Uncertainty estimates and L2 bounds for the Kuramoto-Sivashinsky equation

We consider the Kuramoto-Sivashinsky (KS) equation in one dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam [18], and later improved by Collet, Eckmann, Epstein and Stubbe[1], and Goodman [10] to prove that lim sup t→∞ ||u||2 ≤ CL 3 2 . This result is slightly weaker than that recently announced by Giacomelli and Otto [9], but applies in the presence of an additional linear destabilizing term. We further show that for a large class of functions φx the exponent 3 2 is the best possible from this line of argument. Further, this result together with a result of Molinet[17] gives an improved estimate for L2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.