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Zbl 0818.60046
Molchanov, S.A.; Surgailis, D.; Woyczynski, W.A.
Hyperbolic asymptotics in Burgers' turbulence and extremal processes.
(English)
[J] Commun. Math. Phys. 168, No.1, 209-226 (1995). ISSN 0010-3616; ISSN 1432-0916/e

Summary: Large time asymptotics of statistical solution $u(t,x)$, $$u(t,x) = {\int\sp \infty\sb{-\infty} [(x - y)/t] \exp [(2 \mu)\sp{-1} (\xi (y) - (x-y)\sp 2/2t)] dy \over \int\sp \infty\sb{- \infty} \exp [(2 \mu)\sp{-1} (\xi (y) - (x-y)\sp 2/2t)] dy},$$ of the Burgers' equation $$\partial\sb t u + u \partial\sb x u = \mu \partial\sp 2\sb x u,$$ is considered, where $\xi (x) = \xi\sb L(x)$ is a stationary zero mean Gaussian process depending on a large parameter $L > 0$ so that $\xi\sb L(x) \sim \sigma\sb L \eta (x/L)$ $(L \to \infty)$, where $\sigma\sb L = L\sp 2 (2 \log L)\sp{1/2}$ and $\eta (x)$ is a given standardized stationary Gaussian process. We prove that as $L \to \infty$ the hyperbolicly scaled random fields $u(L\sp 2t, L\sp 2x)$ 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 $\xi (x)$, which corresponds to the zero viscosity solutions. At the physical level of rigor, such asymptotics was considered before by {\it S. N. Gurbatov}, {\it A. N. Malakhov} and {\it A. I. Saichev} [Nonlinear random waves in media without dispersion (1990; Zbl 0753.76004)].
MSC 2000:
*60H15 Stochastic partial differential equations
35Q53 KdV-like equations

Keywords: large time asymptotics; Burgers' equation; stationary Gaussian process; Poisson process

Citations: Zbl 0753.76004

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