Global structure of 2-D incompressible flows
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.
scientific article; zbMATH DE number 1882988