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 ⊂L1t,x([0,1]3) made of 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 ;

2. Ergodic vector fields b are a residual Gδ-set in ;

3. Weakly mixing vector fields b are a residual Gδ-set in ;

4. Strongly mixing vector fields b are a first category set in ;

5. Exponentially (fast) mixing vector fields are a dense subset of .

    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 greater than or equal to 3 is also presented.