Global well-posedness for 2D nonhomogeneous asymmetric fluids with magnetic field and density-dependent viscosity

We study an initial-boundary value problem of two-dimensional nonhomogeneous asymmetric fluids with magnetic field and density-dependent viscosity μ (ρ ) . Applying Desjardins' interpolation inequality and delicate energy estimates, we show the global-in-time existence of a unique strong solution when ∇ μ (ρ ₀) Lq is properly small. Moreover, we prove that the velocity, the micro-rotational velocity, and the magnetic field converge exponentially to zero in H² as time goes to infinity.