Conservation Laws with a Random Source

We study the scalar conservation law with a noisy non-linear source, viz., ut +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 ow in porous media. 0. Introduction. There has recently been a considerable increase in the interest in stochastic partial diierential equations where one either modiies well-known deterministic partial diierential equations by stochastic perturbations or consider random initial data, or both. The motivation for this comes from physical considerations as well as purely mathematical interest. An important example is the Burgers' equation in various stochastic settings. In fact, Burgers studied heuristically the equation (0.1) u t + uu x = u xx ; u(x; 0) = u 0 (x) with white noise initial data as a model of turbulence 5]. Burgers equation (0.1) has over the years been applied to many physical problems. We here mention shortly large scale structures of the universe, 29], dynamics of growing interfaces (adding a force term in (0.1)), 18], 26], transmission of neural signals in squid, 27], chemical oscillations 23], and acoustic turbulence, 10]. We refer to 9] for further discussion of physical applications. The analysis of the stochastic Burgers equation (> 0) is based on the Forsyth{Florin{Hopf{Cole transformation (0.2) u = ?2 x that linearizes (0.1) into the heat equation (0.3) t = xx : Solving the heat equation explicitly and transforming back, one in fact obtains an explicit solution of Burgers equation (0.1). Sinai 31] has studied the inviscid Burgers equation (i.e., = 0) with Brownian motion initial data (see also She, Aurell, and Frisch 30] for numerical results), proving rather detailed results for the solution. Furthermore Sinai has studied asymptotic properties of the the viscous Burgers equation with a random force and/or random initial data, 32], proving that for large t the solution is largely independent of the initial data. Asymptotic properties of Burgers equation with random initial data are also …