Communications in Mathematical Physics

, Volume 275, Issue 1, pp 163–186 | Cite as

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions

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Abstract

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes.

A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with a static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes’ boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.

Keywords

Small Hole Macroscopic Model Stochastic Partial Differential Equation Perforated Domain Static Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of Applied MathematicsChinese Academy of SciencesBeijingChina
  2. 2.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

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