American Institute of Mathematical Sciences

Advanced
April  2001, 7(2): 431-445. doi: 10.3934/dcds.2001.7.431

Global structure of 2-D incompressible flows

Tian Ma 1, and  Shouhong Wang 2,

1. 

Department of Mathematics, Sichuan University, Chengdu
2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405

Revised  November 2000 Published  January 2001

The main objective of this article is to classify the structure of divergence-free vector fields on general two-dimensional compact manifold with or without boundaries. First we prove a Limit Set Theorem, Theorem 2.1, a generalized version of the Poincaré-Bendixson to divergence-free vector fields on 2-manifolds of nonzero genus. Namely, the $\omega$ (or $\alpha$) limit set of a regular point of a regular divergence-free vector field is either a saddle point, or a closed orbit, or a closed domain with boundaries consisting of saddle connections. We call the closed domain ergodic set. Then the ergodic set is fully characterized in Theorem 4.1 and Theorem 5.1. Finally, we obtain a global structural classification theorem (Theorem 3.1), which amounts to saying that the phase structure of a regular divergence-free vector field consists of finite union of circle cells, circle bands, ergodic sets and saddle connections.
Keywords: Compact manifold, global structural classification theorem., Limit Set Theorem.
Mathematics Subject Classification: 34D, 35Q35, 58F, 76, 86A1.
Citation: Tian Ma, Shouhong Wang. Global structure of 2-D incompressible flows. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 431-445. doi: 10.3934/dcds.2001.7.431
[1]

Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639
[2]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
[3]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[4]

Matthew Foreman, Benjamin Weiss. From odometers to circular systems: A global structure theorem. Journal of Modern Dynamics, 2019, 15: 345-423. doi: 10.3934/jmd.2019024
[5]

Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011
[6]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[7]

Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847
[8]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
[9]

Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2
[10]

Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143
[11]

Kuo-Chih Hung, Shao-Yuan Huang, Shin-Hwa Wang. A global bifurcation theorem for a positone multiparameter problem and its application. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5127-5149. doi: 10.3934/dcds.2017222
[12]

Brock C. Price, Xiangsheng Xu. Global existence theorem for a model governing the motion of two cell populations. Kinetic & Related Models, 2020, 13 (6) : 1175-1191. doi: 10.3934/krm.2020042
[13]

Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549
[14]

Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021
[15]

Habibulla Akhadkulov, Akhtam Dzhalilov, Konstantin Khanin. Notes on a theorem of Katznelson and Ornstein. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4587-4609. doi: 10.3934/dcds.2017197
[16]

Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549
[17]

Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517
[18]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367
[19]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011
[20]

Cristina Stoica. An approximation theorem in classical mechanics. Journal of Geometric Mechanics, 2016, 8 (3) : 359-374. doi: 10.3934/jgm.2016011

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy