Burgers Equation with Affine Linear Noise: Dynamics and Stability

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by affine linear white noise. It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space L2([0, 1],R). Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum {λi}i=1 of the linearized cocycle along a stationary solution/equilbrium. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the equilibrium. In the absence of additive space-time noise, we explicitly compute the Lyapunov spectrum of the linearized 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 flow-invariant submanifolds {Si(ω)}i=1 of the energy space L2([0, 1],R) so that on each submanifold Si(ω), the cocycle decays towards the equilibrium with fixed exponential speed less than or equal to λi. Each local manifold Si(ω) is smooth and has finite-codimension i − 1 for each i ≥ 1. On a global level, we show the existence of a flow-invariant random flag in the energy space. The global random flag is characterized by the Lyapunov spectrum of the linearized cocycle. In the presence of a linear drift and with no additive space-time noise, we also give sufficient conditions on the parameters of the stochastic Burgers equation which guarantee uniqueness of the stationary solution or its hyperbolicity. In the general hyperbolic (non-ergodic) case, we establish a local stable manifold theorem near the equilibrium. AMS Subject Classification: Primary 60H15 Secondary 60F10, 35Q30. 1 Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901, USA. Email: [email protected]. The research of this author is supported by NSF award DMS-0705970. 2 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, U.K. Email: [email protected]