Properties of mixing BV vector fields
speaker: Stefano Bianchini (SISSA, Trieste)abstract: We consider the density properties of divergence-free vector fields $b \in L1(0,1,\BV(0,12))$ which are ergodicweakly mixingstrongly mixing: this means that their Regular Lagrangian Flow $Xt$ is an ergodicweakly mixingstrongly mixing measure preserving map when evaluated at $t=1$. Our main result is that there exists a $G\delta$-set $\mathcal U \subset L1{t,x}(0,13)$ made of divergence free vector fields such that \begin{enumerate} \item the map $\Phi$ associating $b$ with its RLF $Xt$ can be extended as a continuous function to the $G\delta$-set $\mathcal{U}$; \item ergodic vector fields $b$ are a residual $G\delta$-set in $\mathcal{U}$; \item weakly mixing vector fields $b$ are a residual $G\delta$-set in $\mathcal{U}$; \item strongly mixing vector fields $b$ are a first category set in $\mathcal{U}$; \item exponentially (fast) mixing vector fields are a dense subset of $\mathcal{U}$. \end{enumerate} 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 ergodicmixing 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.
timetable:
Tue 25 Jan, 10:00 - 10:50,
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