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Global regularity for a class of 2D generalized tropical climate models. (English) Zbl 1412.35249
Summary: 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 D. M. W. Frierson et al. [Commun. Math. Sci. 2, No. 4, 591–626 (2004; Zbl 1160.86303)] 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 \(\theta\). 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 \((-\Delta)^\beta v\) and \((-\Delta)^\gamma\theta\) for \(\beta>1\) and \(\beta+\gamma>\frac{3}{2}\); and second, the global regularity of the system with \((-\Delta)^\beta v\) for \(\beta>\frac{3}{2}\). 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
86A10 Meteorology and atmospheric physics
35R11 Fractional partial differential equations
42B25 Maximal functions, Littlewood-Paley theory
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