On some semilinear, dissipative systems of equations with a small parameter arising in the numerical solution of Navier-Stokes equations, equations of motion of Oldroyd and Kelvin-Voigt fluids. (Russian. English summary) Zbl 0801.76017

Oskolkov, A. P.

Zap. Nauchn. Semin. POMI 202, 158-184 (1992).

Summary: Solutions of the two-dimensional initial-boundary value problem for Navier-Stokes equations are approximated by solutions of the initial- boundary value problem \[ {{\partial v^ \varepsilon} \over {\partial t}}- \nu \Delta v^ \varepsilon+ v_ k^ \varepsilon {{\partial v^ \varepsilon} \over {\partial x_ k}} + {1\over 2} v^ \varepsilon \text{ div } v^ \varepsilon- {1\over \varepsilon} \text{ grad div }w^ \varepsilon =f, \qquad {{\partial w^ \varepsilon} \over {\partial t}}+ \alpha w^ \varepsilon= v^ \varepsilon, \] \[ v^ \varepsilon|_{t=0}= v_ 0^ \varepsilon(x), \quad w^ \varepsilon|_{t=0} =0, \quad x\in\Omega; \qquad v^ \varepsilon|_{\partial\Omega}= w^ \varepsilon |_{\partial\Omega} =0, \quad t\in \mathbb{R}^ +. \] We study proximity of solutions of these problems in appropriate norms and also proximity of their minimal global \(B\)-attractors. Analogous results are valid for the two-dimensional equations of motion of the Oldroyd and Kelvin-Voigt fluids.

MSC:

76D05	Navier-Stokes equations for incompressible viscous fluids
76A10	Viscoelastic fluids
35Q30	Navier-Stokes equations
35Q35	PDEs in connection with fluid mechanics

Keywords:

initial-boundary value problem; proximity of solutions; minimal global \(B\)-attractors