{{\varrho}}u^2_t; w_t) \in L_\infty(0,T; L_2({{\Omega}}) \times L_2({{\Omega}})\times H_\alpha)$ and $u(0)=u_0$;

-   for every $\phi\in {{\cal L}}_T^0$ with $\phi|_{{\Omega}}=b=(b^1;b^2;d)$ the following equality holds: $$\begin{gathered}
                -\!\int_0^T\!\!(v,\phi_t)_{{\cal O}}dt +\nu\!\int_0^T\!\!E(v,\phi)_{{{\cal O}}}  dt -\!\int_0^T\!\! \big[(M_{{\alpha}}w_t,d_t)_{{\Omega}}+{{\varrho}}({\bar{u}}_t,\bar{b}_t)_{{\Omega}}\big]dt
                 \\ +
                {{\gamma}}\int_0^T (M_{{\alpha}}w_t, d)_{{{\Omega}}}dt+ \!\int_0^T\!\!({\Delta}w, {\Delta}d)_{{\Omega}}dt +  \!\int_0^T\!\!a({\bar{u}}, \bar{b})_{{\Omega}}dt  + \!\int_0^T\!\! q(u,b)_{{\Omega}}dt \\
              =  (v_0, \phi(0))_{{{\cal O}}}
              +(M_{{\alpha}}w_1,d(0))_{{\Omega}}+{{\varrho}}({\bar{u}}_1,\bar{b}(0))_{{\Omega}}\\
              +\int_0^T(G_f(t), \phi)_{{{\cal O}}} dt +\int_0^T(G_{sh}(t),b)_{{\Omega}}dt , \label{weak_sol_def}
            \end{gathered}$$ where ${\bar{u}}=(u^1;u^2)$, $\bar{b}=(b^1;b^2)$ and also $$\begin{aligned}
        E(u,\phi)=&\frac 12 \sum_{i,j=1}^3\left(v^j_{x_i} + v^i_{x_j} \right)\left(\phi^j_{x_i} + \phi^i_{x_j} \right), \notag\\
        a({\bar{u}},\bar{b})=&\sum_{i=1}^2 (\nabla u^i,\nabla b^i)_{{\Omega}}+\frac{1+\mu}{1-\mu}({\rm div} u, {\rm div} b)_{{\Omega}}, \label{form-a} \\
        q(u,b)=&
        (k_1 N_{11}+k_2 N_{22}, d)_{{\Omega}}\notag
        \\ & +
        (N_{11}w_{x_1} + N_{12} w_{x_2}, d_{x_1})_{{\Omega}}+(N_{12}w_{x_1} + N_{22} w_{x_2}, d_{x_2})_{{\Omega}}\notag \\
        &+\frac 1{1-\mu} \left[((w_{x_1})^2+\mu (w_{x_2})^2, b^1_{x_1})_{{\Omega}}+ ((w_{x_2})^2+\mu (w_{x_1})^2, b^2_{x_2})_{{\Omega}}\right]
         \notag\\ &
        +(w_{x_1}w_{x_2}, b^1_{x_2}+b^2_{x_1})_{{\Omega}}+ \frac 2{1-\mu}
         (w, \kappa_1b^1_{x_1}+\kappa_1b^2_{x_2})_{{\Omega}}. \notag
        \end{aligned}$$

\[re:weak-sol\]

[**(1)**]{} In the case when we neglect the inertia of longitudinal deformations (${{\varrho}}=0$) the equations in (or in ) become elliptic. However we keep the initial data for the in-plane displacement $(u^1;u^2)$. The point is that the first order evolution for $(u^1;u^2)$ goes from the boundary condition for the fluid velocity in .

[**(2)**]{} It also follows from and from the standard trace theorem that for every weak solution $(v(t);u(t))$ we have that $$\label{smooth_est}
||w(t)||^2_{H^{1/2}_*({{\Omega}})}+||u_t^1(t)||^2_{H^{1/2}_*({{\Omega}})}+||u_t^2(t)||^2_{H^{1/2}_*({{\Omega}})} \le C||{{\nabla}}v(t)||^2_{{\cal O}}$$ for 