Decay of unique global solution for 3D tropical climate model with partial dissipation

Abstract

<abstract><p>In this article, we studied the asymptotic behavior of weak solutions to the three-dimensional tropical climate model with one single diffusion $ \mu\Lambda ^{2\alpha}u $. We established that when $ u_{0}\in L^{1}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3}) $, $ (w_0, \theta_0)\in (L^{2}(\mathbb{R}^{3}))^2 $ and $ w\in L^\infty(0, \infty; W^{1-\alpha, \infty}(\mathbb{R}^3)) $ with $ \alpha\in(0, 1] $, the energy $ \Vert u(t)\Vert_{L^2(\mathbb{R}^3)} $ vanishes and $ \Vert w(t)\Vert_{L^2(\mathbb{R}^3)}+\Vert \theta(t)\Vert_{L^2(\mathbb{R}^3)} $ converges to a constant as time tends to infinity.</p></abstract>