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

Well-posedness of the Stokes-transport system in bounded domains and in the infinite strip