Div-Curl Problems and $\mathbf{H}^1$-regular Stream Functions in 3D Lipschitz Domains

TL;DR

This paper addresses the div-curl problem in 3D Lipschitz domains, establishing existence, uniqueness, and a well-posed construction for stream functions, along with a numerical method demonstrating improved regularity of solutions.

Contribution

It introduces a new well-posed construction for stream functions in 3D Lipschitz domains and provides a numerical method with enhanced regularity of approximations.

Findings

Existence and uniqueness of solutions are established.

A new construction for the stream function is proposed.

Numerical experiments show higher regularity of solutions.

Abstract

We consider the problem of recovering the divergence-free velocity field ${\mathbf U}\in\mathbf{L}^2(\Omega)$ of a given vorticity ${\mathbf F}=\mathrm{curl}\,{\mathbf U}$ on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^3$. To that end, we solve the "div-curl problem" for a given ${\mathbf F}\in{\mathbf H}^{-1}(\Omega)$. The solution is expressed in terms of a vector potential (or stream function) ${\mathbf A}\in{\mathbf H}^1(\Omega)$ such that ${\mathbf U}=\mathrm{curl}\,{\mathbf A}$. After discussing existence and uniqueness of solutions and associated vector potentials, we propose a well-posed construction for the stream function. A numerical method based on this construction is presented, and experiments confirm that the resulting approximations display higher regularity than those of another common approach.