Properties of Mixing BV vector fields

Abstract

We consider the density properties of divergence-free vector fields $b\in L^1([0,1],\text{BV}([0,1{]}^2))$ 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 $\mathcal{U}\subset L_{t,x}^1([0,1{]}^3)$ 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 $d\ge 3$ is also presented.

47 pages

References in corpus (1)

• A Lemma and a Conjecture on the Cost of Rearrangements

Cited by in corpus (3)

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• An alternative approach to Shnirelman's inequality