An improved Liouville-type theorem for the stationary tropical climate model

TL;DR

This work proves a Liouville-type property for the stationary 3D tropical climate model in $\mathbb{R}^3$ under mild integrability assumptions $u\in L^3$, $v\in L^2$, and $\nabla\theta\in L^2$. It develops an energy-based approach combined with a Bogovskii operator to construct appropriate test functions and obtain energy inequalities, then uses an iterative, vanishing-energy argument over expanding balls to deduce $\nabla u,\nabla v,\nabla\theta\in L^2$ and hence constancy of $u$, $v$, and $\theta$. The result shows $u=v=0$ and $\theta$ constant, improving previous results that required $\nabla u,\nabla v,\nabla \theta\in L^2$. The method sharpens Liouville-type conclusions for the tropical climate system and may extend to related fluid models with coupled velocity and temperature fields.

Abstract

In this paper, we study the Liouville-type property for smooth solutions to the steady 3D tropical climate model. We prove that if a smooth solution $(u,v,θ)$ satisfies $u \in L^3 (\mathbb{R}^3)$, $v \in L^2 (\mathbb{R}^3)$, and $\nabla θ\in L^2 (\mathbb{R}^3)$, then $u=v=0$ and $θ$ is constant, which improves the previous result, Theorem 1.3 (Math. Methods Appl. Sci. 44, 2021) by Ding and Wu.

Theorems & Definitions (4)

• Theorem 1
• Remark 1
• Lemma 2
• Remark 2: the Poincaré--Sobolev inequality on annuli