Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes

We introduce and analyze the class ${\mathcal{C}\mathcal{M}}^p$ 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 ${\mathrm{H}}^{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}({\mathbb{R}}^3)$, which arise, for example, in the study of Maxwell’s equations. The sharpness of our theorems is illustrated through several fundamental examples.