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

## TL;DR

This paper constructs divergence-free velocity fields in 2D that can change the Hausdorff dimension of advected measures over time, demonstrating non-conservation of dimension in passive and active scalar systems.

## Contribution

It provides explicit examples of divergence-free velocities that alter the Hausdorff dimension of measures, challenging the notion of dimension conservation in such systems.

## Key findings

- Constructed velocities that increase measure support dimension from 0 to h
- Demonstrated measure-valued solutions with support dimension change over time
- Established existence and uniqueness of Regular Lagrangian Flows for these velocities

## Abstract

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.