کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6415267 | 1335093 | 2009 | 58 صفحه PDF | دانلود رایگان |
In this paper, we consider solutions u(t,x) of the one-dimensional Kuramoto-Sivashinsky equation, i.e.âtu+âx(12u2)+âx2u+âx4u=0, which are L-periodic in x and have vanishing spatial average. Numerical simulations show that for Lâ«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 Lâ«1. More specifically, after an initial layer, it is observed that the spatial quadratic average ã(|âx|αu)2ã of all fractional derivatives |âx|αu of u is bounded independently of L. In particular, the time-space average ãã(|âx|αu)2ãã is observed to be bounded independently of L. The best available result states that ãã(|âx|αu)2ãã1/2=o(L) for all 0⩽α⩽2. In this paper, we prove thatãã(|âx|αu)2ãã1/2=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 ããu2ãã1/2⩽O(L1/3+), which improves the known bounds.
Journal: Journal of Functional Analysis - Volume 257, Issue 7, 1 October 2009, Pages 2188-2245