Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in H1-tau(R2)boolean AND Hs(R2)withs>3 and for any 0<tau <1. Such a result improves upon the ex-isting literature, where the asymptotic stability is proved for initial perturbations belonging at least toH20(R2). More precisely, the aim of the article is threefold.First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity inH1-tau(R2)boolean AND Hs(R2)withs>3 and 0<tau <1. Next, we prove the strong convergence of the Boussi-nesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. Asymmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition pla ykey roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity & Vert;u2(t)& Vert;L infinity(R2)for initial data only in H1-tau(R2)boolean AND Hs(R2)with s>3.