Energy equality for the tropical climate models

Abstract

In this paper, we derive several new energy conservation criterion of weak solutions for both the inviscid primitive tropical climate model and the tropical climate model with partial or full diffusion. For the inviscid primitive tropical climate model, we first establish a Onsager-type sufficient condition on the regularity of the weak solutions to guarantee conservation the total energy, where, the regularity of the weak solutions are measured in terms of the Triebel–Lizorkin type of norms, Ḟp,qs and Besov norms, Ḃp,qs. This result can be understood as a generalization of the notable work of Chae (2006). And than, for the dissipative models, we provide a Shinbrot-type sufficient condition on the weak solutions to ensure the energy conservation. In addition, without the condition of Leray–Hopf class, the energy equality holds if a solution (u,v,θ) to the tropical climate model belongs to L4L4. The result can be regarded as a corresponding extension with different approach in Galdi (2019). To the best of our knowledge, this is the first energy conservation criterion of distributional solutions for this model.