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 $mathcal{H}_{DW}$.
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