Block structure and block stability of two-dimensionalincompressible flows

Tian Ma; Shouhong Wang

doi:10.3934/dcdsb.2006.6.169

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2006, Volume 6, Issue 1: 169-184. Doi: 10.3934/dcdsb.2006.6.169

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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
2.
Department of Mathematics, Indiana University, Bloomington, IN 47405

Received: July 31, 2004

Revised: July 31, 2005

Published: December 2005

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Abstract

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].

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Mathematics Subject Classification: 34D, 35Q35, 58F, 76.

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