Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations

• Local existence, blowup, and other regularity criteria in a special class of solutions of the 2d inviscid Boussinesq equations are studied.
• Local-in-space criteria on smooth initial data leading to finite-time (boundary) blowup is derived.
• Nonzero initial vorticity at the boundary and Dirichlet boundary conditions can lead to finite-time boundary blowup.
• Vanishing of the initial vorticity at particular points in the spatial domain may lead to global-in-time solutions.
• A sufficient and necessary condition for blowup is established.

The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of this system is established. On an infinite strip Ω={(x,y)∈[0,1]×R+}Ω={(x,y)∈[0,1]×R+}, we consider velocities of the form u=(f(t,x),−yfx(t,x))u=(f(t,x),−yfx(t,x)), with scalar temperature θ=yρ(t,x)θ=yρ(t,x). Assuming fx(0,x)fx(0,x) attains its global maximum only at points xi⁎ located on the boundary of [0,1][0,1], general criteria for finite-time blowup of the vorticity −yfxx(t,xi⁎) and the time integral of fx(t,xi⁎) are presented. Briefly, for blowup to occur it is sufficient that ρ(0,x)≥0ρ(0,x)≥0 and f(t,xi⁎)=ρ(0,xi⁎)=0, while −yfxx(0,xi⁎)≠0. To illustrate how vorticity may suppress blowup, we also construct a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of ‖fx(t,⋅)‖L∞([0,1])‖fx(t,⋅)‖L∞([0,1]) are also provided.