248 research outputs found
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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