Nonnegative measures belonging to $H^{-1}(\mathbb{R}^2)$

TL;DR

This paper investigates conditions under which nonnegative Radon measures supported on lines belong to the negative Sobolev space $H^{-1}(R^2)$, with implications for fluid mechanics and vortex sheet modeling.

Contribution

It provides new regularity criteria for measures on lines to be in $H^{-1}(R^2)$ and explores their support properties, including support on sets of Hausdorff dimension zero.

Findings

Measures in $H^{-1}(R^2)$ can be supported on sets of Hausdorff dimension zero.

Derived regularity conditions for nonnegative Radon measures on lines.

Implications for modeling vortex sheets in fluid mechanics.

Abstract

Radon measures belonging to the negative Sobolev space $H^{-1}(\mathbb{R}^2)$ are important from the point of view of fluid mechanics as they model vorticity of vortex-sheet solutions of incompressible Euler equations. In this note we discuss regularity conditions sufficient for nonnegative Radon measures supported on a line to be in $H^{-1}(\mathbb{R}^2)$. Applying the obtained results, we derive consequences for measures on $\mathbb{R}^2$ with arbitrary support and prove elementarily, among other things, that measures belonging to $H^{-1}(\mathbb{R}^2)$ may be supported on a set of Hausdorff dimension $0$. We comment on possible numerical applications.