Abstract
We discuss large-time asymptotics of a suitably rescaled solution of Burgers’ equation with a random stationary piecewice constant initial potential, in the situations when (1) the viscosity μ > 0 is constant, or (2) μ ~ 1/t → 0. In the first case, the rescaled solution of Burgers’ equation converges in distribution to a random solution of the heat equation with a stable noise initial data, or the logarithmic derivative of the latter, depending on the distribution of the random heights and random constancy intervals of the potential. In the second case, the asymptotics is determined by high fluctuations of the potential described by a Poisson statistics, and the limit (velocity) field consists of pure shock waves traveling with random speed and coalescing at collisions. This extends the recent result of Molchanov, Surgailis, Woyczynski [MSW95], obtained for a Gaussian initial potential.
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Surgailis, D. (1997). Asymptotics of Solutions of Burgers’ Equation with Random Piecewise Constant Data. In: Molchanov, S.A., Woyczynski, W.A. (eds) Stochastic Models in Geosystems. The IMA Volumes in Mathematics and its Applications, vol 85. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8500-4_20
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DOI: https://doi.org/10.1007/978-1-4613-8500-4_20
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