Abstract
The statistical properties of solutions of the one-dimensional Burgers equation in the limit of vanishing viscosity are considered when the initial velocity potential is fractional Brownian motion (FBM). We establish the asymptotic power-law order for log-probability of large values, both velocity and shock (amplitude of velocity discontinuity). This confirms the conjecture of U. Frisch and his collaborators. Rigorous results for this problem were previously derived for the case of Brownian motion using Markov techniques. Our approach is based on the intrinsic properties of FBM and the theory of extreme values for Gaussian processes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Avellaneda, Statistical properties of shocks in Burgers turbulence, II: tail probabilities for velocities, shock-strengths and rarefaction intervals,Commun. Math. Phys. 169:N1, 45–59 (1995).
M. Avellaneda and W. E., Statistical properties of shocks in Burgers turbulence,Commun. Math. Phys. 172:N1, 13–38 (1995).
J. M. Burgers,The nonlinear diffusion equation (Reidel, Do Dordrecht, 1974).
S. Gurbatov, A. Malakchov, and A. Saichev,Nonlinear random waves and turbulence in nondispersive media: waves, rays and particles (Manchester Univ Press, New York, 1991).
K. Handa, A remark on shocks in inviscid Burgers’s trbulence. In F. M. N. Fitzmaurice, D. Gurarie, F. Mc Caughan, and W. A. Woyczyński (eds.),Nonlinear waves and weak turbulence (Birkhäuser, Boston, 1993), pp. 239–445.
B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications,SIAM Rev. 10:422–437 (1968).
S. Molchanov, D. Surgailis, and W. Woyczyński, Hyperbolic asymptotics in Burgers’ Turbulence and Extremal Processes,Commun. Math. Phys. 168:209–226 (1995).
S. Orey, Growth rates of Gaussian processes with stationary increments,Bull. Amer. Math. Soc. 77:609–612 (1971).
V. I. Piterbarg and V. P. Prisyzhnyuk, Asymptotic of the probability of large deviation of Gaussian nonstationary process,Prob. Theory and Math. Statistics 18:121–134 (1978).
Ya. Sinai, Statistics of shocks in solutions of inviscid Burgers equation,Commun. Math. Phys. 148:601–622 (1992).
Z. S. She, E. Aurell, and U. Frisch, The inviscid Burgers equation with initial data of Brownian type,Commun. Math. Phys. 148:623–641 (1992).
D. Slepian, The one-sided barrier problem for Gaussian noise,Bell. System Techn. J. 41:463–501 (1962).
M. Vergassola, B. Dubrulle, U. Frisch, and A. Noullez, Burgers’ equation, Devil’s staircases and the mass distribution for large-scale structures,Astron. Astrophys. 289:325–356 (1994).
S. F. Shandarin and Y. B. Zeldovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium,Rev. Mod. Phys. 61:N2, 185–220 (1989).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Molchan, G.M. Burgers equation with self-similar gaussian initial data: Tail probabilities. J Stat Phys 88, 1139–1150 (1997). https://doi.org/10.1007/BF02732428
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02732428