Front speed in the Burgers equation with a random flux

Jan Wehr Jack Xin

Analysis of PDEs mathscidoc:1912.43868

Journal of statistical physics, 88, 843-871, 1997.8
We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known BuckleyLeverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a ColeHopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the
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@inproceedings{jan1997front,
  title={Front speed in the Burgers equation with a random flux},
  author={Jan Wehr, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210322943388432},
  booktitle={Journal of statistical physics},
  volume={88},
  pages={843-871},
  year={1997},
}
Jan Wehr, and Jack Xin. Front speed in the Burgers equation with a random flux. 1997. Vol. 88. In Journal of statistical physics. pp.843-871. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210322943388432.
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