The initial-boundary value problem with a free surface condition for the $\varepsilon$-approximations of the Navier–Stokes equations and some their regularizations

We study the unique solvability in the large on the semiaxis $\mathbb R^2$ of the initial boundary value problems (IBVP) with the boundary slipcondition (the natural boundary condition) for the $\varepsilon$-approximations (0.6)–(0.8), (0.20); (0.13)–(0.15), (0.21), and (0.16–0.18), (0.22) of the Navier–Stokes equations (NSE), of the NSE modified in the sense of O. A. Ladyzhenskaya, and the equations of motion of the Kelvin–Voight fluids. For the classical solutions of perturbed problems we prove certain estimates which are uniform with respect to $\varepsilon$, and show that as $\varepsilon\to0$ the classical solutions of the perturbed IBVP respectively converge to the classical solutions of the IBVP with the boundary slip condition for the NSE, for the NSE (0.11) modified in the sense of Ladyzhenskaya, and for the equations (0.12) of motion of the Kelvin–Voight fluids. Bibliography: 40 titles.