  We consider the inhomogeneous incompressible Navier-Stokes system in a smooth
two or three dimensional bounded domain, in the case where the initial density
is only bounded. Existence and uniqueness for such initial data was shown
recently in [10], but the stability issue was left open. After observing that
the solutions constructed in [10] have exponential decay, a result of
independent interest, we prove the stability with respect to initial data,
first in Lagrangian coordinates, and then in the Eulerian frame. We actually
obtain stability in $L_2({\mathbb R}_+;H^1(\Omega))$ for the velocity and in a
negative Sobolev space for the density.
  Let us underline that, as opposed to prior works, in case of vacuum, our
stability estimates are not weighted by the initial densities. Hence, our
result applies in particular to the classical density patches problem, where
the density is a characteristic function.
