Burgers Equation with Affine Noise: Stability and Dynamics
Seminar Room 1, Newton Institute
We analyze the dynamics of Burgers equation on the unit interval, driven by affine multiplicative white noise. We show that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic 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 nonlinear cocycle near the stationary solution. 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 invariant smooth finite-codimensional
submanifolds of the energy space through the stationary solution. On these invariant manifolds, solutions of Burgers equation decay towards 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 Tusheng Zhang.