On the large time behavior of the 2D inhomogeneous incompressible viscous flows

This paper studies the two-dimensional inhomogeneous Navier–Stokes equations governing stratified flows in a bounded domain under a gravitational potential $f$. Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition $\mathbf{u}{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=\mathbf{0}$, all admissible equilibria are hydrostatic and satisfy $\mathrm{\nabla }{p}_s=-{\rho }_s\mathrm{\nabla }f$. Second, through a perturbative analysis around arbitrary hydrostatic profiles, we show that despite possible transient growth induced by the Rayleigh–Taylor mechanism, the system always relaxes to a hydrostatic equilibrium. Third, we identify a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile of the form ${\rho }_s=-\gamma f+\beta$, with $\gamma >0$ and $\beta >0$. Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space $H^3(\mathrm{\Omega })$.