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Global attractor of atmospheric circulation equations with humidity effect. (English) Zbl 1246.86012

Summary: Global attractor of atmospheric circulation equations is considered in this paper. Firstly, it is proved that this system possesses a unique global weak solution in \(L^2(\Omega, \mathbb{R}^4)\). Secondly, by using \(C\)-condition, it is obtained that atmospheric circulation equations have a global attractor in \(L^2(\Omega, \mathbb{R}^4)\).

MSC:

86A10 Meteorology and atmospheric physics
35Q86 PDEs in connection with geophysics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] H. Luo, “Global solution of atmospheric circulation equations with humidity effect,” submitted. · Zbl 1246.86012
[2] T. Ma and S. H. Wang, Phase Transition Dynamics in Nonlinear Sciences, Springer, New York, NY, USA, 2012.
[3] T. Ma, Theories and Methods in Partial Differential Equations, Academic Press, Beijing, China, 2011 (Chinese).
[4] N. A. Phillips, “The general circulation of the atmosphere: A numerical experiment,” Quarterly Journal of the Royal Meteorological Society, vol. 82, no. 352, pp. 123-164, 1956. · doi:10.1002/qj.49708235202
[5] C. G. Rossby, “On the solution of problems of atmospheric motion by means of model experiment,” Monthly Weather Review, vol. 54, pp. 237-240, 1926.
[6] J.-L. Lions, R. Temam, and S. H. Wang, “New formulations of the primitive equations of atmosphere and applications,” Nonlinearity, vol. 5, no. 2, pp. 237-288, 1992. · Zbl 0746.76019 · doi:10.1088/0951-7715/5/2/001
[7] J.-L. Lions, R. Temam, and S. H. Wang, “On the equations of the large-scale ocean,” Nonlinearity, vol. 5, no. 5, pp. 1007-1053, 1992. · Zbl 0766.35039 · doi:10.1088/0951-7715/5/5/002
[8] J.-L. Lions, R. Temam, and S. H Wang, “Models for the coupled atmosphere and ocean. (CAO I),” Computational Mechanics Advances, vol. 1, no. 1, pp. 5-54, 1993. · Zbl 0805.76011
[9] C. Foias, O. Manley, and R. Temam, “Attractors for the Bénard problem: existence and physical bounds on their fractal dimension,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 8, pp. 939-967, 1987. · Zbl 0646.76098 · doi:10.1016/0362-546X(87)90061-7
[10] B. L. Guo, “Spectral method for solving two-dimensional Newton-Boussinesq equations,” Acta Mathematicae Applicatae Sinica. English Series, vol. 5, no. 3, pp. 208-218, 1989. · Zbl 0681.76048 · doi:10.1007/BF02006004
[11] B. Guo and B. Wang, “Approximate inertial manifolds to the Newton-Boussinesq equations,” Journal of Partial Differential Equations, vol. 9, no. 3, pp. 237-250, 1996. · Zbl 0867.35072
[12] T. Ma and S. Wang, “El Niño southern oscillation as sporadic oscillations between metastable states,” Advances in Atmospheric Sciences, vol. 28, no. 3, pp. 612-622, 2011. · doi:10.1007/s00376-010-9089-0
[13] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997. · Zbl 0871.35001
[14] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, vol. 25 of Studies in Mathematics and Its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1992. · Zbl 0850.68019 · doi:10.1016/S0168-2024(08)70270-4
[15] J.-M. Ghidaglia, “Finite-dimensional behavior for weakly damped driven Schrödinger equations,” Annales de l’Institut Henri Poincaré, vol. 5, no. 4, pp. 365-405, 1988. · Zbl 0659.35019
[16] J.-M. Ghidaglia, “A note on the strong convergence towards attractors of damped forced KdV equations,” Journal of Differential Equations, vol. 110, no. 2, pp. 356-359, 1994. · Zbl 0805.35114 · doi:10.1006/jdeq.1994.1071
[17] J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1988. · Zbl 0642.58013
[18] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lincei Lectures, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0755.47049 · doi:10.1017/CBO9780511569418
[19] Q. Ma, S. Wang, and C. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541-1559, 2002. · Zbl 1028.37047 · doi:10.1512/iumj.2002.51.2255
[20] T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, Science Press, Beijing, China, 2007 (Chinese). · Zbl 1298.35003
[21] T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, Hackensack, NJ, USA, 2005. · doi:10.1142/9789812701152
[22] J. Lü and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 775-858, 2006. · Zbl 1097.94038 · doi:10.1142/S0218127406015179
[23] J. Lü, F. Han, X. Yu, and G. Chen, “Generating 3-D multi-scroll chaotic attractors: a hysteresis series switching method,” Automatica, vol. 40, no. 10, pp. 1677-1687, 2004. · Zbl 1162.93353 · doi:10.1016/j.automatica.2004.06.001
[24] J. Lü, G. Chen, X. Yu, and H. Leung, “Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Transactions on Circuits and Systems. I, vol. 51, no. 12, pp. 2476-2490, 2004. · Zbl 1371.37060 · doi:10.1109/TCSI.2004.838151
[25] J. Lü, S. Yu, H. Leung, and G. Chen, “Experimental verification of multidirectional multiscroll chaotic attractors,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 1, pp. 149-165, 2006. · doi:10.1109/TCSI.2005.854412
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