Homogenization of linear and nonlinear transport equations

Abstract

AbstractWe study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n‐dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two‐dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non‐local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two‐scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.