Solvability of the cohomological equation for regular vector fields on the plane

Abstract

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of X is ``simple enough'' (e.g. if X is polynomial) then X is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of LX and to build an embedding of R^2 into R^4 which rectifies X. Finally we use this embedding to characterize the functions in the image of L

Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane

“…[6] for foliations, see also [11] for a thorough discussion of its action on C ∞ (R 2 )). For example,…”

“…On the other hand, nonsurjective vector fields appear in the context of the geometry of foliations (see [17]) and dynamical systems (see [11,20]). In particular, the issue of the global solvability of the cohomological equations of the type Xu = f is a challenging and difficult problem related to Geometry, Dynamical Systems (cf.…”

Proof of a Gromov conjecture on the infinitesimal invertibility of the metric inducing operators

“…The corresponding PDE L ξ (f ) = g is called cohomological equation, whose solvability has been recently studied in several contexts (see [For99,Nov08,DeL15]).…”

“…Remark 1. It was shown in [DeL11] that, in case of vector fields with no zeros on R 2 , L ξ is surjective if and only if ξ is conjugate to a constant vector field (recall that on the 2-torus even this condition is not enough for the surjectivity, e.g. see [For95]).…”