TY - JOUR T1 - On Instability of the Rayleigh–Bénard Problem Without Thermal Diffusion in a Bounded Domain under $L^1$ -Norm AU - Zhang , Pan AU - Liu , Mengmeng AU - Song , Fangying JO - Annals of Applied Mathematics VL - 3 SP - 261 EP - 279 PY - 2022 DA - 2022/08 SN - 38 DO - http://doi.org/10.4208/aam.OA-2020-0060 UR - https://global-sci.org/intro/article_detail/aam/20878.html KW - Rayleigh–Bénard problem, thermal instability, initial-boundary value problem. AB -
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.