Workshop on Recent developments in incompressible fluid dynamics

Abstract: We consider the density properties of divergence-free vector fields $b \in L^1([0,1],\BV([0,1]^2))$ which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow $X_t$ is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at $t=1$. Our main result is that there exists a $G_\delta$-set $\mathcal U \subset L^1_{t,x}([0,1]^3)$ made of divergence free vector fields such that
1. The map $\Phi$ associating $b$ with its RLF $X_t$ can be extended as a
continuous function to the $G_\delta$-set $\mathcal{U}$;
2. Ergodic vector fields $b$ are a residual $G_\delta$-set in $\mathcal{U}$;
3. Weakly mixing vector fields $b$ are a residual $G_\delta$-set in $\mathcal{U}$;
4. Strongly mixing vector fields $b$ are a first category set in $\mathcal{U}$;
5. Exponentially (fast) mixing vector fields are a dense subset of $\mathcal{U}$.
    The proof of these results is based on the density of BV vector fields such that $X_{t=1}$ 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 $d \geq 3$ is also presented.