American Institute of Mathematical Sciences

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December  2016, 21(10): 3709-3722. doi: 10.3934/dcdsb.2016117

The steady state solutions to thermohaline circulation equations

Chao Xing 1, , Ping Zhou 1, and  Hong Luo 1,

1. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China, China

Received  December 2015 Revised  May 2016 Published  November 2016

In the article, we study the existence and the regularity of the steady state solutions to thermohaline circulation equations. Firstly, we obtain a sufficient condition of the existence of weak solutions to the equations by acute angle theory of weakly continuous operator. Secondly, we prove the existence of strong solutions to the equations by ADN theory and iteration procedure. Furthermore, we study the generic property of the solutions by Sard-Smale theorem and the existence of classical solutions by ADN theorem.
Keywords: regularity., thermohaline circulation equations, Steady state solutions, existence.
Mathematics Subject Classification: Primary: 35Q35, 35A01; Secondary: 34K2.
Citation: Chao Xing, Ping Zhou, Hong Luo. The steady state solutions to thermohaline circulation equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3709-3722. doi: 10.3934/dcdsb.2016117
References:
[1]

H. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulations: a dynamical system approach, Review of Geophysics, 43 (2005), 1-38. doi: 10.1029/2002RG000122.  Google Scholar

[2]

H. Dijkstra and M. Molemaker, Symmetry breaking and overturning oscillations in thermohaline-driven flows, J. Fluid Mech., 331 (1997), 195-232. Google Scholar

[3]

H. Dijkstra and J. Neclin, Imperfections of the thermohaline circulation: Multiple equilibria and flux correction, Journal of Climate, 12 (1999), 1382-1392. doi: 10.1175/1520-0442(1999)012<1382:IOTTCM>2.0.CO;2.  Google Scholar

[4]

L. Evans, Partial Differential Equations, American Mathematical Society, Rhode Island, 1998.  Google Scholar

[5]

C. Foias and R. Temam, Structure of the set of stationary solutions of the Navier-Stokes equations, Communications on Pure and Applied Mathematics, 30 (1977), 149-164. doi: 10.1002/cpa.3160300202.  Google Scholar

[6]

Y. G. Gu and W. J Sun, Nontrivial equilibrium solutions for a semilinear reaction diffusion system, Applied Mathematics and Mechanics, 25 (2004), 1382-1389. doi: 10.1007/BF02438295.  Google Scholar

[7]

S. Henry and P. Hall, Thermohaline convection with two stable regimes of flow, Tellus, 13 (1961), 224-230. Google Scholar

[8]

Z. J. Jiang and S. L. Sun, Functional Analysis, (Chinese)Higher Education Press, China, 2005. Google Scholar

[9]

H. Luo, Steady state solution to atmospheric circulation equations with humidity effect, Journal of Applied Mathematics, 2012 (2012), Art. ID 867310, 18 pp.  Google Scholar

[10]

T. Ma and S. H. Wang, Dynamic transition theory for thermohaline circulation, Physical D, 239 (2010), 167-189. doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[11]

T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, (Chinese) Science Press, Beijing, China, 2007. Google Scholar

[12]

T. Ma and S. H. Wang, Bifurcation Theory and Applications, Nonl. Sci. Ser. A, World Scientific, 2005. doi: 10.1142/5798.  Google Scholar

[13]

T. Ma, Theories and Methods in Partial Differential Equations, (Chinese) Science Press, China, 2011. Google Scholar

[14]

R. Sell G and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. Google Scholar

[15]

R. Temam, Navier-Stokes Equation and Nonlinear Functional Analysis, $2^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.  Google Scholar

[16]

R. Temam, Navier-Stokes equation: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, The Netherlands, 1979.  Google Scholar

[17]

R. Temam, Infinite-dimensional Dynamical Systems In Mechanics And Physics, Applied Mathematics and Science, Vol. 68, New York, Springer, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

O. Thual and J. McWilliams, The Catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models, Geophysical Astrophysical Fluid Dynamics, 64 (1992), 67-95. doi: 10.1080/03091929208228085.  Google Scholar

[19]

E. Tziperman, J. Toggweiler, Y. Fzliks and K. Bryan, Instability of the thermohaline circulation with respect to mixed boundary conditions: Is it really a problem for realistic models?, Journal of Physical Oceanography, 24 (1994), 217-232. doi: 10.1175/1520-0485(1994)024<0217:IOTTCW>2.0.CO;2.  Google Scholar

show all references

References:
[1]

H. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulations: a dynamical system approach, Review of Geophysics, 43 (2005), 1-38. doi: 10.1029/2002RG000122.  Google Scholar

[2]

H. Dijkstra and M. Molemaker, Symmetry breaking and overturning oscillations in thermohaline-driven flows, J. Fluid Mech., 331 (1997), 195-232. Google Scholar

[3]

H. Dijkstra and J. Neclin, Imperfections of the thermohaline circulation: Multiple equilibria and flux correction, Journal of Climate, 12 (1999), 1382-1392. doi: 10.1175/1520-0442(1999)012<1382:IOTTCM>2.0.CO;2.  Google Scholar

[4]

L. Evans, Partial Differential Equations, American Mathematical Society, Rhode Island, 1998.  Google Scholar

[5]

C. Foias and R. Temam, Structure of the set of stationary solutions of the Navier-Stokes equations, Communications on Pure and Applied Mathematics, 30 (1977), 149-164. doi: 10.1002/cpa.3160300202.  Google Scholar

[6]

Y. G. Gu and W. J Sun, Nontrivial equilibrium solutions for a semilinear reaction diffusion system, Applied Mathematics and Mechanics, 25 (2004), 1382-1389. doi: 10.1007/BF02438295.  Google Scholar

[7]

S. Henry and P. Hall, Thermohaline convection with two stable regimes of flow, Tellus, 13 (1961), 224-230. Google Scholar

[8]

Z. J. Jiang and S. L. Sun, Functional Analysis, (Chinese)Higher Education Press, China, 2005. Google Scholar

[9]

H. Luo, Steady state solution to atmospheric circulation equations with humidity effect, Journal of Applied Mathematics, 2012 (2012), Art. ID 867310, 18 pp.  Google Scholar

[10]

T. Ma and S. H. Wang, Dynamic transition theory for thermohaline circulation, Physical D, 239 (2010), 167-189. doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[11]

T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, (Chinese) Science Press, Beijing, China, 2007. Google Scholar

[12]

T. Ma and S. H. Wang, Bifurcation Theory and Applications, Nonl. Sci. Ser. A, World Scientific, 2005. doi: 10.1142/5798.  Google Scholar

[13]

T. Ma, Theories and Methods in Partial Differential Equations, (Chinese) Science Press, China, 2011. Google Scholar

[14]

R. Sell G and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. Google Scholar

[15]

R. Temam, Navier-Stokes Equation and Nonlinear Functional Analysis, $2^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.  Google Scholar

[16]

R. Temam, Navier-Stokes equation: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, The Netherlands, 1979.  Google Scholar

[17]

R. Temam, Infinite-dimensional Dynamical Systems In Mechanics And Physics, Applied Mathematics and Science, Vol. 68, New York, Springer, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

O. Thual and J. McWilliams, The Catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models, Geophysical Astrophysical Fluid Dynamics, 64 (1992), 67-95. doi: 10.1080/03091929208228085.  Google Scholar

[19]

E. Tziperman, J. Toggweiler, Y. Fzliks and K. Bryan, Instability of the thermohaline circulation with respect to mixed boundary conditions: Is it really a problem for realistic models?, Journal of Physical Oceanography, 24 (1994), 217-232. doi: 10.1175/1520-0485(1994)024<0217:IOTTCW>2.0.CO;2.  Google Scholar

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