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Hyperbolic asymptotics in Burgers’ turbulence and extremal processes. (English) Zbl 0818.60046

Summary: Large time asymptotics of statistical solution u(t,x),

u(t,x)= - [(x-y)/t]exp[(2μ) -1 (ξ(y)-(x-y) 2 /2t)]dy - exp[(2μ) -1 (ξ(y)-(x-y) 2 /2t)]dy,

of the Burgers’ equation

t u+u x u=μ x 2 u,

is considered, where ξ(x)=ξ L (x) is a stationary zero mean Gaussian process depending on a large parameter L>0 so that ξ L (x)σ L η(x/L) (L), where σ L =L 2 (2logL) 1/2 and η(x) is a given standardized stationary Gaussian process. We prove that as L the hyperbolicly scaled random fields u(L 2 t,L 2 x) 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 S. N. Gurbatov, A. N. Malakhov and A. I. Saichev [Nonlinear random waves in media without dispersion (1990; Zbl 0753.76004)].

MSC:
60H15Stochastic partial differential equations
35Q53KdV-like (Korteweg-de Vries) equations
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