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- W1569173978 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 $Omega={(x,y)in[0,1]timesmathbb{R}^+}$, we consider velocities of the form $u=(f(t,x),-yf_x(t,x))$, with scalar temperature, $theta=yrho(t,x)$. 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." @default.
- W1569173978 created "2016-06-24" @default.
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- W1569173978 date "2014-08-27" @default.
- W1569173978 modified "2023-09-26" @default.
- W1569173978 title "Blowup in Stagnation-point Form Solutions of the Inviscid 2d Boussinesq Equations" @default.
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- W1569173978 doi "https://doi.org/10.48550/arxiv.1408.6625" @default.
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