  The well-known Constantin-Lax-Majda (CLM) equation, an important toy model of
the 3D Euler equations without convection, can develop finite time
singularities [5]. De Gregorio modified the CLM model by adding a convective
term [6], which is known important for fluid dynamics [10,14]. Presented are
two results on the De Gregorio model. The first one is the global
well-posedness of such a model for general initial data with non-negative (or
non-positive) vorticity which is based on a newly discovered conserved
quantity. This verifies the numerical observations for such class of initial
data. The second one is an exponential stability result of ground states, which
is similar to the recent significant work of Jia, Steward and Sverak [11], with
the zero mean constraint on the initial data being removable. The novelty of
the method is the introduction of the new solution space $\mathcal{H}_{DW}$
together with a new basis and an effective inner product of $\mathcal{H}_{DW}$.
