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We initiate the study of stability of solutions of the 2D inviscidincompressible porous medium equation (IPM). We begin by classifying allstationary solutions of the inviscid IPM under mild conditions. We then provesome linear stability results. We then study solutions of the IPM equationwhich are sufficiently regular perturbations of linearly stable steady states.We prove that sufficiently regular perturbations which are also small must beglobally regular and strongly converge to a steady state. The mechanism behindthe stability is \emph{stratification} as opposed to previous stability resultsbased on dispersion and/or mixing \cite{BedrossianMasmoudi}. More or less, weprove that stratified stationary solutions which do not go against gravity areasymptotically stable.

Asymptotic Stability for some Stratified Stationary Solutions of the Inviscid Incompressible Porous Medium Equation