Seminar Analysis and Mathematical Physics: Martina Zizza (SISSA Trieste)

Mixing properties of BV vector fields

In this talk we tackle the question "How many vector fields are mixing?" analyzing  the density properties of divergence-free BV vector fields which are weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow is a weakly mixing/strongly mixing measure-preserving map when evaluated at time t=1. More precisely we prove the existence of a G_delta-set U in the space L^1_{t,x}([0,1]^3) made of divergence-free vector fields such that:

    1)  weakly mixing vector fields are a residual G_delta-set in U;

    2)  (exponentially fast) strongly mixing vector fields are a dense subset of U.

The proof of these results exploits some connections between ergodic theory and fluid dynamics and it is based on the density of BV vector fields whose Regular Lagrangian Flow is a permutation of subsquares of the unit square [0,1]^2 when evaluated at time t=1. 

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