A New Divergence-Curl Result for Measures. Application to the Two-Dimensional ODE’s Flow

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6398-6421, October 2024.
Abstract. The paper is devoted to divergence-curl results involving a divergence free measure-valued field [math], where [math] is a signed Radon measure on [math] and [math] is a nonvanishing regular vector field in [math], and a gradient measure-valued field [math] on [math], [math]. On the one hand, in a nonperiodic framework we prove that for any open set [math] of [math], the orthogonality condition [math] in [math] implies the equality [math] in [math]. The key ingredient of the proof is based on the existence of a representative in [math] of the bounded variation function [math] in [math]. This result allows us to extend in the setting of ODE’s flows the famous Franks–Misiurewicz theorem, which claims that the Herman rotation set of any continuous two-dimensional flow on the torus [math] is a closed line segment of a line of [math] passing through [math]. Moreover, this nonperiodic divergence-curl result can be applied to a finite almost periodic bounded variation function [math] and to a finite almost periodic measure-valued field [math]. On the other hand, in the periodic case with dimension [math], assuming that [math] is absolutely continuous with respect to Lebesgue’s measure on the torus [math], we prove that if the product [math] is the zero measure on [math], so is the product of the [math]-means [math].