Conservation Laws with a Random Source

摘要

We study the scalar conservation law with a noisy nonlinear source, namely, u_t + f(u)_x = h(u,x,t) + g(u)W(t), where W(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.