Some new Liouville type theorems for 3D steady tropical climate model

Abstract

In this paper, we establish two major classes of Liouville type results for the three-dimensional stationary tropical climate model. The first class is obtained under the assumptions imposed on $u,v,\theta$ whereas the second one relies on the assumptions imposed on $u,v,\nabla\theta$. Using the energy method and an iteration argument, we obtain Liouville type theorems under the condition that Lebesgue norms of the smooth solutions on the annulus satisfy some power-law growth conditions. As a consequence, we show that a smooth solution is trivial provided that it belongs to some Lebesgue spaces or satisfies some decay conditions at infinity. Furthermore, with the aid of a contradiction argument and by developing a systematic framework to handle the energy function associated with the non-trivial solutions, we obtain a logarithmic improvement of our Liouville type theorems. Our new framework is very effective for establishing logarithmic improvements of Liouville type theorems for coupled fluid equations.