  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 the noise to regularize the
evolution by suppressing blow-up in the supercritical regime, or for the noise
to trigger blow-up in the subcritical regime. However, as the noise amplitude
becomes large, the blow-up times in the supercritical regime are 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.
