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

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.

References in corpus (1)

• Small scale formations in the incompressible porous media equation

Cited by in corpus (4)

• Dynamics of density patches in infinite Prandtl number convection
• On the Well-Posedness of a Fractional Stokes-Transport System
• Long-time behavior of the Stokes-transport system in a channel
• Weak Solutions for a non-Newtonian Stokes-Transport System