- 03-359 Guillaume van Baalen, Jean-Pierre Eckmann
- Non-Vanishing Profiles for the Kuramoto-Sivashinsky Equation on
the Infinite Line
Aug 7, 03
(auto. generated ps),
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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
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.