Homogenization of two-dimensional linear flows with integral invariance

We study the homogenization of two-dimensional linear transport equations, u_t + a + (Combining right arrow above sign)(x + (Combining right arrow above sign)/ε) · ∇_{x + (Combining right arrow above sign)} u = 0, where a + (Combining right arrow above sign) is a nonvanishing vector field with integral invariance on the torus T². When the underlying flow on T² is ergodic, we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate. This result unifies and illuminates the previously known results in the special cases of incompressible flows and shear flows. When the flow on T² is nonergodic, the homogenized limit is an average, over T¹, of solutions of linear transport equations with constant coefficients; the convergence here is in the weak sense of W^-1,∞_loc(ℝ¹), and the sharp convergence rate is φ(ε). One of the main ingredients in our analysis is a classical theorem due to Kolmogorov, regarding flows with integral invariance on T², to which we present here an elementary and constructive proof.