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January  2006, 6(1): 169-184. doi: 10.3934/dcdsb.2006.6.169

Block structure and block stability of two-dimensional incompressible flows

Tian Ma 1, and  Shouhong Wang 2,

1. 

Department of Mathematics, Xian Jiaotong University, Xian, China
2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  August 2004 Revised  August 2005 Published  October 2005

We study in this article topological structure of divergence-free vector fields on general two-dimensional manifolds. We introduce a new concept called block structural stability (or block stability for simplicity) and prove that the block stable divergence-free vector fields form a dense and open set. Furthermore, we show that a block stable divergence-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 ideas presented in this article have been applied to studies on structure and its evolutions of the solutions of the Navier-Stokes equations; see [4, 9, 10].
Keywords: Divergence-free vector fields, block structural stability, ergodic sets, structural classification., block structure.
Mathematics Subject Classification: 34D, 35Q35, 58F, 7.
Citation: Tian Ma, Shouhong Wang. Block structure and block stability of two-dimensional incompressible flows. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 169-184. doi: 10.3934/dcdsb.2006.6.169
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