%0 Journal Article
%T Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane
%A Roberto De Leo
%A Todor Gramchev
%A Alexandre Kirilov
%J Mathematics
%D 2010
%I arXiv
%X We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in $C^\infty(\R^2)$ iff the geometrical condition -- the existence of separatrix strips -- holds. Next, for nonsurjective vector fields, we demonstrate that if the RHS $f$ has at most infra-exponential growth in the separatrix strips we can find a global weak solution $L^1_{loc}$ near the boundaries of the separatrix strips. Finally we investigate the global solvability for perturbations with zero order p.d.o. We provide examples showing that our estimates are sharp.
%U http://arxiv.org/abs/1001.2121v1