In this joint work with Roberta Bianchini and Marius Paicu, we address the existence of stably stratified solutions to the Incompressible Porous Media (IPM) equation. In particular, we justify the nonlinear asymptotic stability for initial perturbations in $\dot $H^3$. Moreover, we justify the strong relaxation limit of the Boussinesq system with damped velocity towards the (IPM) equation under a suitable scaling. A crucial part of our analysis relies on the use of an anisotropic Littlewood-Paley decomposition to derive new bounds on the vorticity.
Jérémy Heleine, David Lafontaine
