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

TL;DR

This paper investigates the conditions under which smooth nonsingular vector fields in the plane are globally solvable for the cohomological equation, identifying geometric obstructions and providing solutions near boundaries of separatrix strips.

Contribution

It establishes the equivalence between non-surjectivity of the vector field operator and the existence of separatrix strips, and analyzes solvability for perturbations with zero order p.d.o.

Findings

Vector fields are not surjective iff separatrix strips exist.

Global weak solutions exist near separatrix boundaries under certain growth conditions.

Examples show the sharpness of the derived estimates.

Abstract

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.