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Conservation laws with a random source. (English) Zbl 0885.35069
The main result of this paper is following: Let \(f(u)\) and \(g(u)\) be real Lipschitz functions, \(g(u)\) with bounded support, and let \(h(x,t,u)\) be a Lipschitz function which has bounded support in \(u\). Let \(u_0(x)\) be a (deterministic) essentially bounded function with bounded support which has finitely many extrema. Then, almost surely, there exists a weak solution to the stochastic conservation law: \[ u_t+ f(x)_x= h(x, t,u)+ g(u)W(t),\;u(x,0;\omega)= u_0(x), \] where \(W(t)\) is the white noise process, and the right-hand side is to be interpreted in the Itô sense. Some numerical results motivated by two-phase flow in porous media are presented.

35L65 Hyperbolic conservation laws
35R60 PDEs with randomness, stochastic partial differential equations
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