Some remarks on Liouville type theorems for the 3D steady tropical climate model
Yanyan Dong, Zhibing Zhang
TL;DR
The paper studies Liouville-type theorems for the stationary 3D tropical climate model, consisting of the coupled system $- abla^2 u+(u\cdot\nabla)u+\nabla\pi+\operatorname{div}(v\otimes v)=0$, $- abla^2 v+(u\cdot\nabla)v+\nabla\theta+(v\cdot\nabla)u=0$, $- abla^2 \theta+u\cdot\nabla\theta+\operatorname{div} v=0$, with $\nabla\cdot u=0$, $u,v$ representing velocity modes and $\theta$ the temperature. The authors exploit the special structure of the system and the Poincaré–Sobolev inequality, together with the Bogovskii operator to manage the pressure term, to obtain Liouville-type results under weaker integrability assumptions on $u$, $v$, and $\theta-\overline{\theta}_R$. They extend and improve the Liouville results of Cho–In–Yang by showing that, under sequences $R_j\to\infty$ with appropriate bounds on $X_{\alpha,p,R_j}$, $Y_{\beta,q,R_j}$ and $\overline{Z}_{\gamma,r,R_j}$ (or their primed variants), the only smooth solutions are trivial (with $\theta$ typically constant). The paper also provides corollaries in classical $L^p$ settings (B1–B10 and primed versions) demonstrating the robustness of the approach and broadening the Liouville-type theory for this climate-model system.
Abstract
Observing the special structure of the system and using the Poincar{é}-Sobolev inequality, we establish Liouville type theorems for the 3D steady tropical climate model under certain conditions on $u$, $v$, $\nabla θ$. Our results extend and improve a Liouville type result of Cho-In-Yang (arXiv:2312.17441).