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

Albeverio, Sergio; Rozanova, Olga

doi:10.1090/S0002-9939-09-10020-5

Suppression of unbounded gradients in an SDE associated with the Burgers equation
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by Sergio Albeverio and Olga Rozanova PDF

Proc. Amer. Math. Soc. 138 (2010), 241-251 Request permission

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.

References

H. Risken, The Fokker-Planck equation, 2nd ed., Springer Series in Synergetics, vol. 18, Springer-Verlag, Berlin, 1989. Methods of solution and applications. MR 987631, DOI 10.1007/978-3-642-61544-3
Weinan E, K. Khanin, A. Mazel, and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2) 151 (2000), no. 3, 877–960. MR 1779561, DOI 10.2307/121126
J.P. Bouchaud, M. Mézard, Velocity fluctuations in forced Burgers turbulence, Phys. Rev. E54, 5116 (1996).
V. Gurarie, Burgers equations revisited, arXiv:nlin/0307033v1 [nln.CD] (2003).
S. Albeverio, O. Rozanova, The non-viscous Burgers equation associated with random positions in coordinate space: A threshold for blow up behaviour, to appear in $\rm M^3AS$ (Mathematical Methods and Models in Applied Science)(2009), arXiv:0708.2320v2 [math.AP].
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
Azzouz Dermoune, Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 5, 595–599 (English, with English and French summaries). MR 1649309, DOI 10.1016/S0764-4442(98)85013-1
De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI 10.1006/jdeq.1994.1093
Henning Struchtrup, Macroscopic transport equations for rarefied gas flows, Interaction of Mechanics and Mathematics, Springer, Berlin, 2005. Approximation methods in kinetic theory. MR 2287368