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Divergence of C1 vector fields and nontrivial minimal sets
 

Summary: Divergence of C1
vector fields and nontrivial minimal sets
on 2-manifolds
Konstantin Athanassopoulos
Department of Mathematics, University of Crete, GR-71409 Iraklion, Greece
e-mail : athanako@math.uoc.gr
Abstract
We prove a Bendixson-Dulac type criterion for the nonexistence of nontrivial
compact minimal sets of C1 vector fields on orientable 2-manifolds. As a corollary we
get that the divergence with respect to any volume 2-form of such a vector field must
vanish at some point of any nontrivial compact minimal set. We also prove that all the
nontrivial compact minimal sets of a C1 vector field on an orientable 2-manifold are
contained in the vanishing set of any inverse integrating factor. From this we get that
if a C1 vector field on an orientable 2-manifold has a nontrivial compact minimal set,
then an infinitesimal symmetry is inessential on the minimal set.
MSC: 34C40, 37C10, 37E35, 37B20.
Keywords: C1 vector field, divergence, inverse integrating factor, infinitesimal symmetry,
nontrivial compact minimal set.
1. Introduction
A classical problem in the qualitative theory of 2-dimensional ordinary differential

  

Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete

 

Collections: Mathematics