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

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 ∩ L∞ initial density with finite first moment in R. We show that similar results hold on different domain types. We prove that the system is globally wellposed for L∞ initial data in bounded domains of R and R as well as in the infinite strip R × (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.

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