New bounds for the Kuramoto‐Sivashinsky equation

We show that every L ‐periodic mean‐zero solution u of the Kuramoto‐Sivashinsky equation is on average o ( L ) for L ≫ 1, in the sense that for any T > 0 the space average of | u ( t ) | is bounded by $L \over T$ for any t > T and any L sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this non‐standard perturbation of the Burgers equation is based on a “div‐curl” argument. © 2004 Wiley Periodicals, Inc.