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Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes

Gui-Qiang G. Chen, Franz Gmeineder, Monica Torres

TL;DR

The paper develops a unified analytic framework for curl-based vorticity problems by introducing the curl-measure spaces $\mathscr{CM}^{p}$, where the curl is a finite Radon measure. It proves trace theorems and Stokes-type identities in low-regularity regimes, using maximal-function based notions to identify good manifolds for which these results hold; the theory covers both tangential and transversal variations and extends to divergence-measure fields on manifolds. Key contributions include sharp trace results for $\mathscr{CM}^{\infty}$, a general Stokes theorem for $1\le p<\infty$ via localization, and a boundary-pairing formalism that connects vorticity fluxes to circulation in weak settings. The framework accommodates vortex sheets and other concentration phenomena, provides a rigorous link to Maxwell-type equations in low regularity, and lays groundwork for further generalizations and applications in fluid dynamics and electromagnetism.

Abstract

We introduce and analyze the class $\mathscr{CM}^{p}$ of curl-measure fields that are $p$-integrable vector fields whose distributional curl is a vector-valued finite Radon measure. These spaces provide a unifying framework for problems involving vorticity. A central focus of this paper is the development of Stokes-type theorems in low-regularity regimes, made possible by new trace theorems for curl-measure fields. To this end, we introduce Stokes functionals on so-called good manifolds, defined by the finiteness of manifold-adapted maximal operators. Using novel techniques that may be of independent interest, we establish results that are new even in classical settings, such as Sobolev spaces or their curl-variants $\mathrm{H}^{\mathrm{curl}}(\mathbb{R}^{3})$, which arise, for example, in the study of Maxwell's equations. The sharpness of our theorems is illustrated through several fundamental examples.

Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes

TL;DR

The paper develops a unified analytic framework for curl-based vorticity problems by introducing the curl-measure spaces , where the curl is a finite Radon measure. It proves trace theorems and Stokes-type identities in low-regularity regimes, using maximal-function based notions to identify good manifolds for which these results hold; the theory covers both tangential and transversal variations and extends to divergence-measure fields on manifolds. Key contributions include sharp trace results for , a general Stokes theorem for via localization, and a boundary-pairing formalism that connects vorticity fluxes to circulation in weak settings. The framework accommodates vortex sheets and other concentration phenomena, provides a rigorous link to Maxwell-type equations in low regularity, and lays groundwork for further generalizations and applications in fluid dynamics and electromagnetism.

Abstract

We introduce and analyze the class of curl-measure fields that are -integrable vector fields whose distributional curl is a vector-valued finite Radon measure. These spaces provide a unifying framework for problems involving vorticity. A central focus of this paper is the development of Stokes-type theorems in low-regularity regimes, made possible by new trace theorems for curl-measure fields. To this end, we introduce Stokes functionals on so-called good manifolds, defined by the finiteness of manifold-adapted maximal operators. Using novel techniques that may be of independent interest, we establish results that are new even in classical settings, such as Sobolev spaces or their curl-variants , which arise, for example, in the study of Maxwell's equations. The sharpness of our theorems is illustrated through several fundamental examples.

Paper Structure

This paper contains 35 sections, 47 theorems, 367 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vorticity fluxes
  3. Main results
  4. Vorticity flux problems and interplay with $\mathscr{DM}$-fields
  5. Organization of the paper
  6. Preliminaries
  7. Notation
  8. Differential geometry on $\mathbb R^{n}$
  9. Boundary manifolds, collar theorems, and deformations
  10. Function spaces
  11. Lipschitz and $\operatorname{C}^{1}$-spaces
  12. Extended divergence-measure fields
  13. Level set estimates
  14. The Trace Theorem for $\mathscr{C}\!\mathscr{M}^{p}$-Fields
  15. The tangential trace functional
  16. ...and 20 more sections

Key Result

Theorem 2.1

Let $\Sigma\subset\mathbb R^{n}$ be an oriented $(n-1)$-dimensional $\operatorname{C}^{2}$-manifold oriented by $\nu\colon\Sigma\to\mathbb{S}^{n-1}$ such that $\Sigma=\mathrm{int}(\Sigma)$ and $\overline{\Sigma}$ is compact. Moreover, suppose that, for each $x_{0}\in\Gamma_{\Sigma}$, there exists bo Then there exist both a normal vector field $\mathbf{H}_{\Sigma}\in\operatorname{C}^{0}(\Sigma;\mat

Figures (9)

  • Figure 1: Vorticity fluxes through a surface $\Sigma$ oriented by $\nu_{\Sigma}\colon\Sigma\to\mathbb{S}^{2}$. (a) The situation of vortex sheets $\Sigma$, where the entire vorticity is concentrated on $\Sigma$. (b) The situation as encountered e.g. in whirlwinds.
  • Figure 2: Oscillations close to the boundary.
  • Figure 3: Geometric set-up underlying § \ref{['sec:notionsdiffgeom']}--§ \ref{['sec:bdrymanifolds']}.
  • Figure 4: The geometric situation in § \ref{['sec:stokes']}--§ \ref{['sec:divmeasfieldsmanif']}. The boundary manifolds in tangential direction as indicated in light green correspond to the tangentially varied manifolds as studied in § \ref{['sec:stokes']}; see Theorem \ref{['thm:stokes']}. The boundary manifolds in transversal direction as indicated in light blue correspond to the setting of § \ref{['sec:divmeasfieldsmanif']}; see Theorem \ref{['thm:stokes1st']}. These variations are governed by $\mathcal{M}_{\Sigma,\partial\Omega'}^{\Psi}$ (green) and $\mathcal{M}_{\Sigma,\partial\Omega'}^{\Phi}$ (blue).
  • Figure 6: On the proof of Lemma \ref{['lem:geometrichelp']}.
  • ...and 4 more figures

Theorems & Definitions (137)

  • Definition 1.1: Curl-measure fields
  • Remark 1.2: Generalizations
  • Example 1.3: Gradient fields
  • Theorem 2.1: Smooth integration by parts-formula
  • Remark 2.2
  • Lemma 2.3: Coarea formula on $\operatorname{C}^{1}$-manifolds
  • Definition 2.4: Boundary manifolds
  • Lemma 2.5: Collar-type I
  • Definition 2.6: Transversal shifting
  • Remark 2.7
  • ...and 127 more