Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

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 measure whose support is initially the origin but for positive times has Hausdorff dimension $h$. These velocities are continuous space-time locally Lipschitz except at one point satisfy conditions existence uniqueness Regular Lagrangian Flow sense Di Perna Lions theory. We then construct active scalar systems $\mathbb{R}^3$ with measure-valued solutions initial co-dimension 2 such it only 1. The associated divergence free, sufficiently regular to admit unique Flows. This part motivated by investigation conservation systems. question occurs study vortex filaments three-dimensional Euler equations.