Multiscale Young Measures in almost Periodic Homogenization and Applications

Abstract

We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport equations, Hamilton–Jacobi equations and fully nonlinear elliptic equations. Motivated by the application in nonlinear transport equations, we also prove basic results on flows generated by Lipschitz almost periodic vector fields, which are of interest in their own. In our analysis, an important role is played by the so-called Bohr compactification \({\mathbb {G}^N}\) of \({\mathbb {R}^N}\) ; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behavior for the whole set of \({\mathbb {G}^N}\) -translates of the solutions, which is in the spirit of recent studies on the homogenization of stationary ergodic processes.