Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

Abstract

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions

“…Using (17) and the continuity of h we can pass to the limit in this equality and get (20), which, in view of (18), implies (19) and therefore B(t) is an F t -Wiener process.…”

“…The investigation of the asymptotic behavior for composite materials and flow of fluid in fixed and porous media has been very well established (from mathematical analysis point of view) by the so called homogenization theory, see, for instance, [16][17][18] and the references therein, for homogenization of deterministic partial differential equations. We also mention that there are few results in the stochastic setting, see [19][20][21][22]. However, one of the most important principles on which the theory of homogenization depends is the well posedness of the governing equation.…”

Homogenization of Stochastic Conservation Laws with Multiplicative Noise

“…Concerning homogenization of stochastic partial differential equations, this has not been a frequently researched topic, although the earliest contribution seems to have appeared already in the early 1990's by Bensoussan in [8]. As to more recent publications on this subject, we mention the contributions of Ichihara [37], Sango [55], Mohammed [47], and Mohammed and Sango [48], among others. Consult also references in these papers.…”