Non–uniform decay of solutions to the incompressible magneto–micropolar fluids with/without magnetic diffusion and spin viscosity

Abstract

We are concerned with the long time behavior of solutions to the incompressible magneto–micropolar fluids with/without magnetic diffusion and spin viscosity in two and three dimensions. When the magnetic diffusion is absent, we show that L2–norm of fluid velocity and micro–rotational velocity vanishes and L2–norm of the magnetic field converges to a non–negative constant as time tends to infinity. Moreover, when both magnetic diffusion and spin viscosity are present and the initial data belongs to L2, we prove that the solutions decay to zero without a rate, and this non–uniform decay is optimal. The proofs are based on Fourier splitting method, low–frequency and high–frequency decomposition techniques and delicate energy estimates.