On Instability of the Rayleigh–Bénard Problem Without Thermal Diffusion in a Bounded Domain under $L^1$ -Norm
Year: 2022
Author: Pan Zhang, Mengmeng Liu, Fangying Song
Annals of Applied Mathematics, Vol. 38 (2022), Iss. 3 : pp. 261–279
Abstract
We investigate the thermal instability of a three-dimensional Rayleigh–Bénard (RB for short) problem without thermal diffusion in a bounded domain. First we construct unstable solutions in exponential growth modes for the linear RB problem. Then we derive energy estimates for the nonlinear solutions by a method of a prior energy estimates, and establish a Gronwall-type energy inequality for the nonlinear solutions. Finally, we estimate for the error of $L^1$-norm between the both solutions of the linear and nonlinear problems, and prove the existence of escape times of nonlinear solutions. Thus we get the instability of nonlinear solutions under $L^1$-norm.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aam.OA-2020-0060
Annals of Applied Mathematics, Vol. 38 (2022), Iss. 3 : pp. 261–279
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Rayleigh–Bénard problem thermal instability initial-boundary value problem.