Guillaume van Baalen, Jean-Pierre Eckmann
Non-Vanishing Profiles for the Kuramoto-Sivashinsky Equation on 
the Infinite Line
(132K, ps)

ABSTRACT.  We study the Kuramoto-Sivashinsky equation on the 
infinite line with initial conditions having arbitrarily large limits 
$\pm Y$ at 
$x=\pm\infty$. We show that the solutions have the same limits for all 
positive times. This implies that an 
attractor for this equation cannot be defined in $L^\infty$. 
To prove this, we consider 
profiles with limits at $x=\pm\infty$, and show that initial conditions 
$L^2$-close to such profiles lead to solutions which remain $L^2$-close to 
the profile for all times. Furthermore, the difference between these solutions and the 
initial profile tends to $0$ as $x\to\pm\infty$, for any fixed time 
$t>0$. 
Analogous 
results hold for $L^2$-neighborhoods of periodic stationary solutions. 
This implies that profiles and periodic stationary solutions partition the 
phase space into mutually unattainable regions.