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 \(\mu (\rho )\) $\mu (\rho )$ . Applying Desjardins’ interpolation inequality and delicate energy estimates, we show the global-in-time existence of a unique strong solution when \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\) $\Vert \mathrm{\nabla }\mu ({\rho }_0){\Vert }_{L^q}$ is properly small. Moreover, we prove that the velocity, the micro-rotational velocity, and the magnetic field converge exponentially to zero in \(H^2\) $H^2$ as time goes to infinity.