Attractors for the Kuramoto-Sivashinsky equations Page: 3 of 23
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In this paper, we address the question of constructing an
upper bound of the Hausdorff and Fractal dimensions d^(x) and
dp(x) for attractors X of the Kuramoto-Sivanshinsky equation.
We investigate the large time behavior of the solution u =
u (x i b ) of:
‘!U H v>A2n H An -t \ ■+ |Vu| 2 0
cJ r 2.
uCxO) - u()(x)
ll(x • 1,1' . , l ) - II t X 11 )
I ■ i'n,
wh(>re \> ■() and u^ is I-per i nd i c. Iliis equation occurs in .i large
variety nl physical s i l ua l. i ons ; it. mode I s the format, ion ol
cellular patterns whose temporal behavior becomes chaotic, when
the typical s i/e I ol the cell i s large enough. I he natural
hi function parameter ol the problem i ■■ the adimcusional number I
1 n the former paper [ | , we < me. idereil solutions ol
(0. 1 )"((). 2) with either per i m I i i or Neum.m I loiinda ry ( end i t i me.
We obtained general upper bounds lor d (X). d (X) in terms nl
K I mi | | Vnl M||
11N 0 0 (IM / (
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Nicolaenko, B.; Scheurer, B. & Temam, R. Attractors for the Kuramoto-Sivashinsky equations, article, January 1, 1985; United States. (https://digital.library.unt.edu/ark:/67531/metadc1093990/m1/3/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.