Abstract
We study in this article topological structure ofdivergence-free vector fields on general two-dimensional manifolds.We introduce a new concept called block structural stability(or block stability for simplicity) andprove that the block stable divergence-free vector fields forma dense and open set. Furthermore, we show that a block stabledivergence-free vector field, which we call a basic vector field,is fully characterized by a nice and simple structure,which we call block structure. The results and ideaspresented in this article have been applied to studies on structure and itsevolutions of the solutions ofthe Navier-Stokes equations; see [4, 9, 10].
Highlights
This article is part of a research program initiated recently by the authors to develop a geometric theory of two-dimensional (2-D) incompressible fluid flows in the physical spaces
The study in Area 1) is more kinematic in nature, and the results and methods developed can naturally be applied to other problems of mathematical physics involving divergence-free vector fields; see [4, 7]
To go beyond the instability result as just mentioned, we introduce in this article two new concepts: block structure and block stability
Summary
This article is part of a research program initiated recently by the authors to develop a geometric theory of two-dimensional (2-D) incompressible fluid flows in the physical spaces. 2. Let B ⊂ M be an open set, such that for any x ∈ B, the orbit Φ(x, t) is closed, and any connected component Σ of ∂B is not a single point. Let v ∈ DBr (T M ) be a basic vector field with block decomposition M = ∪Rj=1Ωj + ∪Ki=0Ai. there exists a neighborhood O ⊂ Dr(T M ) of v such that. The perturbed vector field v + w has a decomposition of invariant sets M = ∪Rj=1Ω(j1) + ∪Ki=0A(i1) with Ω(j1) and A(i1) homeomorphic to Ωj and Ai respectively, and each connected component Γ(1) of ∂A(i1) is homological to zero and contains only one saddle point
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