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∞(R2) 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 Lloc1​ 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.Comment: 22 pages, 2 figures, submitted to the PDE volume of the proceedings
of the ISAAC2009 conferenc