Preprint 1996-003

Conservation Laws with a Random Source

Helge Holden and Nils Henrik Risebro

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.

Available as PostScript
Conservation Laws with a Random Source
Helge Holden, <>
Nils Henrik Risebro, <>
Publishing information:
To appear in Appl. Math. Optim.
Submitted by:
<> June 12 1996.

[ 1996 Preprints | All Preprints | Preprint Server Homepage ]
© The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

Conservation Laws Preprint Server <>
Last modified: Wed Jun 12 09:48:07 1996