On the Asymptotic Stability of Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

Tarek M. Elgindi

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

We study the stability of stationary solutions of the two dimensional inviscid incompressible porous medium equation (IPM). We show that solutions which are near certain stable stationary solutions must converge as t → ∞ to a stationary solution of the IPM equation. It turns out that linearizing the IPM equation about certain stable stationary solutions gives a non-local partial damping mechanism. On the torus, the linearized problem has a very large set of stationary (undamped) modes. This makes the problem of long-time behavior more difficult since there is the possibility of a cascading non-linear growth along the stationary modes of the linearized problem. We solve this by, more or less, doing a second linearization around the undamped modes, exploiting a special non-linear structure, and showing that the stationary modes can be controlled.

Original languageEnglish (US)
Pages (from-to)573-599
Number of pages27
JournalArchive for Rational Mechanics and Analysis
Volume225
Issue number2
DOIs
StatePublished - Aug 1 2017

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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