In this paper, we investigate the nonlinear stability and transition threshold for the 3D Boussinesq system in Sobolev space under the high Reynolds number and small thermal diffusion in $\mathbb{ T}\times\mathbb{ R}\times\mathbb{ T} $. It is proved that if the initial velocity $v_{ \rm in}$ and the initial temperature $ \theta_{ \rm in} $ satisfy $ \|v_{ \rm in}-(y,0,0)\|_{ H^{ 2}}\leq \varepsilon\nu, \|\theta_{ \rm in}\|_{ H^{ 2}}\leq \varepsilon\nu^{ 2} $, respectively for some $ \varepsilon>0 $ independent of the Reynolds number or thermal diffusion, then the solutions of 3D Boussinesq system are global in time.
Categories: math.AP
