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| On Instability of the Rayleigh–B´enard Problem without Thermal Diffusion in a Bounded Domain under L1-Norm |
| Pan Zhang,Mengmeng Liu,Fangying Song |
| (School of Mathematics and Statistics, Fuzhou University, Fuzhou,Fujian 350108, China) |
| DOI： |
| Abstract: |
| We investigate the thermal instability of a three-dimensional
Rayleigh–B′enard (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 L1-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 L1-norm. |
| Key words: Rayleigh–B´enard problem, thermal instability, initial-boundary value problem. |