Suppression of unbounded gradients in a SDE associated with the Burgers equation
arXiv:0804.1553
Abstract
We consider the Langevin equation describing a stochastically perturbed by uniform noise non-viscous Burgers fluid and introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of position fixed. We study interrelations between this function and the solution of the non-perturbed Burgers equation. Especially we are interested in the property of the solution of the latter equation to develop unbounded gradients within a finite time. We study the question how the initial distribution of particles for the Langevin equation influences this blowup phenomenon. It is shown that for a wide class of initial data and initial distributions of particles the unbounded gradients are eliminated. The case of a linear initial velocity is particular. We show that if the initial distribution of particles is uniform, then the mean of the velocity for a given position coincides with the solution of the Burgers equation and in particular does not depend on the constant variance of the stochastic perturbation. Further, for a one space space variable we get the following result: if the decay rate of the power-behaved initial particles distribution at infinity is greater or equal then the blowup is suppressed, otherwise, the blowup takes place at the same moment of time as in the case of the non-perturbed Burgers equation.
9 pages, submitted Minor change in the abstract, minor correction in the computations, discussion added, accepted in Proc.Amer.Math.Soc