2014/11/25 by Tarek M. Elgindi, Elgindi, Tarek M.
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP #physics.flu-dyn
paper · pdf · doi:10.48550/arxiv.1411.6958
openalex publication_date 2014/11/25 · arxiv created 2016/12/08 · arxiv updated 2016/12/09 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability results. We then study solutions of the IPM equation which are sufficiently regular perturbations of linearly stable steady states. We prove that sufficiently regular perturbations which are also small must be globally regular and strongly converge to a steady state. The mechanism behind the stability is stratification as opposed to previous stability results based on dispersion and/or mixing \citeBedrossianMasmoudi. More or less, we prove that stratified stationary solutions which do not go against gravity are asymptotically stable.