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.