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

Ying Zeng; Wenjing Yang; Ying Zeng; Wenjing Yang

doi:10.3934/math.20231579

AIMS Mathematics

2023, Volume 8, Issue 12: 30882-30894. doi: 10.3934/math.20231579

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Research article Special Issues

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

Ying Zeng ,
Wenjing Yang ^,

College of Science, China Three Gorges University, Yichang 443002, China

Received: 11 July 2023 Revised: 24 October 2023 Accepted: 06 November 2023 Published: 17 November 2023
MSC : 35B40, 35Q35

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.
- tropical climate model,
- asymptotic behavior,
- energy vanish,
- weak solutions,
- one single diffusion
Citation: Ying Zeng, Wenjing Yang. Decay of unique global solution for 3D tropical climate model with partial dissipation[J]. AIMS Mathematics, 2023, 8(12): 30882-30894. doi: 10.3934/math.20231579

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Abstract

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.

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