Suppression of unbounded gradients in an SDE associated with the Burgers equation

Abstract

We consider the Langevin equation describing a non-viscous Burgers fluid stochastically perturbed by uniform noise. We introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of the position fixed. We study interrelations between this function and the solution of the non-perturbed Burgers equation. We are especially interested in the property of the solution of the latter equation to develop unbounded gradients within a finite time. We study the question of how the initial distribution of particles for the Langevin equation influences this blowup phenomenon. We show 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, it does not depend on the constant variance of the stochastic perturbation. Further, for a one space variable we get the following result: if the decay rate of the even power-behaved initial particles distribution at infinity is greater than or equal to |x| -2 , 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.