Global regularity for a class of 2D generalized tropical climate models |
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Authors: | Bo-Qing Dong Wenjuan Wang Jiahong Wu Zhuan Ye Hui Zhang |
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Institution: | 1. College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;2. School of Mathematical Sciences, Anhui University, Hefei 230601, China;3. Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States of America;4. Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China;5. School of Mathematics and computation Sciences, Anqing Normal University, Anqing 246133, Anhui, China |
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Abstract: | This paper establishes the global existence and regularity of solutions to a two-dimensional (2D) tropical climate model (TCM) with fractional dissipation. The inviscid counterpart of this model was derived by Frierson, Majda and Pauluis 8] as a model for tropical geophysical flows. This model reflects the interaction and coupling among the barotropic mode u, the first baroclinic mode v of the velocity and the temperature θ. The systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances. Mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization. The aim here is the global regularity with the least dissipation. We prove two main results: first, the global regularity of the system with and for and ; and second, the global regularity of the system with for . The proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal. The key tools employed here include the maximal regularity for general fractional heat operators, the Littlewood–Paley decomposition and Besov space techniques, lower bounds involving fractional Laplacian and simultaneous estimates of several coupled quantities. |
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Keywords: | 35Q35 35B40 35B65 76B03 Tropical climate model Global regularity |
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