TY - JOUR

T1 - On the Asymptotic Stability of Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

AU - Elgindi, Tarek M.

N1 - Funding Information:
The author acknowledges the support of NSF Postdoctoral Research Fellowship Award 1402357. He also acknowledges helpful conversations with Jacob Bedrossian, Pierre Germain, and Nader Masmoudi

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00205-017-1090-7

DO - 10.1007/s00205-017-1090-7

M3 - Article

AN - SCOPUS:85017149609

VL - 225

SP - 573

EP - 599

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -