Homogenization of Stochastic Conservation Laws with Multiplicative Noise

Abstract

We consider the generalized almost periodic homogenization problem for two types of stochastic conservation laws with oscillatory coefficients and a multiplicative noise, namely, the nonlinear transport equation and the equation with a stiff oscillating external force. We use the notion of homogenization by noise-approximation, introduced here, which amounts to the homogenization of the equations with an approximate noise as well as showing that the solutions of the approximate equations with an artificial viscosity term converge, as the noise-approximation parameter goes to zero, to the solutions of the original equation with artificial viscosity, whose solutions are shown to converge to the solutions of the original equation as the viscosity parameter goes to zero. Besides, the homogenization limits of the approximate equations are themselves limits of a two-parameter sequence, when one of these parameters, representing viscosity, goes to zero, and whose counterpart limits, obtained when the other parameter, representing the noise-approximation, goes to zero, converge to a well determined limit which we call the homogenization limit by noise-approximation (b.n.a.). In both cases the multiplicative noise is prescribed so that the corresponding stochastic equation has special solutions that play the role of steady-state solutions in the deterministic case and are crucial elements in the homogenization analysis. As a byproduct, our prescription of the multiplicative noise provides a way to justify the noise perturbation of the corresponding deterministic equation. CONTENTS 1. Introduction 2 2. Nonlinear transport, proof of Theorem 1.1 9 3. Stiff oscillatory external force, proof of Theorem 1.2 17 4. Wong-Zakai problem for quasilinear parabolic equations 23 5. A well-posedness result 42 6. Comparison principle & stochastic Kružkov inequality 57 References 60