A general Frobenius' Theorem via the Transport of Currents

Abstract

A classical result in Differential Geometry states that the flows of two smooth vector fields commute if and only if their Lie Bracket vanishes. In this work, we extend this result to a more general setting where one of the vector fields is bounded and Lipschitz, while the other may be a singular vector-valued measure, i.e. a normal 1-current. This result is achieved via the study of two distinct evolutionary PDEs describing the transport of vector quantities (the Vector Advection Equation and the Geometric Transport Equation). Furthermore, we show that a celebrated theorem by Alfv\'en in Magnetohydrodynamics can be interpreted as a suitable time-dependent version of Frobenius' Theorem. Our approach builds on recent advances concerning the Geometric Transport Equation for currents [5, 6].