On some way of the approximation of solutions of initial boundary value problems for Navier–Stokes equations

Solutions of the initial boundary value problem for Navier–Stokes equations are approximated by solutions of the initial boundary value problem
\begin{gather*} \partial_t u(t)+u_k(t)\partial_ku(t)-\nu\Delta u(t)-\frac1\varepsilon\nabla\mathrm{div}\,u(t)+\frac12u(t)\mathrm{div}\,u(t)=f(t),\\ u(0)=u_0\text{ in }\Omega;\quad u(t)=0\text{ on }\partial\Omega. \end{gather*}
We study proximity of solutions of these problems in suitable norms and also proximity of their minimal global $B$-attractors.