Suppression of unbounded gradients in an SDE associated with the Burgers equation
Authors: Sergio Albeverio and Olga Rozanova
Journal: Proc. Amer. Math. Soc. 138 (2010), 241-251
MSC (2000): Primary 35R60
Published electronically: August 31, 2009
MathSciNet review: 2550189
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 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.
- 1. 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
- 2. 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, https://doi.org/10.2307/121126
- 3. J.P. Bouchaud, M. Mézard, Velocity fluctuations in forced Burgers turbulence, Phys. Rev. E54, 5116 (1996).
- 4. V. Gurarie, Burgers equations revisited, arXiv:nlin/0307033v1 [nln.CD] (2003).
- 5. 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 (Mathematical Methods and Models in Applied Science)(2009), arXiv:0708.2320v2 [math.AP].
- 6. 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
- 7. 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, https://doi.org/10.1016/S0764-4442(98)85013-1
- 8. 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, https://doi.org/10.1006/jdeq.1994.1093
- 9. Henning Struchtrup, Macroscopic transport equations for rarefied gas flows, Interaction of Mechanics and Mathematics, Springer, Berlin, 2005. Approximation methods in kinetic theory. MR 2287368
- H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, Second Edition, Springer-Verlag, Berlin, 1989. MR 987631 (90a:82002)
- Weinan E, K. M. Khanin, A. E. Mazel, Ya. G. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2) 151, no. 3, 877-960 (2000). MR 1779561 (2002e:37134)
- 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 (Mathematical Methods and Models in Applied Science)(2009), arXiv:0708.2320v2 [math.AP].
- I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series, and products. 6th ed., Academic Press, San Diego, CA, 2000. MR 1773820 (2001c:00002)
- A. Dermoune, Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics, C. R. Acad. Sci. Paris, Serie I, 326, 595-599 (1998). MR 1649309 (2000f:82071)
- D. Tan, T. Zhang, Y. Zheng, Delta-shock waves as limits of vanishing viscocity for hyperbolic system of conservation laws, J. Diff. Equat., 112, 1-32 (1994). MR 1287550 (95g:35124)
- H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory, Springer-Verlag, Berlin-Heidelberg, 2005. MR 2287368 (2009f:76150)
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Affiliation: Institut für Angewandte Mathematik, Abteilung für Stochastik, Universität Bonn, Wegelerstraße 6, D-53115, Bonn, Germany – and – Hausdorff Center for Mathematics (HCM), SFB611 and Interdisziplinäre Zentrum für Komplexe Systeme (IZKS), Bonn, Germany – and – Research Center BiBoS (Bielefeld Bonn Stochastics), Bielefeld–Bonn, Germany
Affiliation: Mathematics and Mechanics Faculty, Moscow State University, Moscow 119992, Russia
Keywords: Burgers equation, gradient catastrophe
Received by editor(s): April 16, 2008
Received by editor(s) in revised form: April 22, 2009
Published electronically: August 31, 2009
Additional Notes: This work was supported by Award DFG 436 RUS 113/823/0-1 and the special program of the Ministry of Education of the Russian Federation, “The development of scientific potential of the Higher School”, project 2.1.1/1399
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.