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

Details Suppression of unbounded gradients in a SDE associated with the Burgers equation Suppression of unbounded gradients in a SDE associated with the Burgers equation

2008-04-09

arxiv.org/abs/0804.1553v3

Proc. Amer. Math. Soc. 138 (2010), no.1, 241-251

Mathematics

Analysis of PDEs

9 pages, submitted Minor change in the abstract, minor correction in the computations, discussion added, accepted in Proc.Amer

Scientific paper

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 $|x|^{-2},$ 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.

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