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. partial derivative(t)u + partial derivative(x) (1/2 u(2)) + partial derivative(2)(x)u + partial derivative(4)(x)u = 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 <(vertical bar partial derivative(x)vertical bar(alpha)u)(2)> of all fractional derivatives vertical bar partial derivative(x)vertical bar(alpha)u of u is bounded independently of L. In particular, the time-space average <<(vertical bar partial derivative(x)vertical bar(alpha)u)(2)>> is observed to be bounded independently of L. The best available result states that <<(vertical bar partial derivative(x)vertical bar(alpha)u)(2)>>(1/2) = o(L) for all 0 <= alpha <= 2. In this paper, we prove that <<(vertical bar partial derivative(x)vertical bar(alpha)u)(2)>>(1/2) = O(ln(5/3) L) for 1/3 < alpha <= 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. (C) 2009 Elsevier Inc. All rights reserved.