Homogenization of Two-Dimensional Linear Flows with Integral Invariance

Abstract

We study the homogenization of two-dimensional linear transport equations, $u_t + \av(\xv/\ep) \cdot \nabla_{\xv}u = 0$, where $\av$ is a nonvanishing vector field with integral invariance on the torus $T^2$. When the underlying flow on $T^2$ 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 $T^2$ is nonergodic, the homogenized limit is an average, over $T^1$, of solutions of linear transport equations with constant coefficients; the convergence here is in the weak sense of $W^{-1,\infty}_{loc}(\RRa^1)$, and the sharp convergence rate is $\Oe$. One of the main ingredients in our analysis is a classical theorem due to Kolmogorov, regarding flows with integral invariance on $T^2$, to which we present here an elementary and constructive proof.