  We investigate the instability and stability of some steady-states of a
three-dimensional nonhomogeneous incompressible viscous flow driven by gravity
in a bounded domain $\Omega$ of class $C^2$. When the steady density is heavier
with increasing height (i.e., the Rayleigh-Taylor steady-state), we show that
the steady-state is linear unstable (i.e., the linear solution grows in time in
$H^2$) by constructing a (standard) energy functional and exploiting the
modified variational method. Then, by introducing a new energy functional and
using a careful bootstrap argument, we further show that the steady-state is
nonlinear unstable in the sense of Hadamard. When the steady density is lighter
with increasing height, we show, with the help of a restricted condition
imposed on steady density, that the steady-state is linearly globally stable
and nonlinearly locally stable in the sense of Hadamard.
