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

doi:10.1007/s10473-023-0420-0

Abstract

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].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^∗

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