Uncertainty estimates and L2 bounds for the Kuramoto–Sivashinsky equation

Abstract

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 et al (1985 Physica D 16 155–83) and later improved by Collet et al (1993 Commun. Math. Phys. 152 203–14) and Goodman (1994 Commun. Pure Appl. Math. 47 293–306) to prove that This result is slightly weaker than a related result by Giacomelli and Otto (2005 Commun. Pure Appl. Math. 58 297–318), but applies in the presence of an additional linear destabilizing term. We further show that for a large class of functions ϕx the exponent is the best possible from this line of argument. Finally, we mention several related results from the literature on equations related to the KS equation that can be improved using these ideas.