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Hyperbolic asymptotics in Burgers' turbulence and extremal processes

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Abstract

Large time asymptotics of statistical solutionu(t,x) (1.2) of the Burgers' equation (1.1) is considered, whereξ(x)=ξ L(x) is a stationary zero mean Gaussian process depending on a large parameterL>0 so that

$$\xi _L (x) \sim \sigma _L \eta (x/L)(L \to \infty ),$$

where\(\sigma _L = L^2 (2\log L)^{1/2} \) and η(x) is a given standardized stationary Gaussian process. We prove that asL→∞ the hyperbolicly scaled random fieldsu(L 2t, L2x) converge in distribution to a random field with “saw-tooth” trajectories, defined by means of a Poisson process on the plane related to high fluctuations of ξ(x), which corresponds to the zero viscosity solutions. At the physical level of rigor, such asymptotics was considered before by Gurbatov, Malakhov and Saichev (1991).

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, USA

    S. A. Molchanov

  2. Institute of Mathematics and Cybernetics, Lithuanian Academy of Sciences, Vilnius, Lithuania

    D. Surgailis

  3. Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, 44106, Cleveland, Ohio, USA

    W. A. Woyczynski

Authors
  1. S. A. Molchanov
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  2. D. Surgailis
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  3. W. A. Woyczynski
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Communicated by Ya.G. Sinai

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Molchanov, S.A., Surgailis, D. & Woyczynski, W.A. Hyperbolic asymptotics in Burgers' turbulence and extremal processes. Commun.Math. Phys. 168, 209–226 (1995). https://doi.org/10.1007/BF02099589

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