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In this paper, we investigate the well-posedness theory and exponentialstability for the inhomogeneous incompressible Navier-Stokes equation with onlyhorizontal dissipative structure. Due to the lack of the vertical dissipativeterm and appearance of vacuum, it is a highly challenging tricky problem for usto study the well-posedness, stability and large-time behavior problems intwo-dimensional whole space. The local-in-time well-posedness theory issuccessfully established at first because we develop some good estimates forthe density and vorticity to control the nonlinear term. Finally, these goodestimates of density and vorticity help us to establish the global-in-timewell-posedness and exponential stability if the initial velocity is suitablesmall.

Well-posedness and exponential stability of the inhomogeneous  anisotropic incompressible Navier-Stokes equation with far-field vacuum in  two-dimensional whole space