Predictability of the Burgers dynamics under model uncertainty

Abstract

Complex systems may be subject to various uncertainties. A great effort has been concentrated on predicting the dynamics under uncertainty in initial conditions. In the present work, we consider the well-known Burgers equation with random boundary forcing or with random body forcing. Our goal is to attempt to understand the stochastic Burgers dynamics by predicting or estimating the solution processes in various diagnostic metrics, such as mean length scale, correlation function and mean energy. First, for the linearized model, we observe that the important statistical quantities like mean energy or correlation functions are the same for the two types of random forcing, even though the solutions behave very differently. Second, for the full nonlinear model, we estimate the mean energy for various types of random body forcing, highlighting the different impact on the overall dynamics of space-time white noises, trace class white-in-time and colored-in-space noises, point noises, additive noises or multiplicative noises.