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
Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
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 -