Optimal bounds on the Kuramoto–Sivashinsky equation

In this paper, we consider solutions u(t, x) of the one-dimensional Kuramoto–Sivashinsky equation, i.e. ∂tu +∂x 1 2 u2 +∂2 xu +∂4 xu = 0, which are L-periodic in x and have vanishing spatial average. Numerical simulations show that for L 1, solutions display complex spatio-temporal dynamics. The statistics of the pattern, in particular its scaled power spectrum, is reported to be extensive, i.e. not to depend on L for L 1. More specifically, after an initial layer, it is observed that the spatial quadratic average (|∂x |αu)2 of all fractional derivatives |∂x |αu of u is bounded independently of L. In particular, the time-space average (|∂x |αu)2 is observed to be bounded independently of L. The best available result states that (|∂x |αu)2 1/2 = o(L) for all 0 α 2. In this paper, we prove that |∂x |αu 2 1/2 = O ln5/3 L for 1/3 < α 2. To our knowledge, this is the first result in favor of an extensive behavior—albeit only up to a logarithm and for a restricted range of fractional derivatives. As a corollary, we obtain u2 1/2 O(L1/3+), which improves the known bounds. © 2009 Elsevier Inc. All rights reserved.