Convergence of solutions of linear transport equations

The homogenizability of linear transport equations with periodic data is investigated. It can be interpreted in terms of the dynamical system properties of the associated ODEs. De Giorgi conjectured that all equations of this kind are homogenizable. In this paper, the homogenizability is proved is the two-dimensional case for non-vanishing functions (partial results in this direction have been previously proved by other authors). On the other hand, in the three-dimensional case, an example of a non-homogenizable equation is given.