Spectral analysis and global well-posedness for a viscous tropical climate model with only a damp term

Tropical climate model derived by Frierson-Majda-Pauluis in Frierson et al. (2004) has been studied by many works (see, e.g., Li and Titi (2016); Li and Titi (2016); Wan (2016)). In this paper, we consider the Cauchy problem for the viscous tropical climate model with only a damp term and obtain the global well-posedness for the small data. The estimate of ∫0t‖ ∇θ‖L∞2dτ is the main difficulty. To achieve our goal, we need using the spectral analysis to the system and constructing a new energy estimate of u in Besov space.