Properties of Mixing BV Vector Fields

We consider the density properties of divergence-free vector fields b∈L1([0,1],BV([0,1]2)) which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow Xt is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at t= 1 . Our main result is that there exists a Gδ -set U⊂Lt,x1([0,1]3) containing all divergence-free vector fields such that 1.the map Φ associating b with its RLF Xt can be extended as a continuous function to the Gδ -set U ;2.ergodic vector fields b are a residual Gδ -set in U ;3.weakly mixing vector fields b are a residual Gδ -set in U ;4.strongly mixing vector fields b are a first category set in U ;5.exponentially (fast) mixing vector fields are a dense subset of U . The proof of these results is based on the density of BV vector fields such that Xt=1 is a permutation of subsquares, and suitable perturbations of this flow to achieve the desired ergodic/mixing behavior. These approximation results have an interest of their own. A discussion on the extension of these results to d≥ 3 is also presented.