Mathematics

We consider the density properties of divergence-free vector fields b in 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  subset of L1t,x([0,1]3) made of divergence-free vector fields such that 

1. the map 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 behaviour. This is a joint work with Prof. Stefano Bianchini.