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 $\Omega =\{(x,y)\in [0,1]\times {\mathbb{R}}^{+}\}$, we consider velocities of the form $u=(f(t,x),-y{f}_x(t,x))$, with scalar temperature\, $\theta =y\rho (t,x)$. Assuming $f_x(0,x)$ attains its global maximum only at points $x_i^{\ast }$ located on the boundary of $[0,1]$, general criteria for finite-time blowup of the vorticity $-y{f}_{xx}(t,x_i^{\ast })$ and the time integral of $f_x(t,x_i^{\ast })$ are presented. Briefly, for blowup to occur it is sufficient that $\rho (0,x)\ge 0$ and $f(t,x_i^{\ast })=\rho (0,x_i^{\ast })=0$, while $-y{f}_{xx}(0,x_i^{\ast })\ne 0$. 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 ${\Vert f_x(t,\cdot )\Vert }_{L^{\infty }([0,1])}$ are also provided.

Minor typos corrected and streamlined the presentation