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Applied Mathematics and Optimization

, Volume 36, Issue 2, pp 229–241 | Cite as

Conservation laws with a random source

  • H. Holden
  • N. H. Risebro
Article

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.

Key Words

Conservation laws Stochastic partial differential equations Phase transitions 

AMS Classification

35L65 

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Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • H. Holden
    • 1
  • N. H. Risebro
    • 2
  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of OsloOsloNorway

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