Abstract :
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.
