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

TL;DR

This paper proves the global well-posedness of the Stokes-transport system for inhomogeneous fluids in various bounded and unbounded domains, extending known results from the whole space to more complex geometries.

Contribution

It extends the well-posedness results of the Stokes-transport system to bounded domains and the infinite strip, contrasting with the ill-posedness of similar models.

Findings

Global well-posedness in bounded domains of R^2 and R^3

Well-posedness in the infinite strip R x (0,1)

Contrast with ill-posedness of the porous medium equation

Abstract

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.