On the blowup criteria and global regularity for the non-diffusive Boussinesq equations with temperature-dependent viscosity coefficient

In this paper, we consider the Boussinesq equations with temperature-dependent viscosity and zero thermal diffusivity. A blowup criterion of classical solution, depending only on ‖∇μ(θ)‖Lq(0,T;Lp)‖∇μ(θ)‖Lq(0,T;Lp) with p>2p>2 and 1/p+1/q≤1/21/p+1/q≤1/2, is obtained for the two-dimensional equations. This is in particular consistent with the results in Chae (2006), Hou and Li (2005), Lai et al. (2011), where the global regularity of the 2D non-diffusive Boussinesq equations with constant viscosity coefficient (i.e., μ(θ)=Const.) was proved. A Serrin’s type blowup criterion, depending on ‖u‖Ls(0,T;Lr)+‖∇μ(θ)‖Lq(0,T;Lp)‖u‖Ls(0,T;Lr)+‖∇μ(θ)‖Lq(0,T;Lp) with r,p>3r,p>3, 3/r+2/s≤13/r+2/s≤1 and 3/p+2/q≤13/p+2/q≤1, is obtained for the three-dimensional case. We also study the global regularity with large data for the two-dimensional equations in a non-divergence form.