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
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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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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DOI: https://doi.org/10.1007/BF02099589