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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.
}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2020-0060}, url = {http://global-sci.org/intro/article_detail/aam/20878.html} }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.