In this paper, we consider solutions u(t, x) 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 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 | α u)

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... of all fractional derivatives |∂ x | α u of u is bounded independently of L. In particular, the time-space average (|∂ x | α u) 2 is observed to be bounded independently of L. The best available result states that (|∂ x | α u) 2 1/2 = o(L) for all 0 α 2. In this paper, we prove that |∂ x | α u 2 1/2 = O ln 5/3 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 u 2 1/2 O(L 1/3+ ), which improves the known bounds.