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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Burgers Equation with Affine Noise: Stability and
Dynamics - Mohammed\, S (Southern Illinois)
DTSTART;TZID=Europe/London:20100108T153000
DTEND;TZID=Europe/London:20100108T163000
UID:TALK22190AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/22190
DESCRIPTION:We analyze the dynamics of Burgers equation on the
unit interval\, driven by affine multiplicative w
hite noise. We show that the solution field of the
stochastic Burgers equation generates a smooth pe
rfect and locally compacting cocycle on the energy
space. Using multiplicative ergodic theory techn
iques\, we establish the existence of a discrete n
onrandom Lyapunov spectrum of the linearized cocyc
le along a stationary solution. The Lyapunov spect
rum characterizes the large-time asymptotics of th
e nonlinear cocycle near the stationary solution.
In the absence of additive space-time noise\, we e
xplicitly compute the Lyapunov spectrum of the lin
earized cocycle on the zero equilibrium in terms o
f the parameters of Burgers equation. In the ergo
dic case\, we construct a countable random family
of local asymptotically invariant smooth finite-co
dimensional \nsubmanifolds of the energy space thr
ough the stationary solution. On these invariant m
anifolds\, solutions of Burgers equation decay tow
ards the equilibrium with fixed exponential speed
governed by the Lyapunov spectrum of the cocycle.
In the general hyperbolic (non-ergodic) case\, we
establish a local stable manifold theorem near the
stationary solution. This is joint work with Tush
eng Zhang.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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