Mixing and transport in a advection equation of divergence-free vector

Mixing is a fundamental process in both engineering applications and natural fluid flows. Over the past two decades, considerable attention has been devoted to quantifying mixing rates of scalar fields advected by incompressible flows under various physical constraints. In this talk, we turn to the mixing of vector fields and investigate the advection of a divergence-free velocity field by another divergence-free flow subject to an enstrophy constraint. We first present several existence and uniqueness results. These are followed by numerical computations to identify optimal mixing strategies. Finally, we discuss broader implications of our results for the transport of divergence-free vector fields, including connections to enhanced and anomalous dissipation phenomena in turbulent flows.