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

Summary: Large time asymptotics of statistical solution $u\left(t,x\right)$,

$u\left(t,x\right)=\frac{{\int }_{-\infty }^{\infty }\left[\left(x-y\right)/t\right]exp\left[{\left(2\mu \right)}^{-1}\left(\xi \left(y\right)-{\left(x-y\right)}^{2}/2t\right)\right]dy}{{\int }_{-\infty }^{\infty }exp\left[{\left(2\mu \right)}^{-1}\left(\xi \left(y\right)-{\left(x-y\right)}^{2}/2t\right)\right]dy},$

of the Burgers’ equation

${\partial }_{t}u+u{\partial }_{x}u=\mu {\partial }_{x}^{2}u,$

is considered, where $\xi \left(x\right)={\xi }_{L}\left(x\right)$ is a stationary zero mean Gaussian process depending on a large parameter $L>0$ so that ${\xi }_{L}\left(x\right)\sim {\sigma }_{L}\eta \left(x/L\right)$ $\left(L\to \infty \right)$, where ${\sigma }_{L}={L}^{2}{\left(2logL\right)}^{1/2}$ and $\eta \left(x\right)$ is a given standardized stationary Gaussian process. We prove that as $L\to \infty$ the hyperbolicly scaled random fields $u\left({L}^{2}t,{L}^{2}x\right)$ 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 $\xi \left(x\right)$, 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:
 60H15 Stochastic partial differential equations 35Q53 KdV-like (Korteweg-de Vries) equations
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