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

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

2010-01-13

arxiv.org/abs/1001.2121v1

Mathematics

Analysis of PDEs

22 pages, 2 figures, submitted to the PDE volume of the proceedings of the ISAAC2009 conference

Scientific paper

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.

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