# American Institute of Mathematical Sciences

October  2011, 16(3): 687-701. doi: 10.3934/dcdsb.2011.16.687

## Navier--Stokes equations on the $\beta$-plane

 1 Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom

Received  September 2010 Revised  March 2011 Published  June 2011

We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\beta$-plane (i.e. with the Coriolis force varying as $f_0+\beta y$) will become nearly zonal: with the vorticity $\omega(x,y,t)=\bar\omega(y,t)+\tilde\omega(x,y,t),$ one has $|\tilde\omega|_{H^s}^2 \le \beta^{-1} M_s(\cdots)$ as $t\to\infty$. We use this show that, for sufficiently large $\beta$, the global attractor of this system reduces to a point.
Citation: Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\beta$-plane. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 687-701. doi: 10.3934/dcdsb.2011.16.687
##### References:

show all references

##### References:
 [1] D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251 [2] Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189 [3] Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59 [4] Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401 [5] C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253 [6] Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151 [7] Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611 [8] Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 [9] Yutaka Tsuzuki. Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains. Conference Publications, 2015, 2015 (special) : 1079-1088. doi: 10.3934/proc.2015.1079 [10] Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203 [11] Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure & Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635 [12] Tae-Yeon Kim, Xuemei Chen, John E. Dolbow, Eliot Fried. Going to new lengths: Studying the Navier--Stokes-$\alpha\beta$ equations using the strained spiral vortex model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2207-2225. doi: 10.3934/dcdsb.2014.19.2207 [13] Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95 [14] Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103 [15] Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237 [16] Alexei Ilyin, Kavita Patni, Sergey Zelik. Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2085-2102. doi: 10.3934/dcds.2016.36.2085 [17] Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 [18] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [19] Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609 [20] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

2018 Impact Factor: 1.008