Global structure of 2-D incompressible flows
type: Item
instance of: scholarly article
title: Global structure of 2-D incompressible flows
zbMATH Open document ID: 1013.37049
author: Tian Ma
author: Shouhong Wang
published in: Discrete and Continuous Dynamical Systems
publication date: 2003-03-16T00:00:00Z
review text: This article deals with incompressible and regular flows \(v\) on two-dimensional Riemann manifolds with or without boundaries. The results are based on the following analogue of the classical Poincaré-Bendixson theorem: if \(x_0\) is a regular point of a regular vector field \(v\in D_0^r(TM)\) \((r\geq 1)\) on a manifold \(M\) then each limit set \(\alpha (x_0)\) and \(\omega (x_0)\) must be one of the following type: (a) a closed orbit but not a limit cycle; (b) a saddle point; (c) a connected closed domain \(\Omega \subset\overline {M\setminus\overline V}\) with \(\partial\Omega\) consisting of saddle connections having finite length \((D^r_0(TM)\) is the set of all regular \(r\)-differentiable vector fields on \(M\) with \(r\)-differentiable \(v|_{ \partial \Omega}\) on \(\partial\Omega\) and \(\text{div} v=0\); the regularity of \(v\) means that each singular point \(p\) of \(v\) \((v(p)=0)\) is nondegenerate (rank \(Dv(p)=2)\); \(V\) is the set of all closed orbits and centers of \(v)\). The next result is a structural classification theorem: if \(v\in D^r_0(TM)\) \((r \geq 1)\) is a regular vector field on a manifold \(M\), then \(M\) can be decomposed into a finite union of the following invariants sets of \(v\): circle cells (a circle cell \(C\) is the maximal neighbourhoods of center \(p\in M\) such that for any \(x\in C\) the corresponding orbit is closed), circle bands (a circle band \(B\) is a maximal open neighborhood of a closed orbit containing closed orbits, if any connected component of \(\partial B\) is not a single point), saddle connections and ergodic sets (a closed domain \(F\subset M\) is an ergodic set if for any \(x\in F\) \(\omega (x)\) is either a singular point of \(v\) or the whole domain \(F)\). The authors also characterize ergodic sets in the cases when the genus of \(M\) is \(\geq 1\) and when \(M\) is a torus.
Mathematics Subject Classification ID: 37N10
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 1882988
zbMATH Keywords: regular flows
zbMATH Keywords: incompressible flows
zbMATH Keywords: two-dimensional Riemann manifolds
zbMATH Keywords: Poincaré-Bendixson theorem
zbMATH Keywords: classification theorem
zbMATH Keywords: ergodic sets
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
OpenAlex ID: W1998171139
DOI: 10.3934/DCDS.2001.7.431
description: scientific article; zbMATH DE number 1882988