O}}\}_{j,k=1}^{2n}\end{matrix}\right],$$ where ${\mathscr{R}}={\rm diag}\{{{\varrho}}_1,\ldots,{{\varrho}}_{2n}\}$. The first matrix in (\[matr-M\]) is nonnegative and the second one is symmetric and strictly positive (since the functions $\{\psi_i, \phi_j :i=1,\ldots,m, j=1,\ldots,2n\}$ are linearly independent). Therefore system and has a unique solution on some time interval $[0,T']$.

It follows from (\[approx\_sol\]) that $$v_{n,m}(t)=\sum_{i=1}^m \alpha_i(t)\psi_i + N_0[{\partial}_t u_n(t)],$$ where $N_0$ is given by (\[fl.n0\]). This implies the following boundary compatibility condition $$\label{nm-comp}
v_{n,m}(t)={\partial}_t u_n(t)~~ \mbox{on}~~ {{\Omega}}.$$

[*Step 2. A priori estimate and limit transition, ${{\varrho}}>0$.*]{} Multiplying by ${{\alpha}}_k(t)$ and by $\dot\beta_k(t)$, after summation we obtain an energy relation of the form for the approximate solutions $(v_{n,m}; u_n)$ (for a similar argument we refer to [@ChuRyz2011] and also to the classical source [@Vor1957] on (non-interacting) shell evolution). By Proposition \[pr:Korn\] and relation this implies the following a priori estimate: $$\begin{gathered}
\label{a-pri0}
 \sup_{t\in [0,T]}\|v_{n,m}(t)\|_{{\cal O}}^2 + \\
 \sup_{t\in [0,T]}\left[\|M^{1/2}_{{\alpha}}{\partial}_t w_{n}(t)\|_{{\Omega}}^2+ {{\varrho}}\| {\partial}_t {\bar{u}}_{n}(t)\|_{{{\Omega}}}^2
 +\|{\Delta}w_n(t)\|^2_{{\Omega}}+ \|{\bar{u}}_n(t)\|_{1,{{\Omega}}}^2\right]  \\
+\int_0^T\|\nabla v_{n,m}(t)\|_{{\Omega}}^2 dt +  \int_0^T \| {\partial}_t {\bar{u}}_{n}(t)\|_{[H_*^{1/2}({{\Omega}})]^2}^2 dt
 \le C_T
 \end{gathered}$$ for any existence interval $[0,T]$ of approximate solutions, where the constant $C_T$ does not depend on $n$ and $m$. In particular, this implies that any approximate solution can be extended on any time interval by the standard procedure, i.e., the 