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 - <p style="text-align: justify;">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.</p>