The steady state solutions to thermohaline circulation equations

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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

Abstract
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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 and 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.
[2]	H. Dijkstra and M. Molemaker, Symmetry breaking and overturning oscillations in thermohaline-driven flows, J. Fluid Mech., 331 (1997), 195-232.
[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.
[4]	L. Evans, Partial Differential Equations, American Mathematical Society, Rhode Island, 1998.
[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.
[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.
[7]	S. Henry and P. Hall, Thermohaline convection with two stable regimes of flow, Tellus, 13 (1961), 224-230.
[8]	Z. J. Jiang and S. L. Sun, Functional Analysis, (Chinese)Higher Education Press, China, 2005.
[9]	H. Luo, Steady state solution to atmospheric circulation equations with humidity effect, Journal of Applied Mathematics, 2012 (2012), Art. ID 867310, 18 pp.
[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.
[11]	T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, (Chinese) Science Press, Beijing, China, 2007.
[12]	T. Ma and S. H. Wang, Bifurcation Theory and Applications, Nonl. Sci. Ser. A, World Scientific, 2005. doi: 10.1142/5798.
[13]	T. Ma, Theories and Methods in Partial Differential Equations, (Chinese) Science Press, China, 2011.
[14]	R. Sell G and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
[15]	R. Temam, Navier-Stokes Equation and Nonlinear Functional Analysis, $2^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.
[16]	R. Temam, Navier-Stokes equation: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, The Netherlands, 1979.
[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.
[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.
[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.

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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.
[2]	H. Dijkstra and M. Molemaker, Symmetry breaking and overturning oscillations in thermohaline-driven flows, J. Fluid Mech., 331 (1997), 195-232.
[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.
[4]	L. Evans, Partial Differential Equations, American Mathematical Society, Rhode Island, 1998.
[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.
[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.
[7]	S. Henry and P. Hall, Thermohaline convection with two stable regimes of flow, Tellus, 13 (1961), 224-230.
[8]	Z. J. Jiang and S. L. Sun, Functional Analysis, (Chinese)Higher Education Press, China, 2005.
[9]	H. Luo, Steady state solution to atmospheric circulation equations with humidity effect, Journal of Applied Mathematics, 2012 (2012), Art. ID 867310, 18 pp.
[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.
[11]	T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, (Chinese) Science Press, Beijing, China, 2007.
[12]	T. Ma and S. H. Wang, Bifurcation Theory and Applications, Nonl. Sci. Ser. A, World Scientific, 2005. doi: 10.1142/5798.
[13]	T. Ma, Theories and Methods in Partial Differential Equations, (Chinese) Science Press, China, 2011.
[14]	R. Sell G and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
[15]	R. Temam, Navier-Stokes Equation and Nonlinear Functional Analysis, $2^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.
[16]	R. Temam, Navier-Stokes equation: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, The Netherlands, 1979.
[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.
[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.
[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.

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