THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS∗

doi:10.1007/s10473-023-0420-0

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1800-1818.doi: 10.1007/s10473-023-0420-0

THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗

Xia YE, Yanxia XU, Zejia WANG^†

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

收稿日期:2022-02-25 发布日期:2023-08-08
作者简介:Xia YE, E-mail: yexia@jxnu.edu.cn; Yanxia XU,E-mail: xuyanxia-0924@163.com
基金资助:
*Ye's research was supported by the National Natural Science Foundation of China (12061037, 11971209) and the Natural Science Foundation of Jiangxi Province (20212BAB201016). Wang's research was supported by National Natural Science Foundation of China (11861038).

THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗

Xia YE, Yanxia XU, Zejia WANG^†

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

Received:2022-02-25 Published:2023-08-08
About author:Xia YE, E-mail: yexia@jxnu.edu.cn; Yanxia XU,E-mail: xuyanxia-0924@163.com
Supported by:
*Ye's research was supported by the National Natural Science Foundation of China (12061037, 11971209) and the Natural Science Foundation of Jiangxi Province (20212BAB201016). Wang's research was supported by National Natural Science Foundation of China (11861038).

摘要/Abstract

摘要： This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations. It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero, and the convergence rate is established. In addition, we prove that the boundary-layer thickness is of the value $\delta(k)=k^{\alpha}$ with any $ \alpha\in(0,1/4)$ for a small diffusivity coefficient $k>0$, and we also find a function to describe the properties of the boundary layer.

关键词: density-dependent Boussinesq equation, zero thermal diffusivity, convergence rate, boundary layer

Abstract: This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations. It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero, and the convergence rate is established. In addition, we prove that the boundary-layer thickness is of the value $\delta(k)=k^{\alpha}$ with any $ \alpha\in(0,1/4)$ for a small diffusivity coefficient $k>0$, and we also find a function to describe the properties of the boundary layer.

Key words: density-dependent Boussinesq equation, zero thermal diffusivity, convergence rate, boundary layer

Xia YE, Yanxia XU, Zejia WANG. THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1800-1818.

Xia YE, Yanxia XU, Zejia WANG. THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1800-1818.

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THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗

THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS^∗

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