  We consider the Stokes-transport system, a model for the evolution of an
incompressible viscous fluid with inhomogeneous density. This equation was
already known to be globally well-posed for any $L^1\cap L^\infty$ initial
density with finite first moment in $\mathbb{R}^3$. We show that similar
results hold on different domain types. We prove that the system is globally
well-posed for $L^\infty$ initial data in bounded domains of $\mathbb{R}^2$ and
$\mathbb{R}^3$ as well as in the infinite strip $\mathbb{R}\times(0,1)$. These
results contrast with the ill-posedness of a similar problem, the
incompressible porous medium equation, for which uniqueness is known to fail
for such a density regularity.
