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For any $h\in(1,2]$, we give an explicit construction of a compactlysupported, uniformly continuous, and (weakly) divergence-free velocity field in$\mathbb{R}^2$ that weakly advects a measure whose support is initially theorigin but for positive times has Hausdorff dimension $h$.  These velocities are uniformly continuous in space-time and compactlysupported, locally Lipschitz except at one point and satisfy the conditions forthe existence and uniqueness of a Regular Lagrangian Flow in the sense of DiPerna and Lions theory.  We then construct active scalar systems in $\mathbb{R}^2$ and $\mathbb{R}^3$with measure-valued solutions whose initial support has co-dimension 2 but suchthat at positive times it only has co-dimension 1. The associated velocitiesare divergence free, compactly supported, continuous, and sufficiently regularto admit unique Regular Lagrangian Flows.  This is in part motivated by the investigation of dimension conservation forthe support of measure-valued solutions to active scalar systems. This questionoccurs in the study of vortex filaments in the three-dimensional Eulerequations.

Non-conservation of dimension in divergence-free solutions of passive and active scalar systems