Homogenization of Linear Transport Equations with Oscillatory Vector Fields

The L² weak limit as ε ↓ 0 of the solutions of the linear transport equation is studied: \[ {gathered} u_t + { a}( {x/ε } )·∇ _x u = 0, \\ u|t = 0 = u^0 ( {x,x/ε } ), {gathered} \] where ε > 0,x ∈ R² ,t ∈ [ 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 /ε. 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.