Singularity formation in the deterministic and stochastic fractional Burgers equation

Abstract

This study is motivated by the question of how singularity formation and other forms of extreme behavior in nonlinear dissipative partial differential equations are affected by stochastic excitations. To address this question we consider the 1D fractional Burgers equation with additive colored noise as a model problem. This system is interesting, because in the deterministic setting it exhibits finite-time blow-up or a globally well-posed behavior depending on the value of the fractional dissipation exponent. The problem is studied by performing a series of accurate numerical computations combining spectrally-accurate spatial discretization with a Monte-Carlo approach. First, we carefully document the singularity formation in the deterministic system in the supercritical regime where the blow-up time is shown to be a decreasing function of the fractional dissipation exponent. Our main result for the stochastic problem is that there is no evidence for additive noise to regularize the evolution by suppressing singularities in the supercritical regime, or for the noise to trigger singularities in the subcritical regime. However, as the noise amplitude becomes large, the maximum existence time of the solution in the supercritical regime is shown to exhibit an increasingly non-Gaussian behavior. Analogous observations are also made for the maximum attained values of the enstrophy and the times when the maxima occur in the subcritical regime.