almost all $t\in [0,T]$. This estimate does not depend on ${{\varrho}}\ge0$ and provide us an additional regularity estimate for in-plane shell velocities even in the case ${{\varrho}}=0$. Below we use this observation to suggest unified way to prove well-posedness result not distinguishing cases ${{\varrho}}>0$ and ${{\varrho}}=0$ (in contrast with [@Vor1957], see also [@Sed91b] (${{\varrho}}=0$) and [@Sed_1997] (${{\varrho}}>0$)).

[**(3)**]{} Taking in (\[weak\_sol\_def\]) $\phi(t)=\int_t^T\chi(\tau)d\tau\cdot \psi$, where $\chi$ is a smooth scalar function and $\psi$ belongs to the space $$\label{space-W}
\widetilde{V}=\left\{
\psi\in  V \left|  \;
  \psi|_{{\Omega}}=
\beta\equiv(\beta^1;\beta^2;\delta)\in H^1_0({{\Omega}}) \times H^1_0({{\Omega}}) \times {\widehat H^2_{0}({{\Omega}})}\right. \right\},$$ one see that the weak solution $(v(t);u(t))$ satisfies the relation $$\begin{gathered}
           (v(t),\psi)_{{\cal O}}+(M_{{\alpha}}w_t(t),\delta)_{{\Omega}}+{{\varrho}}({\bar{u}}_t(t),\bar{\beta})_{{\Omega}}\\
 = (v_0, \psi)_{{{\cal O}}} + (M_{{\alpha}}w_1,\delta)_{{\Omega}}+{{\varrho}}({\bar{u}}_1,\bar{\beta})_{{\Omega}}\\
-\int_0^t\big[
 \nu E(v,\psi) +({\Delta}w, {\Delta}\delta)_{{\Omega}}+ a({\bar{u}}, \bar{\beta})+{{\gamma}}( M_{{\alpha}}w_t, \delta)_{{{\Omega}}} \\
   + q(u, \beta)  -  (G_f, \psi)_{{{\cal O}}}  -(G_{sh},\beta)_{{\Omega}}\big] d\tau  \label{weak_sol_d2}
        \end{gathered}$$ for all $t\in [0,T]$ and $\psi\in \widetilde{V}$ with $\psi\big\vert_{{\Omega}}=\beta=(\beta^1;\beta^2;\delta)$ and $\bar{\beta}=(\beta^1;\beta^2)$.

As a phase space we use $$\label{space-cH}
{{\cal H}}=
\left\{
\begin{aligned}
\left\{ (v_0;u_0;u_1)\in X\times W\times Y :\; v_0=u_1
~\mbox{on}~ {{\Omega}}\right\}, \qquad {{\varrho}}>0, \\
\left\{ (v_0;u_0;w_1)\in X\times W\times H_\alpha :\; (v_0)^3=w_1
~\mbox{on}~ {{\Omega}}\right\}\qquad {{\varrho}}=0
\end{aligned}
\right.$$ with the norm $$\|(v_0;u_0;u_1)\|_{{\cal H}}^2=\|v_0\|^2_{{{\cal O}}}+\|u_0\|^2_{W}+\|w_1\|^2_{H_\alpha} + {{\varrho}}||(u_1^1, u_1^2)||_{{\Omega}}^2,$$ where $W$ and $Y$ are given by . We also denote by ${\widehat{{{\cal H}}}}$ 