Abstract:We study the scalar conservation law with a noisy non-linear source, viz., $u_t + f(u)_x = h(u,x,t) + g(u)W(t)$, where $W(t)$ is 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.

**Paper:**- Available as PostScript
**Title:**- Conservation Laws with a Random Source
**Author(s):**- Helge Holden, mailto:holden@math.ntnu.no
- Nils Henrik Risebro, mailto:nilshr@math.uio.no
**Publishing information:**- To appear in Appl. Math. Optim.
**Submitted by:**- mailto:holden@math.ntnu.no June 12 1996.

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