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Conservation laws with a random source

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Abstract

We study the scalar conservation law with a noisy nonlinear source, namely,u l + f(u)x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This sequence converges to a (pathwise) solution of the Cauchy problem. The equation can be considered as a model of deterministic driven phase transitions with a random perturbation in a system of two constituents. Finally we show some numerical results motivated by two-phase flow in porous media.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7034, Trondheim, Norway

    H. Holden

  2. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316, Oslo, Norway

    N. H. Risebro

Authors
  1. H. Holden
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  2. N. H. Risebro
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Additional information

This research has been supported by VISTA (a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap, Statoil) and NAVF (the Norwegian Research Council for Science and the Humanities).

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Holden, H., Risebro, N.H. Conservation laws with a random source. Appl Math Optim 36, 229–241 (1997). https://doi.org/10.1007/BF02683344

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  • DOI: https://doi.org/10.1007/BF02683344

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