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SUMMARY:Burgers Equation with Affine Noise: Stability and Dynamics - Moham
 med\, S (Southern Illinois)
DTSTART:20100108T153000Z
DTEND:20100108T163000Z
UID:TALK22190@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We analyze the dynamics of Burgers equation on the unit interv
 al\, driven by affine multiplicative white noise. We show that the solutio
 n field of the stochastic Burgers equation generates a smooth perfect and 
 locally compacting cocycle on the energy space.  Using multiplicative ergo
 dic theory techniques\, we establish the existence of a discrete nonrandom
  Lyapunov spectrum of the linearized cocycle along a stationary solution. 
 The Lyapunov spectrum characterizes the large-time asymptotics of the nonl
 inear cocycle near the stationary solution. In the absence of additive spa
 ce-time noise\, we explicitly compute the Lyapunov spectrum of the lineari
 zed cocycle on the zero equilibrium in terms of the parameters of Burgers 
 equation.  In the ergodic case\, we construct a countable random family of
  local asymptotically invariant smooth finite-codimensional \nsubmanifolds
  of the energy space through the stationary solution. On these invariant m
 anifolds\, solutions of Burgers equation decay towards the equilibrium wit
 h fixed exponential speed governed by the Lyapunov spectrum of the cocycle
 . In the general hyperbolic (non-ergodic) case\, we establish a local stab
 le manifold theorem near the stationary solution. This is joint work with 
 Tusheng Zhang. 
LOCATION:Seminar Room 1\, Newton Institute
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