We present evidence on global existence of solutions of De Gregorio's equation, based on numerical computation and a mathematical criterion analogous to the BealeKato-Majda theorem. Its meaning in the context of a generalized Constantin-LaxMajda equation will be discussed. We then argue that the convection term can deplete solutions of blow-up.
We provide rigorous evidence of the fact that the modified
Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics
describes the geodesic flow on the subgroup of orientation-preserving
diffeomorphisms fixing one point, with respect to right-invariant metric
induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence
of the geodesics in the extended group of diffeomorphisms of Sobolev class
$H^{k}$ with $k\ge 2$.Comment: 24 page
We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result of the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent results obtained in [46,47].
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