Two Approximation Results for Divergence Free Vector Fields

Jesse Goodman, Felipe Hernandez, Daniel Spector

Published 2020-10-27Version 1

In this paper we prove two approximation results for divergence free vector fields. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \in M_b(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{div} F=0$ in the sense of distributions, there exist $C^1$ closed curves $\{\Gamma_{i,l}\}_{\{1,\ldots,n_l\}\times \mathbb{N}}$, with parameterization by arclength $\gamma_{i,l} \in C^1([0,L_{i,l}];\mathbb{R}^d)$, $l \leq L_{i,l} \leq 2l$, for which \[ F= \lim_{l \to \infty} \frac{\|F\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)}}{n_l \cdot l} \sum_{i=1}^{n_l} \dot{\gamma}_{i,l} \left.\mathcal{H}^1\right\vert_{\Gamma_{i,l}} \] weakly-star as measures and \begin{align*} \lim_{l \to \infty} \frac{1}{n_l \cdot l} \sum_{i=1}^{n_l} |\Gamma_{i,l}| = 1. \end{align*} The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \[ \left\{ F \in M_b(\mathbb{R}^d;\mathbb{R}^d): \operatorname*{div}F=0 \right\} \] with respect to the strict topology.