Two Approximation Results for Divergence Free Measures

TL;DR

This paper proves two approximation results for divergence-free measures, showing how such measures can be approximated by measures supported on loops and establishing density of smooth functions in this measure space.

Contribution

It provides new approximation theorems for divergence-free measures, extending the understanding of their structure and density properties in the strict topology.

Findings

Divergence-free measures can be approximated by measures on oriented loops.

Smooth compactly supported functions are dense in divergence-free measure space.

The results extend the approximation of solenoidal charges in the strict topology.

Abstract

In this paper we prove two approximation results for divergence free measures. 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 oriented $C^1$ loops $\Gamma_{i,l}$ with associated measures $\mu_{\Gamma_{i,l}}$ such that \[ F= \lim_{l \to \infty} \frac{\|F\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)}}{n_l \cdot l} \sum_{i=1}^{n_l} \mu_{\Gamma_{i,l}} \] weakly-star in the sense of measures and \[ \lim_{l \to \infty} \frac{1}{n_l \cdot l} \sum_{i=1}^{n_l} \|\mu_{\Gamma_{i,l}}\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)} = 1. \] 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.