American Institute of Mathematical Sciences

Advanced
March  2010, 28(1): 99-130. doi: 10.3934/dcds.2010.28.99

Rotating Boussinesq equations: Dynamic stability and transitions

Chun-Hsiung Hsia 1, , Tian Ma 2, and  Shouhong Wang 3,

1. 

Department of Mathematics, National Taiwan University, Taipei, 10617, Taiwan
2. 

Department of Mathematics, Sichuan University, Chengdu
3. 

Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  September 2009 Revised  February 2010 Published  April 2010

The main objective of this article is to study dynamic transitions and stability for rotating incompressible flows, governed by the rotating Boussinesq equations. It is proved that there are only two types of transitions, Type-I (continuous) and Type-II (jump) transitions, as the Rayleigh number crosses the first real or complex eigenvalues. Specific criteria are given to determine the type of transitions as well.
Keywords: dynamic transition theory, geophysical fluid dynamics, jump transition, periodic solutions., continuous transition, singularity separation, Rotating Boussinesq equations.
Mathematics Subject Classification: 86A10, 76U05, 76E, 35Q35, 37.
Citation: Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Rotating Boussinesq equations: Dynamic stability and transitions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 99-130. doi: 10.3934/dcds.2010.28.99
[1]

Kota Kumazaki. Periodic solutions for non-isothermal phase transition models. Conference Publications, 2011, 2011 (Special) : 891-902. doi: 10.3934/proc.2011.2011.891
[2]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[3]

I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173
[4]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809
[5]

Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609
[6]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
[7]

Alessia Berti, Claudio Giorgi, Angelo Morro. Mathematical modeling of phase transition and separation in fluids: A unified approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1889-1909. doi: 10.3934/dcdsb.2014.19.1889
[8]

Jeffrey J. Early, Juha Pohjanpelto, Roger M. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1571-1586. doi: 10.3934/dcds.2010.27.1571
[9]

Martin Gugat, Michael Herty, Siegfried Müller. Coupling conditions for the transition from supersonic to subsonic fluid states. Networks & Heterogeneous Media, 2017, 12 (3) : 371-380. doi: 10.3934/nhm.2017016
[10]

Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174
[11]

Xiaojun Zhou, Chunhua Yang, Weihua Gui. State transition algorithm. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1039-1056. doi: 10.3934/jimo.2012.8.1039
[12]

Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909
[13]

Fadia Bekkal-Brikci, Giovanna Chiorino, Khalid Boushaba. G1/S transition and cell population dynamics. Networks & Heterogeneous Media, 2009, 4 (1) : 67-90. doi: 10.3934/nhm.2009.4.67
[14]

Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1197-1211. doi: 10.3934/dcdsb.2011.16.1197
[15]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
[16]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[17]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
[18]

Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202
[19]

Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
[20]

Jimmy Garnier, FranÇois Hamel, Lionel Roques. Transition fronts and stretching phenomena for a general class of reaction-dispersion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 743-756. doi: 10.3934/dcds.2017031

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences