Global well-posedness and large-time behavior for anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation

In this paper, we consider the Cauchy problem for the two-dimensional anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation, where the density at infinity is in the state $ \widetilde{\rho} > 0 $. We obtain the global well-posedness of strong solutions and the large-time behavior of the velocity field under the condition that $ \Vert \sqrt{\rho_0} \mathbf{u}_0 \Vert_{L^2} $ is sufficiently small. It is worth noting that our results include the case of vacuum.