On the low regularity of strong solutions to 2D tropical climate model

The tropical climate model derived by Frierson-Majda-Pauluis has received widespread attention in recent years. In the cutting-edge work by Li and Titi [Discrete Contin. Dyn. Syst. 36(8), 4495–4516 (2016)], the global well-posedness for 2D tropical climate model without heat diffusivity has been successfully proved. Meanwhile, an interesting problem “if the initial assumption θ0∈L∞(R2) can be dropped ?” was proposed there. In this paper, we give a positive answer on this issue through proving the uniqueness part in the Lagrangian coordinates. Noticing that the equation of temperature is a transport type equation and taking advantage of the characteristics of particle trajectories Xu(t, y), we discover that the temperature can be directly represented as θ̄=θ̄0+∫0tdivAv̄(x,s)ds while working in the Lagrangian coordinates, which helps us to remove any additional assumptions imposed on the initial temperature in the proof of uniqueness.