Burgers Equation with Multiplicative Noise: Dynamics and Stability

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by multiplicative 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 compute the discrete non-random Lyapunov spectrum {λi}i=1 of the linearized cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. In particular, we construct a countable random family of local asymptotically flow-invariant manifolds {Si(ω)}i=1 so that on each manifold Si(ω), the cocycle decays with fixed exponential speed less than or equal to λi. Each local manifold Si(ω) is smooth and has finite-codimension i − 1 for i ≥ 1. On a global level, we show the existence of a flow-invariant random flag in the energy space L2([0, 1],R). The global random flag is characterized by the Lyapunov spectrum of the linearized cocycle. In the presence of a linear drift, 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 hyperbolic case, we establish a local stable manifold theorem near the zero equilibrium. AMS Subject Classification: Primary 60H15 Secondary 60F10, 35Q30.