03-359 Guillaume van Baalen, Jean-Pierre Eckmann

Non-Vanishing Profiles for the Kuramoto-Sivashinsky Equation on the Infinite Line (132K, ps) Aug 7, 03

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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 $t>0$. 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.

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