Nonlinear Stability of stratified flows at critical regularity: the Incompressible Porous media and Boussinesq equations

We investigate the stabilizing effect of stratification in fluid models, focusing on the two-dimensional Incompressible Porous Media (IPM) and Boussinesq equations. For the stratified IPM system, we establish its global well-posedness for initial perturbations in $\dot{B}^{1}_{2,1}\cap \dot{B}^{2}_{2,1}$, under a smallness assumption in the critical space $\dot B^{2}_{2,1}$. We also study a stratified Boussinesq system with damping and rigorously justify its strong relaxation limit toward the IPM equation, with an explicit rate of convergence for ill-prepared initial data after subtraction of nonlinear initial layers. The analysis combines anisotropic Littlewood--Paley estimates with a

nonlinear change to density coordinates, which removes the problematic vertical transport term and allows us to recover the time-integrable Lipschitz bounds needed to close the critical estimates.