Attractors for the Kuramoto-Sivashinsky equations Page: 4 of 23
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lim ' J | | I)'\i ( y. ) | | ^ds if n- 1
t t o
fim * J | | I)\i ( s ) | | ‘ d.s il n- 2,.'))
In one dimension, tor eve,i solutions of (0.1), (0.2) (including
Neuman conditions), we proved t.lu* uniform boundedness of orbits.
As a conse(|uence:
-1/A 7 ■>/:'
,l||(xl : .I,,! x )
C:;l e I.
( 0 . A )
Mere, we s ign i I i r.ui t ly shiirpen tin* uppi*r bound (0.0) in the one
diinension.il context. In [ ], the m.i i n step in the derivation ot
the ((),(.) . w.e. .i cl.isAic.il lower estimate on the t.rac'* of same
operator nssniinled to linearized flow ol (0.1), (0.2). Our
main improvement, results from extending a remarkable Sobolev
type i ne'ju.i I i I y obtained !iy lieti and Ibirving (see [ 1) in the
context, ol Si nrnd i nger operators. Our ex l.ens i on t .akes I lie
! o I I ow i in| f orm:
m ., ., iv
') | |Ai|> . |'dx- Oslo | (> 11|. . ( x ) ) "dx (0./)
.M J J
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Nicolaenko, B.; Scheurer, B. & Temam, R. Attractors for the Kuramoto-Sivashinsky equations, article, January 1, 1985; United States. (digital.library.unt.edu/ark:/67531/metadc1093990/m1/4/: accessed January 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.