On semilinear dissipative systems of equations with a small parameter that arise in solution of the Navier--Stokes equations, equation of motion of the Oldroyd fluids, and equations of motion of the Kelvin--Voight fluids

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part IX, Tome 202 (1992), pp. 158-184

Résumé

Solutions of the two-dimensional initial boundary-value problem for the Navier–Stokes equations are approximated by solutions of the initial boundary-value problem \begin{gather} \frac{\partial v^\varepsilon}{\partial t}-\nu\Delta v^\varepsilon+v^\varepsilon_kv^\varepsilon_{x_k}+\frac12v^\varepsilon\operatorname{div}v^\varepsilon-\frac{1}{\varepsilon}\operatorname{grad}\operatorname{div}w^\varepsilon=f,\enskip \frac{\partial w^\varepsilon}{\partial t}+\alpha w^\varepsilon=v^\varepsilon,\enskip \nu,\alpha>0 \tag{9} \\ v^\varepsilon|_{t=0}=v_0^\varepsilon(x),\quad w^\varepsilon|_{t=0}=0,\quad x\in\Omega;\quad v^\varepsilon|_{\partial\Omega}=w^\varepsilon|_{\partial\Omega}=0,\quad t\geqslant0, \tag{10} \end{gather} We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global $B$-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin–Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles.