Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes

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.