312 research outputs found
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
Two regularity criteria for the 3D MHD equations
This work establishes two regularity criteria for the 3D incompressible MHD
equations. The first one is in terms of the derivative of the velocity field in
one-direction while the second one requires suitable boundedness of the
derivative of the pressure in one-direction
An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations
We give an example of a well posed, finite energy, 2D incompressible active
scalar equation with the same scaling as the surface quasi-geostrophic equation
and prove that it can produce finite time singularities. In spite of its
simplicity, this seems to be the first such example. Further, we construct
explicit solutions of the 2D Boussinesq equations whose gradients grow
exponentially in time for all time. In addition, we introduce a variant of the
2D Boussinesq equations which is perhaps a more faithful companion of the 3D
axisymmetric Euler equations than the usual 2D Boussinesq equations.Comment: 9 pages; simplified a solution formula in section 4 and added a
sentence on the time growth rate in the solutio
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