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(12u2)+∂x2u+∂x4u=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(ln5/3L) 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.