }\Upsilon dV\!-\!\int _D\!\!\varrho _1\kappa _1\dot{\phi }\dot{\Upsilon }dV\!+ \nonumber \\{} & {} \quad +\int _D\left[ \lambda _{m}^{(1)}\Upsilon _{,m}-\left( g^{(1)}+C\right) \Upsilon \right] dV+ \int _D\left( \varrho _1L^{(1)}-C\right) \left( \dot{\phi }-\Upsilon \right) dV. \end{aligned}$$
(28)

Proof

We adapt the Dirichlet problem (12) by substituting the function w with \(\dot{\phi }\) and denote the solution of the obtained problem with \(\Upsilon \).

The same steps and the same calculations are then used as in the proof of Proposition 2 and the identity (28) is obtained. \(\square \)

Proposition 5

. If \(\chi \) is a solution of the equation (3)\(_2\) then the following equality occurs:

$$\begin{aligned}{} & {} \int _D\left[ \varrho _2\kappa _2\dot{\chi }\ddot{\chi }+\lambda _{m}^{(2)}\dot{\phi }_{,m}-\left( g^{(2)}+C\right) \dot{\chi }\right] dV= \frac{d}{dt}\int _D\varrho _2\kappa _2\dot{\chi }\emptyset dV-\int _D\varrho _1\kappa _2\dot{\phi }\dot{\emptyset }dV+ \nonumber \\{} & {} \quad +\int _D\left[ \lambda _{m}^{(2)}\emptyset _{,m}-\left( g^{(1)}+D\right) \emptyset \right] dV+ \int _D\left( \varrho _2L^{(2)}-D\right) \left( \dot{\phi }-\emptyset \right) dV. \end{aligned}$$
(29)

Proof

We adapt the Dirichlet problem (12) by substituting the function w with \(\dot{\chi }\) and denote the solution of the obtained problem with \(\emptyset \).

The same steps and the same calculations are then used as in the proof of Proposition 2 and the identity (29) is obtained.

Now we can formulate and prove the first basic identity of our study. \(\square \)

Theorem 1

. If \(S=\left\{ v_m, u_m, \varphi _m, \psi _m, \phi , \chi \right\} \) is a solution of the problem \({{\mathcal {P}}}\), then the following equality occurs:

$$\begin{aligned}{} & {} \frac{d}{dt}\!\int _D\!\!\left[ \frac{1}{2}\!\left( \varrho _1\dot{v}_m\dot{v}_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m\dot{\varphi }_n\!+\!\varrho _1\kappa _1\dot{\phi }^2\!+\! \varrho _2\dot{u}_m\dot{u}_m\!+\!I_{mn}^{(2)}\dot{\psi }_m\dot{\psi }_n\!+\!\varrho _2\kappa _2\dot{\chi }^2\right) \!+\!E(S)\right] \!dV \nonumber \\{} & {} \quad -\frac{d}{dt}\int _D\left( \varrho _1\dot{v}_m V_m+I_{mn}^{(1)}\dot{\varphi }_m \Phi _n+\varrho _1\kappa _1\dot{\phi }\Upsilon + \varrho _2\dot{u}_m U_m+I_{mn}^{(2)}\dot{\psi }_m \Psi _n+\varrho _2\kappa _2\dot{\chi }\emptyset \right) dV= \nonumber \\{} & {} \quad -\int _D\left( \varrho _1\dot{v}_m \dot{V}_m+I_{mn}^{(1)}\dot{\varphi }_m \dot{\Phi }_n+\varrho _1\kappa _1\dot{\phi }\dot{\Upsilon }+ \varrho _2\dot{u}_m \dot{U}_m+I_{mn}^{(2)}\dot{\psi }_m \dot{\Psi }_n+\varrho _2\kappa _2\dot{\chi }\dot{\emptyset }\right) dV+ \nonumber \\{} & {} \quad +\!\int _D\!\!\!\left[ \!\sigma _{mn}V_{m,n}\!\!+\!\lambda _{m}^{(1)}\Upsilon _{,m}\!\!-\!\left( \!g^{(1)}\!\!+\!C\!\right) \!\Upsilon \!\!+\!h_m\!\left( V_m\!\!-\!U_m\right) \!\!+\! \tau _{mn}U_{m,n}\!\!+\!\lambda _{m}^{(2)}\emptyset _{,m}\!\!-\!\left( \!g^{(2)}\!\!+\!D\!\right) \!\emptyset \right] \!dV \nonumber \\{} & {} \quad +\!\int _D\left[ \varrho _1F_m^{(1)}\dot{v}_m+\varrho _1G_{m}^{(1)}\dot{\varphi }_m+\varrho _1L^{(1)}\dot{\phi }+ \varrho _2F_m^{(2)}\dot{u}_m+\varrho _2G_{m}^{(2)}\dot{\psi }_m+\varrho _2L^{(2)}\dot{\chi }\right] dV \nonumber \\{} & {} \quad -\int _D\left[ \varrho _1F_m^{(1)} V_m+\varrho _1G_{m}^{(1)}\Phi _m+\varrho _1L^{(1)}\Upsilon + \varrho _2F_m^{(2)} U_m+\varrho _2G_{m}^{(2)}\Psi _m+\varrho _2L^{(2)}\emptyset \right] dV. \end{aligned}$$
(30)

Proof

Equality is obtained immediately if there are add, term by term, the equalities (18), (27), (28), (29) above and take into account relations (9) and (10). \(\square \)

Now we propose to give a simpler form for the identity (30). For this purpose, we introduce the following notations:

$$\begin{aligned} {{\mathcal {S}}}(t)\!= & {} \!\frac{1}{2}\!\int _{D(t)}\!\!\!\left( \! \varrho _1\dot{v}_m\dot{v}_m\!\!+\!I_{mn}^{(1)}\dot{\varphi }_m\dot{\varphi }_n\!\!+\!\varrho _1\kappa _1\dot{\phi }^2\!\!+\! \varrho _2\dot{u}_m\dot{u}_m\!\!+\!I_{mn}^{(2)}\dot{\psi }_m\dot{\psi }_n\!\!+\!\varrho _2\kappa _2\dot{\chi }^2\!\!+\!2E(S)\!\right) \!dV \nonumber \\ f_1(t)\!= & {} \!\!\int _{\!D(t)}\!\!\left[ \!\sigma _{mn}V_{m,n}\!\!+\!\lambda _{m}^{(1)}\Upsilon _{,m}\!\!-\!\left( \!g^{(1)}\!\!+\!C\!\right) \!\Upsilon \!\!+\!h_m\!\left( \!V_m\!\!-\!U_m\!\right) \!\!+\! \tau _{mn}U_{m,n}\!\!+\!\lambda _{m}^{(2)}\emptyset _{,m}\!\!-\!\left( \!g^{(2)}\!\!+\!D\!\right) \!\emptyset \right] dV \nonumber \\ f_2(t)\!= & {} \!\int _{D(t)}\!\!\left[ \varrho _1F_m^{(1)}\dot{v}_m\!+\!\varrho _1G_{m}^{(1)}\dot{\varphi }_m\!+\!\varrho _1L^{(1)}\dot{\phi }\!+\! \varrho _2F_m^{(2)}\dot{u}_m\!+\!\varrho _2G_{m}^{(2)}\dot{\psi }_m\!+\!\varrho _2L^{(2)}\dot{\chi }\right] dV \nonumber \\ f_3(t)= & {} -\int _{D(t)}\left[ \varrho _1F_m^{(1)} V_m+\varrho _1G_{m}^{(1)}\Phi _m+\varrho _1L^{(1)}\Upsilon + \varrho _2F_m^{(2)} U_m+\varrho _2G_{m}^{(2)}\Psi _m+\varrho _2L^{(2)}\emptyset \right] dV. \end{aligned}$$
(31)

For the four integrals in (31) we specified the dependence on t to emphasize the fact that their integrals are calculated at time t.

With the help of the notations (31), the identity (30) receives the following form:

$$\begin{aligned} {{\mathcal {S}}}(t)\!= & {} \!\!\int _{D(t)}\!\!\!\left( \varrho _1\dot{v}_m V_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \Phi _n\!+\!\varrho _1\kappa _1\dot{\phi }\Upsilon \!+\! \varrho _2\dot{u}_m U_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \Psi _n\!+\!\varrho _2\kappa _2\dot{\chi }\emptyset \right) \!dV \nonumber \\{} & {} -\!\int _0^t\!\! \int _{D(\tau )}\!\!\!\left( \varrho _1\dot{v}_m \dot{V}_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \dot{\Phi }_n\!+\!\varrho _1\kappa _1\dot{\phi }\dot{\Upsilon }\!+\! \varrho _2\dot{u}_m \dot{U}_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \dot{\Psi }_n\!+\!\varrho _2\kappa _2\dot{\chi }\dot{\emptyset }\right) \!dVd\tau \nonumber \\{} & {} +\int _0^tf_1(\tau )d\tau +\int _0^tf_2(\tau )d\tau +\int _0^tf_3(\tau )d\tau +{{\mathcal {S}}}(0) \nonumber \\{} & {} -\int _{D(0)}\!\!\!\left( \varrho _1\dot{v}_m \dot{V}_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \dot{\Phi }_n\!+\!\varrho _1\kappa _1\dot{\phi }\dot{\Upsilon }\!+\! \varrho _2\dot{u}_m \dot{U}_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \dot{\Psi }_n\!+\!\varrho _2\kappa _2\dot{\chi }\dot{\emptyset }\right) \!dV. \end{aligned}$$
(32)

We will evaluate the integrals in (31) and (32) so that the identity (32) can be used to obtain a result of the continuous dependence of the solution of the problem \({{\mathcal {P}}}\).

Thus, with the help of the Cauchy-Schwarz inequality and the arithmetic–geometric mean inequality, one obtains:

$$\begin{aligned}{} & {} \int _{D(t)}\!\!\!\left( \varrho _1\dot{v}_m V_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \Phi _n\!+\!\varrho _1\kappa _1\dot{\phi }\Upsilon \!+\! \varrho _2\dot{u}_m U_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \Psi _n\!+\!\varrho _2\kappa _2\dot{\chi }\emptyset \right) \!dV \nonumber \\\le & {} \frac{1}{4} \int _{D(t)}\!\!\!\left( \varrho _1\dot{v}_m \dot{v}_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \dot{\varphi }_n\!+\!\varrho _1\kappa _1\dot{\phi }^2\!+\! \varrho _2\dot{u}_m \dot{u}_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \dot{\psi }_n\!+\!\varrho _2\kappa _2\dot{\chi }^2\right) \!dV \nonumber \\{} & {} +\max \limits _{t\in [0,T]}\int _{D(t)}\!\!\!\left( \varrho _1 V_m V_m\!+\!I_{mn}^{(1)}\Phi _m \Phi _n\!+\!\varrho _1\kappa _1\Upsilon ^2\!+\! \varrho _2 U_m U_m\!+\!I_{mn}^{(2)} \Psi _m \Psi _n\!+\!\varrho _2\kappa _2\emptyset ^2\right) \!dV. \end{aligned}$$
(33)

In the same way, using the notation:

$$\begin{aligned} f_4(t)=\left( \int _{D(t)}\!\!\!\left( \varrho _1 V_m V_m\!+\!I_{mn}^{(1)}\Phi _m \Phi _n\!+\!\varrho _1\kappa _1\Upsilon ^2\!+\! \varrho _2 U_m U_m\!+\!I_{mn}^{(2)} \Psi _m \Psi _n\!+\!\varrho _2\kappa _2\emptyset ^2\right) \!dV\right) ^{1/2}, \end{aligned}$$
(34)

it can be obtained:

$$\begin{aligned}{} & {} -\!\int _0^t\!\! \int _{D(\tau )}\!\!\!\left( \varrho _1\dot{v}_m \dot{V}_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \dot{\Phi }_n\!+\!\varrho _1\kappa _1\dot{\phi }\dot{\Upsilon }\!+\! \varrho _2\dot{u}_m \dot{U}_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \dot{\Psi }_n\!+\!\varrho _2\kappa _2\dot{\chi }\dot{\emptyset }\right) \!dVd\tau \nonumber \\\le & {} \!\!\int _0^t\!\!\left( \!\int _{D(t)}\!\!\!\left( \!\varrho _1\dot{v}_m \dot{v}_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \dot{\varphi }_n\!+\!\varrho _1\kappa _1\dot{\phi }^2\!+\! \varrho _2\dot{u}_m \dot{u}_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \dot{\psi }_n\!+\!\varrho _2\kappa _2\dot{\chi }^2\!\right) \!dV\!\!\right) ^{\!1/2}\!\!\!\!f_4(\tau )d\tau \nonumber \\\le & {} \int _0^t\left( 2{{\mathcal {S}}}(\tau )\right) ^{1/2} f_4(\tau )d\tau . \end{aligned}$$
(35)

Now, the notation (31)\(_2\) are used in order to obtain the following estimate:

$$\begin{aligned}{} & {} \left| f_1(t)\right\| \le f_5(t)\left\{ \int _{D(t)} \left[ \sigma _{mn}\sigma _{mn}+\mu _{mn}\mu _{mn}+\tau _{mn}\tau _{mn}+\nu _{mn}\nu _{mn}+ \right. \right. \nonumber \\{} & {} \quad \left. \left. +\frac{1}{\kappa }\left( h_m^{(1)}h_m^{(1)}+h_m^{(2)}h_m^{(2)}+p_mp_m\right) +\left( g_m^{(1)}+C\right) ^2\left( g_m^{(2)}+D\right) ^2\right] dV\right\} ^{1/2} \nonumber \\{} & {} \quad \le f_5(t)\sqrt{2M\mathcal{S}(t)}, \end{aligned}$$
(36)

in which \(\kappa \) and M are introduced in (11) and it is used the notation:

$$\begin{aligned} f_5(t)= & {} \left\{ \int _{D(t)}\left[ V_{m,n}V_{m,n}+U_{m,n}U_{m,n}+\kappa \left( \Upsilon _{,m}\Upsilon _{,m}+\emptyset _{,m}\emptyset _{,m}\right) + \right. \right. \nonumber \\{} & {} \quad \left. \left. +\left( V_m-U_m\right) \left( V_m-U_m\right) + \Upsilon ^2+\emptyset ^2\right] \right\} ^{1/2}. \end{aligned}$$
(37)

Also, by using the notation:

$$\begin{aligned} f_6(t)= & {} \left\{ \int _{D(t)}\left[ \varrho _1F_{m}^{(1)}F_{m}^{(1)}+ \varrho _1G_{m}^{(1)}G_{m}^{(1)}+ \kappa _1\left( L^{(1)}-C\right) ^2+ \right. \right. \nonumber \\{} & {} \quad \left. \left. +\varrho _2F_{m}^{(2)}F_{m}^{(2)}+ \varrho _2G_{m}^{(2)}G_{m}^{(2)}+ \kappa _2\left( L^{(2)}-D\right) ^2 \right] \right\} ^{1/2}, \end{aligned}$$
(38)

from the notation (31)\(_2\) it is obtained the following estimate:

$$\begin{aligned}{} & {} \left| f_2(t)\right\| \!\le \!f_6(t)\!\!\left\{ \!\int _{D(t)}\!\!\left[ \varrho _1\dot{v}_{m}\dot{v}_{m}\!+\! I_{mn}^{(1)}\dot{\varphi }_m\dot{\varphi }_n\!\!+\!\varrho _1\kappa _1\dot{\phi }^2\!\!+\! \varrho _2\dot{u}_m\dot{u}_m\!\!+\!I_{mn}^{(2)}\dot{\psi }_m\dot{\psi }_n\!\!+\!\varrho _2\kappa _2\dot{\chi }^2 \right] \!\right\} ^{\!1/2} \nonumber \\{} & {} \quad \le f_6(t)\sqrt{2{{\mathcal {S}}}(t)}. \end{aligned}$$
(39)

With the help of the following two notations:

$$\begin{aligned} f_7(t)= & {} 2\sqrt{2}\left( f_4(t)+\sqrt{M}f_5(t)+f_6(t)\right) , \nonumber \\ {{\mathcal {F}}}= & {} 2{{\mathcal {S}}}(0)+ \nonumber \\{} & {} +2 \left| \int _{D(t)}\!\!\!\left( \varrho _1\dot{v}_m V_m\!+\!I_{mn}^{(1)}\dot{\varphi }_m \Phi _n\!+\!\varrho _1\kappa _1\dot{\phi }\Upsilon \!+\! \varrho _2\dot{u}_m U_m\!+\!I_{mn}^{(2)}\dot{\psi }_m \Psi _n\!+\!\varrho _2\kappa _2\dot{\chi }\emptyset \right) \!dV\right| \nonumber \\{} & {} +\max \limits _{t\in [0,T]}\int _{D(t)}\!\!\!\left( \varrho _1 V_m V_m\!+\!I_{mn}^{(1)}\Phi _m \Phi _n\!+\!\varrho _1\kappa _1\Upsilon ^2\!+\! \varrho _2 U_m U_m\!+\!I_{mn}^{(2)} \Psi _m \Psi _n\!+\!\varrho _2\kappa _2\emptyset ^2\right) \!dV \nonumber \\{} & {} +2\int _0^T\left| f_3(t)\right| dt, \hspace{6cm} \end{aligned}$$
(40)

we obtain from the identity (32), in which we consider the estimates (33), (35), (36) and (39), the following integral estimate:

$$\begin{aligned} {{\mathcal {S}}}(t)\le {{\mathcal {F}}}+\int _0^tf_7(\tau )\sqrt{{{\mathcal {S}}}(\tau )}d\tau . \end{aligned}$$
(41)

Now we can formulate and prove the result of the continuous dependence of the solution of the mixed problem in relation to the loads.

Theorem 2

.

It is assumed that the regularity assumptions in Sect. 2 are satisfied.

Then, the following estimate, regarding continuous dependence, is satisfied:

$$\begin{aligned} \sqrt{{{\mathcal {S}}}(t)}\le \sqrt{{{\mathcal {F}}}}+\frac{1}{2}\int _0^tf_7(\tau )d\tau ,\; \forall t\in [0,T]. \end{aligned}$$
(42)

Proof

Considering the notation:

$$\begin{aligned} f_8(t)=\left( {{\mathcal {F}}}+\int _0^tf_7(\tau )\sqrt{{{\mathcal {S}}}(\tau )}d\tau \right) ^{1/2}, \end{aligned}$$
(43)

from (41) it is obtained:

$$\begin{aligned} {{\mathcal {S}}}(t)\le \left( f_8(t)\right) ^2, \; \forall t\in [0,T], \end{aligned}$$
(44)

from where, considering (43), it is easy to deduce that:

$$\begin{aligned} 2f_8(t)\dot{f}_8(t)=f_7(t)\sqrt{{{\mathcal {F}}}(t)}\le f_7(t)f_8(t), \; \forall t\in [0,T], \end{aligned}$$
(45)

from where, by integrating, it is deduced that:

$$\begin{aligned} f_8(t)\le \frac{1}{2}\int _0^tf_7(\tau )d\tau +f_8(0),\; \forall t\in [0,T]. \end{aligned}$$
(46)

Clearly, from (44) and (46) it is deduced the desired result (42) and so the proof of Theorem 2 is concluded. \(\square \)

5 Conclusion

It can be seen that in the case of zero external loads, the estimation (42) receives the simpler form:

$$\begin{aligned} {{\mathcal {F}}}(t)\le 2t^2\int _D\left( \frac{1}{\varrho _1\kappa _1}C^2+\frac{1}{\varrho _2\kappa _2}D^2\right) dV, \;\forall t\in [0,T], \end{aligned}$$

that is, the continuous dependence of the solution is obtained only in relation to the coefficients C and D.

On the other hand, if the hypotheses (5) and (11) are taken into account, with the help of relations (9) and (31)\(_1\) can be deduced that estimation (42) offers a priori evaluations for the fields that follow:

$$\begin{aligned}{} & {} \int _D\dot{v}_m\dot{v}_mdV,\; \int _D\dot{\varphi }_m\dot{\varphi }_mdV,\;\int _D\dot{u}_m\dot{u}_mdV,\; \int _D\dot{\psi }_m\dot{\psi }_mdV,\; \int _D\dot{\phi }^2 dV,\;\int _D\dot{\chi }^2 dV, \nonumber \\{} & {} \int _D v_{m,n} v_{m,n}dV,\; \int _D \varphi _{m,n} \varphi _{m,n}dV,\; \int _D\phi ^2 dV,\;\int _D\chi ^2 dV,\; \int _D\phi _{,m}\phi _{,m} dV,\;\int _D\chi _{,m}\chi _{,m}dV. \end{aligned}$$
(47)

It can be anticipated that the procedure explained above can be adapted to obtain a priori estimates for the following fields as well:

$$\begin{aligned}{} & {} \int _D\dot{v}_{m,n}\dot{v}_{m,n}dV,\; \int _D\dot{\varphi }_{m,n}\dot{\varphi }_{m,n}dV,\; \int _D\dot{\phi }_{,m}\dot{\phi }_{,m} dV, \nonumber \\{} & {} \int _D\dot{u}_{m,n}\dot{u}_{m,n}dV,\; \int _D\dot{\psi }_{m,n}\dot{\psi }_{m,n}dV,\; \int _D\chi _{,m}\chi _{,m}dV. \end{aligned}$$
(48)

Finally, it should be specified that estimates of the form of (47) and (48) can be useful to obtain the continuous dependence of the solution of the mixed problem in relation to the coupling characteristic coefficients.