BLOWUP IN STAGNATION-POINT FORM SOLUTIONS OF THE INVISCID 2D BOUSSINESQ EQUATIONS

Abstract. 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+}, we consider velocities of the form u = ( f (t, x),−y fx(t, x)), with scalar temperature θ = yρ(t, x). Assuming fx(0, x) attains its global maximum only at points x∗i located on the boundary of [0, 1], general criteria for finite-time blowup of the vorticity −y fxx(t, x∗i) and the time integral of fx(t, x∗i) are presented. Briefly, for blowup to occur it is sufficient that ρ(0, x) ≥ 0 and f (t, x∗i) = ρ(0, x∗i) = 0, while −y fxx(0, x∗i) , 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]) are also provided. 1