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

2010/01/13 by R. De Leo, Roberto De Leo, Todor Gramchev +4
Mathematics · #35F05 #35S35 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #math.AP #math.GT #msc:35F05 #msc:35S35

paper · pdf · doi:10.48550/arxiv.1001.2121

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

arxiv created 2010/01/13 · openalex publication_date 2010/01/13 · arxiv updated 2010/01/14 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

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^∞(\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 L¹_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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