Blowup in Stagnation-point Form Solutions of the Inviscid 2d Boussinesq Equations

TL;DR

This paper investigates finite-time blowup phenomena in specific solutions of the 2d Boussinesq equations, providing criteria for singularity formation, conditions for global existence, and explicit solutions illustrating these behaviors.

Contribution

It establishes new criteria for blowup in stagnation-point form solutions of the 2d Boussinesq equations and constructs explicit global solutions to demonstrate vorticity effects.

Findings

Blowup occurs under specific initial conditions involving temperature and velocity derivatives.

Global solutions can be explicitly constructed, showing vorticity can prevent blowup.

Criteria linking initial data and solution behavior are rigorously derived.

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 $u=(f(t,x),-yf_x(t,x))$, with scalar temperature\, $\theta=y\rho(t,x)$. Assuming $f_x(0,x)$ attains its global maximum only at points $x_i^*$ located on the boundary of $[0,1]$, general criteria for finite-time blowup of the vorticity $-yf_{xx}(t,x_i^*)$ and the time integral of $f_x(t,x_i^*)$ are presented. Briefly, for blowup to occur it is sufficient that $\rho(0,x)\geq0$ and $f(t,x_i^*)=\rho(0,x_i^*)=0$, while $-yf_{xx}(0,x_i^*)\neq0$. 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 $\left\|f_x(t,\cdot)\right\|_{L^\infty([0,1])}$ are also provided.