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Blowup in Stagnation-point Form Solutions of the Inviscid 2d Boussinesq Equations
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
, we consider velocities of the
form , with scalar temperature\, .
Assuming attains its global maximum only at points located
on the boundary of , general criteria for finite-time blowup of the
vorticity and the time integral of are
presented. Briefly, for blowup to occur it is sufficient that
and , while . 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 are
also provided.Comment: Minor typos corrected and streamlined the presentatio
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