  For any $h\in(1,2]$, we give an explicit construction of a compactly
supported, uniformly continuous, and (weakly) divergence-free velocity field in
$\mathbb{R}^2$ that weakly advects a measure whose support is initially the
origin but for positive times has Hausdorff dimension $h$.
  These velocities are uniformly continuous in space-time and compactly
supported, locally Lipschitz except at one point and satisfy the conditions for
the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di
Perna 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 such
that at positive times it only has co-dimension 1. The associated velocities
are divergence free, compactly supported, continuous, and sufficiently regular
to admit unique Regular Lagrangian Flows.
  This is in part motivated by the investigation of dimension conservation for
the support of measure-valued solutions to active scalar systems. This question
occurs in the study of vortex filaments in the three-dimensional Euler
equations.
