On a lemma due to Ladyzhenskaya and Solonnikov and some applications

Abstract

We present some applications of a lemma by Ladyzhenskaya and Solonnikov [Determination of solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980) 117–160 (English Transl.: J. Soviet Math. 21 (1983) 728–761)]. Some other results in that paper referring to stationary Navier–Stokes equations are extended to a non-Newtonian fluid, the so-called micropolar fluid. This model depends on the microrotational viscosity ν r which vanishes for a Navier–Stokes fluid. We use the lemma in full to show that, as ν r tends to zero, the solutions of the Ladyzhenskaya–Solonnikov problem converge to the solutions of the corresponding problem for Navier–Stokes equations. In addition, we obtain a similar convergence regarding the Leray problem for micropolar fluids.