Advertisement

Communications in Mathematical Physics

, Volume 172, Issue 1, pp 13–38 | Cite as

Statistical properties of shocks in Burgers turbulence

  • Marco Avellaneda
  • Weinan E 
Article

Abstract

We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity,\(\frac{\partial }{{\partial t}}u\left( {x,t} \right) + \frac{\partial }{{\partial x}}\left( {\frac{1}{2}u\left( {x,t} \right)^2 } \right) = 0\), with Gaussian whitenoise initial data. This system was originally proposed by Burgers[1] as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..[12] to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp{−Cs3} fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.

Keywords

Initial Data Brownian Motion Spatial Location Quantum Computing Cumulative Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Burgers, J.M.: The nonlinear diffusion equation. Dordrecht: Reidel, 1974Google Scholar
  2. 2.
    Chorin, A.J.: Gaussian Fields and Random Flow. J. Fluid Mech.63, 21–32 (1974)Google Scholar
  3. 3.
    Donsker, M., Varadhan, S.R.S.: Asymptoticevaluation of certain Wiener integrals for large time. In: Functional Integration and its Applications, Arthur, A.M., ed., Oxford: Oxford Press, 1975, pp. 15–33Google Scholar
  4. 4.
    Feller, W.: An Introduction to Probability Theory and Its Applications I. 2nd ed., New York: Wiley, 1971Google Scholar
  5. 5.
    Hopf, E.: The partial differential equationu t+uu x=μu xx.Comm. Pure Appl. Math.3, 201–230 (1950)Google Scholar
  6. 6.
    Groenboom, P.: The concave majorant of Brownian motion. Annals of Probability11, No. 4, 1016–1027 (1983)Google Scholar
  7. 7.
    Gurbatov, S., Malakhov, A., Saichev, A.: Nonlinear random waves and turbulence in nondispersive media: Waves, rays and particles. New York: Manchester University Press, 1991Google Scholar
  8. 8.
    Kida, S.: Asymptotíc properties of Burgers turbulence. J. Fluid Mechanics79, 337–377 (1977)Google Scholar
  9. 9.
    McKean, H.P.: Stochastic Integras. New York: Academic Press, 1969.Google Scholar
  10. 10.
    Millar, P.W.:A path decomposition for Markov Processes. Annals of Probability,6, No. 2, 345–348 (1978)Google Scholar
  11. 11.
    Pitman, J.M.: Seminar on Stochastic Processes. Cinlar, E., Chung, K.L., Getoor, R.K., editors, Basel: Birkhauser, 1982, pp. 219–227Google Scholar
  12. 12.
    Shandarin, S.F., Zeldovich, Ya.B.: The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys.,61, No. 2, 185–220 (1989)CrossRefGoogle Scholar
  13. 13.
    She, Z.-S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys.148, 623–641 (1992)Google Scholar
  14. 14.
    Sinai, Ya.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys.148, 640 (1992)Google Scholar
  15. 15.
    Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Society3, 28, 738–768 (1974)Google Scholar
  16. 16.
    Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers' equation Devil's staircases and the mass distribution for large scale structures. Astron. Astrophys.289, 325–356 (1994)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Marco Avellaneda
    • 1
  • Weinan E 
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

Personalised recommendations