Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915525594972160 |
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| author | Chen, Gui-Qiang G. Gmeineder, Franz Torres, Monica |
| author_facet | Chen, Gui-Qiang G. Gmeineder, Franz Torres, Monica |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes Chen, Gui-Qiang G. Gmeineder, Franz Torres, Monica Analysis of PDEs 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. |
| title | Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.26465 |