Lyapunov exponents, entropy and mixing for DiPerna-Lions flows

The main goal of this work is to establish an asymptotic form of Bressan’s mixing conjecture. To this end, we develop an ergodic-theoretic framework for incompressible DiPerna-Lions flows. Lyapunov exponents are defined via an Oseledets-type decomposition and related to metric entropy through a version of Ruelle’s inequality in this low-regularity setting. These tools yield sharp bounds on asymptotic regularity propagation and mixing rates, leading to our main result.