Some new Liouville type theorems for 3D steady tropical climate model
Yan Fang, Zhibing Zhang
TL;DR
The paper addresses Liouville-type rigidity for the 3D stationary tropical climate system consisting of $u$, $v$, and $\theta$ solving $-\Delta u+(u\cdot\nabla)u+\nabla\pi+\mathop{\mathrm{div}}(v\otimes v)=0$, $-\Delta v+(u\cdot\nabla)v+\nabla\theta+(v\cdot\nabla)u=0$, $-\Delta\theta+u\cdot\nabla\theta+\mathop{\mathrm{div}}v=0$, with $\nabla\cdot u=0$. It combines a Giaquinta-type iteration with delicate integral estimates and uses the Bogovskii operator to handle the pressure, introducing scale functionals $X_{\alpha,p,R}$, $Y_{\beta,q,R}$, $Z_{\gamma,r,R}$ under seventeen structural conditions (A1)–(A17) (and related endpoint regimes) to derive Liouville-type conclusions. The main result is that any smooth solution is trivial, i.e., $u=v=\theta=0$, under these criteria, with corollaries recovering $L^p$-integrability and endpoint-decay cases. This work extends Cho–In–Yang (2024) and provides a unified framework for rigidity in coupled fluid-thermodynamic systems in $\mathbb{R}^3$, clarifying the roles of the two baroclinic modes and offering a toolkit for further decay/space-condition analyses in tropical climate dynamics.
Abstract
In this paper, we study the Liouville type theorems for the stationary tropical climate model in three dimension. With the help of the delicate estimates of several integrals and an iteration argument, we establish Liouville type theorems under seventeen different assumptions. As a consequence, we show that a smooth solution is trivial provided that they belong to some Lebesgue spaces or satisfy some decay conditions at infinity. Our results extend and improve the recent work of Cho-In-Yang (2024 Appl. Math. Lett. 153 109039).