Optimal bounds on the Kuramoto–Sivashinsky equation

Abstract: In this paper, we consider solutions of the one-dimensional Kuramoto–Sivashinsky equation, i.e. which are L-periodic in x and have vanishing spatial average. Numerical simulations show that for , 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 . More specifically, after an initial layer, it is observed that the spatial quadratic average of all fractional derivatives of u is bounded independently of L. In particular, the time-space average is observed to be bounded independently of L. The best available result states that for all . In this paper, we prove that for . 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 , which improves the known bounds. [Copyright &y& Elsevier]