American Institute of Mathematical Sciences

Advanced
November  2002, 2(4): 483-494. doi: 10.3934/dcdsb.2002.2.483

Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus

Chuong V. Tran 1, , Theodore G. Shepherd 1, and  Han-Ru Cho 1,

1. 

Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, Canada M5S 1A7, Canada, Canada, Canada

Received  January 2002 Revised  June 2002 Published  August 2002

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain $[0,2\pi]\times[0,2\pi/\alpha]$, where $\alpha\in(0,1]$, with doubly periodic boundary conditions. For the linear problem we employ the classical energy--enstrophy argument to derive some fundamental properties of unstable eigenmodes. From this it is shown that forces of pure $x_2$-modes having wavelengths greater than $2\pi$ do not give rise to linear instability of the corresponding primary stationary solutions. For the nonlinear problem, we prove the equivalence of nonlinear stability with respect to the energy and enstrophy norms. This equivalence is then applied to derive optimal conditions for nonlinear stability, including both the high- and low-Reynolds-number limits.
Keywords: Two-dimensional Navier–Stokes equations, linear stability, asymptotic (global) stability..
Mathematics Subject Classification: 34D, 35Q30, 7.
Citation: Chuong V. Tran, Theodore G. Shepherd, Han-Ru Cho. Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 483-494. doi: 10.3934/dcdsb.2002.2.483
[1]

Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245
[2]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141
[3]

Muriel Boulakia, Anne-Claire Egloffe, Céline Grandmont. Stability estimates for a Robin coefficient in the two-dimensional Stokes system. Mathematical Control & Related Fields, 2013, 3 (1) : 21-49. doi: 10.3934/mcrf.2013.3.21
[4]

Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567
[5]

Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 329-349. doi: 10.3934/dcdss.2020019
[6]

Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, , () : -. doi: 10.3934/era.2021051
[7]

Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559
[8]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
[9]

Frederic Heihoff. Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4703-4719. doi: 10.3934/dcdsb.2020120
[10]

Thierry Gallay. Stability and interaction of vortices in two-dimensional viscous flows. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1091-1131. doi: 10.3934/dcdss.2012.5.1091
[11]

Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Bifurcation and stability of two-dimensional double-diffusive convection. Communications on Pure & Applied Analysis, 2008, 7 (1) : 23-48. doi: 10.3934/cpaa.2008.7.23
[12]

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
[13]

Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan. On the stability of two-dimensional nonisentropic elastic vortex sheets. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021083
[14]

Vu Manh Toi. Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020099
[15]

Qing Yi. On the Stokes approximation equations for two-dimensional compressible flows. Kinetic & Related Models, 2013, 6 (1) : 205-218. doi: 10.3934/krm.2013.6.205
[16]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[17]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[18]

Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991
[19]

Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751
[20]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences