HOMOGENIZATION OF LINEAR TRANSPORT-EQUATIONS WITH OSCILLATORY VECTOR-FIELDS

The L2 weak limit as epsilon down 0 of the solutions of the linear transport equation is studied: u(t) + a (x/epsilon).NABLA(x)u = 0, u\t = 0 = u0(x, x/epsilon), where epsilon > 0, x is-an-element-of R2, t is-an-element-of [0, T], for any T > 0; a is a smooth divergence-free vector field on the two-dimensional unit torus, and has no equilibrium points. The initial condition u0 is assumed to be smooth, compactly supported in x, and periodic in x/epsilon. Using a weak L2 formulation and the idea of rotation number for flows on the two-dimensional torus, the effective equations for the weak limit are found. They depend sensitively on the underlying flow on the two-dimensional torus and its ergodicity.