Uncertainty estimates and<i>L</i>₂bounds for the Kuramoto–Sivashinsky equation

Abstract

We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension\nwith periodic boundary conditions. We apply a Lyapunov function argument\nsimilar to the one first introduced by Nicolaenko, Scheurer, and Temam, and\nlater improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove\nthat ||u||_2 &lt; C L^1.5. This result is slightly weaker than that recently\nannounced by Giacomelli and Otto, but applies in the presence of an additional\nlinear destabilizing term. We further show that for a large class of Lyapunov\nfunctions \\phi the exponent 1.5 is the best possible from this line of\nargument. Further, this result together with a result of Molinet gives an\nimproved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in\nthin rectangular domains in two spatial dimensions.\n