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Hausdorff dimension of regular points in stochastic Burgers flows with Lévy α-stable initial data

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This paper studies statistical properties of shocks for the inviscid Burgers equation with an α-stable Lévy motion initial data. In the absence of analytic results, numerical and computer simulation tools are utilized. Qualitative and quantitative information on the scaling properties of Lagrangian regular points of solutions is obtained and, in particular, their Hausdorff dimension is estimated to be 1/α. This suggestsa possible extension of Ya. Sinai's result for Brownian initial data.

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  1. 1.

    S. Albeverio, S. A. Molchanov, and D. Surgailis, Stratified structure of the Universe and Burgers' equation: A probabilistic approach,Prob. Theory Rel. Fields 100:457–484 (1994).

  2. 2.

    M. Avellaneda and W. E. Statistical properties of shocks in Burgers turbulence,Commun. Math. Phys. 172:13–38 (1995).

  3. 3.

    A. V. Bulinskii and S. A. Molchanov, Asymptotic Gaussianness of solutions of the Burgers equation with random initial data,Teorya Veroyat. Prim. 36:217–235 (1991).

  4. 4.

    J. M. Burgers,The Nonlinear Diffusion Equation (Reidel, Dordrecht, 1974).

  5. 5.

    J. M. Chambers, C. L. Mallows, and B. Stuck, A method for simulating stable random variables,J. Am. Stat. Assoc. 71:340–344 (1976).

  6. 6.

    C. D. Cutler, Some results on the behavior and estimation of the fractal dimensions of distributions on attractors,J. Stat. Phys. 62:651–708 (1991).

  7. 7.

    T. Funaki, D. Surgailis, and W. A. Woyczynski. Gibbs-Cox random fields and Burgers turbulence,Ann. Appl. Prob. 5:701–735 (1995).

  8. 8.

    T. Gotoh and R. H. Kreichnan, Statistics of decaying Burgers turbulence,Phys. Fluids A 5:445–457 (1993).

  9. 9.

    S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev,Nonlinear Random Waves in Dispersionless Media (Nauka, Moscow, 1990).

  10. 10.

    J. Holtzmark, Über die Verbreiterung von Spektrallinien,Ann. Phys. (Leipzig).58:577–630 (1991).

  11. 11.

    Y. Hu and W. A. Woyczynski, An extremal rearrangement property of statistical solutions of the Burgers equation,Ann. Appl. Prob. 4:838–858 (1994).

  12. 12.

    Y. Hu and W. A. Woyczynski, Shock density in Burgers' turbulence, inNonlinear Stochastic PDE's Burgers Turbulence and Hydrodynamic Limit (Springer-Verlag, Berlin, 1995), pp. 201–213.

  13. 13.

    I. A. Ibragimov and K. E. Chernin, On the unimodality of stable laws,Theory Prob. Appl. 4:453–456 (1959).

  14. 14.

    A. Janicki and A. Weron,Simulation and Chaotic Behavior of α-Stable Stochastic Processes (Marcel Dekker, New York; 1994).

  15. 15.

    M. Kanter, Stable densities under change of scale and total variation inequalities,Ann. Prob. 3:697–707 (1975).

  16. 16.

    Y. Kasahara and K. Yamada, Stability theorem for stochastic differential equations with jumps,Stoch. Proc. Appl. 38:13–32 (1991).

  17. 17.

    S. Kida, Asymptotic properties of Burgers turbulence,J. Fluid Mech. 79:337–377 (1977).

  18. 18.

    S. Kwapien and W. A. Woyczynski:Random Series and Stochastic Integrals; Single and Multiple (Birkhäuser; Boston, 1992).

  19. 19.

    B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).

  20. 20.

    S. A. Molchanov, D. Surgailis, and W. A. Woyczynski, Hyperbolic asymptotics in Burgers' turbulence and extremal processes.Commun. Math. Phys. 165:209–226 (1995).

  21. 21.

    M. Rosenblatt, Scale renormalization and random solutions of the Burgers equation,J. Appl. Prob. 24:328–338 (1987).

  22. 22.

    G. Samorodnitsky and M. S. Taqqu,Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman and Hall, London, 1994).

  23. 23.

    S. F. Shandarin and Ya. B. Zeldovich, Turbulence, intermittency, structures in a selfgravitating medium: The large scale structure of the Universe.Rev. Mod. Phys. 61:185–220 (1989).

  24. 24.

    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).

  25. 25.

    Ya. G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation,Commun. Math. Phys. 148:601–621 (1992).

  26. 26.

    D. Surgailis and W. A. Woyczynski, Long range prediction and scaling limit for statistical solutions of the Burgers' equation, inNonlinear Waves and Weak Turbulence (Birkhäuser, Boston, 1993), pp. 313–338.

  27. 27.

    D. Surgailis and W. A. Woyczynski, Burgers' equation with nonlocal shot noise data,J. Appl. Prob. 31A:351–362 (1994).

  28. 28.

    D. Surgailis and W. A. Woyczynski, Scaling limits of solutions of Burgers' equation with singular Gaussian initial data, inChaos Expansions. Multiple Wiener-Itô Integrals and Their Applications, C. Houdré and V. Perez-Abreu, eds. (CRC Press, Boca Raton, Florida, 1994), pp. 145–162.

  29. 29.

    T. Utsu, A catalog of large earthquakes in Japan for the years 1885–1925, inHistorical Seismographs of Earthquakes of the World, W. H. K. Lee, H. Meyers, and K. Shimazaki, eds. (Academic Press, San Diego, California, 1988), pp. 150–161.

  30. 30.

    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).

  31. 31.

    W. A. Woyczynski, Stochastic Burgers' flows inNonlinear Waves and Weak Turbulences (Birkhäuser, Boston, 1993), pp. 279–311.

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Janicki, A.W., Woyczynski, W.A. Hausdorff dimension of regular points in stochastic Burgers flows with Lévy α-stable initial data. J Stat Phys 86, 277–299 (1997). https://doi.org/10.1007/BF02180207

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Key Words

  • Burgers equation
  • shocks
  • Hausdorff dimension
  • α-stable process